Searching the Cosmos: Ripples from Avant-Garde Cosmological Probes

Home , Chris Hirata, Ofer Lahav, Richard Battye

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Paulo Montero-Camacho, B.S., M.S.

Graduate Program in Physics

The Ohio State University

2019

Dissertation Committee:

Christopher M. Hirata, Advisor Barbara Ryden Amy Conolly Eric Braaten c Copyright by

Paulo Montero-Camacho

2019 Abstract

The standard model of cosmology (ΛCDM model) makes several well-justiﬁed assumptions about the underlying physics involved in cosmology from the early Uni- verse up till now. Based on those assumptions the standard model of cosmology makes extremely precise, and successful, predictions about the physical properties of the Universe we live on. Even though the ΛCDM model of Cosmology has proved to be remarkably successful, there are still open questions. This thesis presents the work

I have done to explore the relaxation of a few of those well-justiﬁed assumptions, in order to study the impact they could have on our current understanding of diﬀerent cosmological probes. In Chapter 2, I consider possible sources - within the Physics

Standard Model - of circular polarization in the Cosmic Microwave Background, and resolve the conflict between previous results in the literature. Chapter 3 shows the effects of inhomogeneous reionzation in the Lyman-α forest. Also it studies a first approach at mitigating these effects in near-future Lyman-α data. Finally, I discuss white dwarf star explosions as a constraining mechanism for primordial black holes as Dark Matter candidates in Chapter 4.

ii to my friends, family, and especially to my partner on this journey and the next,

Jiaxin

iii Acknowledgments

I have always been fascinated with learning about cool science and space. When I was a kid, I was interested in learning more about galaxies, stars and the solar system,

I would spend a lot of time looking at the old encyclopedias we had in my parents home. I quite enjoyed the whole learning process, and I believe this gratiﬁcation from the search for new knowledge lead me to my PhD in physics.

In my time at the Scientiﬁc High School of Alajuela, I decided to pursue physics thanks to the wonderful guidance of my physics teacher, Pablo Blanco. However, it took half of my undergraduate career - with the advice of Francisco Frutos, Javier

Bonatti and Rodrigo Carboni - to ﬁnally choose which area in physics I was going to dedicate my time to. I ended up choosing cosmology, because I was really fascinated by the standard model of cosmology. I will always be charmed by how in the beginning there was nothing, and then ... well then there was everything!

My time at The Ohio State University and, in particular, at the Center for Cos- mology and AstroParticle Physics (CCAPP) has been amazing thanks to the friendly and supportive environment. I am thankful to my advisor Chris Hirata not only for being a great advisor and role model in these ﬁve years, but also for providing the right atmosphere to enhance the growth of my abilities as a researcher. I will miss our conversations not only because of his vast knowledge of cosmology but also because of how approachable, considerate and patient he is.

iv These ﬁve years at The Ohio State University would have been very diﬀerent without the support of my colleagues and friends. Xiao Fang and Daniel Martens,

I am so glad we were oﬃce-mates, thank you for the constant support in the good times and the bad ones. Also, thank you for all the advice, discussions and for all the trips! Benjamin Buckman, I am grateful for having you as a friend, and your advice and suggestions have played a role in moving my research forward. To my current oﬃce-mates: Jahmour Givans and Daniella Roberts, thank you for the good times.

To the members of my committee, Barbara Ryden, Amy Connolly and Eric Braaten, thank you for your guidance, but most importantly, thank you for your questions.

My letter writers and collaborators Paul Martini and Klaus Honscheid, I am grateful for all the support and counseling. To Makana Silva, Chenxiao Zeng and Gabriel

Vasquez, I thank you for your work on our projects, and I hope I did a good job advising you.

To the other members of CCAPP, who make it a great place to do science, and have helped my growth one way or another. To Niall MacCrann, Ami Choi, Ashley

Ross and Heidi Wu, thank you for being great postdocs and for asking great questions.

To the other graduate students in CCAPP who I have enjoyed discussing with: Su-

Jeong Lee, Bei Zhou, Matthew Digman, Heyang Long, Brian Clark and Hui Kong.

Finally, to John Beacom and Annika Peter, thank you for being awesome role models.

My family has also been a strong pillar during my studies. My father Carlos has always been interested in my research, and his engaging conversations have been the inspiration for how to overcome a few obstacles. My mother Jeannette taught me the need to be open-minded, and she is partially to blame for how much I enjoy teaching. And ﬁnally, to my brother Carlos Andr´es, his support was pivotal during

v my application to graduate school, and it still is. Without you I would not have made it.

I am also grateful to my friends outside of astrophysics. Ethel Perez and Emilio

Codecido, who helped by distracting me from the stress. To my gaming and soccer buddies, Carlos Avila, Jorge Esquivel, Mariano Esquivel, Scarleth Morales, Alexander

Sandoval and Pablo Sandoval; thank you for the great times, life would not be the same without you - but please stop getting married while I am so far away! To my partner in this journey, Jiaxin Wu, without whom my PhD would have been far worse and less enjoyable. I hope we continue making great memories in the years to come.

To everyone I mentioned above, one way or another you have helped me greatly, and for that I thank you. This work would not have been possible without you.

vi Vita

2013 ...... B.S. University of Costa Rica

2016 ...... M.S. The Ohio State University

Publications

Research Publications

“Slowly Rotating Curzon-Chazy Metric” Montero-Camacho P., Frutos-Alfaro F., Gutierrez-Chaves C., Cordero-Garcia I. Revista de Matem´atica: Teor´ıa y Aplicaciones, Volume 22, Issue 2, 265-274 (2015)

“Approximate Metric for a Rotating Deformed Mass” Frutos-Alfaro F., Montero-Camacho P., Araya M., Bonatti-Gonzalez J. International Journal of Astronomy and Astrophysics, Volume 5, Issue 1, 1-10 (2015)

“Exploring Circular Polarization in the CMB due to Conventional Sources of Cosmic Birefringence” Montero-Camacho P., Hirata C. M. Journal of Cosmology and Astroparticle Physics, Volume 2018, Issue 08, 040 (2018)

“Impact of Inhomogeneous Reionization in the Lyman α forest” Montero-Camacho P., Hirata C. M., Martini P., Honscheid− K. Monthly Notices of the Royal Astronomical Society, Volume 487, Issue 1, 1047-1056 (2019)

Fields of Study

Major Field: Physics

vii Table of Contents

Page

Abstract...... ii

Dedication...... iii

Acknowledgments ...... iv

Vita ...... vii

ListofTables...... xi

List of Figures ...... xii

1. Introduction...... 1

1.1 Cosmology 101 ...... 1 1.1.1 Statistical tools in modern cosmology ...... 7 1.2 Cosmic Microwave Background ...... 9 1.3 The Dark Ages and the epoch of reionization ...... 14 1.3.1 Lyman-α forest ...... 18 1.3.2 21 cm cosmology ...... 19 1.4 Dark Matter suspects ...... 21 1.5 Synopsisofthiswork...... 24

2. Exploring circular polarization in the CMB due to conventional sources of cosmic birefringence ...... 27

2.1 Introduction ...... 28 2.2 General aspects of circular polarization ...... 33 2.3 Birefringence from spin polarized hydrogen atoms from the Cosmic Dawnepoch...... 36 2.3.1 Order of magnitude ...... 39

viii 2.3.2 Detailed calculation ...... 41 2.4 Birefringence from spin polarized hydrogen atoms at recombination 50 2.4.1 Order of magnitude ...... 50 2.4.2 Detailed calculation ...... 53 2.5 Photon-photon scattering ...... 56 2.5.1 Order of magnitude ...... 56 2.5.2 Detailed calculation ...... 58 2.5.3 Comparison with previous calculations ...... 70 2.6 Static non-linear polarizability of hydrogen ...... 72 2.6.1 Order of magnitude ...... 72 2.6.2 Detailed computation ...... 73 2.7 Plasma delay: non-linear response of free electrons ...... 76 2.7.1 Order of magnitude ...... 77 2.7.2 Rigorous calculation ...... 77 2.8 Conclusion ...... 84

3. Impact of inhomogeneous reionization on the Lyman-α forest ...... 87

3.1 Introduction ...... 88 3.2 Conventions and formalism ...... 93 3.3 Simulations ...... 97 3.3.1 Smallboxes...... 98 3.3.2 Largeboxes...... 100 3.4 Assessment of contamination ...... 102 3.4.1 Linear power spectrum (3D) ...... 104

3.4.2 Computation of ψ(zre) ...... 104 3.4.3 Cross-power spectrum of matter and ψ ...... 105 3.4.4 Linear power spectrum (1D) ...... 109 3.5 Constraining the eﬀect with 21 cm cosmology ...... 113 3.6 Discussion ...... 116

4. White Dwarf star survival in an PBH ocean ...... 120

4.1 Introduction ...... 121 4.2 White Dwarf survival ...... 127 4.2.1 Velocity and density proﬁles ...... 128 4.2.2 Thermal eﬀects on WD materials by a passing PBH . . . . 130 4.2.3 Ignition and runaway explosion? ...... 135 4.2.4 Ignition rate and PBH constraints ...... 141 4.3 Conclusion ...... 147

ix 5. Conclusion...... 148

Appendices 150

A. Circular polarization of the Cosmic Microwave Background ...... 150

A.1 Anisotropic 21 cm radiation in expanding media ...... 150 A.2 Irreducible components of the density matrix ...... 154 A.3 Special functions: Spherical harmonics ...... 155 A.4 Source term for polarized atoms from CMB anisotropies ...... 156 A.5 Stokes parameters ...... 160 A.6 Special functions: Spin-weighted spherical harmonics ...... 161 A.7 Line of sight integrals of photon-photon scattering ...... 162 A.8 Plasma delay integrals ...... 165

B. Impact of inhomogeneous reionization in the Lyman-α forest ...... 167

B.1 Special function: Legendre polynomials ...... 167

C. White Dwarf Survival: mapping of hydrodynamical code ...... 168

C.1 Equation of state ...... 168 C.1.1 Physical description ...... 168 C.1.2 Conversion to code units ...... 170 C.2 Carbon dissociation ...... 171

Bibliography ...... 174

x List of Tables

Table Page

1.1 Summary table for the six independent parameters of ΛCDM. Values taken from the Planck Collaboration et al.(2018a)...... 7

2.1 Glossary of physical quantities used in this chapter...... 37

2.2 Summary table for sources of circular polarization at an observed frequency of 100 GHz. The ﬁnal column shows the level of detail of the calculation (A = detailed numerical evaluation of all line of sight factors and power spectra; B = the RMS was determined with all relevant numerical pre-factors; C = the numerical pre-factors in the birefringence were computed, but the full statistics of the line of sight integral were notcomputed)...... 84

3.1 Summary of the different reionization models. Note that since model A has not finished the reionization process by redshift 5.90, its optical depth is not as accurate as the other models. Furthermore, since the volume weighted neutral fraction differs from the mass weighted one at the end of the reionization process, the optical depths are approximations.102

3.2 Bias factors: Flux bias bF, radiation bias bΓ and ratio of bΓ/bF at diﬀerent redshifts...... 105

3.3 Results for the small-scale simulations. Transparency variations in the

IGM, ψ = ∆ ln τ1(Sim. B)/τ1(Sim. A). The number in square brackets is the redshift at which reionization turns on. Note that ∆ ln τ1 is negative if simulation A is more transparent than simulation B. . . . 106

3.4 Percentage deviation of the 3D and 1D Lyman-α power spectrum due to patchy reionization for the diﬀerent reionization models considered. Here we have used k = 0.14 Mpc−1, and we included the bias ratio. . 108

xi List of Figures

Figure Page

1.1 The Cosmic Microwave Background obtained by Planck Collabora- tion et al.(2016). In this image, one can appreciate the temperature anisotropies of the CMB, i.e. the diﬀerences between blue and red, which represent ﬂuctuations of order 10−5. Photo: The Planck Collab- oration and ESA...... 10

1.2 The mechanism that produces linear polarization in the early Universe. Unpolarized light in a quadrupole density ﬁeld Thompson scatters with a free electron. As a result, the outgoing radiation is linearly polarized. Image taken from Hu and White(1997)...... 11

1.3 Power spectra of the CMB. Upper panel: angular auto-power spectrum of temperature and linear polarization of the CMB. The E component of the polarization is the curl-free component and the B is the divergence-free component. Middle panel: cross-power spectrum of E- component of the linear polarization and temperature. Lower panel: lensing potential, i.e. how gravity transforms some of the E-modes into B-modes. Diﬀerent color points belong to diﬀerent experiments. Image taken from Planck Collaboration et al.(2018b)...... 13

1.4 Upper limits for the angular power spectrum of the circular polarization of the CMB (Nagy et al., 2017). Note that these upper limits are even higher than the level of the angular power spectrum of the linear polarization from Figure 1.3...... 15

1.5 The Dark Ages and cosmic reionization in the history of the Uni- verse (after recombination). Image credit: SPHEREx Collaboration, http://spherex.caltech.edu/docs/spherex_NASA_APSv4.slide17.jpg. 16

xii 1.6 Neutral gas distribution at diﬀerent redshifts for a Universe that reionizes at redshift 8. Even though at later redshifts the complex web-like structure is completely destroyed, the distribution of the particles is still being impacted by the thermal relics. Image taken from Hirata (2018)...... 18

1.7 Constraints on the mass range of primordial black holes as dark matter candidates. On the left, we have the Hawking evaporation limit in purple. The CMB constraint on the right limits the high masses. The green WD region corresponds to the constraint we will revisit in Chapter4. Image taken from Carr et al.(2017)...... 24

2.1 Two classes of cosmic birefringence sources: matter-related mechanism (left) and radiation-related mechanism (right). On the left side, the inhomogeneous matter, i.e. spin polarized hydrogen atoms, is responsible for the difference in indices of refraction. The blue arrows represent the spins of the hydrogen atoms. Here the black arrows stand for the radiation field, which can be either the CMB or the 21 cm radiation from other hydrogen atoms. This is the mechanism used in 2.3 and 2.4. In contrast, inhomogeneous radiation produces birefringence§§ through nonlinear response on the right panel, here the red arrows stand for the CMB quadrupole and the x in the circles corresponds to the source of the nonlinearity, e.g. vacuum ( 2.5), bounded electrons ( 2.6) or free electrons ( 2.7)...... §§ §§ 34 §§ 2.2 Spin polarized hydrogen atom with CMB magnetic field. Note that the CMB field is oscillating with frequency ω, which is considerably larger than the hyperfine frequency. We have used red to illustrate the hyperfine contribution, i.e. the spin of the proton. In the absence of hyperfine structure the change in the magnetic moment is perpendicular to the CMB magnetic field, see the blue line in the right figure. Note that we show the precession of the electron spin with respect to the proton spin ignoring the CMB magnetic field in the left panel, however, it should be pointed out that the proton spin also precesses around the electron spin. We illustrated the hyperfine contribution with the red lineintherightfigure...... 40

xiii 2.3 Integrands of Eq. (2.30). Essentially, the left ﬁgure represents the contribution of the linear polarization of the CMB to the rms of the circular polarization. Note that the peak of the distribution is approximately at ` = 1000. In the right panel we have evaluated the power spectrum at z = 20. This is the contribution due to the spin polarized hydrogen atoms. Note that the peak of the distribution is near k 0.1 Mpc−1. 47 ∼ 2.4 Window function, note that we multiply the y axis by a factor of 1036 for convenience...... 48

2.5 Relevant processes for estimating the alignment tensor. The contribution from Lyman-β and Lyman-α are represented by solid lines. The dashed lines represents the Hα transitions. The 2γ transition will play a minor role in the Peebles model calculation...... 53

2.6 Photon-photon scattering, γγ γγ. The case of interest to us is where the two ingoing photons→ and two outgoing photons occupy the same pair of wavenumbers, so that the diagrams can interfere with free propagation and result in a phase shift...... 57

2.7 Circular polarization power spectrum from photon-photon scattering. 69

3.1 Neutral fraction history for the diﬀerent models used in our large boxes. Our models diﬀer only by the number of ionizing photons escaping into the IGM per baryon, with model A having the latest reionization, and model B having the earliest reionization...... 103

3.2 Dimensionless cross-power spectrum of matter and neutral hydrogen fraction as a function of redshift for our diﬀerent models of reionization history. All cross-power spectra have been evaluated at wavenumber k = 0.14 Mpc−1...... 107

3.3 Comparison for the diﬀerent reionization models between the dimensionless cross-power spectrum of ψ and matter, and the dimensionless power spectrum of matter as a function of wavenumber for ﬁxed redshift of observation...... 110

3.4 Comparison between the 1D dimensionless cross-power spectrum of ψ and matter, and the 1D dimensionless power spectrum of matter as a function of wavenumber for ﬁxed redshift of observation...... 112

xiv 3.5 Failure of the linear µk-decomposition to reproduce the quadrupole of the 21 cm power spectrum at redshift 7 for the diﬀerent models. The dashed lines represent the quadrupole from 21cmFAST for the diﬀerent models. In solid lines we have the quadrupole from the linear approximation...... 115

4.1 A schematic diagram for the PBH passing through the WD materials, creating a shock. The materials with impact parameter less than about the accretion radius eventually get eaten by the PBH, while materials farther away get shocked, compressed and heated. The diagram is shown in the rest frame of the PBH...... 132

4.2 The simulated evolution of the logarithm ratio of the positions (i.e., Eulerian divided by Lagrangian radius coordinate), velocity, thermodynamic parameter J, temperature, density and pressure of the 1200 mass shells. The critical radius is at shell 400. The impact gets weaker with time and in mass shells farther out. All the quantities are in our code units, see Appendix C.1...... 134

4.3 Temperature time evolution for three diﬀerent shells. The shells are chosen so that they correspond to the accretion radius for three diﬀer- ent mach numbers: = 2.60 (purple), 3.00 (green) and 3.24 (orange). 135 M 4.4 Final and maximum temperatures reached by each cell as a function of Lagrangian radius. The initial radius for the shells from Fig. (4.3) have been highlighted preserving the color code...... 136

4.5 The minimum primordial black hole mass needed to achieve thermonuclear runaway for speciﬁc WD total mass. For comparison purposes we include the model without considering convection losses by KH instabilities(ingreen)...... 142

4.6 The parameter space where ignition is possible ranging from 0.75 − 1.385 M . Each total WD mass we considered is separated by ∆MWD ≈ 0.05 M while the resolution between each mass shell is ∆m ≈ 0.01 M . The green region is the parameter space where ignition can take place if we ignore the Kelvin-Helmholtz instability but that has

τburn > τKH, while the magenta region has τburn < τKH and hence ignition is robust against this instability. The cyan region is the parameter space where carbon dissocociation into α particles occurs...... 143

xv 4.7 The relationship between the rate of collisions Γ and the minimum PBH

mass mPBH to cause ignition ranging from MWD = 1.00 1.26M . The upper (green) line is when we ignore the Kelvin-Helmholtz− instability time as a criteria for ignition while the lower (purple) line is the rate when we include “burning before instability” as a criteria for ignition. These plots can also be interpreted as the PBH-induced ignition rates

as a function of mPBH...... 144

A.1 Polarization Stokes parameters. The arrows represent the direction of the electromagnetic ﬁelds...... 161

xvi Chapter 1: Introduction

1.1 Cosmology 101

Here we give a brief description of important concepts that will be crucial to understand, before diving into the new physics explored in this Thesis.

Cosmology is the science that studies the origin, evolution and likely end of the

Universe. In order to achieve this, we utilize complex instruments to look for electromagnetic and gravitational waves from ground and space observatories. The data obtained by those observations is then processed with powerful statistical techniques to infer – among other quantities – the energy budget of the Universe.

Amidst the great triumphs of modern cosmology, we have the observation of the accelerated expansion of the Universe (Riess et al., 1998). This expansion is due to an unknown component of the universe, Dark Energy, which constitutes almost seventy percent of the energy density of the Universe (Troxel et al., 2018). Furthermore, we know that the Universe is expanding in all directions (Hubble, 1929; Lemaˆıtre, 1931).

Thus, one could extrapolate to the beginning of the Universe and realize that all that we can – and cannot – see originated form a single singularity, the Big Bang.

The relic radiation from the Big Bang, i.e. the photons from the Cosmic Microwave

Background (CMB), was ﬁrst observed by Penzias and Wilson(1965); Dicke et al.

1 (1965). Based on these observations we have tightly constrained the composition of the Universe, pinned down the start of the epoch of reionization, and ﬁgured out that our Universe is very close to spatially ﬂat (Planck Collaboration et al., 2018a,c).

The standard model of cosmology predicts how the Universe began from initial conditions that are set up by inﬂation, which consists of a brief period of exponential expansion (Guth, 1981). Once inﬂation ends we have the Big Bang, and eventually the primordial “soup” cools down enough to have electrons, protons and photons.

Electrons and protons are coupled by Coulomb interactions, while electrons and photons interact via Thompson scattering. Therefore, all components of our primordial soup are in thermal equilibrium in the early Universe. Later on, the primordial gas has lowered its temperature, allowing for the formation of the first hydrogen atoms; this is known as recombination. Once recombination is finished, photons decouple due to the lack of free electrons to Thompson scatter from – the step immediately before the completion of recombination is known as the surface of last scattering and it corresponds to the CMB. These CMB photons effectively free-stream from the surface of last scattering to us, and as such they are a unique window into the physics of the early Universe.

As the Universe expands, matter perturbations continue to grow due to gravitational instability. Eventually, complex structures form – such as the ﬁrst stars and galaxies – some of these structures will start to emit ionizing radiation. The area surrounding these sources will start to reionize, i.e. hydrogen atoms will split into

H+ and free electrons. These ionized bubbles eventually grow and connect to other ionized regions. The end result is that most of the intergalactic medium (IGM) is ionized. This process is known as cosmic reionization.

2 Even though the standard model of cosmology is extremely successful at predict-

ing how the Universe evolved, we still have unsolved mysteries. Nearly one fourth

of the energy content of the Universe is made of cold dark matter (Planck Collabo-

ration et al., 2018a), and its nature is not understood despite the fact that we have

had observational evidence of dark matter for roughly eight decades (Zwicky, 1937).

Gargantuan eﬀorts have been made towards understanding cold dark matter; never-

theless, it is still unclear if it is even made of one type of particle.

The Universe is homogeneous and isotropic, i.e. all positions and directions are

the same. In addition, the Universe expands as a function of time. In cosmology

we describe these aspects of the Universe with the Friedmann-Lemaitre-Robertson-

Walker (FLRW) metric (Friedmann, 1924; Lemaˆıtre, 1931; Robertson, 1935; Walker,

1937), which is given by

ds2 = dt2 + a(t)2 dχ2 + r(χ)2 dθ2 + sin2 θdφ2 , (1.1) −

where ds is the line element1, χ is the radial coordinate, a is the scale factor that

encodes how the Universe expands, and  χ R sin , Closed Universe   R  r(χ) = χ, Flat Universe (1.2)   χ ˜ χ iR sinh = R sinh , Open Universe iR R˜ We deﬁne R – the radius of curvature – as the radius of the 3-sphere embedded into our 4D spacetime.

As photons traverse the Universe, their wavelength expands; this is known as redshift. Suppose we look at a light source from Earth today ( i.e. atoday = a0 = 1).

1The inﬁnitesimal separation of two events in 4D spacetime.

3 The light we observe λobs has a diﬀerent wavelength that when it was emitted λem.

In fact, λ a 1 obs = 0 = . (1.3) λem aem aem However, observations of emitted light beneﬁt from one additional point. If the spec-

tral line of an atom is known then we can infer what is λem, and since we observe λobs with our instruments, it makes sense to deﬁne the redshift z as

λ 1 + z = obs . (1.4) λem

Substituting Eq. (1.3) in the previous expression, we obtain the relationship between the scale factor and redshift: 1 1 + z = . (1.5) a We use both redshift and the scale factor as proxies for time; however, their exact

relation with time depends on the energy content of the Universe, and therefore on

the expansion history. In order to parametrize that mapping, let us ﬁrst deﬁne the

Hubble function2: 1 da a˙ H(t) = = . (1.6) a dt a

−1 −1 The value of the Hubble function today is the Hubble constant: H0 70 km s Mpc ∼ (Planck Collaboration et al., 2018a) with 1 Mpc = 3.086 1019 km. Eq. (1.6) is useful. × In particular, it tell us that the age of the Universe is inversely proportional to the

Hubble constant. However, the equations of motion of our FLRW metric coupled to

the energy content of the Universe can lead to another important expression as shown

in what follows.

2H(z) is usually known as the Hubble parameter. However, to avoid confusion with the Hubble constant, we will refer to it as Hubble function.

4 The Einstein ﬁeld equations describe how the geometry of a manifold is related

to the mass (and energy) present. The “amount” of mass and energy is quantiﬁed by

the stress-energy tensor, which for an isotropic perfect ﬂuid is given by

ρ 0 0 0  µν 0 P 0 0  T =   , (1.7) 0 0 P 0  0 0 0 P

where ρ is the energy density and P is the pressure. On the other hand, the geometry

enters the fray by computing the Ricci curvature tensor (Rµν) and the scalar of

curvature ( , which is actually the trace of the Ricci tensor). For the ﬂat FLRW R metric in Eq. (1.1), they are given by

a¨ R00 = 3 , (1.8) − a 2 Rij = δij(aa¨ + 2˙a ) and (1.9) " # a¨ a˙ 2 = 6 + , (1.10) R a a

where i, j run over the spatial coordinates.

The Einstein ﬁeld equations – ignoring the cosmological constant term – in natural

units can be written as 1 Rµν gµν = 8πGTµν , (1.11) − 2 R where G is Newton’s constant. We can substitute the geometry, i.e. Eqs. (1.8, 1.9 and 1.10), and the stress-energy tensor Eq. (1.7), to obtain the Friedmann equations

a˙ 2 8πG H2 = = ρ , (1.12) a 3 a¨ 4πG = (ρ + 3P ). (1.13) a − 3

The ﬁrst equation plays the role of conservation of energy.

5 The Universe can have diﬀerent components and therefore a few diﬀerent contri-

butions to the density in Eq. (1.13), e.g. radiation, cold dark matter (CDM), baryonic

matter and “vacuum” (cosmological constant). Note that in an expanding universe,

3 −3 the volume expands as a , thus ρm a . Moreover, in the case of radiation, one ∝ ∝ −1 −4 has to take into account that its wavelength gets shortened a , hence ρrad a . ∝ ∝

In the case of the cosmological constant, we haveρ ˙Λ = 0. Coupling the time dependence of the densities with the Friedmann equations one can obtain an expression for the Hubble parameter

p 3 4 H(z) = H0 Ωm(1 + z) + Ωrad(1 + z) + ΩΛ , (1.14)

2 where Ωx = ρx/ρcr, and ρcr 3H /(8πG) is the total density of a ﬂat Universe. This ≡ 0 density is known as the critical density and has a value of 1.88 10−29h2 g cm−3 with × −1 −1 h = H0/(100 km s Mpc ) 0.7 the dimensionless Hubble constant. ≈ Now that we have introduced the Hubble function and it’s relation to the en-

ergy budget of the Universe, I will introduce the six independent parameters of the

standard model of cosmology or ΛCDM.

2 Ωbh : the physical baryon density, i.e. the matter we actually know what it is • made of. It consists of 5 percent of the energy budget of the Universe. ∼

2 Ωch : the physical dark matter density. Roughly 25 percent of the Universe. • ∼

t0: age of the Universe. •

ns: the scalar spectral index, i.e. the tilt of the primordial power spectrum. •

As: amplitude of the primordial power spectrum. •

6 Table 1.1: Summary table for the six independent parameters of ΛCDM. Values taken from the Planck Collaboration et al.(2018a). Symbol Cosmological parameter Value 2 Ωbh Physical baryon density 0.0223 2 Ωch Physical dark matter density 0.1198

t0 Age of the Universe [Gyr] 13.801

ns Scalar spectral index 0.9652 10 ln(10 As) Amplitude of the primordial power spectrum 3.0430

τre Optical depth to reionization 0.0540

τre: optical depth to reionization. This parameter describes how much of a • nuisance free electrons are to CMB photons as the reionization process occurs.

See table 1.1 for the updated values from the ﬁnal results of the Planck collaboration

(Planck Collaboration et al., 2018a). Of course these six parameters get supplemented with information that we have strong priors for, e.g. dark energy equation of state and curvature of the Universe. This eﬀectively narrows down the type of Universe our ΛCDM model is capable of describing.

All the outstanding predictions of the standard model of cosmology would not have been possible without the right tools to infer the cosmological parameters from the data.

1.1.1 Statistical tools in modern cosmology

In cosmology we use statistical methods due to the intrinsic quantum mechanical nature of the ﬂuctuations that seed into the observable Universe. Therefore, the question of where a galaxy forms or merges is not as valuable as the evolution of the

whole distribution of galaxies. Moreover, the Universe is homogeneous and isotropic

7 – for the most part – hence statistical descriptions of the Universe that surrounds us are naturally the easiest way to understand it.

The two-point correlation function, and its power spectrum, are of particular importance to the ﬁeld of modern cosmology. In terms of random ﬁelds, φ(x), the two-point correlation function is given by

φ(x)φ(x0) = ξ(x x0) = ξ( x x0 ) , (1.15) h i − | − | where we have assumed that our random ﬁelds are statistically isotropic in the last equality, and hence the correlation function can only depend on the magnitude of the diﬀerence of vectors. If the medium is homogeneous – like our Universe – then the correlation function is invariant against translations.

The two-point correlation function quantifies how more likely it is to find a galaxy within a certain radius, in comparison to the probability of finding a galaxy drawn from a random distribution. But how does the correlation functions achieve this?

In practice, the two-point correlation function is a ruler of varying size that tries to answer if there are galaxies at both ends of the ruler. The same is true of the power spectrum with the caveat that instead of rulers now we have waves.

Alternatively, one can rewrite Eq. (1.15) in Fourier space

φ˜(k)φ˜∗(k0) = (2π)3P (k)δ3D(k k0) , (1.16) h i −

3D where δ is the three dimensional Dirac delta distribution, Pφ(k) is the auto-power spectrum of φ, and it contains information on the power at each wavenumber k. The power spectrum is the Fourier Transform of the correlation function, i.e.

Z 3 ik·r Pφ(k) = d k ξ(r)e . (1.17)

8 In cosmology we are primarily focused on Gaussian distributions, not only because of their simplicity, but also because the quantum ﬂuctuations from inﬂation are indeed

Gaussian. For a Gaussian distribution, the two-point correlation function – and therefore the power spectrum – contains all possible information. This is the reason why the two-point correlation function and the power spectrum are such basic pillars of modern cosmology.

Now that we have introduced the fundamental bricks of cosmology, we can move into the more complex relevant topics. In particular, I will present the basic picture of the CMB, cosmic reionization and dark matter candidates.

1.2 Cosmic Microwave Background

The Universe consists of a hot homogeneous primordial soup after the Big Bang.

However, the primordial plasma cools while the Universe expands. Around 389000 years after the Big Bang (z = 1059), the temperature decreased to roughly 3000 K, which allowed for protons and electrons to form neutral hydrogen (HI). Before recombination, photons and electrons were tightly coupled due to the photons scattering oﬀ the free electrons. The primordial soup lies in gravitational wells due to the density perturbations, hence gravity is trying to pull it down. On the other hand, radiation pressure counteracts this force, i.e. radiation behaves like the restoration force in a spring while gravity is trying to compress the plasma. In fact, as a result what we have is acoustic oscillations in our primordial soup. However, Thomson scattering requires the presence of free electrons. Hence after recombination the CMB photons are free, and provide us with a unique window – actually a map, see Figure 1.1 – to gaze at the early universe.

9 Figure 1.1: The Cosmic Microwave Background obtained by Planck Collaboration et al.(2016). In this image, one can appreciate the temperature anisotropies of the CMB, i.e. the diﬀerences between blue and red, which represent ﬂuctuations of order 10−5. Photo: The Planck Collaboration and ESA.

When I look at images of the the CMB, I usually emphasize the diﬀerence in colors in Figure 1.1, i.e. the focus is on the anisotropies in the early Universe. However, those diﬀerences are of order 10−5. Therefore, the Universe is extremely isotropic.

Everywhere we point a telescope, we see a black body radiation with temperature

TCMB = T0(1 + z) , (1.18)

where T0 = 2.73 K is the temperature of the CMB today.

The inhomogeneities in the density ﬁeld coupled to the presence of free electrons results in the CMB photons getting linearly polarized. This process can be explained as follows. Light propagating in the x direction will have the electric and magnetic

10 Figure 1.2: The mechanism that produces linear polarization in the early Universe. Unpolarized light in a quadrupole density ﬁeld Thompson scatters with a free electron. As a result, the outgoing radiation is linearly polarized. Image taken from Hu and White(1997).

ﬁelds in the perpendicular directions, i.e. the yz-plane. This unpolarized light prop- agates in a quadrupole density ﬁeld. Hence from the perspective of a free electron, unpolarized light coming from the x direction is colder than unpolarized light coming from the z direction. Thomson scattering only allows the transverse radiation to pass while stopping any radiation that is parallel to the outgoing direction. Therefore, the outgoing CMB photon in the y direction will have the hot horizontal intensity and cold vertical intensity from the z-incoming radiation and x-incoming radiation respectively, and hence it is linearly polarized. Note that this mechanism is incapable of generating circular polarization.

11 The relic radiation from the CMB is a prime window to the physics of the early

Universe, since CMB photons eﬀectively free-stream on their way to our telescopes,

hence, conserving the information from 389000 years after the Big Bang. We convert

this data into a power spectrum of the temperature and polarization anisotropies of

the CMB, see Figure 1.3.

The power spectra in Figure 1.3 is quite the gold mine. In particular, the position

of the ﬁrst acoustic peak is sensitive to the overall energy content of the Universe.

2 Therefore, if one considers constant physical matter density (Ωmh = const) and one decreases the curvature of the Universe then the first acoustic peak will move to smaller angles – or higher `. The current position of the first acoustic peak (` 150) ∼ is consistent with a flat Universe. The second acoustic peak – in conjunction with the first one – tells us about the baryon physical density, because more baryons is the analog of adding more mass to a spring (that is hanging down). Thus, the odd peaks of the oscillation, which correspond to the maximal compression of the primordial soup or the maximal expansion of our spring, are going to be enhanced with respect to the even ones that correspond to the maximum rarefaction of the plasma or maximal compression of the spring, since that length is the same regardless of the mass attached to the spring. Of course another effect of adding more baryons is that we decrease the frequency of the oscillations (in our spring analogy ω m−1), hence the acoustic peaks ∝ will be slightly shifted into lower angular scales. Furthermore, adding more baryons also leads to significant damping of the higher order peaks (Hu and White, 1996).

Again turning to our spring analogy, what is happening is similar to introducing a velocity dependant friction, i.e. more baryons imply more air friction. Hence, one

12 Angular scale

90◦ 1◦ 0.2◦ 0.1◦ 0.05◦

103 CMB- TT Planck WMAP ACT 102 SPT ACTPol SPTpol POLARBEAR BICEP2/Keck 101 BICEP2/Keck/ WMAP/Planck ] 2 K µ

[ CMB- EE `

D 100

1 10−

2 10− CMB- BB

3 10−

100 ] 2 K µ

[ 0 TE `

D 100 − 200 π − 2 / 1.5 φφ ` C

2 1.0 0.5 + 1)] ` (

` 0.0 [ 7

10 0.5 − 2 150 500 1000 2000 3000 4000 Multipole `

Figure 1.3: Power spectra of the CMB. Upper panel: angular auto-power spectrum of temperature and linear polarization of the CMB. The E component of the polarization is the curl-free component and the B is the divergence-free component. Middle panel: cross-power spectrum of E-component of the linear polarization and temperature. Lower panel: lensing potential, i.e. how gravity transforms some of the E-modes into B-modes. Different color points belong to different experiments. Image taken from Planck Collaboration et al.(2018b). 13 should expect an exponential damping that will be significant for the higher order

peaks.

Recent CMB measurements – e.g. Planck Collaboration et al.(2016) – have been

designed to explore the linear polarization of the CMB, which is a spin-2 quantity.

Thus, our current and near future instruments are at a disadvantage regarding their

capability to observe the circular polarization of the CMB. Nevertheless, under the

assumptions of the standard cosmological model, CMB photons are not circularly

polarized and hence it is natural to neglect it. However, current CMB experiments –

coupled to creative thinking – have been used to establish upper limits on the circular

polarization of the CMB (Mainini et al., 2013; Nagy et al., 2017), see Figure 1.4. These

upper limits are important to justify the strategies employed by instruments designed

to measure linear polarization of the CMB.

Although there is no primordial circular polarization with conventional physics –

e.g. without changing the early universe Lagrangian – one can imagine CMB photons

can acquire a small degree of circular polarization due to diverse propagation eﬀects.

This parameter space will be studied in Chapter2.

1.3 The Dark Ages and the epoch of reionization

After recombination (zrec = 1059, Trec = 2.73(1 + zdec) = 2894 K), hydrogen is formed and only a small fraction of the electrons are free. This epoch of the life of the Universe is known as the Dark Ages, because CMB photons do not interact with matter and hence our usual electromagnetic probes do not provide any information about what is happening. Hence the Universe was truly dark. The Dark Ages span a redshift range of 1059 < z < 10 (when the Universe is 0.4 Gyr old), where the ∼

14 Figure 1.4: Upper limits for the angular power spectrum of the circular polarization of the CMB (Nagy et al., 2017). Note that these upper limits are even higher than the level of the angular power spectrum of the linear polarization from Figure 1.3.

lower limit should be taken as an approximation because really the transition is set by cosmic reionization. But what can we be sure about the Dark Ages? In this epoch, the Universe is mainly dark matter, CMB photons, neutral hydrogen and helium.

However, there is another significant source of photons. Neutral hydrogen can emit photons of wavelength 21 cm due to the hyperfine transition, see 1.3.2. Furthermore, § we know that matter will cluster and form structures due to gravitational collapse and this has profound consequences. For instance, this mechanism is what leads to the creation of the first stars and galaxies.

15 Figure 1.5: The Dark Ages and cosmic reionization in the history of the Universe (after recombination). Image credit: SPHEREx Collaboration, http://spherex.caltech.edu/docs/spherex_NASA_APSv4.slide17.jpg.

As the Universe evolves to lower redshifts, the structure formation process con- tinues. Therefore more sources of ultraviolet (UV) photons appear. The gas around these regions gets heated up by the UV radiation. As a result it gets ionized, i.e. hydrogen breaks down into protons and free electrons. Despite our best eﬀorts, we currently do not know when reionization occurred. Nevertheless, we estimate that the half way point of reionization, i.e. when the Universe is half ionized, corresponds to zre 8 (Planck Collaboration et al., 2018a). ∼ Although our current understanding of cosmic reionization requires some kind of UV sources, it is not known what exactly reionizes the Universe. However, we have three “suspects”: quasars3, star-forming galaxies or a large population of faint

galaxies. Even though the precise nature – or contribution – of the UV sources is not

3A quasar is a very luminous active galactic nucleus. In other words, a colossal black hole surrounded by a gargantuan accretion disk.

16 clear, one thing is certain: reionization is not a homogeneous process. The neutral gas surrounding the UV sources is ionized, making ionized hydrogen (HII) bubbles.

Hence reionization does not happen everywhere at the same time; instead reionization is an inhomogenous process. As a result, free electrons are mainly in ionized regions while neutral hydrogen atoms are in not yet ionized regions. Therefore, interactions between them are severely diminished, and thus higher temperatures can be achieved in inhomogeneous reionization models.

As reionization occurs, the Intergalactic Medium (IGM) is heated to temperatures of order 104 K(McQuinn, 2016), which leaves an imprint on the IGM even though the structure is mostly destroyed by the ionization fronts. The small-scale structure post reionization would no longer be gravitationally bound (masses smaller than the

Jeans mass) but the relaxation time is cosmological, and thus the structure might survive for a considerable amount of time (as can be seen from the comparison in

Figure 1.6).

Examining this imprint comes with signiﬁcant technical challenges. The small- scale structure goes non-linear prior to reionization, hence one must be able to resolve what is happening in the small-scale structure if one hopes to model these thermal relics (Hirata, 2018). Besides, one must account for the inhomogeneous nature of reionization. In Chapter3, I show a way of studying this imprint.

Having established the main concepts behind inhomogeneous reionization, I will now focus on two of the prime probes for the high redshift4 IGM, which consequently are our primary tools to learn about reionization.

4Here high redshifts stands for z > 4.

17 Figure 1.6: Neutral gas distribution at diﬀerent redshifts for a Universe that reionizes at redshift 8. Even though at later redshifts the complex web-like structure is completely destroyed, the distribution of the particles is still being impacted by the thermal relics. Image taken from Hirata(2018).

1.3.1 Lyman-α forest

The Lyman-α line is a transition from hydrogen that corresponds to an atom in the 2p state emitting a photon and decaying to the ground state, 1s. The wavelength of the emitted photon is 121.6 nm and hence it is in the UV range. The lifetime of this transition is approximately 10−8 s.

Hydrogen is the most abundant element throughout the history of the Universe.

Between us and distant quasars, there are a plethora of neutral hydrogen clouds.

These clouds absorb photons that have Lyman-α wavelengths when they reach the clouds. In the simplest scenario where there is only one cloud in the way, the transmission ﬂux from the quasar will have one absorption spike. However, once the other

18 clouds get accounted for there is a forest of absorption spikes, this is why we refer to

this phenomena as the Lyman-α forest.

The distribution of hydrogen clouds traces the overall distribution of matter. This feature makes the forest a valuable tool in cosmology since it can be used to study the matter power spectrum. In addition, the Lyman-α forest can be used to put a constraint on the sum of the neutrino masses, because it is sensitive to the underlying total matter density.

The typical Lyman-α forest measurement is usually between the range 2 < z < 5,

where the upper limit is motivated by statistics – how many measurements – and

the lower limit is due to the wonderful fact that our atmosphere protects us against

UV radiation. Sensitivity to higher redshifts allows the Lyman-α forest to be used to

study the state of the IGM just after reionization ended (z 6 or when the Universe ∼ is 0.9 Gyr). Nevertheless, as we will show in Chapter3, there are also disadvantages –

or avenues to new interesting physics! – to sensitivity to the imprint from reionization.

1.3.2 21 cm cosmology

Here I will only describe the absolute basics of 21 cm cosmology, since a full

description is not the focus of my PhD research. See Pritchard and Loeb(2012) for

a comprehensive review of 21 cm cosmology.

The ground state of hydrogen 1s splits into two states when one considers the spin

of the proton, i.e.

F = L + S + I = S + I , (1.19)

where the orbital angular momentum L is zero for s-orbitals. Note that since the spin

of the nucleus I is the spin of the proton, mI can only have the values 1/2 just as mS. ±

19 Consequently, F = 0, 1 . Therefore the previous ground state of neutral hydrogen { } splits into a triplet state with slightly higher energy and a true singlet ground state.

Atoms can transition from one state to another if they acquire energy to ﬂip their

spin. This translates into a transition wavelength of approximately 21 cm. Hence

this radiation – in the rest frame of the atom – would be microwave radiation.

This transition is a forbidden transition since the lifetime corresponds to 1014 s; ∼ however, in the Dark Ages we have two conditions to our advantage: colossal amount

of neutral hydrogen and plenty of time before reionization. Thus, the Universe does

have a considerable population of 21 cm photons. Similarly to the Lyman-α forest,

since in the Dark Ages neutral hydrogen fell into the potential wells from the dark

matter halos, 21 cm cosmology is another tracer of the distribution of total matter in

the Universe.

One should imagine 21 cm cosmology not like a direct analog of the CMB, but

instead as a full 3D map of neutral hydrogen during the Dark Ages and reionization.

In order to achieve this map, one of the key concepts that is needed is the distribution

of how many hydrogen atoms are in which level. We parametrize this with the spin

temperature as follows n T F =1 = 3 exp ∗ , (1.20) nF =0 −Ts where T∗ is the temperature of the transition and the factor of three comes from the degeneracy of the triplet state. The actual observable from the 21 cm transition is in fact the temperature brightness Tb. This temperature can be seen in emission or absorption against the CMB temperature depending on the population of the hyperﬁne states, i.e. Tb Ts TCMB. This is why the spin temperature is so crucial, ∝ − we only have a signal when it diﬀers from the CMB temperature. Of course the

20 brightness temperature is also proportional to the amount of neutral hydrogen since they are the source of the 21 cm photons.

There are mainly two exciting avenues in the ﬁeld of 21 cm cosmology. First, one could attempt to directly measure the global signal – in other words Tb(z). This is exactly what the EDGES collaboration claimed to have measured recently (Bowman et al., 2018). Besides, one could attempt to observe the ﬂuctuations of the temperature brightness, δTb(z), to compute their power spectrum. This is completely analogous to measuring the CMB temperature or the angular power spectrum of the temperature

ﬂuctuations.

In Chapter2, we will show how the anisotropic radiation of 21 cm photons from neutral hydrogen atoms can be used to generate circular polarization in the CMB.

Furthermore, I study how 21 cm cosmology can help disentangle the eﬀects of patchy reionization in the Lyman-α forest in Chapter3.

1.4 Dark Matter suspects

Here I will only describe the very basics of a few dark matter candidates. The purpose is to provide background on the popular “suspects”. For a more detailed description of dark matter candidates, see (Roszkowski et al., 2018) (WIMP perspective) or Buckley and Peter(2018) (astrophysical perspective).

A good dark matter candidate has the following surprising properties: it is dark, and it behaves like matter. It is dark in the sense that it can only have weak interactions with other Standard Model particles, otherwise we would already found what dark matter is. Besides it must interact gravitationally just like regular matter. These two conditions have profound consequences. One must look for particles outside of

21 the Particle Physics Standard Model for potential candidates. Another important property is that any dark matter candidate must survive for cosmological timescales.

The WIMP (Weakly Interacting Massive Particle) satisﬁes the conditions described above. In fact, WIMPs are the most popular dark matter candidate due to the beauty of their formation mechanism. In the early Universe, dark matter was part of the primordial soup, and it was in thermal equilibrium with the rest of the plasma, i.e. dark matter production and annihilation were balancing out. However, the Universe expands and cools. As a consequence, WIMPs eventually freeze-out of equilibrium when the annihilation rate falls, because WIMPs cannot ﬁnd each other as easily as when the temperature was higher than the mass of the WIMP. Ultimately,

WIMPs will have even more trouble ﬁnding their antiparticle to annihilate and then they will reach a constant number of particles.

Most dark matter candidates – certainly the WIMP – introduce a new elementary particle capable of surviving till today. Primordial Black Holes (PBHs) are in that sense quite conventional, since they do not require the introduction of any physics outside the Standard Model. Furthermore, they are solutions to the Einstein ﬁeld equations described primary by two parameters: the mass of the PBH and its spin.

There are several diverse formation mechanisms of PBHs in the early Universe; however, the essence of PBH formation is simple. PBHs are formed by the gravitational collapse of the surrounding region caused by a non-linear density perturbation.

Another attractive feature of PBHs as dark matter candidates is that their allowed mass is restricted. Their minimum mass is set by Hawking radiation, i.e. if

PBHs are the dark matter, then they must survive till the present day, thus they cannot evaporate completely due to Hawking radiation. In contrast, the upper limit

22 is strongly constrained by the CMB. This constraint is based on the eﬀects of having

accreting PBHs in the recombination epoch. Accreting PBH do not have a black

body spectrum and will leave their imprint in the CMB power spectra.

The discovery of gravitational waves by Abbott et al.(2016) revitalized the en-

thusiasm of the community for PBHs as dark matter candidates. In principle, there

are two open mass-windows for PBH to be the totality of dark matter, i.e. fPBH =

30 ρPBH/ρDM = 1. The ﬁrst window is the solar mass window (M = 1.99 10 kg), × where a lot of recent work has been directed towards realizing if this window still exists

22 −8 or not. The second window is the sub-lunar (Mmoon = 7.34 10 kg = 3.69 10 M ) × × mass window5. In Figure 1.7, the two windows can be seen if one focuses only on the solid black line which comes from gravitational lensing constraints6.

In Chapter4, I will elaborate on a quite explosive constraint mechanism for PBHs.

We will consider the survival of White Dwarf stars (WD) if they live in a PBH ocean.

The question we will be interested in is whether the passage of a single asteroid-mass

PBH can heat up the star through dynamical friction to then cause a runaway nuclear

explosion. Note that there are two kinds of possible constraints from this scenario.

First, we observe a certain abundance of WDs in space. Hence we can constrain the

number of events that lead to explosion of the WD due to PBHs. On the other hand,

the explosion itself can potentially be used to constrain PBHs since it would impact

the inferred type 1a supernovae rates. In Chapter4, we will only consider the former

type of constraint.

5As a result of Montero-Camacho et al.(2019a) and Sugiyama et al.(2019), this window should be renamed the asteroid-mass window. 6The reason it is sensible to ignore the constraint titled “NS” – capture of PBH by neutron stars – in Figure 1.7 is due to the astrophysical uncertainties involving the physics of the capture mechanism. (In fact, in Montero-Camacho et al.(2019a), we argue that this constraint cannot currently constrain PBHs with asteroid masses.)

23 Figure 1.7: Constraints on the mass range of primordial black holes as dark matter candidates. On the left, we have the Hawking evaporation limit in purple. The CMB constraint on the right limits the high masses. The green WD region corresponds to the constraint we will revisit in Chapter4. Image taken from Carr et al.(2017).

1.5 Synopsis of this work

In this Thesis I present three independent projects that increase our understanding of modern cosmology. Given the distinct diﬀerences of motivations and physics involved in these projects, here I will provide a big picture description and explain the importance of each of them.

As described in 1.2, the CMB is not expected to have any degree of circular po- § larization. Hence, theoretical and observational eﬀorts assume – and take advantage – of no circular polarization in the CMB photons. Nevertheless, after the surface of last

24 scattering, CMB photons could potentially acquire a small degree of circular polar-

ization due to propagation eﬀects. The source of the propagation eﬀect can be varied

and therefore one must consider a wide range of mechanisms in order to calculate

the expected signals of circular polarization in the CMB. In Chapter 2, we consider

two classes of mechanisms for birefringence. One is alignment of the matter to pro-

duce an anisotropic susceptibility tensor: the hydrogen spins can be aligned either

by density perturbations or CMB anisotropies themselves. The other is anisotropy of

the radiation ﬁeld coupled to the non-linear response of the medium to electromag-

netic ﬁelds: this can occur either via photon-photon scattering (non-linear response

of the vacuum); atomic hyperpolarizability (non-linear response of neutral atoms); or

plasma delay (non-linear response of free electrons). Ultimately, the ﬁnal objective

is to determine whether any of the mechanisms produce a non-negligible degree of

circular polarization in the CMB. A non-zero signal would become a plausible source

for systematic errors, which must be taken into account in instruments with high

sensitivity, and a potentially new channel for new physics.

In contrast to the CMB, our understanding of both the Dark Ages and the epoch of

reionization is still under development, see 1.3. Currently the Lyman-α forest stands § as our primary tool to learn about cosmic reionization at high redshifts (2 < z < 5),

see 1.3.1. Furthermore, as pointed out in 1.3.2, the 21 cm signal from the hyperﬁne § § splitting of the hydrogen atom will open a new window into the Dark Ages and the

epoch of reionization in the near future. In Chapter 3, we compute the impact of

inhomogeneous reionization in the Lyman-α forest, and show that it leaves a strong signal. Given the strength of this signal, we also focus on possible mitigation schemes, such that upcoming surveys – e.g. the Dark Energy Spectroscopic Instrument (DESI)

25 (DESI Collaboration et al., 2016) – can achieve their science goals and potentially increase our knowledge of cosmic reionization through an unexpected route. We suggest the use of 21 cm cosmology in order to mitigate the eﬀect of inhomogeneous reionization in the Lyman-α forest.

One of the big remaining mysteries in modern cosmology corresponds to what actually are the constituents of the Universe, i.e. what are dark matter and dark energy. In Chapter 4, we consider PBHs as candidates for the dark matter. In particular, we constrain the contribution of PBHs to the total amount of dark matter via the survival of White Dwarf stars to the passage of a PBH through them.

26 Chapter 2: Exploring circular polarization in the CMB due to conventional sources of cosmic birefringence

Here I will present our paper on Exploring circular polarization in the CMB due to conventional sources of cosmic birefringence (Montero-Camacho and Hirata, 2018).

The original abstract is given below:

The circular polarization of the cosmic microwave background (CMB) is usually taken to be zero since it is not generated by Thomson scattering. Here we explore the actual level of circular polarization in the CMB generated by conventional cosmological sources of birefringence. We consider two classes of mechanisms for birefringence.

One is alignment of the matter to produce an anisotropic susceptibility tensor: the hydrogen spins can be aligned either by density perturbations or CMB anisotropies themselves. The other is anisotropy of the radiation ﬁeld coupled to the non-linear response of the medium to electromagnetic ﬁelds: this can occur either via photon- photon scattering (non-linear response of the vacuum); atomic hyperpolarizability

(non-linear response of neutral atoms); or plasma delay (non-linear response of free electrons). The strongest eﬀect comes from photon-photon scattering from recombination at a level of 10−14 K. Our results are consistent with a negligible circular ∼

27 polarization of the CMB in comparison with the linear polarization or the sensitivity

of current and near-term experiments.

The original authors are: Paulo Montero-Camacho & Christopher M. Hirata.

2.1 Introduction

The past two decades have seen enormous progress in cosmology thanks to rapid

advances in the observational data. Most of our knowledge of the early Universe comes

from the cosmic microwave background (CMB), including in particular the tempera-

ture and linear polarization anisotropies. Using these observations cosmologists have

been able to constrain inﬂationary models, tightly constrain the composition of the

Universe (e.g. the ratio of photons, baryons, dark matter, and neutrinos), pin down the

epoch of reionization, realize that our Universe is very close to spatially ﬂat, and even

explore the late Universe via secondary anisotropies. However, the CMB anisotropies,

while enormously useful, suﬀer from the limitation of cosmic variance: with only one

sky, we can access a ﬁnite number of independent modes in the CMB, and thus there

is a fundamental minimum statistical error (Durrer, 2008). The Planck mission has

already reached the cosmic variance limit in temperature over a wide range of angular

scales (Planck Collaboration et al., 2016).

The limitation posed by cosmic variance might be reduced by exploring other as-

pects of the CMB; these also often carry not only more information, but information on otherwise inaccessible physics. For instance, one could examine the spectral distortions of the CMB, i.e. the deviations from a perfect blackbody spectrum. These encode information on the thermal history of the Universe – e.g. energy injection from epochs well before last scattering, which would be otherwise invisible – as well as very

28 small-scale perturbations that are Silk-damped in the CMB anisotropies and destroyed

by non-linear evolution in galaxy surveys (Chluba et al., 2012). Today, the“deﬁnitive”

constraint on spectral distortions is from the COBE/FIRAS experiment, however pro-

posed experiments using modern technology (such as PIXIE (Kogut et al., 2011) and

PRISM (Andr´eet al., 2014)) could make dramatic improvements. One could also

consider sources of frequency-dependent anisotropy. For example, Rayleigh scatter-

ing from neutral atoms leads to small frequency dependent distortions to the CMB

power spectrum (Yu et al., 2001; Lewis, 2013). In terms of observations, future ex-

periments like PRISM and PIXIE may be able to detect this signal (Alipour et al.,

2015). Another idea is to use the time evolution of the CMB anisotropies (Lange and

Page, 2007), which probes the radial direction at the surface of last scattering instead

of giving simply a 2D slice.

This paper considers the circular polarization of the CMB. In radiative transfer

problems, circular polarization is often the result of a two-step process: ﬁrst, linear

polarization is generated (e.g. by selective absorption or emission, or scattering), and

then a phase delay is induced between the x and y axes (e.g. passage through a birefringent medium; reflection off a tilted surface). In this case, circular polarization traces the overall geometry of the setup as well as properties of the medium. Examples include circular polarization of starlight (Martin, 1972) or of diffuse radiation in star- forming regions (Kwon et al., 2014), and the circular polarization observed from solar system planets (Swedlund et al., 1972; Kemp et al., 1971). An alternative source of circular polarization (relevant for both foregrounds and new physics) involves magnetized media, where the intrinsic emission can be circularly polarized due to the preferred handedness of particle trajectories, as occurs in masers (Deguchi and

29 Watson, 1986; Watson and Wyld, 2001), GRB afterglows (Wiersema et al., 2014;

Matsumiya and Ioka, 2003), and (we expect) in the diﬀuse synchrotron emission

from our Galaxy. In AGN jets, both mechanisms are considered as candidates as a wide range of physical conditions and observed polarization properties can occur

(Brunthaler et al., 2001; Homan et al., 2009, 2001; Beckert and Falcke, 2002). Finally, there is ongoing discussion on the origin of circularly polarized radio emission from pulsars (Melrose, 2003; Karastergiou et al., 2003; Mitra et al., 2009). The circular polarization of the CMB is usually assumed to be zero in the context of standard cosmology, though it could be a channel for new physics.

The interest in the circular polarization of the CMB has been steadily growing over the recent past. This rising enthusiasm in circular polarization has also motivated the search and study of new possible sources of circular polarization in the

CMB, e.g. Zarei et al.(2010) where the authors show that the CMB picks up a small circular polarization using a background magnetic ﬁeld and by considering physics outside of the standard model. Mohammadi(2014) argues that CMB photons can acquire circular polarization because the anisotropies of the cosmic neutrino background acts as a birefringent medium, although we do not expect such a process from conventional physics.7 De and Tashiro(2015) suggest that the CMB photons could become circularly polarized from propagating through a magnetized plasma by means

7 The problem of a photon beam (A) experiencing birefringence by interacting with a neutrino beam (B) is similar in concept to the photon-photon scattering problem treated in 2.5, except that a W and a charged lepton appear in the loop (Mohammadi, 2014). Without loss of§ generality, the interaction can be treated in the center-of-mass frame, where beams A and B are collinear. Under the little group of rotations that ﬁx the momentum vectors, the birefringence terms that convert linear to circular polarization in beam A (nQ and nU in the notation of our 2.2) are spin 2. If beam B consists of spin-s particles, its density matrix contains components of spin§ up to 2s; to have the needed spin 2 component, the spin of beam B must be at least 1. Thus consideration of the symmetry group allows circular polarization to be sourced when beam B is a photon beam (spin 1 1), but not when B is a neutrino beam (spin 2 ). This argument remains true regardless of the neutrino masses and PMNS matrix, and whether the neutrino has a Majorana or Dirac mass term.

30 of Faraday conversion. King and Lubin(2016) explore additional sources of circular

polarization in the CMB and conclude that Population III stars are the strongest

source. They also discuss the detectability of this signal. In addition, there has been

considerable recent eﬀort directed towards observation of circular polarization; see

Refs. Mainini et al.(2013); Nagy et al.(2017) for recent upper limits on the CMB

circular polarization.

Here we explore the possibility of circular polarization in the CMB due to cosmic

birefringence at high redshift. (Astrophysical mechanisms and foregrounds at low

redshift are outside the main scope of this paper, though we do comment on them

brieﬂy.) We restrict ourselves to conventional physics in the standard cosmological

scenario. We focus on order-of-magnitude estimates because our main goal is to

identify if an eﬀect is strong enough to require further examination; nevertheless, we

keep factors of 2, π, etc. in the derivation of the indices of refraction because such eﬀects can “multiply out” to give large factors even if these are in principle order unity. We study diﬀerent possible sources for cosmic birefringence and estimate their respective levels of circular polarization near recombination.

Conversion of linear to circular polarization can occur in a medium where the two axes (x and y) have diﬀerent indices of refraction. There are two classes of ways this can occur in cosmology. One is a medium where the structure of the matter has a preferred axis; in the gas phase, this would come from alignment of the atomic spins by an external radiation ﬁeld. We will consider this mechanism both during the recombination epoch (where the dominant alignment comes from the CMB anisotropies

The linear-to-circular conversion from an anisotropic neutrino background is thus not expected in conventional physics.

31 acting on excited hydrogen atoms through the Hα line) and in the epoch of Cos- mic Dawn (where the alignment comes from scattering of 21 cm radiation). The other involves the fact that the response of a medium to electromagnetic ﬁelds is not linear. In non-linear electrodynamics, the presence of an anisotropic radiation back- ground8 makes the medium birefringent. We consider three sources of non-linearity: the non-linearity induced by the ionized plasma component (where the non-linearity arises from the ﬁnite displacements of the electrons); the non-linearity induced by the atoms (where the non-linearity arises from the fact that the hydrogen atom potential is not a harmonic oscillator); and the non-linearity of the vacuum (photon-photon scattering, where the non-linearity arises from virtual electron-positron pairs).

Of these candidate primordial mechanisms, we ﬁnd that photon-photon scattering at recombination produces the strongest circular polarization. This mechanism has received the greatest attention in the recent past (Motie and Xue, 2012; Sawyer,

2015; Ejlli, 2016; Sadegh et al., 2017), although we have had to correct some of the calculations in the literature.

This paper is organized as follows. We start by studying some general facts of circular polarization in the CMB in 2.2. We then proceed to consider each effect § in turn; in every case, we make an order of magnitude estimate and then proceed to a more detailed calculation. We examine the effect of birefringence due to spin- polarization of the hydrogen atoms in 2.3 (for the Cosmic Dawn epoch) and 2.4 (for § § the recombination epoch). We consider photon-photon scattering at recombination in 2.5. In 2.6, we estimate the cosmic birefringence due to the static non-linear § § 8An external magnetic field with net alignment would also do – it involves the same couplings but is a DC rather than AC field. This is not present in the standard picture of the early Universe, but is relevant to secondary sources of circular polarization and foregrounds.

32 polarizability of hydrogen. Then we proceed to explore the birefringence produced

by plasma delay in 2.7. We conclude in 2.8. We use SI units throughout. § § Unless stated otherwise9 throughout this paper we used the (Plick) Planck cos-

2 mology (Planck Collaboration et al., 2016). Speciﬁcally, H0 = 67.26, Ωbh = 0.02222,

2 Ωcdmh = 0.1199, Ωm = 0.316 and zre = 8.8.

2.2 General aspects of circular polarization

In this section, we explore some aspects of circular polarization in the CMB that are common to all of the mechanisms in this paper.

Most CMB polarization is produced by Thomson scattering, which interconverts quadrupolar anisotropy and linear polarization. However, symmetry considerations prevent Thomson scattering from converting either temperature or linear polarization perturbations into circular polarization; this would have to come from propagation eﬀects. Here the eﬀect of interest is birefringence. The general description of birefringence, for light propagating on the z-axis in a medium of low density, is to introduce an index of refraction tensor,

1 n = δ + (χ + χ ), (2.1) ij ij 2 e,ij m,ij where χe and χm are the electric and magnetic susceptibilities respectively and only the x and y components of the tensor are considered (there is no longitudinal polarization). This can be decomposed as

n + n n + in n = I Q U V . (2.2) nU inV nI nQ − − 9The line of sight computation for photon-photon scattering in 2.5 relies on an older version of the Planck cosmology. §

33 H H x x H H H x x H

Figure 2.1: Two classes of cosmic birefringence sources: matter-related mechanism (left) and radiation-related mechanism (right). On the left side, the inhomogeneous matter, i.e. spin polarized hydrogen atoms, is responsible for the diﬀerence in indices of refraction. The blue arrows represent the spins of the hydrogen atoms. Here the black arrows stand for the radiation ﬁeld, which can be either the CMB or the 21 cm radiation from other hydrogen atoms. This is the mechanism used in 2.3 and 2.4. In contrast, inhomogeneous radiation produces birefringence through§§ nonlinear response on the right panel, here the red arrows stand for the CMB quadrupole and the x in the circles corresponds to the source of the nonlinearity, e.g. vacuum ( 2.5), bounded electrons ( 2.6) or free electrons ( 2.7). §§ §§ §§

34 Here nQ is (half of) the diﬀerence of indices of refraction on the x and y-axes, and

nU represents the half-diﬀerence of indices of refraction on the two diagonal axes.

The component nI is the polarization-averaged index of refraction, which is not of

interest as it induces no phase shift. Finally, nV is the diﬀerence in indices of re-

fraction between the two circular polarizations; it is non-zero only for media that

are not time-reversible (e.g. magnetized), and is responsible for Faraday rotation; it

does not convert linear to circular polarization. Since the background cosmology is

homogeneous and isotropic, nQ and nU must originate from perturbations in either

the matter or the radiation.

If we write ∆n as the diﬀerence of the two eigenvalues of n, we have a phase shift

between the two eigenvectors given by

Z ω ω ∆χ ∆φ = φx φy = ∆n drproper ∆n . (2.3) − c ≈ c 1 + z

Here ω is the proper frequency of the CMB photons, z is the redshift and χ is the comoving distance (see Table 2.1 for a glossary of physical quantities used throughout this work). Conversion from pure linear to pure circular polarization occurs in the idealized circumstance that nV is negligible, the phase shift is π/2 and the incident ± linearly polarized wave makes an angle of π/4 to the principal directions of the birefringent material. In the cosmological context, the phase shifts are 1 and instead we are concerned with conversion from pure linear polarization to mostly linear polarization with a small admixture of circular polarization. The circular polarization so induced is V = 2U∆φ Cooray et al.(2003), where U is the input diagonal linear polarization in the frame chosen to align with the principal axes of the medium (nQ > 0, nU = 0). In the more general case, where we allow anisotropy on an arbitrary axis,

35 the equation of radiative transfer Beckert and Falcke(2002), becomes

dV 2ω Z 2ω dχ = (nQU nU Q) or V = (nQU nU Q) . (2.4) drproper c − c − 1 + z

We proceed to classify the two kind of cosmic birefringence that will be treated in this work in terms of the type of mechanism that generates the anisotropy nQ, nU = 0. 6 In Fig. 2.1 we illustrate the two classes of sources: matter-related and radiation-

related. Our ﬁrst two mechanisms are matter-related and both involve the spin-

polarization of the hydrogen atoms; however, the mechanism for the alignment of the

spins is not the same for both cases. In the Cosmic Dawn era ( 2.3) the atoms are § aligned by 21 cm radiation, whereas in the recombination era ( 2.4) the spins are § aligned by Balmer line (mostly Hα) radiation. For the radiation-related cases, the

anisotropic radiation ﬁeld is always the perturbed CMB, but the nonlinear behavior

in the radiation can be generated by the vacuum by electron positron pairs as in the

case of photon-photon scattering ( 2.5), by bounded electrons in hydrogen atoms like § in the static nonlinear polarizability of hydrogen ( 2.6) or by free electrons as in the § plasma delay ( 2.7). § Having discussed the origin of the circular polarization, brieﬂy highlighted its

importance in astrophysics, and simpliﬁed our problem to identifying possible sources

of birefringence, we proceed to study spin polarized hydrogen atoms as a source of

birefringence in 2.3. § 2.3 Birefringence from spin polarized hydrogen atoms from the Cosmic Dawn epoch

In this section we will explore spin polarized hydrogen atoms as a possible source

of cosmic birefringence. Hydrogen atoms in the Cosmic Dawn epoch can become

36 Table 2.1: Glossary of physical quantities used in this chapter. Symbol SI unit Physical quantity z 1 Redshift 1 ω s− Angular frequency of the CMB photons χ m Comoving distance 1 µe JT− Magnetic moment of the electron 3 nH m− Number density of hydrogen atoms 3 nnl m− Number density of hydrogen atoms for the state nl 1 ωhf s− Angular frequency of the hyperfine transition 1 2 ωme s− Electron mass in angular frequency units, mec /~ 1 ω s− Angular frequency of the Lyman α photons Lyα − ρnm 1 Components of the density matrix jm 1 Irreducible components of the density matrix P(0) E¯n J Energy eigenvalues of the unperturbed hamiltonian 1 µi JT− Magnetic dipole moment in the i-direction F 1 Total angular momentum quantum number M 1 Total magnetic quantum number 2 1 αij C m V− Components of the atomic electric polarizability tensor m 2 αij JT− Components of the atomic magnetic polarizability tensor Q, U, V K Stokes parameters (in temperature units) T = Tγ K Temperature of the CMB Ts K Spin temperature τ 1 Optical depth Cl 1 Angular power spectrum of ∆T/T 3 Pδ m Matter power spectrum 3 Pζ m Primordial curvature power spectrum 1 H s− Hubble expansion rate 2 1 1 1 Jα m− s− Hz− sr− Flux of Lyman α photons on the blue side of the line ˜ − Sα,(2) 1 Correction factor for the Lyman-α line shape 1 Γ s− Einstein A-coefficient for the Lyman α 2p − γ2p Hz HWHM of the Lyman α line 1 − Ax y s− Einstein A-coefficient for the transition from x to y → | i | i xe 1 Ionization fraction Xi 1 Fraction of hydrogen atoms in level i 1 Λ s− Decay rate for the two-photon decay 3 1 αB m s− Case B recombination coefficient B 3 ani J m− Anisotropic magnetic energy density of the CMB U E 3 ani J m− Anisotropic electric energy density of the CMB U 2 Ae T− Euler-Heisenberg interaction constant iso χe 1 Isotropic contribution to the electric susceptibility iso χm 1 Isotropic contribution to the magnetic susceptibility ani,e χij 1 Anisotropic contribution to the electric susceptibility ani,m χij 1 Anisotropic contribution to the magnetic susceptibility sYlm 1 Spin-weighted spherical harmonic 3 4 arad J m− K− Radiation energy density constant A 2 1 αNL,ij C m V− Anisotropic non-linear components of the polarizability 3 1 d /dΩ~ J m− sr− Energy density in ambient electromagnetic waves U 4 4 γ J m V− Second-order hyperpolarizability of the hydrogen atom

37 spin-polarized in the presence of an anisotropic radiation background, which here is primarily the 21 cm radiation ﬁeld generated by other nearby atoms Venumadhav et al.(2014). Since the line is narrow and is formed in an expanding medium, the line proﬁle-weighted intensity and the probability of re-absorption depend on the

Einstein coeﬃcients, the local density, and the velocity gradient Sobolev(1960). 10

In an unperturbed universe, the resulting radiation in the 21 cm line is isotropic, but in the presence of velocity shear (direction-dependent velocity gradient) it has a quadrupole anisotropy. For pedagogical purposes we elaborate on the mechanism that leads to this anisotropic radiation in Appendix A.1. This anisotropic radiation unevenly populates the triplet state of the hyperﬁne F = 1 level as illustrated in

Figures 1 and 2 in Ref. Venumadhav et al.(2014).

Thus the 21 cm anisotropic radiation is continuously sourcing the anisotropy of the spins of the hydrogen atoms. The alignment of the hydrogen spins in steady state is the result of the balance between the aligning eﬀect (anisotropic component of the 21 cm radiation) and the randomizing eﬀect from collisions, Lyman-α radiation and the isotropic component of 21 cm radiation. Spin-polarized hydrogen is then a source of birefringence because the magnetic polarizability of a hydrogen atom in the

F = 1 level is diﬀerent on the axes parallel to and perpendicular to the total angular momentum.

As a starting point we will compute an order of magnitude estimate for both the birefringence and the phase shift generated because of this eﬀect. Note that

10The line width itself does not matter as long as it is narrow. This is because as the line profile gets narrower, there are two competing effects: (i) the path length over which the photon can be reabsorbed gets smaller; but (ii) the cross section increases over a small range of frequencies centered on ν21 as appropriate for a δ-function. Thus there is still a finite probability to reabsorb the redshifted 21 cm radiation by the surrounding neutral hydrogen atoms.

38 this mechanism only applies to hydrogen – the other abundant element in the early

Universe, helium, is a spin singlet and cannot be polarized.

2.3.1 Order of magnitude

We start with a semiclassical order of magnitude estimate of this eﬀect at 1 + z =

20. The diﬀerence in indices of refraction should be proportional to the number density of hydrogen atoms nH, the magnetic permeability and the magnetic polarizability,

∆n nHµ0∆αm. This polarizability should depend on the polarization of the hy- ∼ drogen atoms, which is described by irreducible spherical tensor components of the density matrix: using the conventions of Ref. Venumadhav et al.(2014), there is a spin-0 part (net population of the F = 1 level), spin-1 part (net vectorial polarization of the atomic spins), and a spin-2 part (“headless vector” alignment of the atomic spins). The relevant part is only the spin-2 part of the density matrix 2m, which P has the correct symmetry properties to generate nQ and nU since Q and U are spin-2 quantities.

The anisotropy of the magnetic polarizability should scale linearly with the atomic polarization: ∆n nHµ0 2m∆αm pp, where ∆αm pp is the anisotropic polarizability ∼ P | | for perfectly polarized atoms. This perfectly polarized polarizability can be approxi- mated by the change in magnetic moment from the magnetic ﬁeld in the time 1/ω, ∼ where ω is the CMB photon frequency. The change in angular momentum is µeB/ω, ∼ where µe is the magnetic moment of the electron, and the change in magnetic moment

2 5 2 is diﬀerent by a factor of µe/~. Hence ∆αm pp µe/~ω = αc 0/(ωmeω), where α ∼ | ∼ is the ﬁne structure constant, c is the speed of light, 0 is the vacuum permittivity

39 Hyperﬁne axis

S~e iωt B~ CMB e− S~p ∝

CMB axis

HH xˆ xˆ

Figure 2.2: Spin polarized hydrogen atom with CMB magnetic field. Note that the CMB field is oscillating with frequency ω, which is considerably larger than the hyperfine frequency. We have used red to illustrate the hyperfine contribution, i.e. the spin of the proton. In the absence of hyperfine structure the change in the magnetic moment is perpendicular to the CMB magnetic field, see the blue line in the right figure. Note that we show the precession of the electron spin with respect to the proton spin ignoring the CMB magnetic field in the left panel, however, it should be pointed out that the proton spin also precesses around the electron spin. We illustrated the hyperfine contribution with the red line in the right figure.

2 and ωme = mec /~ is the mass of the electron in frequency units. Then the classical

3 2 birefringence is given by ∆n αnHc /(ω ω) 2m. ∼ me × P Nevertheless, this simple estimate is incorrect, since the change in angular momentum (and hence magnetic moment) is perpendicular to the magnetic ﬁeld and

m hence the diagonal (or, more generally, symmetric) contribution to αij vanishes (see

Fig. 2.2). Hence one gets no contribution to nQ or nU . However, if we consider the coupling to the spin of the proton i.e. the hyperﬁne structure, then the electron spin would now try to precess around the proton spin, and vice versa. This interaction

40 results in a small component of the magnetic moment of the electron aligned with the

magnetic ﬁeld (see IVB of Ref. Hirata et al.(2017) for a detailed description of how § this works). Hence, the electron magnetic moment follows an ellipse with major axis sourced by the CMB anisotropies and minor axis sourced by the hyperﬁne splitting.

Therefore, one should multiply the previous estimate for the birefringence by the ratio of the axes ωhf /ω to include the correct geometrical factor. Evaluating the diﬀerent ∼ parameters at redshift 1 + z = 20,

3 αnHc −36 ∆n 2 ωhf 2m 10 , (2.5) ∼ (ωmeω) P ∼ where we have used the results of Ref. Venumadhav et al.(2014) to estimate the

−5 alignment tensor at the desired redshift of 1 + z = 20 ( 2m 10 ) and ω 6 P ∼ ≈ × 1012 rad s−1.

Applying Eq. (2.3) for a photon with λ 1 mm today (λ 5 10−5 m at ∼ ∼ × 1 + z = 20), and with a proper distance of approximately 100 Mpc (the Hubble

length at 1 + z = 20), we ﬁnd a phase shift of

∆φ 10−7. (2.6) ∼ Note that the spin alignment direction will vary along the line of sight, so this should

be thought of as an upper limit to the integrated phase shift.

2.3.2 Detailed calculation

We now turn to the detailed calculation of the CMB circular polarization induced

by propagation through spin-polarized neutral gas. There are three steps to this cal-

culation. First, we apply quantum mechanics to determine the anisotropic magnetic

polarizability of the atoms and hence index of refraction of the gas. Then we in-

voke previous computations of the spin-polarization in the Cosmic Dawn epoch to

41 relate this to the cosmological density perturbations. Finally we use the statistics

of large-scale structure (speciﬁcally the Limber equation, which deals with projected

quantities) to compute the RMS circular polarization.

In the Schr¨odinger picture, the Hamiltonian for a hydrogen atom in an oscillating magnetic ﬁeld is given by

(0) (1) (0) −iωt ∗ iωt = + = B0jµje B µje , (2.7) H H H H − − 0j where ω is the frequency from the CMB photons and µj is the magnetic dipole moment.

We suppose that the unperturbed state of the atom is Ψ(0) , and that this is an | i (0) eigenstate of with energy E0. Using ﬁrst-order perturbation theory, we ﬁnd that H the perturbation to the state is

−iωt ∗ iωt (1) X B0je B0je (0) Ψ = + n µj Ψ , (2.8) | i (0) (0) h | | i n En E0 ~ω En E0 + ~ω − − − (0) where En is the unperturbed energy of state n . The perturbed dipole moment is | i −iωt (0) (0) X (0) (0) B0je µi = Ψ µi Ψ + Ψ µi n n µj Ψ h i h | | i h | | ih | | i (0) n En E0 ~ω − − ∗ iωt B0je + (0) + c.c., (2.9) En E0 + ~ω − where “c.c.” indicates a complex conjugate. We can then identify the magnetic

m −iωt polarizability tensor at positive frequency, αij , as the coeﬃcient of B0je in µi:

(0) (0) (0) ∗ (0) ∗ m X Ψ µi n n µj Ψ Ψ µi n n µj Ψ αij = h (0)| | ih | | i + h | (0)| i h | | i . (2.10) n En E0 ~ω En E0 + ~ω − − − Finally, we are interested in the average response over an ensemble of atoms in the

F = 1 level, described by a density matrix ρMM 0 . To carry out this average, we make

42 the replacement Ψ(0) Ψ(0) ρ: | ih | → " 0 m X ρMM 0 1,M µj n n µi 1,M αij = h (0) | (0)| ih | | i n,M,M 0 En EF =1 ~ω − − 0 ∗ ∗ # ρM 0M 1,M µj n n µi 1,M + h (0) | |(0)i h | | i . (2.11) En EF =1 + ~ω − The magnetic dipole moment has three main ingredients – contributions from orbital motion, electron spin and nuclear spin:

e e eg e µ = L S + p I S, (2.12) −2me − me 2mp ≈ −me

where we have dropped the ﬁrst term since we are working with electrons in s-orbitals

and the third term since mp me. Using Eq. (2.12) and the Clebsch-Gordan co- eﬃcients, we can construct any of the matrix elements needed for the polarizability

tensor.

(0) (0) We furthermore use that EF =0 EF =1 = ~ωhf . Then − − 2 m e ~ 1 1 αxx 2 + (ρ11 ρ−1,1 + ρ−1,−1 ρ1,−1). (2.13) ≈ 8m ωhf ω ωhf + ω − − e − − −

Taking the limit of ω ωhf , the xx component of the susceptibility is given by 2 e µ0 ωhf χxx nH 2 2 (ρ11 ρ−1,1 + ρ−1,−1 ρ1,−1). (2.14) ≈ 4me ω − − Similarly, 2 e µ0 ωhf χyy nH 2 2 (ρ11 + ρ−1,1 + ρ−1,−1 + ρ1,−1). (2.15) ≈ 4me ω At last, we can compute the birefringence using Eq. (2.14) and Eq. (2.15):

3 3 αnHc ωhf 2π αnHc ωhf 2nQ nxx nyy 2π 2 2 (ρ−1,1+ρ1,−1) = 2 2 ( 22+ 2,−2), (2.16) ≡ − ≈ − ωmeω −√3 ωmeω P P

where ωme is the mass of the electron in frequency units, and in the last step we

changed from the density matrix to the irreducible components of the density matrix

following Ref. Venumadhav et al.(2014) (see Appendix A.2).

43 The alignment tensor for hydrogen atoms in the range of redshifts we are currently

interested in is given in the Fourier domain Venumadhav et al.(2014) as

r ˆ 1 T∗ T τ ˆ 4π ˆ 2m(k) = 1 δm(k) Y2m(k), (2.17) P 20√2 T − Ts 1 + xα,(2) + xc,(2) 5

where kˆ is the direction of the wavevector of the radiation, τ is the optical depth of

11 the neutral hydrogen gas, Ts is the spin temperature , T∗ = 68 mK is the hyperfine splitting in temperature units, δm is the density contrast and the xx parametrize the rates of depolarization by optical pumping and collisions. We neglect the primordial magnetic field term since we are focusing specifically on conventional physics in this study. The derivation of Eq. (2.17) is presented in Ref. Venumadhav et al.(2014), but the reader who wishes to follow the basic ingredients without as much mathematical formalism may consult the abbreviated treatment in Appendix A.1.

We may now compute the phase shift using Eq. (2.3):

dφ c dφ = d ln a aH dχ ω = (nxx nyy) H − 3 2π αc ωhf nH = 2 ( 2,2 + 2,−2) −√3 ωme ωH P P 1/2 −7 1 + z 2,2 + 2,−2 100 GHz = 1.4 10 P −P5 , (2.18) − × 20 10 νtoday

where we obtain our estimate for the alignment tensor from the expression computed

by Venumadhav et al. in Venumadhav et al.(2014). Note that the pre-factor in

Eq. (2.18) is an over-estimate of the net phase shift since the sign of 2,m will change P along the line of sight.

11The spin temperature parametrizes the diﬀerence of population of hydrogen atoms in the hy- perﬁne F = 1 and F = 0 levels.

44 To obtain the net circular polarization due to the passage through the neutral

medium, we need to write out the full line-of-sight integral and then perform a sta-

tistical study. Therefore in what follows we are no longer constrained to a single

redshift. The conversion integral, Eq. (2.4), can be written as

Z Z 2ω0 2ω0 V = UnQdr QnU dr. (2.19) c − c

Substituting in Eqs. (2.16) and (2.17), we ﬁnd that

Z ˆ ˆ V = UφQ QφU , where φQ = p W (r)[Y22(k) + Y2,−2(k)]δm(rnˆ)dr (2.20) − ˆ (and similarly for φU ). Here k is the direction of the wave-vector of the radiation and

2 2π αω0c ωhf p = − 2 and (2.21) √3 ωme r 1 4π nH T∗ TCMB τ W (r) = 2 1 . (2.22) 20√2 5 ω TCMB − Ts 1 + xα,(2) + xc,(2)

[Note that Eq. (2.20) contains both a real-space line of sight integral and density ˆ ﬁeld, and a Fourier-space operator, Y2m(k), that depends on the direction of the

Fourier modes. The latter is intended to operate on the density ﬁeld in the sense

of taking the inverse Fourier transform, multiplying by the stated function of k, and

Fourier-transforming back.]

The optical depth for resonant 21 cm absorption is given by Eq. (87) of Venumad- hav et al.(2014)

2 3 3 π c nH(z)Ahf (3 4 00(z)) 3c nH (z)Ahf T∗ τ(z) = 3 − P 3 , (2.23) H(z)ωhf [1 + (1/H(z))(dvk/drk)] ≈ 4ωhf H(z) Ts

where the Ahf is the Einstein coeﬃcient and 00 3/4 3T∗/(16Ts) is the occupancy P ≈ − of the excited state. Note that the Hubble ﬂow in the denominator indicates the

path length available for a 21 cm photon to resonate with nearby neutral hydrogen

45 atoms before its frequency is shifted out of resonance by the Hubble ﬂow. Finally, τ

is used here multiplying a perturbation, so to linear order we may use its value in the

unperturbed Universe.

The total variance of V is related to that of the phase shifts via

2 2 2 2 V = (UφQ QφU ) = 2 Q φ , (2.24) rms h − i h ih Qi

where we have used the symmetry relations that Q2 = U 2 , QU = 0, and similarly h i h i h i for the phase shifts.

Here Q2 is simply the linear polarization of the primary CMB; it is given by h i 1 Z d` Q2 = `2(CEE + CBB) . (2.25) h i 4π ` ` `

The variance of the conversion angle φ2 can be obtained from its power spectrum: h Qi Z φQ 2 2 ` dr ˆ 2 C = p W (r)Pδ k = G(`) , (2.26) ` r r2 | |

ˆ ˆ where Pδ is the matter power spectrum, ` is the direction in the sky, and G(`) = ˆ ˆ Y22(`) + Y2,−2(`) with Y`,m are the spherical harmonics, see Appendix A.3. Normally

φQ we average C` over azimuthal directions, so we make the replacement

1 Z 2π 2 15 ˆ 2 π G(`) G , φ dφ = . (2.27) | | → 2π 0 2 16π

Changing variables from proper distance to redshift using dr = c dz/H(z) gives

Z φ 15 ` c dz C Q = p2 W 2(r)P k = , z , (2.28) ` 16π δ r r2 H(z) and hence

Z Z Z 2 1 2 φQ d` 15 2 2 2 ` dk c dz φ = ` C = p W (r) k Pδ k = , z . (2.29) h Qi 2π ` ` 32π2 r k H(z)

46 300 0.5 0.45

2 250 0.4 K] µ 0.35 200 0.3 ) [ BB l 150 0.25 0.2 P(k) [Mpc] + C 2 EE l 100 k 0.15

( C 0.1 2

l 50 0.05 0 0 0.0001 0.001 0.01 0.1 1 10 10 100 1000 k [Mpc-1] l

Figure 2.3: Integrands of Eq. (2.30). Essentially, the left ﬁgure represents the contribution of the linear polarization of the CMB to the rms of the circular polarization. Note that the peak of the distribution is approximately at ` = 1000. In the right panel we have evaluated the power spectrum at z = 20. This is the contribution due to the spin polarized hydrogen atoms. Note that the peak of the distribution is near k 0.1 Mpc−1. ∼

Combining this with Eq. (2.25), we conclude that the RMS circular polarization is

15p2 Z d` Z Z ` dk c dz V 2 = `2(CEE + CBB) W 2(r) k2P k = , z . rms 64π3 ` ` ` δ r k H(z) (2.30)

Here in principle the integrals over ` and k range over all scales, although they are dominated by the peak of the CMB polarization power spectrum (` 103) and the ∼ k2-weighted matter power spectrum (k 0.5 Mpc−1), respectively. This is shown in ∼ Figure 2.3. The redshift integral extends over the Cosmic Dawn epoch – from the

beginning of the Lyman-α coupling (when Ts drops below Tγ) through reionization

(when there are no more neutral hydrogen atoms); here we take the range 8.8 < z <

34.

47 5

0 ] 2 s

-3 -5

[cm -10 36

-15 W(z)*10 -20

-25 10 15 20 25 30 35 z

Figure 2.4: Window function, note that we multiply the y axis by a factor of 1036 for convenience.

We ﬁrst focus in the `-integral. We obtain the angular power spectrum for the

linear polarization from CLASS Blas et al.(2011),

15p2 Z Z ` dk c dz V 2 = 252 (µK)2 W 2(r) k2P k = , z . (2.31) rms 64π3 δ r k H(z)

In order to compute the RMS of the circular polarization from Eq. (2.31) we employ 21cmFAST Mesinger and Furlanetto(2007); Mesinger et al.(2011), with the

Plick cosmology from Planck and mostly all the default parameters from the code12,

to directly obtain the spin temperature and the Lyman-α ﬂux (which is needed for

the rate of depolarization by optical pumping) as functions of redshift. Then one can

compute the optical depth of the neutral hydrogen gas with Eq. (87) of Venumadhav

et al.(2014) (i.e. Eq. (2.23)). Moreover, using Eqs. (98-99) of Venumadhav et al.

(2014) and the Lyman-α ﬂux, we compute the rate of randomizing spins by collisions

12We use Population III stars as the sources for early heating.

48 xc,(2) and Lyman-α photons xα,(2). In addition, the matter power spectrum was

extracted from CLASS with the Planck cosmological parameters (see Figure 2.3)

also we plotted the window function in terms of redshift in Figure 2.4. We get

−16 Vrms = 3.1 10 K. (2.32) ×

Our resulting circular polarization amplitude may be sensitive to the reheating

and reionization history chosen in the 21cmFAST realization, since these control the

window function W . The peak of W occurs at the era when the Lyman-α coupling | |

turns on (xα of order 1) since the window function is suppressed both in the limit of ∼

xα 1 (where Ts TCMB) and the limit where xα 1 (where Ts and τ are constant ≈

but xα,(2) is large). In our ﬁducial model this happens at z = ztrans 17, and this ∼

is before the redshift zheat when X-ray heating is signiﬁcant. If we vary the Lyman-

α coupling transition redshift ztrans, but retain the assumption that ztrans > zheat

2 – so that the gas kinetic temperature Tk (1 + z) in accordance with adiabatic ∝ 1/2 expansion – then we have W (ztrans) (1 + ztrans) , so over reasonable (factor of ∝ 2) variations in the Lyman-α coupling redshift there are only minor changes in the ∼ implied window function and circular polarization properties. If X-ray heating took

place before Lyman-α coupling, then 1 TCMB/Ts is suppressed (it may ﬂip sign, but −

since 1/Ts is bounded, 1 TCMB/Ts can never be as strong in emission as it is for | − | unheated gas in absorption), and also τ is suppressed – thus the window function W

can be suppressed. We thus conclude that it would be diﬃcult to increase the circular

polarization signal by more than a factor of a few with standard physics13, but with

13The EDGES experiment Bowman et al.(2018) has recently reported a stronger absorption dip in 21 cm radiation than would be expected even for no X-ray heating and Ts Tk (but see, e.g., Ref. Hills et al.(2018)). If conﬁrmed, this would imply a larger optical depth ≈τ and greater circular polarization than conventional scenarios. Since this paper is focused only on mechanisms

49 X-ray heating it could be signiﬁcantly suppressed. In any case, the Cosmic Dawn

circular polarization signal is small both compared to potentially observable signals

and to other sources of circular polarization.

Having done the analysis of the circular polarization produced by the spin polar-

ized hydrogen atoms at the Cosmic Dawn epoch, 8.8 < z < 34; we proceed to explore

the same eﬀect near recombination, 1 + zrec 1000. ∼ 2.4 Birefringence from spin polarized hydrogen atoms at recombination

Birefringence from spin-polarized atoms should exist not just during reionization

but also during the epoch of recombination. The main diﬀerence is that the spin

alignment is not coming from the 21 cm radiation, but rather from the much more

intense Balmer radiation (primarily Hα) present during recombination.

2.4.1 Order of magnitude

Aside from the actual mechanism behind the alignment of the spins, which is

mainly present in the expression for 2m, the dependence of the cosmic birefringence P on the physical parameters from the last section should remain intact. Thus we can use Eq. (2.5) here with only two caveats. First, redshift factors must be evaluated at recombination, and we must construct an expression for the alignment tensor

3 αnHc ωhf nx ny 2 2m. (2.33) − ≈ (ωmeω) P

The main issue is how to estimate 2m. This is determined by a balance between the P isotropic component of the radiation bath, which acts to randomize the orientations that operate in the conventional cosmological model without adding new physics, we do not consider this further here.

50 of hydrogen spins (with a rate R) and the anisotropic components, which act to align the spins (with a rate S2m):

˙ 2m = R 2m + S2m. (2.34) P − P

The major randomizing processes that we would consider are Lyman-α scattering and

21 cm absorption/emission.14 The rate of Lyman-α scattering per hydrogen atom is

similar to the 2p 1s decay rate, since 2p is mostly populated and de-populated via → 9 −1 −14 −5 −1 Lyman-α: R(Lyα) ALyαx2p (10 s )(10 ) 10 s . The rate of 21 cm ∼ ∼ ∼ absorption/emission is the natural rate 3 10−15 s−1 enhanced by the stimulated ×

emission factor for the Rayleigh-Jeans tail of the CMB, Tγ/T? (3000 K)/(68 mK) = ∼ 4 104: thus R(21 cm) 10−10 s−1. Thus, in terms of spin-randomizing processes, × ∼ we can neglect the 21 cm interaction in comparison with Lyman-α.

Let us now turn to the contributions to the alignment, S2m: one can expect

the atoms to be polarized if the radiation ﬁeld in any of the major atomic lines is

anisotropic. We would ﬁrst consider the Lyman series, since this interacts directly

with the ground state, however the optical depth is so large (of order 109 for Lyman-

α) that the radiation is extremely isotropic. Thus we instead consider the Balmer

series, which is optically thin and has anisotropies equal to the continuum background

−5 radiation, i.e. of order Θ2m 10 . At z 1100, the CMB blackbody peak is at λ 1 ∼ ∼ ∼ µm, and hence of the Balmer lines, Hα will dominate the atom-CMB interactions.

The alignment mechanism would be 2s 3p 1s (see Fig. 2.5). The alignment rate → →

−hνHα/kBT 7 −1 −14 −3 −5 −15 −1 would be S2m AHαx2se Θ2m (10 s )(10 )(10 )(10 ) 10 s . ∼ ∼ ∼ 14Rayleigh scattering does not ﬂip the hydrogen spins, since unlike Lyman-α scattering it is non- resonant. Thus the energy denominators 1/(Einit + hν Eexc) in the scattering amplitude are approximately the same for all spin states, and the scattering− amplitude is a Kronecker delta in the spin states. A similar argument applies to the 1s 2s two photon transition. −

51 We should also consider the anisotropy in the 21 cm line, coming from the Rayleigh-

Jeans tail of the CMB anisotropies. If the CMB has an anisotropy Θ2m, then we would ˙ expect that in steady state (i.e. setting 2m = 0 in Eq. 2.34) and considering only the P

21 cm line, that the solution would be a spin temperature that diﬀers by ∆Ts = Θ2mTs depending on which excited state is used to deﬁne the spin temperature. This leads

−5 to 2m(steady state) Θ2mT?/Ts 2 10 Θ2m (since Ts Tγ in the recombination P ∼ ∼ × ≈ −5 −20 −1 epoch), and hence S2m(21 cm) 2 10 R(21 cm)Θ2m 2 10 s . Thus we are ∼ × ∼ × justiﬁed in ignoring alignment by 21 cm in comparison to Hα.

Figure 2.5 shows the key radiative processes involving hydrogen atoms in energy

levels up through the third shell. We have seen that the dominant randomizing term

(R in Eq. 2.34) is through Lyman-α scattering with R 10−5 s−1, while the dominant ∼

aligning term (S2m) involves Hα absorption by the n = 2 level, with the n = 3

−15 −1 intermediate level, followed by Lyβ emission to the n = 1 level, here S2m 10 s . ∼ −10 This leads to 2m 10 . We would then obtain a birefringence of P ∼

−40 nx ny 10 , (2.35) − ≈

where we have evaluated the redshift dependent parameters at 1 + z = 1000, namely

8 −3 15 −1 nH 2 10 m and ω 10 rad s (approximate angular frequency of a CMB ≈ × ∼ photon). Using Eq. (2.3) for a photon of 1 µm at the desired redshift, and a proper

path length of the horizon size at recombination 100 kpc (physical), we would have ∼ a phase shift of

∆φ 10−12. (2.36) ≈

52 3s 3p 3d

L 2s β 2p

2γ Lα 1s

Figure 2.5: Relevant processes for estimating the alignment tensor. The contribution from Lyman-β and Lyman-α are represented by solid lines. The dashed lines represents the Hα transitions. The 2γ transition will play a minor role in the Peebles model calculation.

2.4.2 Detailed calculation

We now perform a more detailed calculation of the circular polarization induced to the CMB by spin-polarized neutral hydrogen atoms at recombination, with an emphasis on estimating the alignment tensor. The key radiative processes are shown in Fig. 2.5; we note that 2m is both sourced by the CMB (hence Hα) anisotropy, P but that there is also the randomizing eﬀect of Lyman-α scattering.

First, let us deal with the randomization by Lyman-α photons. From Venumadhav et al.(2014), we ﬁnd that

˙ 2 ˜ 2m Lyα = 0.601 6πλ γ2pSα,(2)Jα 2m, (2.37) P | − × Lyα P

˚ where λLyα = 1215 A is the wavelength of a Lyman-α photon, Jα is the ﬂux of ˜ Lyman alpha photons on the blue side of the line, Sα,(2) corresponds to correction factors for the detailed frequency dependence of Lyman-α ﬂux (we approximate this

53 as 1 since the corrections are small at high temperature), and γ2p is the half-width at

half maximum (HWHM) of the 2p states, given by

Γ2p 7 −1 γ2p = 5.0 10 s , (2.38) 4π ≈ × where Γ2p is taken to be the natural width (i.e. the Einstein coeﬃcient for 2p 1s → decay). We use the cosmological recombination code HyRec Ali-Ha¨ımoud and Hirata

−2 −1 −1 −1 (2011) to compute the Lyman-α ﬂux, Jα = 0.353 m s Hz sr at 1 + z = 1000.

Therefore, the randomizing eﬀect is given by

˙ −6 −1 2m Lyα (2.97 10 s ) 2m. (2.39) P | ≈ − × P

The most direct aligning process is a combination of Hα excitation and Lyman-β decay: 2s 3p 1s. As previously noted, the “source”S2m for this process involves the → → abundance of atoms in 2s; the excitation rate to 3p (the Einstein coeﬃcient A3p→2s times the photon phase space density e−hνHα/kBT ); the branching fraction for 3p decays to 1s, A3p→1s/(A3p→1s + A3p→2s); and the CMB anisotropy Θ2m. We also expect a factor of hνHα/kBT in the Wien tail for conversion of the fractional temperature perturbation into a fractional anisotropy perturbation. The hardest part is the numerical pre-factor, which contains a long chain of Clebsch-Gordan coeﬃcients, and is computed in Appendix A.4. Incorporating this into Eq. (2.39) gives:

d −6 −1 1 A3p→2sA3p→1s (x1s 2m) = (2.97 10 s ) 2m + x2s Θ2m dt P − × P 16√10π A3p→1s + A3p→2s hνHα e−hνHα/kBT . (2.40) × kBT

Note the small pre-factor √1 0.011 in the source term for polarized atoms, 16 10π ≈ which arises from “order unity” eﬀects that turn out to be small. We note that

there are other excitation chains that start with anisotropic Hα radiation and end

54 with polarized atoms in 1s (e.g. 2p 3d 2p 1s; 2s 3p 2s 1s+2γ) but we have → → → → → → considered here only the most direct chain since there is some de-polarization at each step and we expect even smaller contributions from longer chains that end in 1s. We also ignore Hβ and higher lines as they are farther into the Wien tail (hence less abundant) than Hα. We will thus proceed using Eq. (2.40).

Now we only need to obtain x2s to determine the strength of the alignment. The fraction of hydrogen atoms in the 2s level can be computed using the Peebles model of recombination Peebles(1968):

2 −~ωLyα/T kB αBnHxe + (Λ + Λα)x1e x2 = 4 , (2.41) Λ + Λα + 4β

−1 where ωLyα is the frequency of the Lyman-α photons, Λ = 8.2 s is the decay rate for the two-photon decay, xe 0.049423 is the ionized fraction at z = 1000 obtained using ' −1 HyRec,Λα 20.2 s is the Lyman-α decay rate with optical depth suppression, ≈

αB is the case B recombination coeﬃcient, and β is the thermal photoionization rate from excited states. Then

−15 x2 5.209 10 . (2.42) ≈ ×

1 −15 Hence the fraction of hydrogen atoms in the 2s level is x2s x2 = 1.302 10 . ≈ 4 × We thus rewrite Eq. (2.40) as

d −6 −1 −13 −1 (x1s 2m) = (2.97 10 s ) 2m + (7.29 10 s )Θ2m. (2.43) dt P − × P ×

We furthermore ignore the left hand side since the change with time is small (probably

−13 −1 of order H 2m, where the Hubble rate H 10 s ), so the alignment tensor is given P ∼ −12 by 2m 2.45 10 . P ≈ ×

55 Finally, we follow the steps used in Eq. (2.18) to compute the relative phase shift

per Hubble time generated by the spin-polarized hydrogen atoms at recombination:

dφ ω = (nxx nyy) d ln a H − 3 2π αc ωhf nH = 2 ( 2,2 + 2,−2) √3 ωme ωH P P 1/2 −13 1 + z 2,2 + 2,−2 100 GHz = 1.9 10 P P −12 . (2.44) × 1000 2.45 10 νtoday × Note that Eq. (2.44) is not in agreement with the initial order-of-magnitude estimate, since it turns out that Lyman-α scattering lowers the atomic polarization to a

level far below the CMB anisotropies. Also, the numerical pre-factors in the transition

from 2s 3p 1s signiﬁcantly lower the expected signal. Converting the above phase → → shift into circular polarization we obtain V 5.2 10−19 K. This value is extremely ≈ × small even in the context of the other signals studied in this paper. Note that in

comparison to the signal from spin polarized atoms at low z this signal is a factor of

1000 smaller.

2.5 Photon-photon scattering

In this section photon-photon forward scattering is treated as a possible source

of cosmic birefringence. This is the non-linear behavior of the vacuum that emerges

from the creation and annihilation of virtual electron-positron pairs (see Fig. 2.6).

We expect this process to be most important shortly after recombination, when the

radiation density is high.

2.5.1 Order of magnitude

We start using order of magnitude and dimensional analysis to obtain a rough es-

timate for the birefringence and subsequent phase shift of the vacuum. The diﬀerence

56 Figure 2.6: Photon-photon scattering, γγ γγ. The case of interest to us is where the two ingoing photons and two outgoing photons→ occupy the same pair of wavenumbers, so that the diagrams can interfere with free propagation and result in a phase shift.

in indices of refraction should be proportional to the anisotropic energy density of the CMB ani. Also, since the lowest-order photon-photon scattering diagram has 4 U 2 4 vertices and 4 electron propagators, we expect that it is proportional to α /me. The

factors of ~ and c can be inferred from dimensional analysis:

2 3 α ~ −35 nx ny 2 ani 10 , (2.45) − ≈ mec mec U ∼

where we have obtained the anisotropic energy density by multiplying the usual

−5 blackbody energy density evaluated at 1 + z = 1000 by a factor of 10 ( ani = U 4 10−7 J m−3). We have written this equation with the reduced Compton wave- ×

lengthλ ¯C = ~/(mec) for emphasis. By means of Eq. (2.3), and for a photon of 1 µm at 1 + z = 1000 with a path

length of 100 kpc physical, we ﬁnd a phase shift of

∆φ 10−7. (2.46) ∼

57 2.5.2 Detailed calculation

To obtain a more accurate estimate, we use the Euler-Heisenberg Lagrangian

equations of motion to construct an expression for the birefringence via the vacuum

current. We are interested in the eﬀect of photon-photon scattering in two beams of

radiation with electric ﬁelds EA and EB, and magnetic ﬁelds BA and BB, respectively.

The focus of attention is in the eﬀective index of refraction that aﬀects beam A, i.e.

the birefringence felt by A in the presence of B.

The Euler-Heisenberg Lagrangian (see e.g. Ref. Zavattini et al.(2012)) is

" 2 2 2# Ae E 2 E B EH = 2 B + 7 · , (2.47) L µ0 c − c

2 3 2 where Ae 2α λ¯ /(45µ0mec ). The equations of motion, after including the classical ≡ C EM Lagrangian, are

( 2 ) E 2 ρvac = 0∇ E = 0Ae 2 ∇ B E 7 [∇ (E B)] B (2.48) · − · c2 − − · · and

( 2 1 ∂E 2 E 2 Jvac = ∇ B 0 = Ae ∇ 2 B B µ0 × − ∂t −µ0 × c −

2 ) ∂ E 2 ∂ +70∇ [(E B)E] 20 B E 70 [(E B)B] , (2.49) × · − ∂t c2 − − ∂t ·

where “ρvac” and “Jvac” denote the charge and current densities associated with the

polarization of the vacuum.

We next treat the vacuum polarization as a perturbation; the unperturbed electric

ﬁeld is then the incident ﬁeld of the two interacting photon beams A and B:

i(kA·r−ωAt) i(kB ·r−ωB t) E = EAe + EBe + c.c., (2.50)

and similarly for the magnetic ﬁeld.

58 One can use the current to obtain the birefringence that aﬀects beam A. The

relevant terms are those with the same spatial and time dependences as beam A, i.e.

ei(kA·r−ωAt). These will also have dependences on the ﬁeld amplitudes of the form ∝ ∗ ∗ EAEBEB or EABBBB. Furthermore, one can split the vacuum current into isotropic and anisotropic contributions:

( iso A A A ani,m A ) χ ijkk B + ijkk χ B i(kA·r−ωAt) iso A ani,e A m j k j kl l Ji ie 0ωA(χe Ei + χij Ej ) + . 3 − µ0 (2.51)

Expanding the electric and magnetic ﬁelds in Eq. (2.49), and keeping only the relevant

terms for birefringence we obtain the expressions for the anisotropic susceptibilities

A χani,e e 4 EBEB∗ + EBEB∗ + 7 BBBB∗ + BBBB∗ and ij ' − c2 j i i j j i i j ani,m B B∗ B B∗ B B∗ B B∗ χ Ae 4 B B + B B + 7 E E + E E , (2.52) ij ' j i i j j i i j where ... indicates an average over the B beam when we replace the monochromatic plane waves with realistic stochastic background radiation.

The anisotropic part of the index of refraction is then given by

1 ani,m ani,m ani,e ani,e nxx nyy = χ χ + χ χ − 2 xx − yy xx − yy 1 B B∗ B B∗ B B∗ B B∗ = 3Ae E E E E B B B B . (2.53) c2 x x − y y − x x − y y In contrast to the other circular polarization sources, here we will take into consideration the contribution due to both temperature and polarization, since the contribution of temperature anisotropies vanishes. Hence

B B∗ = B B∗ + B B∗ , (2.54) Xi Xj Xi Xj Temp Xi Xj Pol where X E,B (no cross terms). Since the temperature contribution is given by ∈ { } ! Z d 1 kBkB B B∗ = i j 2Ω, (2.55) Xi Xj Temp CX U δij 2 d S2 dΩ 2 − kB

59 2 with CE = c µ0/2 and CB = µ0/2, the contribution due to temperature anisotropies

to Eq. (2.53) vanishes because the magnetic and electric terms cancel identically.

However, the polarization contribution does not cancel. In what follows we adopt

the polarization vectors used in Ref. Venumadhav et al.(2014):

1 ˆ ˆ ± = (θ iφ). (2.56) ∓√2 ±

We expand the electric and magnetic ﬁelds using annihilation and creation operators:

Z 3 r d k X Uk EB(x, t) = ei(k·x−ωt)a (k) α + h.c. and i (2π)3 2 α i α 0 Z 3 r d k X Uk BB(x, t) = ei(k·x−ωt)a (k)[kˆ α] + h.c. , (2.57) i (2π)3 2 c2 α i α 0 × where Uk = ~ck represents the energy of the photon with wavevector k. The annihilation and creation operators obey the standard commutation relations.15 The photon

occupation, including polarization, is described by a phase-space density matrix fαβ:

† 0 3 0 a (k), aβ(k ) = (2π) δ(k k )fβα(k). (2.58) h α i −

This can be decomposed in Stokes parameters (see Appendix A.5 for a brief review

on Stokes parameters); again we follow the normalization in Ref. Venumadhav et al.

(2014): fI + fV fQ ifU fαβ = − − . (2.59) fQ + ifU fI fV − − In order to get the polarization contribution, we need the linear polarization terms

fQ and fU of beam B (i.e. the background CMB). In our case, beam A is propagating

on the z-axis. The remaining linear polarization contribution to the birefringence has

15 3 We normalize these so that [a (k), a† (k0)] = (2π) δ(k k0)δ and [a (k), a (k0)] = α β − αβ α β [aα† (k), aβ† (k0)] = 0.

60 the form

B B∗ B B∗ 2 E E E E 2fQ(1 + cos θ) cos 2φ 4fU cos θ sin 2φ and x x − y y ≈ − B B∗ B B∗ 2 B B B B 2fQ(1 + cos θ) cos 2φ + 4fU cos θ sin 2φ. (2.60) x x − y y ≈ −

Using the previous relations we can ﬁnally construct an expression for the anisotropic

index of refraction:

r Z r Z 48Ae π Uk 2 2ˆ h n o nxx nyy 2 k dk d k fQ Re 2Y22 + 2Y2,−2 − ' 0c 5 2 n oi +fU Im 2Y22 +2 Y2,−2 r π 4 E 96 Aeµ0aradT Re a , (2.61) ≈ 5 CMB 22

E where sYlm are the spin-weighted spherical harmonics (see Appendix A.6), a2m are the local quadrupole moments (` = 2) of the E-mode polarization, and we have used

E E∗ the simpliﬁcation that a22 = a2,−2. The rate of change of relative phase per Hubble time due to photon-photon scattering is given by

dφ ω = (nxx nyy) d ln a H − r 2 3 4 128π π α λ¯C νaradTCMB E = 2 Re a22 15 5 mec H ν 1 + z 7/2 Re aE = 8.7 10−8 today 22 , (2.62) × 100 GHz 1000 10−6

where we have estimated the contribution due to the local quadrupole of the polar-

ization with the expected CMB polarization at recombination, i.e of order 10−6, and

substituted in Ae.

Note that Eq. (2.62) is consistent with Eq. (2.46). However, Eq. (2.62) is due to

the local polarization anisotropies in contrast to the birefringence sourced by tem-

perature anisotropies used in the order of magnitude section. The reason is that the

61 temperature anisotropy contribution to the birefringence vanishes, which could not

have been guessed from order-of-magnitude arguments; instead only the local po-

larization anisotropies generate birefringence. Further, note that the more detailed

computation includes the dependence in the speciﬁc spin weighted spherical harmonic

that causes the birefringence.

The signal from photon-photon scattering is the strongest source of circular polar-

ization in the CMB studied in this work (although it is small), hence there is value in

asking what will happen to our signal if we do the full integral over the line of sight

and the Fourier modes contributing to both the initial linear polarization and the

birefringence, and including geometrical suppression factors. This is particularly im-

portant since – as discussed in 2.5.3 – some previous results have been too optimistic § because of order-unity factors.

As mentioned before, this section deviates from the cosmology used in the rest of the paper because it relies on extensive computations performed in an older cosmological model. Here we used the 2013F(CY) cosmology from Ref. Planck Collaboration

2 2 et al.(2016), namely Ω bh = 0.02230, Ωcdmh = 0.1188, H0 = 67.8, Ωm = 0.308.

The difference in cosmology should have a tiny effect since these parameters differ by

< 3% from those used in the rest of this paper.

In order to analyze the evolution of this signal we use our result for the birefringence of photon-photon scattering Eq. (2.61) in Eq. (2.4) as we did in the previous line of sight computation for the birefringence from spin polarized hydrogen atoms from the Cosmic Dawn ( 2.3). Our objective is the same – to compute Vrms – but here we §§ focus ﬁrst on computing the power spectrum of the circular polarization since now both the source of linear polarization and the birefringence are coming from z 1000 ∼

62 and hence are strongly correlated (so Vrms cannot be obtained by pointwise multi-

plication of two independent ﬁelds). We start by writing the CMB polarization as

E E P = Q + i U, and by approximating nQ U nU Q Re a Im P Im a Re P . − ' 2,−2 − 2,−2

(We have used rotational symmetry to infer that nU should be derived from nQ by

E E replacing cos 2φB with sin 2φB, i.e. by replacing Re a2,−2 with Im a2,−2.) Then we can

obtain the new circular polarization of the CMB at a comoving distance s0 from the

last scattering surface as

Z s0 4 E ∗ V A ds (1 + z) Im a2,−2 P , (2.63) ' 0

E where a2m and P are evaluated at position s along the line of sight, and we extract all non-integrating parameters to deﬁne

r π 4 −1 −38 ν0 −1 A 96 Aeµ0aradT c ω0 = 1.11 10 m . (2.64) ≡ 5 0 × 100 GHz

We write both the polarization and the local quadrupole moments in Fourier space.

In what follows we use the distant observer approximation (kr 1) and assume that the surface of last scattering is narrow. Under these conditions we can rewrite both

the quadrupole moments and the polarization, which are spin-2 quantities, by rotating

them with the help of spherical harmonics. Then, for the local quadrupole moment

in Eq. (2.63) we have

r 4π aE∗ (k ) = D2∗ (φ , θ , 0)¯aE∗(k ) = Y ∗ (kˆ )¯aE∗(k ). (2.65) 2,−2 1 −2,0 k1 k1 20 1 5 22 1 20 1

Here un-barred quantities are represented in the line-of-sight frame (with the z- ˆ direction toward the observer), and barred quantities are the wave vector frame (k1

in thez ¯-direction and the direction to the observer in thex ¯z¯-plane, withx ¯ < 0), and

63 ` 16 Dm0,m(α, β, γ) is the usual Wigner rotation matrix Wigner(2012). We have used

E the fact that we have scalar perturbations, so of thea ¯2m, only m = 0 is non-zero, hence we do not need a summation over m in Eq. (2.65). Note that the k1 mode is responsible for birefringence.

We now need the CMB linear polarization for light propagating in the direction eˆz (i.e. toward the observer) from the last scattering surface. Assuming the last scattering surface was narrow, this will be dominated by the quadrupole (` = 2) just after last scattering. We then write:

2 r X 5 P (k , eˆ ) = aE (k ) Y (eˆ ) = aE (k ) 2 z 2m 2 2 2m z 4π 2,−2 2 − m=−2 − r 5 2 E ˆ E = D (φk , θk , 0)¯a (k2) = Y22(k2)¯a (k2) , (2.66) − 4π −2,0 2 2 20 − 20

where barred frame is now the wave frame for k2, and we have collapsed the sum

using the fact that only the m = 2 spin-weighted spherical harmonic is non-zero − p and it evaluates to 5/4π. Note that for Eq. (2.66) to hold the assumption of narrow

surface of last scattering is vital. Now it is clear what are the additional diﬃculties

regarding this particular line of sight evolution with respect to the previous line of

sight computation from 2.3.2. Here we have the CMB quadrupole moments entering § from two diﬀerent avenues: directly from the polarization, and from the birefringence.

Furthermore, we switch the local quadrupole moments for appropriate polarization

17 E transfer functions and primordial curvature perturbations by usinga ¯20(c η, k) =

16The Wigner D-matrix applies to active rotation by the three Euler angles in the order γ around z, β around y, and α around z. In this application, we start with a wave vector k1 pointed toward

the observer; the wave vector and the associated barred basis vectors are then rotated by φk1 and

then θk1 . Due to our choice of thex ¯-axis, no further rotation is necessary. 17We follow the convention of CLASS for the transfer functions, e.g. T E = 5 (G(0) + G(2)), see ` − 4√6 Eq. (B.11) and related expressions in Ref. Tram and Lesgourgues(2013) or even Ma and Bertschinger (1995) for more information.

64 E T20(c η, k)ζ(k). Then, Eq. (2.63) becomes n Z 3 Z 3 Z ¯ d k1 d k2 4 E E V (r⊥) = A Im ds (1 + z) T (c ηLSS + s, k1) T (c ηLSS, k2) − (2π)3 (2π)3 20 20 o ˆ ∗ ˆ ∗ Y22(k2)Y (k1)ζ (k1)ζ(k2) exp i(k2⊥ k1⊥) r⊥ ik1ks , (2.67) 22 − · − q ¯ 4π where we have deﬁned A = 5 A. Moreover, the complex exponentials come from the Fourier transformations of both the polarization and the quadrupole moment.

Since the k2 is the mode from the polarization, it should be in the xy-plane (surface of last scattering) where the line of sight is in the z-direction. Consequently, the exponential only has a contribution from k2⊥ r⊥. On the other hand, the exponential · of k1 has a radial term because it cares about the propagation in the medium, i.e. it depends on the distance from the surface of last scattering to the intersection between the wavefront and the line of sight.

The 2D power spectrum of the circular polarization as seen by a distant observer is

2 0 ˜ ∗ ˜ 0 (2π) δ(k⊥ k )PV (k⊥) = V (k⊥)V (k ) − ⊥ h ⊥ i Z Z 2 2 0 ∗ 0 0 0 = d r⊥ d r V (r⊥)V (r ) exp ik⊥ r⊥ ik r ⊥h ⊥ i { · − ⊥ · ⊥}

= I1 + I2 + I3 + I4, (2.68) where we have expanded the imaginary part as Im X = (X X∗)/(2i), and used − 0 Eq. (2.67) to write V (r⊥) in terms of an integral over k1 and k2, and V (r⊥) in

0 0 terms of an integral over k1 and k2. We then separated the terms in the fol-

∗ ∗ 0 0 lowing way: I1 contains the contribution from ζ(k1)ζ (k2)ζ (k )ζ(k ) , I2 con- h 1 2 i ∗ 0 ∗ 0 ∗ ∗ 0 0 tains ζ(k1)ζ (k2)ζ(k )ζ (k ) , I3 contains ζ (k1)ζ(k2)ζ (k )ζ(k ) and I4 contains h 1 2 i h 1 2 i ∗ 0 ∗ 0 ζ (k1)ζ(k2)ζ(k )ζ (k ) . We explicitly show the form of these integrals in Ap- h 1 2 i pendix A.7.

65 We use Wick’s theorem to rewrite the 4-point functions into the relevant products of two point functions for our line of sight calculation. Generally the 4-point functions simplify to (4 1)!! = 3 terms, however translation invariance guarantees that one − of these will be zero for k⊥ = 0. Therefore, every I integral becomes two integrals, 6 which for the sake of clarity we will refer to as J integrals. To illustrate we show steps of the procedure for I1:

∗ ∗ 0 0 ζ(k1)ζ (k2)ζ (k )ζ(k ) = h 1 2 i ∗ 0 ∗ 0 0 ∗ ∗ 0 ζ(k1)ζ (k ) ζ (k2)ζ(k ) + ζ(k1)ζ(k ) ζ (k2)ζ (k ) h 1 ih 2 i h 2 ih 2 i 6 3 0 3 0 3 0 3 0 = (2π) Pζ (k1)Pζ (k2) δ (k1 k )δ (k2 k ) + δ (k1 + k )δ (k2 + k ) . − 1 − 2 2 1 (2.69)

Then the I1 integral becomes

¯2 Z Z Z Z Z A 2 dk1,k dk2,k 0 4 0 4 I1 = d k1,⊥ ds ds (1 + z) (1 + z ) Pζ (k1)Pζ (k2) 4 2π 2π × 2 0 n E E 0 ∗ ˆ ˆ ˆ ∗ ˆ δ (k⊥ k ) T (c ηLSS, k1) T (c ηLSS + s )Y (k2)Y22(k1)Y22(k2)Y (k1) − ⊥ 20 20 22 22 × 0 E 2 ik1,k(s−s ) E E 0 [T (c ηLSS, k2)] e + T (c ηLSS + s, k1) T (c ηLSS + s , k2) 20 20 20 × 0 o E E ik1,ks −ik2,ks ∗ ˆ 2 ˆ 2 T20(c ηLSS, k1)T20(c ηLSS, k2)e e [Y22(k2)] [Y22(k1)]

= J1 + J2 . (2.70)

Note that we have already integrated over the k2,⊥ with the 2-D delta functions coming directly from the exponential factors in Eq. (2.68). For the case of I1 this means that k2,⊥ = k1,⊥ + k⊥. Moreover, we can separate the integrals in Eq. (2.70)

0 with respect to s and s since they are decoupled, e.g. J1 becomes

¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k 2 E 2 J1 = (2π) δ (k⊥ k ) F (k1, k1,k) [T (c ηLSS, k2)] 4 − ⊥ (2π)2 2π 2π | | 20 ∗ ˆ ˆ ˆ ∗ ˆ Y (k2)Y22(k1)Y22(k2)Y (k1)Pζ (k1)Pζ (k2), (2.71) × 22 22 66 where we deﬁne F (kmagnitude, kparallel) as

Z ηz∼100 4Ω 4 E ik1,kc(η−ηLSS) m F (k1, k1,k) = c dη T20(η, k1) e 2 2 2 , (2.72) H Ω η 4Ωr ηLSS 0 m −

with Ωm being the density of matter in the Universe, Ωr the radiation density, and

H0 the Hubble constant. The full list of integrals J1...J8 is provided in Appendix A.7,

Eqs. (A.43–A.50). Note that in the J1...J4 integrals, k2,⊥ = k1,⊥ + k⊥, but in the

J5...J8 integrals, k2,⊥ = k1,⊥ k⊥. − As a computational strategy, we focus ﬁrst in the F functions because we already

had separated them from the remaining 4D integral; by pre-computing and tabulating

them the 4D integral can be computed much faster. In order to tackle the object in

Eq. (2.72) one needs to tabulate and interpolate the appropriate polarization transfer

function in terms of conformal time and wavenumber. We obtain our table for the

interpolation of the transfer function running CLASS Tram and Lesgourgues(2013)

for k values ranging from 0.0001 to 1 Mpc−1 with a sampling of ∆k = 0.0005 Mpc−1.

Further, we interpolate with the help of the 2D cubic interpolation function from the

GNU Scientiﬁc Library Galassi et al.(2002).

Once the F functions have been obtained one can focus in the remaining 4D integrals. We restrict all the involved wavenumbers18 to span from 0.0005 to 0.3 Mpc−1.

The choice of the upper limit is justiﬁed by CMB observations, in the sense that we want to guarantee that the peak of the CMB linear polarization (` 103) is ∼ included. The comoving angular diameter distance at surface of last scattering is

D = 1.4 104 Mpc, thus the maximum wavenumber is related to the maximum mul- × −1 tipole by `max = kmaxD. Our choice of kmax = 0.3 Mpc corresponds to `max = 4200, well into the CMB damping tail. We use Cartesian coordinates for simplicity and

18 i.e. k1, , k1, , and k2, . ⊥ k k 67 for the symmetries of the problem to be replicated by the numerical discretization.

−1 We take a sampling of ∆k1x = ∆k1y = 0.0005 Mpc with an oﬀset from zero of

−1 19 0.00025 Mpc in order to avoid sampling the points k1,⊥ = 0 or k2,⊥ = 0. We only integrate in the upper (k1x, k1y)-plane because most integrands are even in k1y, except for the terms that are odd and hence vanish. Of course then one needs to rewrite the spherical harmonics in terms of k1x, k1y, k1,k and k2,k. Moreover, the variance is a function of only the magnitude of k⊥, because the Universe is homogeneous and isotropic. Therefore, we can choose k⊥ = k⊥xˆ without losing generality, and k2,⊥ = k1,⊥ k⊥xˆ. ± new There is still one signiﬁcant simpliﬁcation. By changing variables k1 = k2

new and k2 = k1 for J5,J6,J7 and J8, we can analytically cancel the imaginary parts and reduce the number of integrals. Nevertheless, instead of assuming the analytical cancellation we decided to use the imaginary component as a null test, which provides a simple conﬁrmation that our code produces sensible results.

The power spectrum of the circular polarization is related to the RMS of the circular polarization signal by

Z d2k V 2 = ⊥ P (k ). (2.73) RMS (2π)2 V ⊥

Nevertheless, this power spectrum has never been studied before, so we decided to sample a few diﬀerent k⊥ with the objective of learning the behavior of this function both at low and high k, see Fig. 2.7. We do not observe acoustic oscillations. However, we do note that the power spectrum has some expected features. At small k, we have

2 PV (k⊥) k . This occurs because we have a convolution integral –which is usually ∝ ⊥ 19 ˆ The integrals are well-behaved as analytic functions at these points, e.g. Y22(k1) goes to zero at k1, = 0, however the polar coordinates are ill-deﬁned and the if statements needed to handle this case⊥ would slow down the computation.

68 1x10-28

1x10-29

1x10-30

1x10-31 π )/2 ⊥ -32 (k 1x10 V P ⊥ 2 k 1x10-33

1x10-34

1x10-35

1x10-36 0.001 0.01 0.1 -1 k⊥ =l/D(zA) [Mpc ]

Figure 2.7: Circular polarization power spectrum from photon-photon scattering.

analytic as k 0 – and in this case the circular polarization at k⊥ = 0 must be → zero since then the observer line of sight, k1, and k2 are co-planar, and reﬂection symmetry prevents circular polarization from being generated. On the other hand, at large k⊥ where we enter the CMB damping tail region, the power spectrum falls oﬀ.

Finally, integrating the power spectrum and taking the square root we get

−14 VRMS 1.3 10 K. (2.74) ' ×

Note that this value is smaller than the estimate from the detailed calculation without

−13 the line of sight suppression based on Eq. (2.62), namely VRMS 1.4 10 K. The ∼ × diﬀerence of one order of magnitude between both methods is reasonable, especially

69 when taking into account that in Eq. (2.62) we have ignored the radiation contribution to the Hubble parameter.

2.5.3 Comparison with previous calculations

As highlighted before our results are in apparent contradiction with Ref. Motie and Xue(2012). Their Eq. (32) reports an upper limit for the phase shift of 10 −13, six orders of magnitude less than our order of magnitude estimate. However, numerical evaluation of their expression (with factors of ~ and c inserted) gives

6 2 4 Z 1000 5 π α kBT0 kBT0 (1 + z) −6 ∆φFC 2 CQ dz 1 CQ 10 , (2.75) ' 120ζ(3) mec ~ 0 H(z) ≈ ∼ where CQ is the ratio of linear polarization intensity to the total intensity, and we evaluated the expression with the same parameters as Ref. Motie and Xue(2012):

−1 −1 Ωm = 0.3, ΩΛ = 0.7 and H0 = 75 km s Mpc . Equation (2.62) is consistent with this upper limit.

Our calculation also gives a smaller result than Ref. Sadegh et al.(2017). In our notation, their Eq. (12) for circular polarization generation in beam A by a background beam B reads:

2 4 Z 3 ˙ 8α ~ d kB 2 ˆ ˆ fV (kA) = 4 5 fU (k) 3 (ωAωB) f2(kB, kA)fI (kB), (2.76) 45mec ωA (2π) 2ωB where f2 is a term with trigonometric factors. However, if this equation is to be ˆ ˆ ˆ Lorentz invariant, then so long as kA = kB, one can boost to a frame where kA = eˆ3 6 ˆ and kB = eˆ3. In this frame, we must have f2 = 0, since the right-hand side is of − spin 2 when rotating around the z-axis, but the left-hand side is of spin 0. Therefore on the right-hand side one should have the polarization rather than the intensity of the background beam B, which is weaker by a factor of 10−6. Thus the V 10−2 µK ∼ 70 found in Ref. Sadegh et al.(2017) should be reduced to 10 −8 µK, in line with the

order of magnitude of our estimate.

Our result is also in conﬂict with the result from Ref. Sawyer(2015). They report

−9 −9 a circular polarization of V 10 3 10 K, which is smaller than our Vrms ∼ ∼ × ∼ 1.3 10−14 K, i.e. a difference of + 4.89 dex. We track down the differences to several × different contributing factors. First, we use a reference frequency of 100 GHz while

the thermal average frequency for the CMB photons is 154 GHz. This results in ∼ a diﬀerence of + 0.19 dex. We also found some suppression from doing the full line-

of-sight integral as opposed to an order of magnitude calculation using the Hubble

length; this gives a diﬀerence of + 0.9 dex because Ref. Sawyer(2015) did an order

of magnitude calculation. Also, in our order-of-magnitude estimate we use a linear

polarization20 of P 10−6 2.73 µK while Sawyer used P > 54.6 µK, based on ∼ ∼ the argument that one would select the highest-polarization region; this explains

+ 1.3 dex. In contrast to Sawyer’s phase shift of 1.1 10−6 we found ∆φ 10−7. × ∼ Part of this diﬀerence comes from their Eq. (22), which we numerically evaluate as

2.7 10−11 instead of 4 10−10; this explains a + 0.58 dex diﬀerence. (Note that there × × remains a + 0.46 dex diﬀerence in the phase shifts, which are likely due to factors of order unity on both sides.) In addition, there is a factor of + 1.6 dex coming from how Ref. Sawyer(2015) treated the relation between the standard deviation and the circular polarization, in particular in the jump from their Eq. (24) to V 10−9. ∼ In conclusion, we believe the underlying physics in the calculation of Ref. Sawyer

(2015) is the same as in our calculation, and we can identify the main causes of the

20The real RMS CMB linear polarization is approximately 4.47 µK.

71 diﬀerence between both results. The detailed calculation leading to Eq. (2.74) is the most rigorous implementation yet of this physics.

2.6 Static non-linear polarizability of hydrogen

In the previous section we considered the polarization of the vacuum. However, the post-recombination Universe was ﬁlled with a dilute gas of atoms, which also lead to photon-photon scattering due to their non-linear polarizability. Generally atoms are far more non-linear than the QED vacuum at CMB frequencies, but their ﬁlling fraction is tiny. Therefore it is not clear whether their contribution to photon-photon scattering is more or less important.

In this section, we consider the birefringence produced by the static non-linear polarizability of hydrogen given an anisotropic ambient medium. This process is most important shortly after the last scattering epoch, when the CMB has had time to free- stream and hence is anisotropic, but when the radiation density is still high. We will focus on the effect of the second order hyper-polarizability, which is the lowest-order effect that gives a diagram with two photons entering and two photons leaving. The two final-state photons occupy the same quantum states as the initial-state photons.

We further focus on hydrogen, which is both more abundant than helium and has lower excitation energies.

2.6.1 Order of magnitude

We start with an order of magnitude estimate of both the birefringence and the phase shift using dimensional analysis. The diﬀerence in indices of refraction must be

72 proportional to the number density of hydrogen atoms nH, the second order hyper- polarizability21 γ and the anisotropic energy density of the electric ﬁeld of the CMB,

E . The electrostatic constant 0 may also appear. Thus, the order of magnitude of Uani the birefringence is E γnH ani −37 nx ny 2U 10 , (2.77) − ≈ 0 ≈

8 −3 E −7 where we have substituted at z = 1000 that nH 2 10 m , and = 4 10 J ≈ × Uani × m−3 (i.e. the energy density of a blackbody at 2725 K, multiplied by 10−5 to take the anisotropic part). The hyper-polarizability γ of a hydrogen atom should in principle be of order 1 in Bohr units, or 6.2 10−65 J m4 V−4.22 In fact, in atomic units, × we have γ = 1333.125 Sewell and Coulson(1949) or 8 .3 10−62 J m4 V−4. Using × Eq. (2.3), and for a photon with wavelength 1 mm today (1 µm at z = 1000), and ∼ a path length of order 100 kpc physical (the horizon size at recombination), we ﬁnd

∆φ 10−9. (2.78) ≈ 2.6.2 Detailed computation

Next, we do a more accurate order of magnitude estimate of the birefringence produced by the non-linear polarizability of hydrogen. We place a hydrogen atom in two beams of radiation, A and B, with electric ﬁelds EA and EB respectively. Our interest is in the eﬀective index of refraction that applies to beam A.

21This is a coeﬃcient in the Taylor expansion of the dipole moment of the hydrogen atom: p = 1 2 αdE + 6 γE E + ... . 22 The Bohr unit of electric ﬁeld is 1 Hartree per elementary charge per Bohr radius, or Eat = 11 1 18 5.1 10 V m− . The Bohr unit of energy is the Hartree, or Uat = 4.4 10− J. Therefore the × 4 65 ×4 4 Bohr unit of second-order hyper-polarizability is U /E = 6.2 10− J m V− . at at ×

73 One can start with the energy of the hydrogen atom, which can be expanded in

even powers of the applied electric ﬁeld:

1 2 1 4 U = U0 αdE γE + ... ; (2.79) − 2 − 24

then the electric dipole moment is

1 2 p = ∇EU = αdE + γE E + ..., (2.80) − 6

where in this case the applied electric ﬁeld is

−iωAt −iωB t E = EAe + EBe + c.c.. (2.81)

The ﬁrst term gives rise to the ordinary linear index of refraction, but no birefringence.

For birefringence the important term is the second one in Eq. (2.80), particularly the

−iωAt terms that have an e time dependence but also contain EB. These terms are

1 B B∗ A B B∗ A B B∗ A −iωAt pi γ E E E + E E E + E E E e . (2.82) 3 3 j j i j i j i j j

Here ... will indicate an average over the beams B when we replace the single monochromatic plane waves B with a stochastic radiation background. This implies an eﬀective nonlinear contribution to the polarizability tensor

A ∂pi NL,e−iωAt 1 B B∗ B B∗ B B∗ αNL,ij = | A = γ Ek Ek δij + Ej Ei + Ei Ej . (2.83) ∂Ej 3

The anisotropic part of the index of refraction is

nHI A A γnHI B B∗ B B∗ nxx nyy = (αNL,xx αNL,yy) = Ex Ex Ey Ey , (2.84) − 20 − 30 − where nHI is the density of neutral hydrogen atoms.

In the presence of a background radiation ﬁeld, we need to compute the expec-

B B∗ tation values Ex Ex . For any single beam of radiation, the mean square electric

74 B 2 23 ﬁeld is E = /(20). For unpolarized radiation traveling in direction Ω, the | | U fraction of the mean square electric ﬁeld that is in the x-component is 1 (1 Ω2). 2 − x Therefore, we have Z B B∗ 1 d 1 2 2 Ex Ex = U (1 Ωx) d Ω, (2.85) S2 20 dΩ 2 −

where d /dΩ is the energy density in ambient electromagnetic waves per unit solid U angle (i.e. in J m−3 sr−1), and similarly for the y-direction. Then Eq. (2.84) reduces

to Z γnHI d 2 2 2 nxx nyy = 2 U (Ωx Ωy) d Ω − − 120 S2 dΩ − r Z γnHI 32π d 2 = 2 Re U Y22(Ω) d Ω − 120 15 S2 dΩ r 32π γnHI 4 = 2 aradTCMB Re a22, (2.86) − 135 0

where arad is the radiation energy density constant, a22 is the local quadrupole moment of the CMB, and there is a factor of 4 in the last step coming from converting temperature anisotropies to energy density anisotropies.

The rate of change of relative phase per Hubble time is related to the rate of change per unit comoving distance (Eq. 2.3):

dφ c dφ = d ln a aH dχ ω = (nxx nyy) H − r 32π (1 + z)ωtoday γnHI 4 = 2 aradTCMB Re a22 − 135 H 0 13/2 −9 νtoday 1 + z Re a22 = 1.2 10 xHI , (2.87) − × 100 GHz 1000 10−5

23 1 2 Since electric field energy density is 2 0E , we would at first write down 2 /0. However there are two more factors of two in the denominator: one from the fact that onlyU half of the energy density in an electromagnetic wave is in the electric field, and one from our convention in Eq. (2.81) that “EB” is only the positive-frequency part of the wave.

75 where we used the standard baryon abundance and denote by xHI the fraction of the hydrogen that is neutral.

This is in agreement with the order-of-magnitude calculation, but includes the dependence on the specific spherical harmonic components of the radiation that causes the birefringence. The peak of the birefringence effect occurs at recombination, due to the power-law suppression at lower z and the suppression of the neutral fraction and the CMB anisotropy at higher z. In any case, the peak effect is only at the 10−9 ∼ level. In terms of circular polarization Eq. (2.87) converts into V 1.0 10−15 K. ≈ × 2.7 Plasma delay: non-linear response of free electrons

Plasma delay is another non-linear polarization process, this time using the free electrons instead of the virtual pairs in the vacuum ( 2.5) or the hydrogen atoms § ( 2.6). It relies on the fact that recombination is not complete: there are still some § free electrons xe > 0. These electrons are much less abundant than hydrogen atoms, however their excitation energy is zero so for CMB photon energies much less than the excitation energy of hydrogen we expect that electrons can be much more eﬃcient at producing phase shifts than an equal number of hydrogen atoms.

Plasma delay can be described by two beams of light, one of frequency ωA and the perturbing beam with frequency ωB, both incident on a free electron. Plasma delay produces birefringence in the presence of an anisotropic radiation field. This effect should be stronger around recombination since the CMB is already anisotropic, there is still a significant fraction of ionized hydrogen, and the radiation density is still high. Note that this effect also has two incoming photons and two outgoing photons.

Again, the interest is in the eﬀective index of refraction that aﬀects beam A.

76 2.7.1 Order of magnitude

As usual we start by doing the order of magnitude for the birefringence produced

by the eﬀect in consideration. In order to compute this, note that by dimensional

2 analysis we expect factors of e /0, and combinations of factors of c, mass of the

electron me and frequencies of both the perturbing beam ωB and the original beam

ωA. Moreover, nx ny must be proportional to the number density of electrons ne and − the anisotropic energy density of the perturbing beam (here the CMB). Birefringence

should be generated by the third order susceptibility (in the sense that the responsible

term in the current contains three factors of electric ﬁeld), so we expect a factor of

2 2 2 (e /0) instead of e /0, and three factors of the mass of the electron. This leads to

4 E e anine −40 nx ny 2 U3 4 2 10 , (2.88) − ≈ 0meω c ≈ −3 15 −1 where ne = xenH = 9.9 cm at 1 + z = 1000, and we approximate ω 10 rad s , ∼ and E = 4 10−7 J m−3. Note that from dimensional analysis alone one could Uani × only say that the denominator must have a factor of frequency to the fourth power;

when we derive the equations of motion we will learn what combinations of ωA and

2 ωB actually appear. We will ﬁnd that the correct expression contains (ωB ωA) in − the denominator, and deal with the apparent divergence in 2.7.2. § Next, we use Eq. (2.3) for a photon with λ 1 µm at 1 + z = 1000, and with the ∼ usual path length of 100 kpc (physical) relevant for the recombination epoch, we get

∆φ 10−11. (2.89) ≈ 2.7.2 Rigorous calculation

In this subsection we will provide a more rigorous calculation of the CMB circular

polarization produced by free electrons. Along the way, we will ﬁnd and regularize a

77 singularity when ωB = ωA. Again, we are interested in the eﬀective index of refraction felt by EA.

We start with the equations of motion for the electrons with incoming electric

ﬁeld

E = EAei(kA·r−ωAt) + EBei(kB ·r−ωB t) + c.c., (2.90) and magnetic ﬁeld

B = BAei(kA·r−ωAt) + BBei(kB ·r−ωB t) + c.c., (2.91)

A ˆA A R + with B = k E /c, etc. We take ωB = ω + i with 0 , so that beam B had × B → zero amplitude in the distant past. The displacement of the electrons obeys ¨ e 1 ˙ ˙ ξi = Ei + ξjEi,j + ξjξkEi,jk + ijkξjBk + ijkξjξ`Bk,` . (2.92) −me 2 The solution of this equation of motion can be expanded as

(1) (2) (3) ξi = ξi + ξi + ξi + ... , (2.93)

(n) where ξi denotes terms containing n powers of the electric or magnetic ﬁelds. We can use the solution for the displacement to obtain the current density and

hence the susceptibility. The part of the current density we want is that with a

i(kA·r−ωAt) e dependence, and containing two powers of EB or BB. This piece can be

written in Fourier space as: Z Z ˜ 3 −ikA·r 3 −ikA·x ˙ −ikA·ξ Ji(kA) = d r e Ji(r) = d x e neeξie , (2.94)

where we have used r to denote the Eulerian position of the electrons and x to

denote their Lagrangian position. In the last integral in Eq. (2.94), ne thus denotes

the unperturbed (Lagrangian) electron number density. We deﬁne

L ˙ −ikA·ξ Ji (x) = neeξie , (2.95)

78 ˜ L so that Ji(kA) is the Fourier transform of Ji in Lagrangian space, e.g.

Z ˜ 3 −ikA·x L Ji(kA) = d x e Ji (x) . (2.96)

Conceptually, we can understand the exponential factor in Eq. (2.95) as being asso-

ciated with the retarded time for a “downstream” observer measuring beam A.

L Since the problem is translation-invariant, we may evaluate Ji at the origin x = 0 without any loss of generality. Taking the previous result we can break the current

into terms that act as isotropic and anisotropic electric and magnetic susceptibilities;

the third order terms (superscript (3)) are:

1 L(3) −ikA·ξ ˙ (3) ˙ A (1) ˙(2) A A (1) (1) ˙(1) A (2) ˙(1) J = ene[e ξi] = ene ξi ik ξ ξ k k ξ ξ ξ ik ξ ξ i − j j i − 2 j k j k i − j j i iso ˙ ani,e ˙ 1 iso A 1 ani,m A = 0χe Ei + 0χij Ej + χm ijkBk,j + ijkχk` B`,j. (2.97) µ0 µ0

The ﬁrst order solution of Eq. (2.92) is simply

A B A∗ B∗ (1) e E E E E ∗ i i(kA·r−ωAt) i i(kB ·r−ωB t) i −i(kA·r−ωAt) i −i(kB ·r−ωB t) ξi = 2 e + 2 e + 2 e + ∗ 2 e , me ωA ωB ωA ωB (2.98)

R where kB k . The absence of magnetic ﬁelds is due to the electron being static ' B at this order. The terms in the Lagrangian current density, that are relevant for the

birefringence studied here, have two electric or magnetic ﬁelds from the perturbing

beam (no mixed ones) and either a electric or magnetic ﬁeld from the unperturbed

beam. Additionally, these terms must contain e−iωAt. Likewise, the relevant terms

79 for birefringence of the third order term are given by

" ∗ ∗ ∗ e3 EB EAEBkBkA EBEAEB kBkA EABBBB (3) i j k j k + i j k j k + k k i ξi 3 2 2 2 ∗ 2 ∗ 2 ∗ 3 −m ω − ω (ωA + ωB) ω (ω ωA) ωA(ωA ω ) e A B B B − − B A B B∗ A B B∗ A A B B∗ A ! Ek Bi Bk k` E` Ej Bk k` Ej E` Bk + + ijk 2 ∗ ∗ 2 ωA(ωA + ωB) ωBωB − ωB ωB ∗ ∗ !# EBEB BAkB EB EBBAkB i l k j i l k j i(kA·r−ωAt) + jlk ∗ 2 + 2 e . (2.99) ω (ωA ωB) ωB(ωA + ωB) B − Therefore, the anisotropic electric susceptibility coming from third-order terms, keep-

ing only those with the correct spatial and time dependence to interfere with the

original beam A, are given by

4 ( ani,e nee B A B B∗ 1 1 χij 3 2 kj kk Ei Ek ∗ 2 ∗ 2 ∗ 2 ' 0m ω ωAω (ωA + ωB) − ω (ω ωA) e A B B B − 1 1 1 ∗ 2 + 2 2 ∗ 2 −ωAωB(ω ωA) ω (ωA + ωB) − ωAω (ωA + ωB) B − B B B B∗ 1 Bi Bj 1 2 ∗ ∗ −ωAωB(ωA ωB) − ωA ωA(ωA ωB) − ) − 1 + , (2.100) ωA(ωA + ωB) where ... implies an average over the beam B, whose amplitude is stochastic since the CMB radiation is thermal. Therefore we need to integrate the susceptibility contributions both over direction of propagation and over frequency. We ﬁrst perform the angular integration, keeping only the temperature perturbations and neglecting the smaller polarization signal:

Z B B B B∗ 1 d 1 ki kk 2 Ei Ek = U δik 2 d Ω. (2.101) 20 S2 dΩ 2 − kB

A similar relation holds for the magnetic ﬁelds with the replacement 1/0 µ0. →

Choosing kA = (0, 0, kA), and still working in the monochromatic case, the anisotropic

80 electric susceptibility is given by

4 ( r Z ani,e ani,e nee kAkB 2π d 1 χxx χyy = 3 2 U Re Y32(Ω) ∗ 2 − 0me − 0ωA 105 S2 dΩ ωAωB (ωA + ωB) 1 1 1 + ∗ 2 ∗ 2 ∗ 2 2 2 −ωB (ωA ωB) − ωBωA(ωA ωB) ωB(ωA + ωB) − − 1 1 2 ∗ 2 2 ∗ d Ω −ωAωB(ωA + ωB) − ωBωA(ωA ωB) r − 2π Z d +µ0 U Re Y22(Ω) 15 S2 dΩ ) 1 1 2 3 + 3 ∗ d Ω . (2.102) × ω (ωA + ωB) ω (ωA ω ) A − B In an analogous way, we obtain the anisotropic magnetic susceptibility

4 (r Z ani,m ani,m nee 2π d χxx χyy = 2 3 2 U Re Y22(Ω) − 0mec 15 S2 dΩ 1 1 2 ∗ 2 + ∗ ∗ 2 d Ω × ω ωB(ωA + ωB) ωBω (ωA ω ) B B − B r Z 2π kB d + U Re Y32(Ω) 105 kA S2 dΩ ) 2 2 2 2 + ∗ ∗ 2 d Ω (2.103) × ωAωB(ωA + ωB) ωAω (ωA ω ) B − B Combining both contributions we obtain the anisotropic index of refraction in the monochromatic scenario

4 ( r Z nee 2π d ωB ∗ 2 nxx nyy = 2 3 2 U Re Y32(Ω)W (ωA, ωB, ωB) d Ω − 0mec − 105 S2 dΩ ωA r Z ) 2π d ∗ 2 + U Re Y22(Ω)WB(ωA, ωB, ωB) d Ω , (2.104) 15 S2 dΩ where we deﬁne two frequency window functions W and WB: 1 1 1 ( ∗ ) W ωA, ωB, ωB 2 ∗ 2 ∗ 2 ∗ 2 ≡ ωAω (ωA + ωB) − ω (ω ω ) − ωAωB(ω ωA) B B B A B − 1 1− 1 + 2 2 ∗ 2 2 ∗ ωB(ωA + ωB) − ωAωB(ωA + ωB) − ωAωB(ωA ωB) 2 2 − 2 ∗ ∗ 2 , (2.105) −ωAωB(ωA + ωB) − ωAω (ωA ω ) B − B 81 and

∗ 1 1 WB(ωA, ωB, ωB) 3 + 3 ∗ ≡ ωA(ωA + ωB) ωA(ωA ωB) 1 − 1 + ∗ 2 + ∗ ∗ 2 . (2.106) ω ωB(ωA + ωB) ωBω (ωA ω ) B B − B Now we consider the full distribution of frequencies instead of treating B as a mono-

chromatic wave; this leads to

4 " r r # nHxee 2π 2π B nxx nyy = 2 3 2 (Re a32) IT(ωA) + (Re a22) IT (ωA) , (2.107) − 0mec − 105 15

where a32 is the local octopole moment of the CMB, a22 is the local quadrupole moment, xe is the ionization fraction, and we have deﬁned the frequency integrals

Z ∞ R R 3 ωB ∗ d ~ωB R IT(ωA) = W (ωA, ωB, ω ) T dω (2.108) B 3 3 ωR/k T B 0 ωA dT 4π c (e~ B B 1) − and

Z ∞ R 3 B ∗ d ~ωB R I (ωA) = WB(ωA, ωB, ω ) T dω . (2.109) T B 3 3 ωR/k T B 0 dT 4π c (e~ B B 1) − In Eqs. (2.108, 2.109), the quantity in square brackets is d /dΩ/dωR, which depends U B on the CMB temperature T ; its derivative T d/dT is the conversion factor from CMB

anisotropy units (∆T/T ) to energy anisotropy d∆ /dΩ/dωR. U B

We may furthermore expand IT as IT = I1 I2 I3 + I4 I5 + I6 I7 I8 and − − − − − IB = IB IB + IB + IB using the terms in Eqs. (2.105) and (2.106). We explicitly T 1 − 2 3 4 show these integrals in Appendix A.8, and show only one of the involved integrals

here for pedagogical reasons – in particular, to illustrate how we handle the i terms.

For example

∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I2 = R , (2.110) 3 3 ~ω /kBT 2 R 2 R 2 4π c kBT ωA 0 (e B 1) (ω i) (ω ωA i) − B − B − − 82 where the exponential factor comes from the black body spectrum.

In order to simplify the integrals we utilize a combination of integration by parts and principal value, e.g.

( R) ∞ Z ∞ 0( R) R F ωB F ωB dωB 0 R I2 = R + PV R + iπF (ωB = ωA), (2.111) −(ω ωA i) 0 ω ωA B − − 0 B − R where the prime indicates differentiation with respect to ωB, and 3 R 2 R ~ωB /kBT R ~ ωB e 2i F (ωB) R 1 + . (2.112) 3 3 ~ω /kBT 2 R ≡ 4π c kBT ωA (e B 1) ωB − We proceed to numerically integrate the principal value with special care for the redshift dependence in the temperature and frequency of the CMB photons. Fur- thermore, we include the imaginary terms for completion here, even though these relate to differential absorption on the two axes, which does not convert linear to circular polarization. We finally take the limit of 0. We obtain the follow- → ing values around recombination (1 + z = 1000), for T = 2.73 (1 + z) K and

R 11 −1 ω = ωA = 2π 10 (1 + z) rad s B ×

−62 −1 2 IT = (9.23 35.2i) 10 kg m s (2.113) − × and

IB = ( 5.78 13i) 10−62 kg m−1 s2. (2.114) T − − × We estimate the order of magnitude of the birefringence due to plasma delay by replacing the previous values into Eq. (2.107) to get ∆n ( 7.31 0.24i) 10−40. ≈ − − × Moreover, the subsequent phase shift is given by

dφ ω = (nxx nyy) d ln a H − 4 r r ! e nHxeω 2π 2π B = 2 3 2 Re a32 Re IT + Re a22 IT 0mec H − 105 15

5/2 r r −101 + z 2π Re a32 2π Re a22 = 2.4 10 xe + .(2.115) − × 1000 − 105 10−5 15 10−5 83 Table 2.2: Summary table for sources of circular polarization at an observed frequency of 100 GHz. The ﬁnal column shows the level of detail of the calculation (A = detailed numerical evaluation of all line of sight factors and power spectra; B = the RMS was determined with all relevant numerical pre-factors; C = the numerical pre-factors in the birefringence were computed, but the full statistics of the line of sight integral were not computed). Birefringence source Circ. Polarization Level of V (µK) detail Photon-Photon scattering 1 10−8 A × Static non-linear polarizability of hydrogen 1 10−9 C × Spin polarized H atoms (low z, aligned by 21 cm) 3 10−10 B × Plasma delay 3 10−11 C × Spin polarized H atoms (high z, aligned by Hα) 5 10−13 C ×

Note that this result is consistent with Eq. (2.89) estimated in the previous section.

Nevertheless, Eq. (2.115) is more rigorous because it includes the speciﬁc spherical harmonics that are involved in this eﬀect. In addition, it addresses the singularity that occurs at second order or higher in the equations of motion when ωA = ωB.

In the end, the circular polarization produced, V 2.6 10−17 K, is small even ≈ × in comparison to the other sources of circular polarization studied in this work.

2.8 Conclusion

We have summarized our results for the circular polarization of the CMB due to the conventional sources of cosmic birefringence considered in this work in Table 2.2.

We see that of the cosmological signals, all are very small compared to the linear polarization. The largest of the cosmological contributions comes from photon-photon scattering during the epoch of recombination (z 1000 in our model). The predicted ∼ RMS circular polarization for our ﬁducial model is 13 fK; this may vary somewhat for

84 more accurate computations, but in any case the signal is tiny compared to present

observational sensitivities or even compared to dominant foregrounds such as Galactic

synchrotron emission, which at ν = 10 GHz could potentially reach circular polarization levels of 10−8 K King and Lubin(2016). At higher frequencies, the Galactic ∼ foreground situation is improved: the circular polarization from synchrotron is expected to fall as V ν−3.5 but is still large compared to the expected signal, so 4 ∝ ∼ pK at 100 GHz.24 Note that thermal dust emission – the dominant foreground at

> 100 GHz – would be expected to have very low circular polarization, even though

the grains are aligned, because circularly polarized emission requires a phase delay

between the x and y axes. This phase delay and hence a net circular polarization

can be imprinted on the thermal radiation if the grain is rotating (see e.g. 4.2 of §

Ref. Martin(1972)), but this is very weak (of order Ω grain/ω) will only lead to circu-

lar polarization if the grain angular momenta have net vectorial alignment (e.g. spin

parallel to the ambient magnetic ﬁeld more or less likely than anti-parallel), which

is not necessarily expected in radiative grain alignment theory Lazarian and Hoang

(2007).25 Thermal magnetic dipole emission from grains Draine and Lazarian(1999)

could yield a stronger circular polarization signal, but again only if vectorial alignment

of the grains (this time between the permanent magnetic moment and the ambient

magnetic ﬁeld) is achieved. We leave investigation of all of these possibilities to future

24The foreground temperature scaling is in Rayleigh-Jeans units, so requires a factor of 1.29 to convert to blackbody temperature at 100 GHz. The fractional circular polarization V/I 1/γ 1/2 ∼ ∝ ν− , so circular polarization will have another 1/2 power of frequency relative to temperature. − 25In the Larmor precession angle-averaged calculation – 7 of Ref. Lazarian and Hoang(2007) – right-handed grains will prefer to spin one direction and§ left-handed grains the other direction. A racemic mixture would not show vectorial alignment, but would still have the “headless vector” alignment that produces linear polarization.

85 work; in any case, it is clear that there is at least one foreground (synchrotron) that is far larger than the conventional cosmological signals.

In conclusion, none of the conventional sources of circular polarization studied in this work are likely to be important in the foreseeable future. However, one could study non-standard sources of circular polarization as a channel for new physics.

For example: Lorentz invariance violations Colladay and Kosteleck`y(1998); Zarei et al.(2010), primordial magnetic ﬁelds Zarei et al.(2010), non-commutative gauge theories Aschieri et al.(2003), and scattering with the cosmic neutrino background

Mohammadi(2014); see Table 1 in King and Lubin(2016) for a summary of these sources of circular polarization. We view these searches as more promising in light of the tiny contribution of conventional cosmological eﬀects.

86 Chapter 3: Impact of inhomogeneous reionization on the Lyman-α forest

In this Chapter, I will present our paper on the impact of inhomogeneous reionization on the Lyman-α forest (Montero-Camacho et al., 2019b). The original abstract is given below:

The Lyman-α forest at high redshifts is a powerful probe of reionization. Mod- eling and observing this imprint comes with significant technical challenges: inhomogeneous reionization must be taken into account while simultaneously being able to resolve the web-like small-scale structure prior to reionization. In this work we quantify the impact of inhomogeneous reionization on the Lyman-α forest at lower redshifts (2 < z < 4), where upcoming surveys such as DESI will enable precision measurements of the flux power spectrum. We use both small box simulations capable of handling the small-scale structure of the Lyman-α forest and semi-numerical large box simulations capable of representing the effects of inhomogeneous reionization. We find that inhomogeneous reionization could produce a measurable effect on the Lyman-α forest power spectrum. The deviation in the 3D power spectrum at

−1 zobs = 4 and k = 0.14 Mpc ranges from 19 – 36%, with a larger eﬀect for later reionization. The corrections decrease to 2.0 – 4.1% by zobs = 2. The impact on

87 the 1D power spectrum is smaller, and ranges from 3.3 – 6.5% at zobs = 4 to 0.35

– 0.75% at zobs = 2, values which are comparable to the statistical uncertainties in current and upcoming surveys. Furthermore, we study how can this systematic be constrained with the help of the quadrupole of the 21 cm power spectrum.

The original authors are Paulo Montero-Camacho, Christopher M. Hirata, Paul

Martini and Klaus Honscheid.

3.1 Introduction

In the standard model of cosmology, the diﬀuse gas in the Universe has under- gone a complex thermal history marked by several important transitions. At early times, the gas was hot, dense, ionized, and fully coupled to the thermal radiation in the early Universe. As cosmic recombination occurred at z 1100, the gas became ∼ transparent; perturbations from this epoch are directly observable to us as ﬂuctu-

ations in the cosmic microwave background (CMB) temperature and polarization.

During the subsequent epoch – the cosmic Dark Ages – perturbations in the gas and

dark matter grew by gravitational instability, which eventually went non-linear and

began to form collapsed objects. Some of these objects emitted ionizing radiation,

triggering cosmic reionization. This transition, currently believed to have occurred

at zre = 7.7 0.8 (Planck Collaboration et al., 2018a), also must have re-heated the ± intergalactic medium (IGM) to temperatures up to a few 104 K after the passing of × an ionization front (McQuinn, 2016; D’Aloisio et al., 2018). After reionization it is

possible to probe the IGM using the absorption of neutral hydrogen in quasar spectra

from overdense regions – the “Lyman-α forest”. Each quasar spectrum probes a 1-

dimensional skewer through the Universe, with the fraction of ﬂux transmitted at each

88 observed wavelength being inversely related to the gas density at the corresponding

redshift26.

In addition to probing the astrophysics of the IGM, the Lyman-α forest has become an important tool for observational cosmology. It probes a redshift range (2 < z < 5) where conventional large-area galaxy surveys are very diﬃcult due to the enormous luminosity distance and the redshifting of bright nebular emission lines into the infrared. In contrast, while bright background quasars are rare, each one provides information from the many structures along their line of sight. The Lyman-α forest

is also in some ways simpler to model than galaxies: most of the forest consists of

moderately overdense gas, where the physics is dominated by gravitational instability

rather than feedback processes (Cen et al., 1994; Hernquist et al., 1996; Hui et al.,

1997; Machacek et al., 2000; Luki´cet al., 2015); see McQuinn(2016) for a recent

review. The Lyman-α forest has been used to study reionization of hydrogen and

helium (Fan et al., 2002; Cen et al., 2009; McQuinn et al., 2009; Pritchard et al.,

2010; Becker et al., 2011; Compostella et al., 2013; Mesinger et al., 2015; Greig et al.,

2015; McGreer et al., 2015; Choudhury et al., 2015; Bouwens et al., 2015; Nasir et al.,

2016; O˜norbe et al., 2017, 2018; Walther et al., 2018b); the matter power spectrum

(particularly on small scales that are too non-linear in galaxy surveys; Viel et al. 2008;

Weinberg et al. 1998, 2003); constrain cosmological parameters and the late evolution

of the universe (Hannestad et al., 2002; Seljak et al., 2005, 2006; Croft et al., 2006;

Chabanier et al., 2018); and measure the neutrino mass via its eﬀect on the growth

of structure (Palanque-Delabrouille et al., 2015; Y`eche et al., 2017).

26More precisely, the optical depth is proportional to the neutral hydrogen density, which in turn depends on the density and temperature of the gas and the photoionization rate. This is then smeared by the Doppler width of the Lyman-α line.

89 Motivated by both cosmology and astrophysics, there has been a growing number

of observed Lyman-α forest sightlines in the range 2 < z < 5 (Bechtold, 1994;

Croft et al., 1998, 1999, 2002; Rauch, 1998; McDonald et al., 2000, 2006; Lidz et al.,

2010; Slosar et al., 2011, 2013; Eisenstein et al., 2011; Dawson et al., 2013; Font-

Ribera et al., 2014; Delubac et al., 2015; du Mas des Bourboux et al., 2017; Bautista

et al., 2017; Irˇsiˇcet al., 2017; Walther et al., 2018a; Lee et al., 2018). A range of

statistical measures of the Lyman-α forest have been used; for cosmology purposes,

the most common has been the 1D correlation function or power spectrum (that is,

the power spectrum of individual skewers, treated as 1D random ﬁelds). With the

Baryon Oscillation Spectroscopic Survey (BOSS), the number of sightlines became

great enough to extract cosmological parameters from the 3D correlation function of

the Lyman-α forest (that is, using correlations between diﬀerent lines of sight). BOSS

also measured cross-correlations between Damped Lyman-α systems (DLA) and the

Lyman-α forest (Font-Ribera et al., 2012), and measured Baryon Acoustic Oscillations

(BAO) in the Lyman-α forest (Slosar et al., 2013). The number of observed Lyman-

α forest will soon be dramatically enhanced with the Dark Energy Spectroscopic

Instrument (DESI; DESI Collaboration et al., 2016).

The current paradigm for reionization is that it is an extended and inhomogeneous

process, with “bubbles” of ionized gas forming in regions with more sources of ionizing

photons, and the end of reionization is when these bubbles overlap and approach a

ﬁlling factor of unity. Inhomogeneous reionization leaves its imprint in the Lyman-α

forest even after reionzation ends, because the thermal state of the gas depends on

when it was reionized (Miralda-Escud´eand Rees, 1994) and because the thermal his-

tory itself aﬀects the distribution of the gas at the Jeans scale. Because the attractor

90 evolution of the IGM temperature causes the memory of reionization to fade with

time (see McQuinn and Upton Sanderbeck, 2016, for a detailed explanation), as well

as the complication of helium reionization turning on at later redshifts (z 3.5), ∼ most studies on the eﬀect of reionization in the Lyman-α forest have been at the highest redshifts that have a suﬃcient number of sightlines (Hui and Haiman, 2003;

Trac et al., 2008; Furlanetto and Oh, 2009; Lidz and Malloy, 2014; D’Aloisio et al.,

2015; Keating et al., 2018).

The goal of this paper is to compute a first estimate of the effect of inhomogeneous reionization in the Lyman-α forest at the lower redshifts (2 < z < 4) relevant to precision cosmology programs such as DESI. The remnants of hydrogen reionization are expected to be small, but in the DESI era even small effects in the Lyman-α forest power spectrum will be important. Due to the enormous dynamic range in spatial scales, we use two types of simulations. Our approach here is analogous to the recent work by Oñorbe et al.(2018), where the effect of inhomogeneous reionization in the

1D Lyman-α forest power spectrum was computed at higher redshifts (4 < z < 6).

Our small-scale simulations use full hydrodynamics to follow the dynamics of gas down to below the Jeans scale, but with an ionization history set by hand. These use the same modiﬁcation of Gadget-2 (Springel, 2005) that Hirata(2018) used to study the streaming velocity eﬀect in the Lyman-α forest. The large-scale boxes use a

semi-numerical approach (21cmFAST; Mesinger and Furlanetto 2007; Mesinger et al.

2011) to model the ionized bubbles and their correlation with large-scale structure.

Besides estimating the change ∆PLyα(k, z) in Lyman-α forest power spectrum due

to reionization eﬀects, we consider how 21 cm observations could be used to predict

∆P (k, z) and mitigate this systematic eﬀect. The H i 21 cm hyperﬁne transition is a

91 potentially powerful probe of reionization, and global measurements of the signal have

already ruled out some models (Monsalve et al., 2017; Singh et al., 2017, 2018). We

ﬁnd that there is a quantitative relation between ∆PLyα(k, z) and the full history of

the cross-correlation of the matter and the ionization fraction. The latter is related to

the redshift-space distortion in the 21 cm power spectrum (Barkana and Loeb, 2005;

Mao et al., 2012). We show that the simplest approach – using linear pertubation

theory to map the ` = 2 quadrupole of the 21 cm signal into a matter-ionization fraction cross-power spectrum and using this to predict ∆PLyα(k, z) – is not accurate.

We thus recommend that future work consider mitigation using models for the 21 cm

redshift space distortion that go beyond linear theory (e.g., Mao et al., 2012).

We expect the eﬀect studied here to be of importance for current and near-future

Lyman-α experiments. In particular, DESI will measure the three dimensional (3D) and one dimensional (1D) power spectrum of the Lyman-α forest (DESI Collaboration

et al., 2016), both of which will be altered by inhomogeneous reionization. The

1D power spectrum contains the correlation of the spatial structure of the neutral

hydrogen regions along the line of sight of the same quasar (McDonald et al., 2005,

2006; Palanque-Delabrouille et al., 2013), i.e. the individual skewers. In contrast, the

3D spectrum has correlations between diﬀerent lines of sight (Blomqvist et al., 2015;

Bautista et al., 2017; Font-Ribera et al., 2018). The fractional eﬀect ∆P (k, z)/P (k, z)

of inhomogeneous reionization is larger on large scales (smaller k) than at smaller

scales, and is larger in the 3D Lyman-α forest power spectrum than in 1D (because

large k in 3D can map into small k in 1D but not the other way around). The 1D

power spectrum is aﬀected by non-linearities, even for small kk, due to small-scale

92 processes that govern the evolution of the IGM (Palanque-Delabrouille et al., 2013;

Arinyo-i-Prats et al., 2015).

This paper is organized as follows. We introduce the relevant formalism in 3.2. § We describe the two types of simulations we use in 3.3. Then we proceed to assess § the possible contamination of the Lyman-α flux in 3.4, and confirm that for this § systematic the 3D Lyman-α power spectrum is more affected than the 1D. In 3.5 § we explore the use of 21 cm observables to address this contamination, and find

that non-linear eﬀects must be taken into account to appropriately utilize the link

between the imprint of inhomogeneous reionization in the Lyman-α forest and the 21

cm quadrupole. We summarize our results and discuss directions for future work in

3.6. § 3.2 Conventions and formalism

Throughout this paper we use the ΛCDM cosmological parameters from Planck

2015 ‘TT + TE + EE + lowP + lensing + ext’ (Planck Collaboration et al., 2016),

2 2 −1 −1 namely: Ωbh = 0.02230, Ωmh = 0.14170, H0 = 67.74 km s Mpc , σ8 = 0.8159

and ns = 0.9667.

For convenience throughout this paper we use the dimensionless power spectrum

when we refer to a given power spectrum, i.e. no Mpc3 unless explicitly stated.

However, in the case of the 21 cm power spectrum the k3P (k)/2π2 have units of

2 mK . Furthermore, we deﬁne all ﬂuctuations as δp = p/p¯ 1, with the only exception −

of the neutral hydrogen fraction ﬂuctuations where δx = xHI x¯HI. HI −

The fluctuations of the Lyman-α forest transmitted flux are described by δF = (F − F¯)/F¯, where F¯ is the mean observed flux and the transmitted flux is normalized such

93 that 0 F 1. Then at linear order27 the flux fluctuations should be proportional to ≤ ≤ the matter fluctuations. Furthermore, taking into account redshift space distortions

2 due to velocity gradients we have δF = bF(1+βFµ ) δm, where δm is the matter density contrast, bF is the usual Lyman-α ﬂux bias, βF is the redshift distortion parameter and µ = cos θ = kk/k the angle to the line of sight. Moreover, the corresponding power spectrum of δF is denoted by PF(k, µ).

In this work we explore the eﬀect of inhomogeneous reionization in the Lyman-α transmission,

2 δF = bF (1 + βFµ ) δm + bΓ ψ(zre), (3.1)

¯ where bΓ is the radiation bias deﬁned by bΓ = ∂ ln F /∂ ln τ1 (Arinyo-i-Prats et al.,

2015; Hirata, 2018). Here τ1 is the optical depth that must be assigned to a patch of

4 gas with mean density ∆b 1 + δb = 1 and temperature T = 10 K in order for the ≡ mean transmitted ﬂux F¯ to match observations.28 This bias parameter is needed here due to the way the post-processing of our small-scale simulations works (see 3.3). As § is common in Lyman-α forest studies, we vary the normalization of τ1 (or equivalently, we vary the ionizing background) until we obtain the correct mean transmitted ﬂux

F¯. Thus, when varying any aspect of the simulation – here the redshift of reionization

– the quantity we measure in the simulation is ∆ ln τ1, and the corresponding change in ﬂux δF has an additional factor of bΓ. In addition, we deﬁne

τ1(zre) ψ(zre) = ∆ ln τ1(zre, z¯re) = ln , (3.2) τ1(¯zre)

27See Arinyo-i-Prats et al.(2015) for a description of the linear biasing coeﬃcients and their range of validity. 28This is the same usage as in Hirata(2018).

94 which parametrizes the variations in the transparency for an opacity cube that sud-

denly reionizes at zre relative to an opacity cube that reionizes atz ¯re. The sign convention is such that ψ(zre) > 0 if gas reionized at zre is more transparent (has higher F ) than gas reionized atz ¯re.

The corresponding 3D Lyman-α ﬂux power spectrum is given by

3D 2 2 2 P (k, µ, zobs) b (1 + βFµ ) Pm(k, zobs) F ≈ F 2 + 2bF bΓ (1 + βFµ )Pm,ψ(k, zobs), (3.3)

where Pm is the matter power spectrum and Pm,ψ is the cross-power spectrum of matter and ψ. Note that we have neglected the third term in equation (3.3), the auto-power spectrum of ψ, because it is second order in ψ.

In principle the problem is now reduced to devising a way to model Pm,ψ(k, z) and,

since it involves physics from diﬀerent scales, we will divide and conquer. We begin

by re-writing the cross-power spectrum Pm,ψ(k, z) in a form that is straightforward

and numerically stable to compute from 21cmFAST boxes – in particular, that can

be computed from a sequence of redshift outputs during reionization. We see that

3 (3) 0 (2π) δ (k k )Pm,ψ(zobs, k) − Z 0 0 D E 3 0 −ik ·r ˜∗ 0 = d r e δm(zobs, k)ψ(zre(r ), zobs) R3 Z Z zmin 0 3 0 −ik0·r0 D˜∗ ∂ψ(z , zobs) = d r e δm(zobs, k) 0 3 − R zmax ∂z 0 0 E 0 Θ(z zre(r )) dz × − Z zmax 0 ∂ψ 0 ˜∗ 0 0 0 D(zobs) = dz 0 (z , zobs) δm(z , k)˜xHI(z , k ) 0 . (3.4) − zmin ∂z h i D(z )

Here we write ψ˜ as an explicit Fourier transform of the real space in the second line.

We then introduce the step function and apply the fundamental theorem of calculus

95 0 0 to write ψ(zre(r , zobs)) as negative an integral of its derivative from zre(r ) to zmax.

(Here zmin and zmax span the range of redshifts for reionization. The ψ(zmax, zobs) term has no spatial dependence and drops out for k = 0.) In the last equality, we 6 ˜ 0 have used the ratio of growth functions to extrapolate δm from z to zobs. Also we have

0 0 0 0 used the fact that xHI(r , z ) = Θ(z z(r )), and then wrote the neutral hydrogen − fraction ﬁeld in Fourier space. Thus:

Z zmax ∂ψ D(zobs) Pm,ψ(zobs, k) = Pm,xHI (z, k) dz . (3.5) − zmin ∂z D(z)

We integrate from zmax = 34.7 to zmin = 5.90, such that in our default model we

cover from the almost-fully-neutral stage through the end of reionization. Hence we

have computed the cross-power spectrum of ψ and matter by using the cross-power

spectrum of matter and neutral fraction to describe how matter and bubble spatial

structure are correlated at a given redshift, i.e. how patchy reionization enters the

fray. Furthermore, Eq. (3.5) directly depends on the change of the transparency of the

IGM with respect to the redshift of reionization. We compute Pm,xHI by modifying the

code available in 21cmFAST(Mesinger et al., 2011). It is important to highlight that

in fact what gets computed from this procedure is the dimensionless power spectrum

∆m,xHI and hence ∆m,ψ.

We have not included ﬂuctuations in the low-redshift ionizing background, which

will also change the ﬂux power spectrum and have been calculated elsewhere (Pontzen,

2014; Gontcho A Gontcho et al., 2014). To lowest order, the ionizing background

eﬀects should be added to the inhomogeneous reionization eﬀects. The ionizing back-

ground eﬀect is most prominent at large scales; for example, in the default model of

Pontzen(2014) at zobs = 2.3, the change in power spectrum in the range of scales of

3D 3D −1 −1 interest to us is ∆P (k)/P (k) 2∆bHI/bHI 0.08(k/0.14 Mpc ) (see Figure F F ∼ ∼ − 96 2 of Pontzen 2014).29 This is similar in magnitude to the reionization eﬀect considered here.

Furthermore, implementation of the eﬀects of He ii reionization in the IGM (Becker

et al., 2011; Walther et al., 2018b) may be crucial to correctly compute the impact in

the Lyman-α forest of H i reionization. Besides, our simulations use simpliﬁed models

of hydrogen reionization, and the implementation of several key physics will be the

scope of future work (e.g. pre-heating before reionization, variations in the thermal

and reionization histories). See 3.3 and 3.6 for more details. § § 3.3 Simulations

To predict the eﬀect of reionization on the Lyman-α forest, we need two types of

simulations.

The “small box” simulation uses a box size that is smaller than the typical scale

of a reionization bubble (L = 2.55 Mpc), and can resolve structures down to below the Jeans scale. Its purpose is to determine how the transmitted ﬂux of the Lyman-α forest depends on the reionization redshift zre, i.e., to determine the function ψ(zre).

The small boxes are assumed to reionize instantaneously, and to determine the func- tional form ψ(zre) we run a sequence of boxes with zre stepped over the interesting range (here 6–12). This way we account for correlations between density and diﬀerent reionization scenarios. The small boxes have high resolution, since the thermal state of the IGM depends on the way in which small-scale structures are disrupted following reheating (Hirata, 2018). These boxes do not need to self-consistently determine

29 1 The k− dependence at scales small compared to the photon mean free path is the result of the expansion of S(k).

97 where the ionizing sources are since reionization is externally triggered at a particular redshift.

In contrast, the “large box” simulations are large compared to the scale of reionization bubbles. We choose L = 300 Mpc such that the simulations have enough statistical power. Their purpose is to predict the spatial structure of the bubbles

(reionization) and their correlation with matter. The simulations have a prescription for placing ionizing sources, and a fast approximation scheme to track the fate of ionizing photons.

It is only with the combination of the two types of boxes that we can compute all of the ingredients in 3.2. Note that these simulations incorporate different physical § ingredients and hence we will use two completely separate codes. An important caveat of our simulation strategy is that we miss higher-order correlation functions between the reionization field (on large scales) and the small-scale density field – due to the small-scale boxes being hydro simulations.

3.3.1 Small boxes

We follow Hirata(2018) in the methodology of the small box simulations, but we only use the biggest box size studied in their work. In particular, we use a modiﬁed version of Gadget-2 (Springel et al., 2001; Springel, 2005), which is a smoothed particle hydrodynamics (SPH) code. In this modiﬁed Gadget-2 we include the most relevant heating and cooling processes for the gas. Gadget-2 comes with adiabatic expansion/contraction (including Hubble expansion) and shock heating already implemented. We add Compton heating/cooling for neutral gas (with residual ionization) and Compton cooling for the ionized gas; see Eq. (17) in Hirata(2018). Also for

98 ionized gas we include recombination cooling, photoionization heating, free-free cool-

ing, and He ii line cooling; see Eqs. (19–23) in Hirata(2018). Following the default

treatment in Hirata(2018), reionization is treated with an uniform post-reionization

4 temperature of T4,re = 2 (or Tre = 2 10 K) everywhere. Thereafter, photoionization × heating is implemented with an energy injection of 4.2 eV per H i ionization and 7.2

eV per He i ionization.

Each simulation we used in Gadget-2 has the same box size, L = 1728 h−1 =

2551 kpc and the same number of particles, 2 (384)3. The dark matter particle mass × 3 3 in the simulations is 9.72 10 M , while the gas particle mass is 1.81 10 M . × × The only diﬀerence among them is when reionization turns on. We simulate eight

realizations in order to reduce the statistical error due to the limited box size by a

factor of √8.

As in Hirata(2018) we start all the simulations at recombination, zdec = 1059 with

a modiﬁed version of N-Gen-IC (the default initial condition generator in Gadget-

2) to enable streaming velocities between the baryons and dark matter (Tseliakhovich

and Hirata, 2010). The boxes are evolved with neutral gas physics until reionization,

at which point the temperature is reset to 2 104 K, and the box is evolved further × with singly ionized (H+/He+) primordial gas physics. There is no He ii reionization

in these simulations. These simulations were tested for convergence in 5.3 of Hirata § (2018), including varying the box size to test the eﬀects of missing large-scale power.

The missing variance at large-scales for our small-scale boxes with L = 2551 kpc

is σ2(z = 2.5) = 0.77. For comparison, Hirata(2018) also ran the “II-C” boxes

2 with L = 1275 kpc or σ (z = 2.5) = 1.32. For zre = 7 vs. 8 and 9 vs. 8, and for

2.5 zobs 4.0, ∆ ln τ1 changing by < 10% or < 2σ. ≤ ≤

99 These simulations generate a map of optical depth with arbitrary normalization

(Hirata, 2018). The requirement to reproduce the observed mean ﬂux sets the nor-

malization. Furthermore, the required normalization is reported in the form of τ1, and hence we use τ1 to show the dependence of the transmission of the Lyman-α forest with the redshift of reionization.

All of the small box simulations were done on the Ruby cluster at the Ohio Su- percomputer Center (Ohio Supercomputer Center, 2015).

3.3.2 Large boxes

We include the physics of reionization-bubble-scale by using semi-numerical simulations, specifically we use 21cmFAST (Mesinger et al., 2011). We utilize all the default parameters in 21cmFAST with the exceptions of a few parameters that we change for different alternatives scenarios in order to examine different reionization histories, and the use of our chosen background cosmology. The box size for each one of these simulations is L = 300 Mpc. We run eight realizations for each of the three different reionization scenarios with the same physical setup but with different random seeds to diminish the simulation variance. Furthermore, we generate snapshots that track – among other parameters – the density, neutral fraction of hydrogen and the 21 cm temperature fields for 34.70 z 5.90, with a step size of 2%, i.e., ≥ ≥ z< = (z> + 1)/1.02 1. Therefore, in total we have 84 snapshots (each with 3 de- − pendent variables) for the computation of the cross-power spectrum of matter and neutral fraction, and the 21 cm quadrupole per realization. In order to compute the cross-power spectra we only modify “delta ps” in 21cmFAST, such that it can take

100 both a density and neutral fraction snapshot. Furthermore, to be able to compute

the 21 cm quadrupole we include the ability to compute PT (k, µ) instead of PT (k).

To explore diﬀerent reionization histories we change the number of ionizing pho-

tons escaping into the IGM per baryon in collapsed structures, i.e. HII_EFF_FACTOR

in 21cmFAST, which represents the ionizing eﬃciency

ζ = Nγ/b fescf∗ fb , (3.6)

where fb is the baryon fraction of a halo in units of the cosmic mean value Ωb/Ωm, f∗ is the fraction of baryons from the halos that form stars, Nγ/b corresponds to the number of ionizing photons produced per stellar baryon, and fesc is the fraction of produced ionizing photons which escape into the IGM (see Dayal and Ferrara 2018 for more details regarding the physics behind Eq. (3.6)). In contrast to our approach here, O˜norbe et al.(2018) used both ζ and the minimum halo mass for producing ionizing photons as free parameters for their hybrid simulations.

We define a default model with optical depth to reionization equal to the current best fit value of the recent final results of Planck 2018 ‘TT +TE+EE+lowE+lensing’

(Planck Collaboration et al., 2018a), i.e. τ = 0.054, which corresponds to an ionizing eﬃciency of ζ = 25.30 In our default model this optical depth corresponds to a redshift of reionization of zre = 7.61, which corresponds to when the mean neutral hydrogen fraction is 0.5. For comparison purposes we chose alternative models with later (“model A”) and earlier (“model B”) reionization than our default model. We select these alternative scenarios so that they encompass the range of models consistent with the new Planck results and other reionization probes at 1σ (see Bouwens ∼ ± 30The main diﬀerence between the Planck 2015 and 2018 results is the value of the optical depth which in 2015 was τ = 0.066. We keep all other cosmological parameters equal to the cosmology from Planck 2015 used throughout this paper.

101 Table 3.1: Summary of the different reionization models. Note that since model A has not finished the reionization process by redshift 5.90, its optical depth is not as accurate as the other models. Furthermore, since the volume weighted neutral fraction differs from the mass weighted one at the end of the reionization process, the optical depths are approximations.

Reionization model τ zre ζ A: later reionization 0.0512 7.22 20.9 Default: Planck 2018 reionization 0.0548 7.61 25 B: earlier reionization 0.0615 8.34 35

et al. 2015 for a compact list of these probes). We choose our model A such that

the volume weighted neutral faction at z = 5.9 approximately matches the 1σ upper limit extracted from the dark pixel measurements reported in McGreer et al.(2015),

31 x¯HI 0.11. This corresponds to roughly ζ = 20.9, and by construction the reion- ≤ ization process has not been completed by the end of our large simulations.32 Model

B has ζ = 35 with zre = 8.35, which is 1σ away from the Planck optical depth. See

Fig. 3.1 and Table 3.1 for a description of the models.

3.4 Assessment of contamination

The goal of this work is to quantify the eﬀect of inhomogeneous reionization in

the Lyman-α ﬂux. In principle to estimate the contamination to the signal we simply

require both terms in the RHS of equation (3.3), i.e. the linear theory term and the

non-linear extension.

31We note that our model A might be considered as conservative in light of recent high redshift constraints of the hydrogen neutral fraction, e.g., xHI = 0.88 at z = 7.6 (Hoag et al., 2019) and x > 0.76 at z 8 (Mason et al., 2019). HI ∼ 32The lower τ from Planck was reported too late to change the small-box simulations to run to lower z.

102 1

0.8

0.6 model A

HI default x 0.4 model B

0.2

0 6 7 8 9 10 11 12 13 14 z

Figure 3.1: Neutral fraction history for the diﬀerent models used in our large boxes. Our models diﬀer only by the number of ionizing photons escaping into the IGM per baryon, with model A having the latest reionization, and model B having the earliest reionization.

103 3.4.1 Linear power spectrum (3D)

We start by computing the linear term. We simplify this process by assuming that

µ = 0, i.e. perpendicular to the line of sight so that one can ignore redshift space

distortions. We compute the matter power spectrum from the density boxes generated

in 21cmFAST with our chosen cosmology. For the sake of illustrating the contamination

we choose a redshift of observation of 2.5 and a wavenumber of 0.14 Mpc−1, both

values are typical of Lyman-α measurements. Then, we have

3D −1 2 3 PF (zobs = 2.5, k = 0.14 Mpc ) = 995.3bF (zobs) Mpc . (3.7)

We extract the Lyman-α ﬂux bias from Table 1 in McQuinn and White(2011).

We summarize the relevant values in Table 3.2. Since we are using the results from

McQuinn and White(2011) we will be consistent with their assumption that the redshift distortion parameter is equal to unity, i.e. βF = 1 throughout our work.

3.4.2 Computation of ψ(zre)

We use our smaller simulations to calculate how the transparency of the Lyman-

α forest depends on when reionization occurs. Our small-box simulations ( 3.3.1) § compute Lyman-α transmission by the procedure described in detail in 4.4 of Hirata § (2018). Here we brieﬂy describe the strategy used, and the methodology for extracting

ψ from our simulations.

Each run of our small-scale simulations generates eight snapshots of optical depth

maps starting at z = 5.5. They are separated by decrements of ∆z = 0.5. We run

our small simulations for seven diﬀerent redshifts of reionization, starting at redshift

six, i.e. zre = 6, 7, 8, 9, 10, 11, 12 . We chosez ¯re = 8 as our reference redshift of { }

104 Table 3.2: Bias factors: Flux bias bF, radiation bias bΓ and ratio of bΓ/bF at diﬀerent redshifts. z bF bΓ bΓ/bF 2.0 -0.12 -0.084 0.70 2.5 -0.18 -0.146 0.81 3.0 -0.27 -0.237 0.88 3.5 -0.37 -0.356 0.96 4.0 -0.55 -0.505 0.92

reionization, although due to the deﬁnition of ψ as a change in ln τ1, the choice of

reference is simply a constant oﬀset in ψ with no eﬀect on Pm,ψ(k).

Once all the relevant snapshots have been generated, one can ﬁnally compute how the transparency of the Lyman-α forest depends on the redshift of reionization.

We obtain ψ from Eq. (3.2) by comparing the τ1 from Simulation B with redshift of reionization zre to the τ1 obtained for Simulation A with redshift of reionization z¯re = 8. We present our results for ψ and its redshift dependence in Table 3.4.2. In what follows, we linearly interpolate ψ between the redshifts in the table.

3.4.3 Cross-power spectrum of matter and ψ

In Eq. (3.5) we accounted for the inhomogeneous nature of reionization by including the cross-power spectrum of matter and hydrogen neutral fraction. We obtain this cross-power spectrum from our modiﬁed version of 21cmFAST. We plot the dimensionless cross-power spectrum of matter and neutral fraction for the diﬀerent models in

Fig. 3.2. Note that Pm,xHI is not only negative, because overdense regions will ion- ize ﬁrst resulting in an anti-correlation between matter density and neutral fraction.

105 Table 3.3: Results for the small-scale simulations. Transparency variations in the IGM, ψ = ∆ ln τ1(Sim. B)/τ1(Sim. A). The number in square brackets is the redshift at which reionization turns on. Note that ∆ ln τ1 is negative if simulation A is more transparent than simulation B.

5 Box size Sim. A Sim. B 10 ∆ ln τ1 × [ckpc] = 2 0 = 2 5 = 3 0 = 3 5 = 4 0

106 z . z . z . z . z . 2551 [8] [6] 6064 553 6910 895 9772 973 14616 1068 22522 1476 ± ± ± ± ± 2551 [8] [7] 2112 235 2399 367 3612 436 5769 443 9275 550 ± ± ± ± ± 2551 [8] [9] -878.8 166 -649.6 252 -1083 315 -2548 299 -4682 379 ± ± ± ± ± 2551 [8] [10] -1300 303 -407.5 406 -847.8 560 -2950 617 -6344 659 ± ± ± ± ± 2551 [8] [11] -1425 413 -52.51 582 -318.9 724 -2626 780 -6610 899 ± ± ± ± ± 2551 [8] [12] -1552 482 199.7 674 283.5 850 -2128 912 -6400 1007 ± ± ± ± ± 0.005 0 -0.005 HI -0.01 m,x -0.015 model A ) P

2 default -0.02 model B /2 π 3 -0.025 (k -0.03 -0.035 -0.04 5 10 15 20 25 30 35 z

Figure 3.2: Dimensionless cross-power spectrum of matter and neutral hydrogen fraction as a function of redshift for our diﬀerent models of reionization history. All cross-power spectra have been evaluated at wavenumber k = 0.14 Mpc−1.

The absolute value peaks around the redshift of reionization, because both before and

after reionization the perturbations in xHI go to zero.

We determine the impact of inhomogeneous reionization on the Lyman-α ﬂux from

both our calculations of how the transparency of the IGM depends on the redshift

of reionization 3.4.2, and the cross-correlation of mater and the neutral hydrogen § fraction. Using Eq. (3.5) evaluated at k = 0.14 Mpc−1 and observing at redshift 2.5

3 for our default model, we obtain Pm,ψ = 26.7bF(zobs)bΓ(zobs) Mpc . Therefore we − see that the contamination to the linear term in Eq. (3.7), ignoring the bias factors, is approximately 5.36%. We ignored the bias ratio to quantify this change, however as seen in Table 3.2 both bias factors can signiﬁcantly aﬀect the deviation. We tabulated

107 Table 3.4: Percentage deviation of the 3D and 1D Lyman-α power spectrum due to patchy reionization for the diﬀerent reionization models considered. Here we have used k = 0.14 Mpc−1, and we included the bias ratio. −1 Simulation Ratio of 2 (bΓ/bF) Pm,ψ/Pm 100% with k = 0.14 Mpc × × × zobs = 2.0 zobs = 2.5 zobs = 3.0 zobs = 3.5 zobs = 4.0 Model A 3D 4.09 0.47 5.56 0.98 9.85 1.45 19.6 1.87 35.9 2.63 Default 3D 3.39 ± 0.42 4.35 ± 0.87 7.82 ± 1.32 16.4 ± 1.69 31.0 ± 2.31 Model B 3D 1.96 ± 0.32 1.84 ± 0.66 3.45 ± 1.05 9.10 ± 1.34 19.3 ± 1.73 Model A 1D 0.75 ± 0.08 0.92 ± 0.17 1.81 ± 0.28 3.63 ± 0.34 6.53 ± 0.42 Default 1D 0.61 ± 0.08 0.68 ± 0.16 1.36 ± 0.26 2.92 ± 0.31 5.45 ± 0.37 Model B 1D 0.35 ± 0.06 0.26 ± 0.12 0.52 ± 0.21 1.52 ± 0.26 3.27 ± 0.29 ± ± ± ± ±

the deviation of the Lyman-α power spectra, taking into account the role of the bias

factors, for wavenumber 0.14 Mpc−1 and diﬀerent redshifts of observation in Table 3.4.

We obtained the radiation bias directly from our GADGET 2 simulations by computing

¯ 33 bΓ = ∂ ln F /∂ ln τ1 (see Table 3.2).

In Fig. 3.3 we plot a comparison of the strength of the eﬀect by looking at the ratio of the 3D dimensionless cross-power spectrum of the reionization eﬀect and the mass auto-power spectrum as a function of wavenumber. The hierarchical structure in redshift of observation for all the models is anticipated, independent of the scale.

The higher the redshift, the larger the eﬀect of thermal relics from reionization on the

IGM. That is because there is not enough time to relax onto the usual temperature- density relation. In addition, the ratio is larger at large scales, which is expected since it couples to the reionization bubble scales.

33Note that the bias factors obtained from our simulations seem to be slightly higher than the ones reported in Hirata(2018); nevertheless, we consider these good enough for a first estimation of the effects of inhomogeneous reionization in the Lyman-α flux.

108 The error bars in Fig. 3.3 and Table 3.4 were computed by combining the errors

from our large-box and small-box simulations. Because both simulations are indepen-

dent of each other – i.e. the error sources are orthogonal – and both enter “linearly”

into the computation of Pm,ψ, one can average over the realizations of the small-box simulations and then compute the sample variance on the mean for the large-box simulations, and vice-versa. Finally, the complete error bars are computed by adding the two variances in quadrature, i.e.   N =8 Nj =8 1 X 1 Xi 1 X f(ψ , F ) = f ψ , F , (3.8) 64 i j 8 i 8 j ij i=0 j=0 v 2  2 u N =8 ! Nj =8 u 1 Xi 1 X error = uSV F , ψ + SV ψ , F , (3.9) t j 8 i  i 8 j ⇒ i=0 j=0

where f stands for our algorithm to compute ∆P/P , SV stands for sample variance

on the mean, the ψi represent the small-scale realizations and Fj are the large-scale

realizations.

3.4.4 Linear power spectrum (1D)

Once we have the 3D power spectrum (the cross-correlation between diﬀerent

skewers) we can compute the 1D Lyman-α power spectrum (the cross-correlation

between pixels of the same skewer) which is given by averaging over the perpendicular

direction to the line of sight, i.e.

Z ∞ 1D dk⊥ 3D PF (k, zobs) = k⊥PF (k, zobs) 0 2π Z ∞ 2 dk⊥ 2 2 3D = bF k⊥(1 + µ ) Pm (k, zobs) 0 2π Z ∞ dk⊥ 2 3D +2 bF bΓ k⊥(1 + µ )Pm,ψ(k, zobs) 0 2π 2 1D 1D = bFPm (k, zobs) + 2 bF bΓPm,ψ(k, zobs). (3.10)

109 0.5 zobs = 2.0 0.4 zobs = 2.5 zobs = 3.0

3D 0.3 zobs = 3.5 m zobs = 4.0

/ P 0.2 3D

m, ψ 0.1 P 0

-0.1 0.01 0.1 1 k [Mpc-1]

(a). Model A: Later reionization.

0.5 zobs = 2.0 0.4 zobs = 2.5 zobs = 3.0

3D 0.3 zobs = 3.5 m zobs = 4.0

/ P 0.2 3D

m, ψ 0.1 P 0

-0.1 0.01 0.1 1 k [Mpc-1]

(b). Default model: Planck 2018 reionization.

0.5 zobs = 2.0 0.4 zobs = 2.5 zobs = 3.0

3D 0.3 zobs = 3.5 m zobs = 4.0

/ P 0.2 3D

m, ψ 0.1 P 0

-0.1 0.01 0.1 1 k [Mpc-1]

(c). Model B: Earlier reionization.

Figure 3.3: Comparison for the diﬀerent reionization models between the dimensionless cross-power spectrum of ψ and matter, and the dimensionless power spectrum of matter as a function of wavenumber for ﬁxed redshift of observation.

110 −1 Since our 21cmFAST simulations use a kmax 3.20 Mpc , we set this wavenumber as ≈ the upper limit of the integrals over the perpendicular direction.

1D 1D We report the percentage of the deviation, i.e. the ratio of 2(bF/bΓ)P /P m,ψ m × 1D 1D 100% from our simulations in Table 3.4. Furthermore, we plot the ratio of Pm,ψ/PF as a function of wavenumber for the diﬀerent redshifts of observation that we explored in

Fig. 3.4. We use the 1D Lyman-α power spectrum from BOSS (Palanque-Delabrouille et al., 2013). From Fig. 3.4 and Fig 3.3 we conﬁrm that the eﬀect of reionization is stronger for the Lyman-α 3D power spectrum than for the Lyman-α 1D power spectrum, as one could have expected due to the integration smoothing the deviation. Moreover, the 1D ratio is also consistent with the redshift hierarchical structure

(within the error bars). The error bars shown in Fig. 3.4 have been computed in the same way we described in 3.4.3, and hence we have ignored the error bars from § the BOSS data, since they are small compared to the uncertainties in the reionization models. We note that our results for the eﬀect of inhomogeneous reionization in the

1D Lyman-α forest power spectrum at zobs = 4.0 are slightly smaller than the ones reported in O˜norbe et al.(2018) – see their Fig. (7) – but the trend at large scales are consistent and the diﬀerences in numerical values can be attributed to model dependence.

Finally, as could have been foreseen, the imprint from reionization aﬀects the

Lyman-α forest the most for our model of late reionization (model A) since the IGM has the least time to “thermalize” for zre 7.25. In contrast, the early reionization ≈ model suﬀers the lowest eﬀect due to having more time to dilute the thermal relics, and achieve the usual temperature-density relation. Furthermore, the overall magnitude

111 0.06 zobs = 2.0 0.05 zobs = 2.5 z = 3.0 0.04 obs 1D zobs = 3.5 m 0.03 zobs = 4.0 / P

1D 0.02 m, ψ

P 0.01

-0.01 0.1 1 k [Mpc-1]

(a). Model A: Later reionization.

0.06 zobs = 2.0 0.05 zobs = 2.5 z = 3.0 0.04 obs 1D zobs = 3.5 m 0.03 zobs = 4.0 / P

1D 0.02 m, ψ

P 0.01

-0.01 0.1 1 k [Mpc-1]

(b). Default model: Planck 2018 reionization.

0.06 zobs = 2.0 0.05 zobs = 2.5 z = 3.0 0.04 obs 1D zobs = 3.5 m 0.03 zobs = 4.0 / P

1D 0.02 m, ψ

P 0.01

-0.01 0.1 1 k [Mpc-1]

(c). Model B: Earlier reionization.

Figure 3.4: Comparison between the 1D dimensionless cross-power spectrum of ψ and matter, and the 1D dimensionless power spectrum of matter as a function of wavenumber for ﬁxed redshift of observation.

112 of the eﬀect is surprisingly strong. Hence, it will be crucial to correctly account for

this systematic signal in both near-term and future Lyman-α forest surveys.

3.5 Constraining the eﬀect with 21 cm cosmology

In the previous section we described a potentially very signiﬁcant systematic for

Lyman-α forest measurements, one that is particularly important for later reionization

scenarios. In this section we will illustrate how this systematic could become an

interesting link between Lyman-α and 21 cm cosmology.

In this section we show how non-parametric mitigation of the thermal imprint of reionization in the Lyman-α forest is possible through the use of the “linear µk- decomposition” scheme (Barkana and Loeb, 2005). Unfortunately, as we will illustrate, non-linear effects are significant and hence in future work we will continue our efforts of non-parametric mitigation by employing the“quasi-linear µk-decomposition”

(Mao et al., 2012).

Under the linear µk-decomposition the 3D 21 cm power spectrum is given by

(Mao et al., 2012)

3D 2 4 PT (k) = Pµ0 (k) + Pµ2 (k) µ + Pµ4 (k) µ , (3.11)

where each Pµ(k) = Pµ(k) is angle-averaged over constant k-shells. Moreover, in h i

the limit of Ts TCMB, which corresponds to the range where Pm,x is important HI (see Fig. 3.2), Eq. (3.11) becomes

2 0 ( ) = ( ¯ ( )) ( ), (3.12) Pµ k, z δTb z PδρHI k, z

2 2 ( ) = 2( ¯ ( )) ( ) and (3.13) Pµ k, z δTb z PδρHI ,δρH k, z

2 4 ( ) = ( ¯ ( )) ( ), (3.14) Pµ k, z δTb z PδρH k, z

113 ¯ where δρHI = δρH + δxHI + δxHI δρH and δTb is the mean of the brightness temperature.

In the limit of Ts TCMB we have 3 ¯ 3c A10T∗n¯HI(z) δTb(z) = 3 32πνhf (1 + z)H(z) 2 2 1/2 Ωbh Ωmh 1 + z 26.6¯xHI mK , (3.15) ≈ 0.0223 0.1417 10

where A10 is the Einstein coefficient of the hyperfine transition, T∗ is the 21 hyperfine transition in temperature units and νhf is the 21 cm frequency.

Rewriting Eq. (3.11) into `-multipoles we have 3D Pµ2 Pµ4 2 6 P (k, z) = P 0 + + + P 2 + P 4 L (µ) T µ 3 5 3 µ 7 µ 2 8 + P 4 L (µ), (3.16) 35 µ 4

where L` are the Legendre polynomials (see Appendix B.1). Note that in our notation

the quadrupole term can be expanded as `=2 2 ¯ 2 2 20 PT = (δTb) Pm,xHI + Pm , (3.17) 3 x¯HI 7

where we have taken advantage of the fact that hydrogen traces the matter distribu-

tion, and the extra factor of neutral hydrogen fraction comes from the way we have

deﬁned our perturbations on the neutral fraction, i.e. δx = xHI x¯HI. HI − Hence the cross-power spectrum of matter and fraction of neutral hydrogen atoms

is related through Eq. (3.17) to the ` = 2 multipole component of the 21 cm power

spectrum. Therefore with a measurement of the 21 cm power spectrum one can

in principle constrain the cross-power responsible for the systematic imprint in the

Lyman-α forest.

The ﬁrst step for our mitigation scheme to be successful is to be able to reproduce

the 21 cm power spectrum. As was quantiﬁed in Fig. 10 of Mao et al.(2012), the

114 20 ]

2 model A 15 default [mK 10 model B 5 0

T, l = 2 -5 ) P 2 -10 /2 π

3 -15 (k -20 0.1 1 k [Mpc-1]

Figure 3.5: Failure of the linear µk-decomposition to reproduce the quadrupole of the 21 cm power spectrum at redshift 7 for the diﬀerent models. The dashed lines represent the quadrupole from 21cmFAST for the diﬀerent models. In solid lines we have the quadrupole from the linear approximation.

error in the linear method can easily reach thirty percent or more for the smallest scales. Thus one should expect signiﬁcant errors in the quadrupole of the 21 cm power spectrum computed with the linear approximation.

We test the accuracy of the linear decomposition, i.e. we compare the right-hand side of Eq. (3.17) with the output from the 21cmFAST simulations. We show the failure of the linear µk-decomposition in Fig. 3.5 for the default model at redshift 7. The linear decomposition fails similarly for the other models.

115 Unfortunately, we see considerable errors in the decomposition. The main component of the discrepancy lies in the absence of non-linear eﬀects (higher order corre-

lations involving δxHI and δm). This is not unexpected, as Lidz et al.(2007) and Mao et al.(2012) had previously reported significant errors for the linear method due to absence of higher order correlations. In Fig. 3.5 we see the effects of non-linear matter clustering and non-linear velocity perturbations at z = 7, which cause the failure of the linear scheme. However, Mao et al.(2012) also provided an alternative, the quasi-linear µk-decomposition. In the regime we are interested in, the quasi-linear method rewrites Eqs. (3.12–3.14) into Eqs.(64–66) in Mao et al.(2012). In future work we will continue our efforts on extracting the quadrupole of the 21 cm power spectrum armed with the quasi-linear decomposition.

Even though the µk-decomposition fails to reproduce the quadrupole of the 21 cm power spectrum, one can still see the correct trends at large scales. Speciﬁcally, the quadrupole from the earlier reionization model is larger than the one from the default model, and they are both larger than the quadrupole from the later reionization model. We take this trend as a positive sign that future work with the quasi-linear decomposition may be able to correctly mitigate the eﬀect of patchy reionization in the Lyman-α forest.

3.6 Discussion

Hydrogen reionization is one of the deﬁning events in the thermal and dynamical history of the IGM. It heats up the IGM to > 104 K, and by increasing the Jeans mass, it disrupts pre-existing small-scale structure. We have seen that the thermal and dynamical eﬀects persist for cosmological timescales, and that the transmission

116 of the Lyman-α forest even at zobs < 3 is sensitive to the reionization model. Since reionization is believed to have occurred inhomogeneously – by expanding and over- lapping ionized “bubbles” correlated with large-scale structure – this dependence on reionization redshift zre translates into a spatial modulation of the Lyman-α forest

and hence a correction to the power spectrum.

The magnitude of the eﬀect is largest in the 3D power spectrum at the highest

observed redshifts and on large scales (which are better matched to the scale of

reionization bubbles). For example, in the 3D power spectrum at zobs = 4 and

k = 0.14 Mpc−1, we ﬁnd corrections of 19–36% depending on the reionization model

chosen. At lower zobs, the eﬀect of reionization is reduced, declining to 2.0–4.1% at zobs = 2. For the 1D Lyman-α power spectrum the deviation is signiﬁcantly smaller:

−1 again at k = 0.14 Mpc we estimate 3.3–6.5% at zobs = 4, declining to 0.35–0.75%

at zobs = 2. The corrections due to reionization are small, but we should remember

that the 1D power spectrum is already measured at very high signal-to-noise ratio:

−1 for example at zobs = 2.2 and k = 0.116 Mpc , BOSS+eBOSS have a statistical

error of 1.2% per bin34 (Chabanier et al., 2018). These statistical errors will shrink

further in the DESI era.

In principle, measurements of diﬀuse 21 cm radiation from the epoch of reion-

ization can constrain the reionization model, one of the key inputs in calculating

the correction to the Lyman-α power spectrum. We are particularly interested in

`=2 the 21 cm quadrupole PT (k, z), since it is sensitive to the speciﬁc power spectrum

Pm,xHI (k, z) that we need. Unfortunately, the simplest implementation of this idea

– the “linear µk decomposition” theory – is not accurate in the range of parameter

34The bin size is ∆z = 0.2 and ∆k/k = 0.03, and corresponds to the second row of Table 4 in Chabanier et al.(2018).

117 space we need. In future work we will investigate other correction schemes, includ-

ing models with corrections to linear theory (building on past work, e.g., Mao et al.

2012), and schemes where reionization models are parameterized (e.g., source ionizing

eﬃciency, minimum mass, IGM clumping parameters) and then 21 cm observations

are used to constrain the parameters rather than directly infer power spectra.

Another avenue for future work is to incorporate some of the other physical ef-

fects that may interact with inhomogeneous hydrogen reionization. One of the most

important may be He ii reionization, which is believed to have occurred around z 3 ∼ and resulted in an additional energy injection into the IGM. This has likely been seen

in the thermal evolution of the IGM inferred from the Lyman-α forest (e.g., Becker

et al., 2011; Walther et al., 2018b). This additional energy injection can reduce the

sensitivity of the low-redshift IGM to its initial thermal state, e.g., it can reduce

∂ ln T (z = zobs)/∂ ln T (z = zre) (see, e.g., the discussion in Hirata 2018 in the context of streaming velocities), though the change depends on the timeline and the relative contributions of EUV and X-ray radiation. Previous studies have also found that

He ii reionization introduces its own imprints on the Lyman-α forest (McQuinn et al.,

2009; Compostella et al., 2013; Greig et al., 2015). In any case, it appears that at the level of precision of interest for modern cosmological Lyman-α forest studies, the

IGM may not have relaxed from inhomogeneous hydrogen reionization before helium reionization takes place.

A second issue is that we have taken only a simpliﬁed model for hydrogen reionization itself: we have ignored X-ray heating prior to hydrogen reionization, and we have neglected variations in the reheat temperature (e.g., due to spatial variation of the ionization parameter). The choice of modeling of these issues led to only minor

118 changes in the simulations by Hirata(2018), so we did not consider them in this paper, but only a few alternative models were tested and more should be explored.

In addition, our small-scale simulations do not include ﬂuctuations in the photoionization rate due to the clustering of the ionizing sources (Pontzen, 2014; Gontcho A

Gontcho et al., 2014), which will become especially relevant for the lower redshifts

(z < 4).

In conclusion, we have found that inhomogeneous hydrogen reionization results in an imprint on the Lyman-α forest power spectrum, even at “low” redshifts 2 < zobs <

4. The eﬀect is present despite the “attractor” nature of the IGM temperature-density relation, because of the ﬁnite relaxation time and the low redshift of reionization favored by Planck. It can range from 1% at small scales and low redshifts, up to tens of percents in the large-scale 3D power spectrum at zobs > 3.5. While we have not ∼ yet developed a robust mitigation strategy, there are several clear paths forward on both the theory/simulation front, and with additional observations to help constrain reionization.

119 Chapter 4: White Dwarf star survival in an PBH ocean

In this chapter, I will present one of the sections of our paper on revisiting con-

straints on sub-lunar-mass primordial black holes as dark matter candidates35. The

original abstract is given below:

As the only dark matter candidate that does not invoke a new particle that survives

to the present day, primordial black holes (PBHs) have drawn increasing attention

recently. Up to now, various observations have strongly constrained most of the mass

range for PBHs, leaving only small windows where PBHs could make up a substantial

fraction of the dark matter. Here we revisit the PBH constraints for the asteroid-

−17 −12 mass window, i.e., the mass range 3.5 10 M < mPBH < 4 10 M . We × × revisit 3 categories of constraints. (1) For optical microlensing, we analyze the ﬁnite

source size and diﬀractive eﬀects and discuss the scaling relations between the event

rate, mPBH and the event duration. We argue that it will be diﬃcult to push the

existing optical microlensing constraints to much lower mPBH. (2) For dynamical capture of PBHs in stars, we derive a general result on the capture rate based on phase space arguments. We argue that survival of stars does not constrain PBHs, but that disruption of stars by captured PBHs should occur and that the asteroid-mass

35This paper is currently in preparation for publishing in the Journal of Cosmology and Astropar- ticle Physics.

120 PBH hypothesis could be constrained if we can work out the observational signature

of this process. (3) For destruction of white dwarfs by PBHs that pass through the

white dwarf without getting gravitationally captured, but which produce a shock that

ignites carbon fusion, we perform a 1+1D hydrodynamic simulation to explore the

post-shock temperature and relevant timescales, and again we ﬁnd this constraint to

be ineﬀective. In summary, we ﬁnd that the asteroid-mass window remains open for

PBHs to account for all the dark matter.

The original authors are Paulo Montero-Camacho, Xiao Fang, Gabriel Vasquez,

Makana Silva and Christopher M. Hirata.

4.1 Introduction

In the ΛCDM model, roughly ΩDM 0.26 of the Universe’s total energy den- ' sity is made of dark matter (DM) Planck Collaboration et al.(2016), whose nature remains enigmatic even though evidence for its existence was ﬁrst reported over 80 years ago Zwicky(1937). Since then a wide range of astrophysical observations have pointed toward the existence of dark matter Bertone et al.(2005), such as the galactic rotation curves Rubin and Ford(1970), the baryon density constrained by the Big

Bang Nucleosynthesis (see Iocco et al.(2009) for a recent review), the anisotropies in the Cosmic Microwave Background (CMB) and the inhomogeneities in the large-scale structure Dodelson(2003); Davis et al.(1985); Springel et al.(2006); Planck Collab- oration et al.(2016). In addition, gravitational lensing provides direct measurements of the mass distribution, hence becoming a powerful probe of DM, which is further developed into several techniques: the strong gravitational lensing by massive galaxy clusters Taylor et al.(1998), the weak gravitational lensing of galaxies by galaxies

121 and large-scale structure Refregier(2003); Lewis and Challinor(2006); Kitching et al.

(2014); Fu et al.(2014); Hildebrandt et al.(2017); DES Collaboration et al.(2017);

Troxel et al.(2017), and CMB lensing Planck Collaboration et al.(2016); Das et al.

(2014); van Engelen et al.(2012). Also see Buckley and Peter(2017) for a recent

review of gravitational probes of DM physics.

Since dark matter cannot be composed of any of the Standard Model (SM) par-

ticles, the dominant paradigm is that it is a new type of particle. Such a particle

would need to be stable over the lifetime of the Universe, have suﬃciently weak in-

teractions with the Standard Model particles that it has not yet been discovered, and

have a viable production mechanism in the early Universe. Considerable eﬀorts have

been devoted to looking for DM with direct and indirect detection methods using

particle experiments Klasen et al.(2015); Marrod´anUndagoitia and Rauch(2016);

Penning(2017); Baudis(2016). The scenario that has received the greatest attention

is the thermal WIMP scenario (see Arcadi et al.(2017); Roszkowski et al.(2017) for

recent reviews), in which the dark matter is a massive particle that was in thermal

equilibrium in the early Universe when the temperature was much higher than the

WIMP mass. The comoving number density of WIMPs decreases exponentially as

the temperature drops below the WIMP mass and eventually WIMPs are so diluted

that they can not annihilate with each other eﬃciently, thus “freezing out”. A sym-

metry (e.g., R-parity in supersymmetric models) protects the WIMP from decay into

SM particles. However, there are many other Beyond the Standard Model candidates for dark matter, each with its own phenomenology and observational/experimental signatures.

122 Primordial black holes (PBHs) have been considered as an alternative scenario for

DM for almost ﬁfty years Hawking(1971); Carr and Hawking(1974); Carr(1975);

Chapline(1975). PBHs would be formed in the early Universe by gravitational col-

lapse sourced by an order unity perturbation that makes the surrounding region col-

lapse into a black hole (see Carr(2005) for a brief review of diverse PBH formation

mechanisms; there are many possibilities, but a source of perturbations beyond the

extrapolation of the inﬂationary power spectrum is needed). There are several reasons

for interest in PBHs as DM candidates. First, they are solutions to general relativity,

and thus are the only DM scenario that does not invoke a new elementary particle

that survives to the present day. Second is the related fact that their properties are

highly constrained: in general relativity, a PBH’s properties are determined by its

mass mPBH and spin parameter a?,PBH, and there are no additional free parameters needed to determine its interactions with visible matter. (For most of the constraints on PBHs, it is mPBH rather than a?,PBH that matters.) Finally, the parameter space for PBHs is inherently bounded at both the high-mass and low-mass ends. An obvious maximum mass is set by the observed astrophysical objects that are made of dark matter. A bound on the minimum mass is that in order to survive to the present,

PBHs must be massive enough to not have completely evaporated via Hawking radi-

−19 ation Carr and Hawking(1974), which requires mPBH > 2.5 10 M MacGibbon × (1991).

Recent discoveries of binary black hole mergers with 10 M Abbott et al.(2016, ∼ 2017b,a,c) have posed challenges to stellar evolution theories, reviving the enthusiasm about PBHs as DM in that mass range Bird et al.(2016); Sasaki et al.(2016); Clesse and Garc´ıa-Bellido(2017); Eroshenko(2016); Raccanelli et al.(2016); Ali-Ha ¨ımoud

123 et al.(2017); Ali-Ha ¨ımoud(2018). However, there are signiﬁcant constraints on PBHs

in various mass ranges. The CMB spectral distortions and anisotropies strongly

constrain monochromatic masses of PBH due to the non-blackbody spectrum from

the additional energy injection by accreting PBHs Ali-Ha¨ımoud and Kamionkowski

(2017); Poulin et al.(2017); Nakama et al.(2018); Abe et al.(2019). (For spinning

black holes, there may also be a superradiance constraint Pani and Loeb(2013).)

The sizes and velocity dispersions of stellar clusters at the cores of various ultra-

faint dwarf galaxies have been proposed to independently constrain the stellar-mass

PBH fraction due to the dynamical heating (e.g. Brandt, 2016; Koushiappas and

Loeb, 2017). Halo wide binaries can constrain the mass distribution of potential PBHs

because they are susceptible to gravitational perturbations due to encounters Monroy-

Rodr´ıguez and Allen(2014). The impact of Poisson noise from PBHs in the small-

scale Lyman-α forest power spectrum has been recently utilized to constrain massive

PBHs (mPBH > 60M ) Murgia et al.(2019). Both the Poisson noise and accretion methods have promise for future 21 cm observations Tashiro and Sugiyama(2013);

Gong and Kitajima(2017). Furthermore, microlensing searches Paczynski(1986) have constrained PBH masses in the planetary-to-stellar-mass window by monitoring stars in the Large Magellanic Cloud (MACHO Alcock et al.(2000, 2001), EROS-2

Tisserand et al.(2007)) and the Galactic Bulge Niikura et al.(2019b). Lower masses have been constrained by monitoring stars in the Kepler field Griest et al.(2014) and stars in M31 observed by Subaru/HSC Niikura et al.(2019a). It has also been shown in Zumalacárregui and Seljak(2018) that one can constrain PBH masses in the range 0.01M from lensing of Type Ia SNe light curves due to PBHs. At the low ≥ masses, there are constraints from bursts of Hawking radiation from the final stages

124 of the PBH evaporation Page and Hawking(1976); Ackermann et al.(2018), and from the γ-ray background Carr et al.(2010). Normally these constraints are plotted in the (mPBH, fPBH) plane, where fPBH = ΩPBH/ΩDM is the fraction of the dark matter in PBHs. Note that all these constraints have been made assuming monochromatic mass functions, although considering extended mass functions do not relax those constraints Green(2016);K uhnel¨ and Freese(2017); Lehmann et al.(2018).

There is also an asteroid-mass window for PBH dark matter, where the mass is

−17 too large for constraints from Hawking radiation (> 5 10 M ) but too small for ∼ × −12 current optical microlensing surveys (< 4 10 M ). In this work we will revisit ∼ × the PBH constraints in this mass range and correct or strengthen them. The relevant constraints are from the femtolensing and picolensing of GRBs; microlensing of stars in M31 Niikura et al.(2019a); the dynamical capture of PBHs by stars, including neutron stars (NS) and white dwarfs (WD) Capela et al.(2013a,b); Pani and Loeb

(2014); Capela et al.(2014); and WD survival since passage of a PBH could ignite carbon fusion and destroy the WD Graham et al.(2015). We note that recently asteroid-mass PBHs have been considered as a possible explanation of the positron cosmic ray excess Takhistov(2019), adding another motivation for constraining this mass window.

Gammay ray bursts (GRBs) are interesting for asteroid-mass PBHs because of their cosmological distance (hence large lensing probability) and the short wavelength of electromagnetic radiation (so that the Einstein radius can exceed the Fresnel radius even for low-mass lenses). Femtolensing occurs when a GRB source is strongly lensed by an intervening PBH and the two light paths interfere with each other, leaving a

125 signature in the GRB energy spectrum – although the lensed images are not individu-

ally resolved Gould(1992). Picolensing occurs when the magnification varies on AU ∼ scales in the observer plane, and hence the fluence of gamma rays is different as seen at different interplanetary spacecraft Nemiroff and Gould(1995). Upper limits on PBH dark matter are obtained from non-observation of these phenomena. There have been

−16 −14 reported constraints on PBHs in the 2.5 10 M < mPBH < 5 10 M range × × from femtolensing Barnacka et al.(2012), however due to ﬁnite source size eﬀects this range of PBH masses is now allowed Katz et al.(2018). There is one picolensing constraint that includes BATSE + Ulysses Marani et al.(1999), although it used an older cosmological model and depends on an uncertain source redshift; it is not clear whether with modern parameters fPBH = 1 would be excluded for any value of mPBH.

Thus, at present, there is not an excluded range of mPBH from GRBs.

This paper is devoted to a thorough re-consideration of one other (non-GRB) constraint on the asteroid-mass PBH window. We focus our eﬀorts on ignition of white dwarfs, where we use 1D hydrodynamics to follow WD material following passage of a PBH, instead of relying on order-of-magnitude arguments as in previous work.

In 4.2, we examine whether the collision between a PBH and a WD will lead § to thermonuclear explosions of the WD. Going beyond the calculations in Graham et al.(2015), we carefully calculate the trajectory of a PBH passing through a WD and the involved micro-physics present during the passage. In particular, we perform a 1 + 1D Lagrangian hydrodynamic simulation to track the thermodynamics of the surrounding WD materials and to ﬁnd whether a shock forms. We then consider the conditions for a shock to start a runaway explosion based on the comparison of nuclear burning to conduction times Timmes and Woosley(1992), and discuss

126 the uncertainties associated with hydrodynamic instabilities following PBH passage

(which are very diﬀerent from the considerations in Type Ia supernova ignition).

4.2 White Dwarf survival

Here we will treat the problem of PBH transit through a carbon/oxygen white dwarf. We will answer two questions: can the passage of a PBH cause enough localized heating by dynamical friction to ignite the carbon; and, can this ignition provoke a runaway explosion. We expand on the work of Ref. Graham et al.(2015) with the next logical step, i.e., using 1+1D (cylindrical radius + time) hydrodynamic simulations to follow the shock heating during the passage of the PBH. We split this problem into three pieces. First we compute the density profile of the WD and the velocity profile of the incoming PBH. Next, we use a 1+1D Lagrangian hydrodynamic code to study the possible shocks and track the evolution of thermodynamic quantities in the neighborhood of the PBH trajectory, where the initial conditions for the simulations are set by the profiles obtained in the first step. Finally, we follow the arguments in Ref. Timmes and Woosley(1992) to find out if the shock meets the conditions to start a runaway thermonuclear explosion, e.g., by comparing the specific nuclear energy generation rate to the rate of energy loss due to conduction, and the burning timescale to the Kelvin-Helmholtz instability timescale. (Note that it is not clear if the presence of instabilities will definitively destroy the flame propagation; this would require a full 3+1D simulation and is beyond the scope of this paper.)

As pointed out in Ref. Graham et al.(2015), the survival of a WD provides two different ways to constrain PBHs. One could put constraints on the abundance of PBHs simply by the fact that we observe WD in a certain mass range. Furthermore, since

127 the passage of PBH could explode WD with masses lower than the Chandrasekhar

mass, one could look at the rate of Type Ia supernova to place constraints. Here we

will consider only the ﬁrst type of constraint, since it is not clear whether a successful

ignition of the type considered here would lead to a recognizable Type Ia supernova.

4.2.1 Velocity and density proﬁles

To determine the velocity and density proﬁle of a white dwarf star, we numerically

integrated the equation of state assuming zero temperature (since electron degeneracy

pressure dominates) Shapiro and Teukolsky(1986). The composition of a typical

carbon/oxygen (C/O) white dwarf consists of free electrons with carbon and oxygen

ions; we assume here a 50:50 mixture. Neglecting ion-ion interactions, once can

average the atomic number of both atoms, i.e. Z¯ = 7. The ion density and pressure

can be represented by the Fermi parameter x = pF /mec:

Z pF 2 2 2 nemB mB 3 2 p c 2 mec ρ = = 2 3 x and P = 3 2 2 2 4 1/2 4πp dp = 3 φ(x), Ye 3π λeYe 3h 0 (p c + mec ) λe (4.1) where λe = ~/(mec) is the electron Compton wavelength, and φ(x) is given by Shapiro and Teukolsky(1986)

1 2x2 φ(x) = x(1 + x2)1/2 1 + lnx + (1 + x2)1/2 . (4.2) 8π2 3 −

We also applied electrostatic corrections to the pressure and density in our calculation.

At zero temperature, the ions form a lattice that maximizes the inter-ion separation.

Each ion is surrounded by a uniform distribution of electrons such that each spherical

shell, known as Wigner-Seitz cell, is neutral Shapiro and Teukolsky(1986). Due to

the electron-electron and electron-ion interactions in each of the cells, the pressure

and density are reduced by a few percent with respect to the case where electrostatic

128 corrections are ignored. Additionally, we also applied a correction term due to deviations of the electron distribution from uniformity. This provides a decrease of the density and pressure, but by a signiﬁcantly smaller contribution Salpeter(1961). To obtain the density and velocity proﬁles of the star, we use the metric for a spherically symmetric star:

2Gm(r)−1 ds2 = e2Φ(r)c2dt2 + 1 dr2 + r2dθ2 + r2 sin2 θdφ2 , (4.3) − − rc2 where Φ(r) is some scalar function that acts as the “gravitational potential” in the classical limit and m(r) is the mass up to some radius. We numerically solve the

Tolman-Oppenheimer-Volkoﬀ (TOV) equations using the aforementioned equation of state.

We ultimately need the rate of collisions between PBHs and white dwarfs. To determine this, we need the escape velocity at any shell, and the maximum speciﬁc angular momentum ˜l that a PBH can have and still reach that shell. We restrict ourselves to mPBH MWD; it follows that the length scale of the accretion ﬂow and associated shock structure around the PBH as it passes through the white dwarf

2 2 (RA = 2GmPBH/v GmPBH/v ) are much smaller than the radius of the white ∼ esc 2 dwarf itself ( GMWD/v ). It also follows that we can treat the PBH as a test ∼ esc particle in the TOV spacetime generated by the white dwarf.

Assuming we are in the rest frame of a ﬂuid mass element of the white dwarf along the equatorial plane (θ = π/2), we can write the equation of motion for a PBH starting at rest at inﬁnity and radially falling through the interior of the white dwarf.

By solving for the maximum radial velocity, we arrive at the formula for the escape velocity:

p 2Φ(r ) vesc = c 1 e min . (4.4) − 129 For reasonable parameters, the initial velocity of an incoming DM particle is small

compared to vesc, so in the interior of the WD we may take the actual velocity to be the escape velocity, v vesc. We can also compute the Mach number = vesc/cs ≈ M where cs is the speed of sound through the interior of the white dwarf. The speciﬁc angular momentum is then

r v(r ) ˜l = min min . (4.5) p 2 2 1 v (rmin)/c −

For our range of white dwarfs ranging from 0.75 M 1.385 M , we calculated the − density proﬁle, and thus, the escape velocity, Mach number, and angular momentum

per unit mass. In 4.2.2 and 4.2.3, we will calculate the minimum PBH mass for §§ ignition and the rate of collisions assuming the PBH will pass through the white

dwarf, reach a minimum radius, and then leave the star.

4.2.2 Thermal eﬀects on WD materials by a passing PBH

Given the kinematics of PBHs passing through WDs, we are now able to calculate

the thermal eﬀects generated on small scales, which will be crucial to determining

whether part of the WD can be shocked and signiﬁcantly heated up, and consequently,

whether a runaway thermonuclear explosion can occur.

We set up a local 1+1D Lagrangian hydro simulation (see e.g., Benson, 1992,

for a review) to capture the evolution of thermal properties of the ﬂuid around the

PBH trajectory. In the rest frame of the PBH, the ﬂuid is initially steadily ﬂowing

towards the +z direction with velocity v. In the simulation, we scale quantities by

deﬁning cs,∞ = ρ∞ = Rc = 1, where cs is the sound speed, ρ is the density of

the WD. We use subscript to denote the initial background values, which are ∞

equivalent to the values at inﬁnite impact parameter r = . Rc = 2GmPBH/(cs,∞v) ∞ 130 is the critical radius where the inward radial velocity of the ﬂuid parcel due to PBH’s

gravity is equal to the sound speed. Another important scale is the accretion radius

2 RA = 2GmPBH/v = Rc/ . Fluid parcels with impact parameters b smaller than M

RA will be strongly deﬂected from their trajectory. In fact, ﬂuid parcels with b less

than a critical value bc will be eaten by the PBH, thus irrelevant to us. bc is given by s m˙ bc = = √αRA , (4.6) πρ∞v

2 3 wherem ˙ = 4πα(GmPBH) ρ∞/v is the mass accretion rate, and the parameter α is approaching 1 at 1 limit (Bondi and Hoyle, 1944; Hunt, 1971)36. We will M only care about the ﬂuid outside of the accretion radius RA, since material at smaller impact parameter falls into the black hole and it does not matter whether it burns.

See Figure 4.1 for the physical picture.

Computations could in principle be done in the frame of the PBH (so that we work with radius r and downstream position z), or alternatively we could work in the initial frame of the material so that time since passage t = z/v is the independent

variable instead of z. In the high Mach number limit, it is more convenient to take

the latter perspective; except in the vicinity of the PBH (near RA), the ﬂow will

everywhere be supersonic, the partial diﬀerential equations should be hyperbolic, and

the time evolution picture is appropriate.

Our code tracks the evolution of 1200 concentric mass shells following passage of

the PBH at t = 0; by default, it uses time steps ∆t = 4 10−3 (in the code unit ×

tcode = Rc/cs) and runs up to 5000 steps. The code tracks the Eulerian radius r of

2 each shell, its radial velocity vr = dr/dt, and the specific energy s + v /2. The ≡ r 36Note that our definition of α is consistent with Hunt(1971) and is half of that defined in Bondi and Hoyle(1944).

131 WD material

(1) R O · A PBH r

z Shock

Figure 4.1: A schematic diagram for the PBH passing through the WD materials, creating a shock. The materials with impact parameter less than about the accretion radius eventually get eaten by the PBH, while materials farther away get shocked, compressed and heated. The diagram is shown in the rest frame of the PBH.

evolution in each time step is calculated with adaptive step sizes δt = 1 min dr , r , 40 {| dv | vsp } q 5 5 where vsp = v + ( 1)s is the spread speed. We implement artiﬁcial viscosity | | 3 3 − by subtracting a smooth term, m Ψ∆v, from the momentum ﬂux, where ∆v is the

diﬀerence in velocity of the adjacent cells, and Ψ is a function that determines how

much damping is needed at each boundary by computing the ﬁnite velocity diﬀerence.

The mass shells are chosen in a way that the impact parameters are logarithmically

spaced and the outer radius of the 400th shell has initial radius Rc. Initially, the WD

temperature T is much lower than the Fermi temperature TF , so the ﬂuid is treated

as a relativistic degenerate Fermi gas with zero temperature (i.e. γ = 4/3 polytrope).

2 The initial pressure is then Pinit = csρ∞/γ, and the speciﬁc internal energy is given

by in,init = Pinit/[ρ∞(γ 1)]. Initially we set each shell’s Eulerian position to equal its −

132 9 Lagrangian position (so the density is 1), the speciﬁc internal energy is 4 (chosen to agree with cs,∞ = 1), and the initial radial velocity is vr = 1/r (the result computed − in the impulse approximation).

During the evolution, the ﬂuid may be compressed by the PBH, and T may rise non-negligibly comparing to TF , where the constant polytropic equation of state no longer holds and the heat capacity of ions may also be important. To run a Lagrangian hydrodynamic code, it is necessary to derive an equation of state function P (ρ, ).

Our implementation of the equation of state is described in Appendix C.1, and is facilitated by use of the auxiliary variable J = µe/kT (which goes to + for perfectly ∞ degenerate material at zero temperature, and 0 in the limit that the leptons become a pair plasma). The equation of state includes ion thermal pressure (the ideal gas law), as well as relativistic electrons and positrons as appropriate for x 1. The current implementation does not yet include radiation pressure.

In Figure 4.2 we show the evolution of the 1200 mass shells. Naturally, the region closer to the PBH reaches higher temperature, followed by a rapid drop as time passes and the material expands radially. Furthermore, one must ignore shells inside the accretion radius since they will get “eaten”. We ﬁnd that the maximum temperature is achieved right at the accretion radius, which corresponds to the 209-th shell if

= 3. In Fig. (4.3), we show the temperature evolution of that shell and two M additional shells that have = 2.60 (234-th shell, outer radius r = 0.385Rc) and M

3.24 (194-th shell, r = 0.305Rc). The temperature quickly reaches its maximum, then swiftly decreases and approaches an asymptotic ﬁnal temperature – in terms of the

Fermi energy – of 0.4 EF. Additionally, in Fig. (4.4), we present the maximum and ≈ ﬁnal temperature as functions of the mass shells’ initial radii.

133 ln(rf/ri) Velocity 1.0 500 0.4 500

0.5 400 0.2 400

300 0.0 300 0.0

200 0.2 200 0.5 time/4.0000E-03 time/4.0000E-03

100 100 0.4 1.0

0 0 1.5 0 250 500 750 1000 0 250 500 750 1000 shell no. shell no.

log10J 8 temperature 0.8 500 500 0.7 6 400 400 0.6 4 0.5 300 300 2 0.4 200 200 0 0.3 time/4.0000E-03 time/4.0000E-03 0.2 100 2 100

0.1 4 0 0 0.0 0 250 500 750 1000 0 250 500 750 1000 shell no. shell no.

Density 2.0 pressure 5 500 500

1.8 400 400 4

1.6 300 300 3 1.4

200 200 2 1.2 time/4.0000E-03 time/4.0000E-03

100 100 1 1.0

0 0 0.8 0 0 250 500 750 1000 0 250 500 750 1000 shell no. shell no.

Figure 4.2: The simulated evolution of the logarithm ratio of the positions (i.e., Eu- lerian divided by Lagrangian radius coordinate), velocity, thermodynamic parameter J, temperature, density and pressure of the 1200 mass shells. The critical radius is at shell 400. The impact gets weaker with time and in mass shells farther out. All the quantities are in our code units, see Appendix C.1.

134 0.8 shell = 194 0.7 shell = 209 0.6 shell = 234 ]

F 0.5 0.4

[E T 0.3 0.2 0.1 0 0.01 0.1 1 10

t [tcode]

Figure 4.3: Temperature time evolution for three diﬀerent shells. The shells are chosen so that they correspond to the accretion radius for three diﬀerent mach numbers: = 2.60 (purple), 3.00 (green) and 3.24 (orange). M

We test convergence of our hydro simulations by running a lower time resolution

simulation with double the time stepsize, i.e. ∆t = 8 10−3 and up to 2500 steps. × We ﬁnd that at each step the fractional diﬀerences of the thermodynamic quantity J,

and the temperature are within 10−3, and within 10−4 for the pressure.

4.2.3 Ignition and runaway explosion?

The destruction of a WD will occur only if the energy generation due to carbon

fusion in the downstream material exceeds the losses due to conduction. In the case

of burning in the “wake” of a PBH passage, there may be an additional requirement

that the energy injection timescale due to burning be faster than hydrodynamic in-

stabilities that mix the material with cooler plasma and thus suppress the energy

generation rate.

135 1.4

1.2 ]

F 1

0.8 Tmax 0.6 Tfnal

[E T 0.4

0.2

0 0.1 1 10

rinitial [Rc]

Figure 4.4: Final and maximum temperatures reached by each cell as a function of Lagrangian radius. The initial radius for the shells from Fig. (4.3) have been highlighted preserving the color code.

To determine whether the carbon fusion is ignited, we focus on the major nuclear

reactions, i.e. ,

 20  Ne + α + 4.621 MeV 12C + 12C 23Na + p + 2.242 MeV , (4.7) →  23Mg + n 2.598 MeV − which have yields 0.56/0.44/0.00 for T9 < 1.75, 0.50/0.45/0.05 for 1.75 < T9 < 3.3,

9 and 0.53/0.40/0.07 for 3.3 < T9 < 6.0, where T9 T/10 K(Caughlan and Fowler, ≡

1988). We further assume no signiﬁcant change in the branching ratio above T9 =

6.0. These considerations allow us to calculate the mean energy release for each temperature range as Q¯ = 3.574, 3.190, 3.164 MeV per reaction, respectively. The

reaction rate is given by Caughlan and Fowler(1988) as

5/6 ! 26 T9a 84.165 −3 3 3 −1 −1 λ(T9) = 4.27 10 3/2 exp 1/3 2.12 10 T9 cm mol s , (4.8) × T9 − T9a − ×

136 for 0.001 < T9 < 10, where T9a T9/(1 + 0.0396T9), and we assume the formula ≡ holds for higher temperatures. The speciﬁc nuclear energy generation rate ˙nuc is

2 ¯ 12 then given by ˙nuc = ρY12Qλ, where Y12 = 1/28 is the mole number of C per gram of the material. Written in c.g.s. units, we have

Q¯ MeV N ˙ = ρY 2 A λ erg g−1 s−1 . (4.9) nuc 12 MeV erg 2

7 −3 For temperature T9 = 5.59, density ρ = 2.57 10 g cm , we have ˙nuc = 3.83 × × 1026 erg/g/s.

Adding the screening eﬀect of the electrons can enhance the nuclear reaction rate

(Eq. 4.9) due to less repulsion, hence smaller distances, between nearby nuclei. To estimate this enhancement, we reproduce the procedure present in the alpha-chain reaction networks37. For our purposes we are interested in the regime of degenerate screening, so we adopt the method for strong screening, which is appropriate when density is high. The method and physics are described in Itoh et al.(1977); Jancovici

(1977); Alastuey and Jancovici(1978); Itoh et al.(1979). Taking this screening en-

7 −3 hancement into account, and for typical values of T9 = 5.59 and ρ = 2.57 10 g cm , × we obtain

H12 26 −1 −1 ˙nuc screen = ˙nuce = 5.07 10 erg g s , (4.10) | × where the screening function evaluates to H12 = 0.28 for these values.

Ignition occurs when the energy generation rate is higher than the energy loss rate, which is dominated by thermal conduction for white dwarf interiors (Kawaler et al., 1996). We use the electron thermal conductivity of Ref. Yakovlev and Urpin

(1980) (see discussion in Ref. Timmes and Woosley(1992)). For a mass shell with

37We use the 19 isotope chain, http://cococubed.asu.edu/code_pages/burn_helium.shtml, with the code described in Weaver et al.(1978).

137 Lagrangian radius r, the speciﬁc conductive energy loss rate is given by

2D T ˙ = cond , (4.11) cond ρr2 where Dcond is the thermal conductivity, predominantly contributed by the electron-

2 2 ion scattering, i.e. , Dcond = π kBT ne− /(3m∗νei), in which the eﬀective electron mass

2 m∗ = me√1 + x , the electron-ion collision frequency νei and the Coulomb integral

Λei are determined by

2 2 ¯ " 1/3 1/2 2 # 4α mec ZΛei 2 1/2 2 ¯ 3 3 x νei = (1 + x ) , Λei = ln πZ + 2 , 3π~ 3 2 Γe − 2(1 + x ) (4.12)

1/3 respectively, where Γe = α~c(4πne− /3) /(kBT ) is the dimensionless plasma coupling parameter and α = 1/137 is the ﬁne-structure constant. Therefore, we can obtain the speciﬁc conductive energy loss rate as a function of density and temperature of the WD material.

We must also compare the burning timescale to the hydrodynamic instability timescale to determine whether a runaway explosion might occur. The burning timescale is estimated as

cpT (cp,ion + cp,e− )T τburn = = , (4.13) ˙nuc ˙nuc

where cp is the speciﬁc heat capacity at constant pressure which is mainly contributed

by ions and partially degenerate electrons. The ion part is well-described by the ideal

8 ¯ gas, i.e. , cp,ion = 5 /2 = (2.079 10 /A) erg/g/K , where is the gas constant. R × R The electron part can be estimated as

" 2 2# 2 " 2 2# π kBT π kBT π kBT cp,e− = cv,e− 1 + = 1 + . (4.14) 3 EF 2 R EF 3 EF

138 At zero temperature limit, the electrons do not contribute. However, after being

shocked, the temperature can get so high that the electron contribution may become

dominant.

The hydrodynamic instability in this case arises from the shear between neigh-

boring shells. After the PBH passes by, shells with smaller impact parameters are

expected to be “dragged along” behind the PBH, i.e., they should have vz < 0 in the

frame of the WD. The Kelvin-Helmholtz (KH) instability develops on a timescale of

−1 τKH ∇vz , where the z-direction is along the PBH velocity. Fortunately, even ∼ | |

though our code is 1+1D, we can estimate vz using simple physical arguments. In the

rest frame of the PBH, and assuming a time-steady ﬂow, a ﬂuid parcel has conserved

speciﬁc total energy , i.e. H

1 2 P u + v PBH frame + Φ + , (4.15) H ≡ 2 | ρ where u is the speciﬁc internal energy, v PBH frame is the velocity of the parcel in | 2 the PBH frame, and Φ is the gravitational potential. In our case, v PBH frame = | 2 2 ( + vz) + v . In terms of the code units, initially we have u = 9/4, v = , Φ = 0, M r M

and P/ρ = 3/4. After the PBH has long passed, assuming the parcel has uf ,Pf , ρf ,

velocity vz, and using the conservation law, we obtain M − 1 Pf vz = uf + 3 , (4.16) − ρf − M 2 where vz , and we have neglected the v term. We run the simulation up M z to 5000 time steps when the ﬁnal velocity gradients have stabilized. The minimal

KH instability timescale is achieved near the accretion radius, where τKH 0.1 tcode. ∼ −11 For comparison a typical value of the instability timescale is 6.14 10 s for cs = × 8 −1 23 4.23 10 cm s , = 2.58, and mPBH = 4.53 10 g. In contrast, for T9 = 5.59 × M × 139 and ρ = 2.57 107 g cm−3, the burning timescale is 2.09 10−10 s. Therefore, for × × these particular values, convection might be able to destabilize the ﬂame, preventing

runaway explosion.

At very high temperatures, possible endothermic reactions may serve as an addi-

tional way of halting the ignition. For a WD of mass 1.385 M the shock may heat the

local ﬂuid up to 4.1 1010 K, while our fuel, 12C, breaks into α particles with a bind- ∼ × ing energy of B = 7.4 MeV Ajzenberg-Selove(1990); above 300 keV, dissociation is ∼ thermodynamically favored. The tabulated rate coeﬃcient for 12C+γ(+γ +...) 3α → Caughlan and Fowler(1988) rises to 10 10 s−1 (so a dissociation time of 10−10 s, comparable to the shock passage time of a PBH) at T = 1.4 1010 K. In order to avoid this × issue, and the consequent need to follow the reaction network, we simply mark the

region exceeding T = 1.4 1010 K in Figure 4.6 as the “carbon dissociation” region. × Note that this occurs only for very massive white dwarfs.

Loss of energy to neutrino cooling may in principle further reduce the chance

of WD explosions at very high temperatures and densities. At temperatures of > ∼ few 109 K and relevant densities of up to 109 g/cm3, the dominant neutrino × ∼ cooling mechanism is e+ + e− ν +ν ¯ (see Fig. 1 of Ref. Yakovlev et al.(2001)). → The cooling time gets shorter at higher temperature, but even in the relativistic

2 24 9 non-degenerate limit (T TF mec /kB), the cooling rate is Q = 3.6 10 T × 10 erg/cm3/s (Eq. 30 of Ref. Yakovlev et al.(2001)), which – given an energy density

11 4 26 4 3 of ρu = aradT = 2.1 10 T erg/cm , including pairs as well as photons in the 4 × 10 −5 energy density – implies a cooling time of tcool = u/Q = 57T10 s. Thus we expect neutrino losses to be insigniﬁcant on the timescales for ﬂow near a PBH.

140 In Fig. 4.5, we plot the minimum required PBH mass for thermal runaway explosion produced by dynamical friction from the passage of a PBH through a WD of a given total mass. The minimum PBH mass is in principle a function of both

MWD and the mass shell m(r) where we attempt ignition; in this ﬁgure, we take the smallest value of the minimum PBH mass. We show a lower curve (“no KH”) that ignores the Kelvin-Helmholtz instability, and an upper curve (“KH”) that requires the burning time to be shorter than the Kelvin-Helmholtz instability time. Again, the upward-sloping trend at larger masses is due to the inner shells getting to the temperature needed for carbon dissociation.

In Fig. 4.6, we show the range of parameter space where ignition can occur. We considered a range of white dwarf masses from 0.75–1.385 M with each subsequent

WD having a mass larger by ∆MWD = 0.05 M except that we also considered a 1.32,

1.34, 1.36, and a 1.385 M WD into our analysis. For each WD, we considered ignition in each mass shell where the resolution between mass shells is ∆m = 0.01 M . We considered both requiring and not requiring τburn < τKH; including this criterion signiﬁcantly restricts the parameter space of where ignition occurs. In regions where

T 1.4 1010 K, carbon dissociation occurs; we are not able to determine whether ≥ × ignition occurs in this case, but a careful analysis of this parameter space would be of great interest in the future.

4.2.4 Ignition rate and PBH constraints

To constrain the fraction of PBHs that could be dark matter, we ﬁrst need to calculate the rate of collisions between a WD and a PBH. The rate, most generally,

141 1024 no KH

[g] KH

1023

22 PBH 10 min m 1021 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

MWD [Msolar]

Figure 4.5: The minimum primordial black hole mass needed to achieve thermonuclear runaway for speciﬁc WD total mass. For comparison purposes we include the model without considering convection losses by KH instabilities (in green).

is deﬁned as

Z Z ρDM 3 ρDM ˜ 2 P (v∞) 3 Γ = fPBH P (v∞)σ(v∞)v∞ d v∞ = πfPBH [l(rmin)] d v∞, mPBH mPBH v∞ (4.17)

where v∞ is the velocity of the PBH at “inﬁnity”, P (v∞) is the probability, and

σ(v) is our cross-section. In the last equality, we assume the initial PBH velocity

is small compared to the escape velocity from the surface of the WD (5260 km/s

2 ˜ ˜ for MWD = 0.75 M ), so that σ = πb where b l(rmin)/v∞ and l(rmin) is the max ≈ maximum speciﬁc angular momentum that leads to ignition.

We assume an oﬀset Maxwellian velocity distribution,

2 2 2 −3/2 −(v−v¯) /σrms P (v) = (2πσrms) e . (4.18)

142 1.4 Without convection With convection 1.2 Carbon dissociation

1.0

D ) 0.8 M W

M = ( m

m 0.6

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 MWD (M )

Figure 4.6: The parameter space where ignition is possible ranging from 0.75 − 1.385 M . Each total WD mass we considered is separated by ∆MWD 0.05 M ≈ while the resolution between each mass shell is ∆m 0.01 M . The green region is the parameter space where ignition can take place if≈ we ignore the Kelvin-Helmholtz instability but that has τburn > τKH, while the magenta region has τburn < τKH and hence ignition is robust against this instability. The cyan region is the parameter space where carbon dissocociation into α particles occurs.

143 MWD = 1.00 M MWD = 1.06 M 10 1 10 1

) 2 ) 2

1 10 1 10 r r y y G G ( 10 3 ( 10 3 H H B B P P f f / / 10 4 10 4

10 5 10 5 1021 1022 1023 1024 1021 1022 1023 1024 min mPBH (g) min mPBH (g)

MWD = 1.10 M MWD = 1.15 M 10 1 10 1

) 2 ) 2

1 10 1 10 r r y y G G ( 10 3 ( 10 3 H H B B P P f f / / 10 4 10 4

10 5 10 5 1021 1022 1023 1024 1021 1022 1023 1024 min mPBH (g) min mPBH (g)

MWD = 1.21 M MWD = 1.26 M 10 1 10 1

) 2 ) 2

1 10 1 10 r r y y G G ( 10 3 ( 10 3 H H B B P P f f / / 10 4 10 4

10 5 10 5 1021 1022 1023 1024 1021 1022 1023 1024 min mPBH (g) min mPBH (g)

Figure 4.7: The relationship between the rate of collisions Γ and the minimum PBH mass mPBH to cause ignition ranging from MWD = 1.00 1.26M . The upper (green) line is when we ignore the Kelvin-Helmholtz instability− time as a criteria for ignition while the lower (purple) line is the rate when we include “burning before instability” as a criteria for ignition. These plots can also be interpreted as the PBH-induced ignition rates as a function of mPBH.

144 This gives ( q Z P (v ) 1 v¯ 2 −1 ∞ 3 π σ v¯ σ d v∞ = erf , (4.19) v∞ v¯ √2 σ → v¯−1 v¯ σ 38 where erf is the error function. Taking σrms = vgal/√2 andv ¯ = vgal (so for a white

dwarf orbiting in the Galactic disk), the end result is that

˜2 ρDM l Γ = 2.65fPBH . (4.20) mPBH vgal

2 We use parameters relevant to the Galactic disk: vgal = 225 km/s and ρDMc = 0.4

GeV/cm3 Read(2014).

We calculated the rate of collisions given in Eq. (4.20) for each mass shell for each white dwarf in our mass range. However, to demonstrate our results graphically, we plotted six diﬀerent WD masses in Fig. 4.7 ranging from 1.00–1.26 M with incriments of about ∆MWD 0.05M . Since ignition was always possible with larger ≈

PBH masses past a lower cutoﬀ, we calculated Γ/fPBH using the minimum PBH mass that causes ignition. This would provide the largest possible rate for igniting that shell because Γ 1/mPBH. As long as the minimum PBH mass to ignite a shell is an ∼ increasing function of r, we can also interpret this plot as an rate for PBH-induced ignitions as a function of mPBH. (This is because at a given mPBH, the indicated shell is the outermost shell in the WD that can be ignited, and hence the rate of PBH- induced ignitions is the rate of collisions reaching that shell.) For the most massive white dwarfs, there is a turnover due to carbon dissociation where the minimum PBH mass actually increases at small r; this region is not of interest since shells farther out would have ignited anyway.

38We use the conventional normalization that erf = 1. This integral can be easily performed by putting v¯ on the z-axis, and turning the integral in∞v into spherical polar coordinates. The φ and θ integrals are then trivial. The v integral is then a Gaussian-type integral with limits not at , which is the deﬁning form for the error function. ±∞

145 −3 −1 In all cases, we found Γ/fPBH few 10 Gyr for the lower masses (up to ∼ × −2 −1 1.1M ), rising up to few 10 Gyr at MWD = 1.26M , when we don’t impose ∼ ×

the τburn < τKH criterion, and the cases where we do impose this criterion were at

least an order of magnitude less. Thus even the massive WDs survive for longer than

a Hubble time, and more typical white dwarfs at MWD < M can survive for many ∼ Hubble times. In particlar, the survival of the 1.28 0.05 M white dwarf used for ± the constraint in Ref. Graham et al.(2015) – RX J0648.04418, which lies in the Milky

Way’s disk – does not exclude fPBH = 1 at any mPBH.

As noted in Ref. Graham et al.(2015), there is a potential constraint on PBHs based on the rate of Type Ia supernovae if the ignition of a WD by a PBH can be shown to lead to a visible explosion rather than some other outcome (e.g., collapse, or explosion with insuﬃcient 56Ni to be visible), and if the right environment can be

found with suﬃcient dark matter and a suﬃcient density of high-mass C/O white

dwarfs. However, we note that our minimum PBH masses to trigger an explosion in

the conservative case where we require burning before mixing by Kelvin-Helmholtz

−12 instability (Fig. 4.5) are 5 10 M , and a potential WD constraint will not ≈ × probe below this. This is the same as the lower limit of the HSC M31 microlensing constraint, and thus the latter limit cannot be extended by the WD method if the τKH criterion for ignition turns out to be necessary. If future work shows that this criterion is not necessary, then there is the potential to shrink the window for asteroid-mass

PBH dark matter by one order of magnitude.

146 4.3 Conclusion

In this work, we have revisited the WD survival constraint on asteroid-mass PBHs as a dark matter candidate. This potential constraint on asteroid-mass PBHs is based on the possibility that in transiting a white dwarf, PBHs may ignite the carbon and lead to destruction of the white dwarf, even if the PBH is not captured. We have simulated the thermal evolution of the materials along the trajectory before and after the shock passes, and examine whether the carbon ignition and the subsequent runaway explosion occur. We conclude that at present there is not a constraint from white dwarf survival.

Following this result, the re-analysis of the GRB femtolensing constraints Katz et al.(2018) and the other results in Montero-Camacho et al.(2019a), the window for

−17 low-mass PBHs to be all of the dark matter (fPBH = 1) extends from 3.5 10 M to × −12 4 10 M . This is bounded from below by γ-ray background constraints on Hawk- × ing radiation Carr et al.(2010) and from above by the revised HSC M31 microlensing constraints Niikura et al.(2019a).

147 Chapter 5: Conclusion

when you have eliminated the impossible, whatever remains, however improbable, must be the truth - Sir Arthur Conan Doyle

In this thesis, I have presented my work on computational and theoretical cosmology based on looking beyond the traditional “assumptions” of the ΛCDM model of cosmology. In Chapter2, I explored physical mechanisms that could generate circular polarization in the CMB; however, the produced circular polarization signals are all quite small. Although reionization is now known to be an inhomogeneous process, its impact in the Lyman-α forest – especially at low redshifts – was not considered to be signiﬁcant. However, in Chapter3, I showed the imprint of inhomogenous reionization in the Lyman-α forest and its eﬀect is considerably strong – especially at high redshifts.

Finally, Chapter4 considered and attempted to constrain asteroid-mass PBHs as dark matter candidates. Nevertheless, the rates for PBH destroying a WD due to dynamical friction were small compared to the age of the Universe, henceforth this mechanism is unable to place a constraint. This result coupled to the other ﬁndings

148 in Montero-Camacho et al.(2019a) imply an extended open asteroid-mass window for PBHs to be the totality of the dark matter.

Naturally, there are still uncertainties present in our calculations that invite further investigation. For example, in Chapter4, we extrapolated the ﬁtting formulas for nuclear energy generation to higher temperatures; a more careful consideration of the rates, including a full reaction network for the higher temperatures, will be important in the future. However, we do note that the temperature rapidly falls after the shock passes, and most carbon burning occurs within the range of validity of the formulae.

Furthermore, our analysis of the impact of inhomogeneous reionization in the Lyman-

α forest was a ﬁrst study. Luckily, there is work that I plan to do to further expand on the impact of inhomogeneous reionization both in Lyman-α absorption as in emission

(as well as other lines).

In conclusion, I believe that cosmology will surely beneﬁt from relaxing its most fundamental bricks and understanding how the rest of its foundations react to these changes.

149 Appendix A: Circular polarization of the Cosmic Microwave Background

This appendix presents results relevant to Chapter2.

A.1 Anisotropic 21 cm radiation in expanding media

Peculiar velocities and the expansion of the universe modify the emission and ab-

sorption process for photons in a narrow spectral line such as 21 cm. In what follows,

we want to illustrate the way these particular pieces are present in the calculation in

2.3. The purpose here is to provide a simpliﬁed calculation with only the essential § physical ingredients, and with the minimal amount of formalism. For the full detailed

calculation an interested reader should consult Ref. Venumadhav et al.(2014).

The key physics to comprehend here is what happens to a spectral line in a moving

medium, and how velocity gradients imply anisotropic 21 cm radiation. For simplicity,

we will compute the alignment of the hydrogen spins for the special case of kˆ in

the zˆ direction (other choices are related by symmetry). Spectral lines are usually represented by δ-functions; nevertheless, there are broadening eﬀects, and these must be taken into account when doing line radiative transfer to obtain a ﬁnite result. For

21 cm radiation, the most important local broadening source is Doppler broadening due to the thermal motion of the hydrogen atoms, and the line center is shifted in

150 accordance with the (position-dependent) bulk velocity of the gas. We work in the

optically thin limit τ 1, where we can approximate the emission of 21 cm radiation as isotropic, and then obtain the direction-dependent absorption using line radiative

transfer. (It is true that the resulting polarized atoms will produce anisotropic 21 cm

emission, which in turn imprints an additional anisotropy in the absorption, but this

is at the next higher order in τ.)

Without loss of generality, let us consider the absorption of 21 cm radiation by gas at the origin, and work in the reference frame where the bulk velocity of the gas at the origin is zero. We need to know the cross section for absorption of radiation that was emitted by gas at a position r. The emitting gas has a bulk velocity v given by p vi = (Hδij + ∂jvp,i)rj and random velocities of σ = kBTkin/mH (root-mean-square

per axis, for a hydrogen atom, at kinetic temperature Tkin). This cross section (units:

m2) is given by

c 1 l2 = exp [( + )ˆ ˆ ]2 , (A.1) σcs N p 2 Hδij ∂jvp,i ninj ν21 2π(2σ2) −2(2σ )

2 2 R 2 −1 where N = 3c Ahf /(8πν21) = σabs(ν) dν is the normalization (units: m s ), Ahf is the hyperﬁne transition Einstein coeﬃcient, and we have broken the position into a

path length (l) and a direction nˆ: r = lnˆ. The extra factor of two in the variance

comes from taking into account that there is thermal broadening in both the emission

2 2 2 and absorption processes, i.e., σem + σabs = 2σ . The factor of c/ν21 is the conversion factor from frequency to velocity (since the Gaussian is normalized when integrated

over velocity).

151 On the other hand, the rate of absorption per unit volume per unit solid angle

−3 −1 dΓabs/dΩ (units: cm s ) for an absorption cross section σcs is

Z ∞ dΓabs 1 nHIx1 dl nHIx0 Γemσcs , (A.2) dΩ ≈ 4π 0 − 3

where xi is the fraction of hydrogen atoms in the i state (either ground state F = 0

or excited state F = 1). The second term takes into account stimulated emission (it

1 3 is treated as negative absorption). To lowest order in T?, we have x0 = 4 + 16 T?/Ts

3 3 39 −3 −1 and x1 = T?/Ts. The net emission rate (units: cm s ) is approximately 4 − 16 given by

1 1 3 TCMB Γem = nHIAhf x1 1 + T /T 3x0 T /T nHIAhf 1 , e ? CMB 1 − e ? CMB 1 ≈ 4 − Ts − − (A.3) where we included spontaneous and stimulated (by the CMB) emission minus absorption, and carried out the usual algebraic simpliﬁcations.

We now make the assumption that a particular hydrogen atom only sees a small part of the total perturbations, i.e., we consider only modes of wavelengths much larger than the Jeans length, which in this case is of order σ/H. (This will be true for the modes where the gas is actually clustering with the dark matter.) We further ˆ ˆ recall that for linear growth, the velocity gradient is related to δ via ∂jvp,i = Hkikj δ. − Finally, we recall that the Sobolev optical depth is

3 3c Ahf nHIx1 N c nHIx1 τ = 3 nHIx0 = nHIx0 . (A.4) 8πν21H − 3 H ν21 − 3

39 These are the solutions to the total abundance constraint x0 + x1 = 1 and the Boltzmann T?/Ts condition x1/x0 = 3e− expanded to ﬁrst order in T /Ts. ∗

152 With all of these replacements, Eq. (A.2) becomes

Z ∞ dΓabs 1 nHIx1 3 TCMB c 1 = dl nHIx0 nHIAhf 1 N p 2 dΩ 4π 0 − 3 4 − Ts ν21 2π(2σ ) 2 l 2 exp [(Hδij + ∂jvp,i)ˆninˆj] × −2(2σ2) 1 nHIx1 3 TCMB c 1 = nHIx0 nHIAhf 1 N 8π − 3 4 − Ts ν21 (Hδij + ∂jvp,i)ˆninˆj 1 3 TCMB 1 = τ nHIAhf 1 8π 4 − Ts (δij + ∂jvp,i/H)ˆninˆj 3 τn A T HI hf 1 CMB 1 + δ(kˆ nˆ)2 . (A.5) ≈ 4π 8 − Ts ·

(The last line uses the ﬁrst-order expansion of the reciprocal.) With the help of

Eq. (73) from Venumadhav et al.(2014) and with nˆ along the z axis, we expand the

dot product and obtain the perturbation to the absorption rate:

r dΓabs 3τnHIAhf TCMB 2 4π ˆ = 1 cos θ δ Y20(k) . (A.6) dΩn pert 8 − Ts 5

We next need the excitation rate to each of the three excited states, m = 1, − m = 0, and m = +1. To obtain this, we integrate the anisotropic absorption rate

with the appropriate electric dipole antenna patterns. In the case of m = 0, the

electric dipole is aligned with the z axis and we have

Z r 1 dΓabs 3 2 nHIτAhf TCMB 4π ˆ 1 Γabs(m = 0) = sin θ dΩ = 1 δ Y20(k) . 3 dΩ 8π 8 − Ts 5 5 (A.7)

3 2 1 Here 8π sin θ is the beam pattern (normalized to integrate to 1), and the factor of 3

1 is introduced because for an isotropic distribution 3 of the excitations go to m = 0. Similarly for m = 1 the angular integrals give a factor of 2 . ± 5 Finally, we need to take the excitation rate to each state, and determine the resulting spin alignment. From Eq. (13) in Venumadhav et al.(2014), the “20” polarization

153 1 moment is 20 = √ (ρ11 2ρ00 + ρ−1−1). There is a conversion from excitation rate P 2 − −3 −1 (cm s ) to dimensionless probability of tlife/nHI, where tlife is the lifetime of the

atom in the excited state before it is disturbed. Thus:

1 tlife h i 20 Γabs(m = +1) 2Γabs(m = 0) + Γabs(m = 1) , (A.8) P ∼ √2 nHI − − where the lifetime is given by the stimulated emission adjusting for small corrections from both collisions and Lyman-α pumping, e.g. t−1 = A TCMB (1 + x + x ).40 life hf T∗ α c Hence combining the absorption rates into Eq. (A.8) we get

r 1 τ T∗ Tγ 4π ˆ 20(k) = 1 δ(k) Y20(k) . (A.9) P 20√2 1 + xα + xc Tγ − Ts 5

As expected, the velocity gradient enters Eq. (2.17), but does so through the factor of

density perturbations since we used the linear relation between velocity and density

modes.

A.2 Irreducible components of the density matrix

Here I follow the prescription in Venumadhav et al.(2014) to map from density

matrix elements to irreducible components. The procedure is simple. It only requires

Eq. (13) in Venumadhav et al.(2014):

p X 1−m2 1 j 1 jm = 3(2j + 1) ( 1) ρm1m2 , (A.10) P − m2 m m1 m1,m2 − where the quantity with the parenthesis is the Wigner 3j symbol.

40 The factors xα,(2) and xc,(2) diﬀer from xα and xc because they take into account the fact that Lyman-α scattering or collisions can change the polarization state of an excited hydrogen atom, while leaving it in the excited hyperﬁne state. This is treated in detail in Ref.(Venumadhav et al., 2014) and will not be repeated here.

154 It follows that the relevant irreducible components for our work are given by

22 = √3ρ−1,1 (A.11) P

2,−2 = √3ρ1,−1 . (A.12) P A.3 Special functions: Spherical harmonics

The spherical harmonics Y`,m are functions that represent the spatial (angular) dependence of solutions to the Laplace equation in spherical coordinates. Further- more, spherical harmonics are a complete set of orthogonal functions, which can map scalar functions into the surface of a sphere. Hence one can expand a function f(Ω) as ∞ ` X X f(Ω) = almY`m(Ω) . (A.13) `=0 m=−` The spherical harmonics are orthonormal, i.e.

Z π Z 2π ∗ Y`m(Ω)Y`0m0 (Ω)dΩ = δ``0 δmm0 , (A.14) 0 0 where δ``0 is the Kronecker delta function. Also, they satisfy

∗ m Y = ( 1) Y`,−m . (A.15) `m −

The last property we will present here is how spherical harmonics rotate. For more details about spherical harmonics, see F. W. J. Olver et al. (2018). For a rotation R around the origin we have

` 0 X ` ∗ Y`m(r ) = D 0 ( ) Y`m0 (r) , (A.16) mm R m=−`

` where Dmm0 is the Wigner D matrix.

155 A.4 Source term for polarized atoms from CMB anisotropies

This appendix considers the source term in Eq. (2.40). The main text motivates

the “order of magnitude” of this term, however a detailed treatment of the transition

probabilities to each level in the 2s 3p 1s scattering process is needed in order → → to get the correct numerical pre-factor for this process. Note that the reported pre- factor is √1 0.011, so in fact the suppression due to numerical pre-factors is very 16 10π ≈ signiﬁcant even if the factors are in some sense “of order unity.” This is a common

phenomenon in polarization problems.

Let us consider a hydrogen atom with the electron in the 2s orbital in a random

spin state, located in a CMB background with quadrupole moment Θ2m. For simplicity, we consider here the case where only Θ20 = 0, so that the radiation ﬁeld is 6

axisymmetric around the z-axis, i.e. T = Tγ[1 + Θ20Y20(θ, φ)]. This axisymmetry

also prevents any oﬀ-diagonal terms in the density matrix between states of diﬀerent

total magnetic quantum number, which simpliﬁes our arguments. Our objective is to

suppose that the atom undergoes 2s 3p 1s scattering (by absorbing an Hα photon → → and then emitting a Lyβ photon), and then compute the density matrix of the ﬁnal

1s state.

The ﬁrst step is to recognize that the excitation goes to either the 3pz orbital

(orbital quantum number ml = 0) or the 3px, 3py orbitals (ml = 1), but with ± a random spin state. The relative probabilities of going to the diﬀerent orbitals is determined by the phase space densities of photons where the receiving electric dipole is aligned as an antenna on the diﬀerent axes (the z-axis vs. in the xy-plane). An

156 electric dipole on the z axis sees a beam-pattern-averaged temperature of

Z Θ20 Tz = Tγ 1 + Θ20 B(θ, φ)Y20(θ, φ) sin θ dθ dφ = Tγ 1 , (A.17) − √20π

3 2 where B(θ, φ) = 8π sin θ is the beam pattern normalized to integrate over the surface to unity. Similarly, electric dipoles on the Tx and Ty axes see a temperature of Θ20 Txy Tx = Ty = Tγ 1 + . (A.18) ≡ √80π

Now the phase space density for the exciting Hα photons for absorption with the electric ﬁeld on the z-axis is 1 1 1 1 hνHα fz = hν /k T = hν /k T 1 −hν /k T Θ20 , (A.19) e Hα B z 1 e Hα B γ 1 − √20π 1 e Hα B γ kBTγ − − − where we have performed a Taylor expansion. The phase space density for excitations

with electric ﬁelds in the xy-plane is similar but with the replacement of the factor

1/√20π with +1/√80π. We thus conclude that the probabilities for excitation to − the 3p orbitals are

1 1 1 1 1 P→m =−1 : P→m =0 : P→m =1 = δp : + δp : δp, (A.20) l l l 3 − 2 3 3 − 2

where

1 1 hνHα δp = −hν /k T Θ20. (A.21) −3√20π 1 e Hα B γ kBTγ − Our next step is to note that ﬁne+hyperﬁne structure splits 3p into 12 quantum states (3 orbital states, 2 electron spin, 2 nuclear spin). The excitation puts the atom into orbital states in proportion to Eq. (A.20), and random electron and nuclear spin states, but ml is not a conserved quantum number: instead, we should take

our probabilities and project them into the j; FMF basis using the Clebsch-Gordan | i coeﬃcients. (Here again j is the electron total angular momentum, F is the total

157 angular momentum including nuclear spin, and MF is its z-projection.) The four ml = 0 states can be expressed in the ml, ms,MI basis as | i q q q 0, 1 , 1 = 1 3 ; 21 1 3 ; 11 1 1 ; 11 , | 2 2 i 2 | 2 i − 6 | 2 i − 3 | 2 i q q q q 0, 1 , 1 = 1 3 ; 20 + 1 3 ; 10 1 1 ; 10 1 1 ; 00 , | 2 − 2 i 3 | 2 i 3 | 2 i − 6 | 2 i − 6 | 2 i q q q q 0, 1 , 1 = 1 3 ; 20 1 3 ; 10 + 1 1 ; 10 1 1 ; 00 , and | − 2 2 i 3 | 2 i − 3 | 2 i 6 | 2 i − 6 | 2 i q q q 0, 1 , 1 = 1 3 ; 2 1 + 1 3 ; 1 1 + 1 1 ; 1 1 . (A.22) | − 2 − 2 i 2 | 2 − i 6 | 2 − i 3 | 2 − i

Because the fine and hyperfine splittings in 3p are larger than the natural state width, the phases of the different j and F states will be randomized before the atom decays.

Thus, if the excitation goes to the ml = 0 orbital with the 4 spin states chosen at random, then the probabilities for the excited states in the j; FMF basis are: | i

Probability 1 for each of 3 ; 20 and 3 ; 10 ; • 6 | 2 i | 2 i

Probability 1 for each of 3 ; 21 and 3 ; 2 1 ; • 8 | 2 i | 2 − i

Probability 1 for each of 1 ; 11 , 1 ; 10 , 1 ; 00 , and 1 ; 1 1 ; • 12 | 2 i | 2 i | 2 i | 2 − i

Probability 1 for each of 3 ; 11 and 3 ; 1 1 ; • 24 | 2 i | 2 − i

Probability 0 for each of 3 ; 22 and 3 ; 2 2 . • | 2 i | 2 − i

We now consider the Lyman-β decay to the 1s level. The Clebsch-Gordan coeﬃ- cents and the Wigner-Eckart theorem can then be used to determine the amplitudes for decay of each of these states to the four 1s : FMF states. The decay rate for | i 1 0 0 (3p)j; FMF (1s) ; F M is proportional to (see Eq. B2 of Ref. Hirata(2006) | i → | 2 F i

158 and use the Wigner-Eckart theorem):

1 2 1 0 0 0 1 j 2 P[ (3p)j; FMF (1s) 2 ; F MF ] (2j + 1)(2F + 1) 1 | i → | i ∝ 2 0 1 1 2 1 j F X 2 2 F 0M 0 , 1q FM , F 0 1 1 F F × 2 q=−1 |h | i| (A.23) where P stands for probability of that decay, the curly brackets are Wigner 6j symbols and the last object is the square of a Clebsch-Gordan coeﬃcient. A lengthy but straightforward calculation gives the ﬁnal probabilities for the 1s states starting from excitation to 3p ml = 0:

0 0 1 P (F = 0,MF = 0) = 4 ,

P (F 0 = 1,M 0 = 1) = P (F 0 = 1,M 0 = 1) = 5 , and F − F 24 0 0 1 P (F = 1,MF = 0) = 3 . (A.24)

The polarization of the ﬁnal atomic state in this case is given by Eq. (13) of Ref. Venu- madhav et al.(2014):

1 ﬁnal = [P (F 0 = 1,M 0 = 1) 2P (F 0 = 1,M 0 = 0) + P (F 0 = 1,M 0 = 1)] P20 √2 F − F F − 1 = . (A.25) −12√2

2 Note that this is the polarization for pure excitation to 3p ml = 0, i.e. if δp = 3 ; since the ﬁnal polarization must be proportional to δp given Eq. (A.20), we must have

ﬁnal 1 1 1 hνHα 20 = δp = −hν /k T Θ20. (A.26) P −8√2 24√40π 1 e Hα B γ kBTγ − Finally, the rate of sourcing polarized atoms through 2s 3p 1s scattering is ﬁnal → → P20 times the rate at which atoms scatter through this channel (in units of atoms per

159 available H nucleus per unit time) is:

1 [Rate of 2s 3p 1s] = x2s3A3p→2s → → ehνHα/kBTγ 1 × − A3p→1s −hν /k T , (A.27) A3p→1s + A3p→2s/(1 e Hα B γ ) − where the factor of 3 comes from the ratio of statistical weights of 3p:2s; the ﬁrst exponential factor is the blackbody phase space density of exciting photons; and the last factor is the branching fraction for 3p 1s. Finally, multiplying Eq. (A.26) by → Eq. (A.27), and taking the Wien limit e−hνHα/kBTγ 1 (relevant for Hα at the surface of last scattering), we get the source term in Eq. (2.40):

1 A3p→2sA3p→1s hνHα −hνHα/kBTγ S20 = x2s e Θ20. (A.28) 16√10π A3p→1s + A3p→2s kBTγ

(Note that since the choice of z-axis did not matter, this equation must hold for all

m = 2, 1, 0, 1, 2.) − − A.5 Stokes parameters

The Stokes parameters are a set of values that conveniently deﬁne the polarization

of light. The Stokes parameters – for a light wave propagating in thez ˆ-direction –

are given by:

I = E2 + E2 , (A.29) | x| | y | Q = E2 E2 , (A.30) | x| − | y | ∗ U = 2 Re ExE , (A.31) { y } ∗ V = 2 Im ExE , (A.32) − { y }

where I, Q, U and V describe the total intensity of the radiation, the diﬀerence in the intensity of light in thex ˆ-direction against they ˆ-direction, the diﬀerence of intensity

160 Figure A.1: Polarization Stokes parameters. The arrows represent the direction of the electromagnetic ﬁelds.

between the diagonals, and the amount of circular polarization, respectively. See the

diagram A.1.

Q and U are spin-2 quantities that represent the amount of linear polarization.

However, the total intensity I and the circular polarization V are scalars. Note that unpolarized light is described by Q = U = V = 0. See Jackson(1999) for a more detailed description.

A.6 Special functions: Spin-weighted spherical harmonics

Spin-weighted spherical harmonics map vector ﬁelds (instead of scalars) into the sphere. As such they are a generalization of the regular spherical harmonics. In fact, one can obtain spin-weighted spherical harmonics from spherical harmonics by

161 applying the operator to raise the angular momentum. Therefore, the traditional

spherical harmonics are just the spin 0 weighted spherical harmonics, 0Y`m = Y`m.

The spin-weighted spherical harmonics are deﬁned as follows:

s `−s m (` + m)!(` m)!(2` + 1) 2` θ X ` s sY`m(Ω) = ( 1) − sin − − 4π(` + s)!(` s)! 2 r − r=0 ` + s θ ( 1)`−r−seimφ cot2r+s−m . (A.33) × r + s m − 2 −

∗ s+m With the phase convention sY = ( 1) −sY`,−m. Furthermore, they have their `m − own orthogonality relation:

Z π Z 2π sY`m(Ω)sY`0m0 (Ω)dΩ = δ``0 δmm0 . (A.34) 0 0

Finally, I will provide the value of the single spin-weighted spherical harmonic

used in our work:

1r 5 Y (θ, φ) = (1 + cos θ)2e−2iφ (A.35) 2 2,−2 8 π r 1 5 2 2iφ 2Y22(θ, φ) = (1 cos θ) e . (A.36) 8 π −

For more information on spin-weighted spherical harmonics and their connection

to regular spherical harmonics see Dray(1985).

A.7 Line of sight integrals of photon-photon scattering

Here we show the exact form of the integrals I1, I2, I3 and I4 used for the line

of sight suppression computation of the circular polarization from photon-photon

scattering in 2.5. Also, we show the expansion of the 4-point functions into 2-point § functions used for obtaining the J integrals.

162 ˜ We write the 2-point function of the CMB circular polarization V (k⊥), Eq. (2.68), as four contributions from diﬀerent combinations of the 4-point functions of the curvature perturbations involved. Also, we make use of Eq. (2.63) to get

A¯2 Z d3k Z d3k Z Z d3k0 Z d3k0 Z I = 1 2 ds 1 2 ds0 (1 + z)4 (1 + z)04 1 4 2π 2π 2π 2π 0 0 E E E 0 0 exp ( ik s + ik1,ks) T (c ηLSS + s, k1) T (c ηLSS, k2) T (c ηLSS + s , k ) × − 1,k 20 20 20 1 E 0 ∗ ˆ ˆ ˆ0 ∗ ˆ0 ∗ ∗ 0 0 T (c ηLSS, k ) Y (k2)Y22(k1)Y22(k )Y (k ) ζ(k1)ζ (k2)ζ (k )ζ(k ) × 20 2 22 2 22 1 h 1 2 i 2 2 0 0 0 δ (k⊥ + k1,⊥ k2,⊥)δ (k k k ), (A.37) × − 2,⊥ − 1,⊥ − ⊥ ¯2 Z 3 Z 3 Z Z 3 0 Z 3 0 Z A d k1 d k2 d k1 d k2 0 4 04 I2 = ds ds (1 + z) (1 + z) − 4 2π 2π 2π 2π 0 0 E E E 0 0 exp (ik s + ik1,ks) T (c ηLSS + s, k1) T (c ηLSS, k2) T (c ηLSS + s , k ) × 1,k 20 20 20 1 E 0 ∗ ˆ ˆ ∗ ˆ0 ˆ0 ∗ 0 ∗ 0 T (c ηLSS, k ) Y (k2)Y22(k1)Y (k ) Y22(k ) ζ(k1)ζ (k2)ζ(k )ζ (k ) × 20 2 22 22 2 × 1 h 1 2 i 2 2 0 0 0 δ (k⊥ + k1,⊥ k2,⊥)δ (k k k ), (A.38) × − 1,⊥ − 2,⊥ − ⊥ ¯2 Z 3 Z 3 Z Z 3 0 Z 3 0 Z A d k1 d k2 d k1 d k2 0 4 04 I3 = ds ds (1 + z) (1 + z) − 4 2π 2π 2π 2π 0 0 E E E 0 0 exp ( ik s ik1,ks) T (c ηLSS + s, k1) T (c ηLSS, k2) T (c ηLSS + s , k ) × − 1,k − 20 20 20 1 E 0 ˆ ∗ ˆ ˆ0 ∗ ˆ0 ∗ ∗ 0 0 T (c ηLSS, k ) Y22(k2)Y (k1)Y22(k )Y (k ) ζ (k1)ζ(k2)ζ (k )ζ(k ) × 20 2 22 2 22 1 h 1 2 i 2 2 0 0 0 δ (k⊥ + k2,⊥ k1,⊥)δ (k k k ), and (A.39) × − 2,⊥ − 1,⊥ − ⊥ A¯2 Z d3k Z d3k Z Z d3k0 Z d3k0 Z I = 1 2 ds 1 2 ds0 (1 + z)4 (1 + z)04 4 4 2π 2π 2π 2π 0 0 E E E 0 0 exp (ik s ik1,ks) T (c ηLSS + s, k1) T (c ηLSS, k2) T (c ηLSS + s , k ) × 1,k − 20 20 20 1 E 0 ˆ ∗ ˆ ∗ ˆ0 ˆ0 ∗ 0 ∗ 0 T (c ηLSS, k ) Y22(k2)Y (k1)Y (k )Y22(k ) ζ (k1)ζ(k2)ζ(k )ζ (k ) × 20 2 22 22 2 1 h 1 2 i 2 2 0 0 0 δ (k⊥ + k2,⊥ k1,⊥)δ (k k k ). (A.40) × − 1,⊥ − 2,⊥ − ⊥

163 Furthermore, using Wick’s theorem, the 4-point functions can be expanded in

0 0 0 terms that connect the two triangles k⊥ = k1 k2 and k = k k , which are − ⊥ 1 − 2

∗ ∗ 0 0 ∗ 0 ∗ 0 6 ζ(k1)ζ (k2)ζ (k )ζ(k ) = ζ (k1)ζ(k2)ζ(k )ζ (k ) = (2π) Pζ (k1)Pζ (k2) h 1 2 i h 1 2 i 3 0 3 0 3 0 3 0 δ (k1 k )δ (k2 k ) + δ (k1 + k )δ (k2 + k ) , × − 1 − 2 2 1 (A.41)

∗ 0 ∗ 0 ∗ ∗ 0 0 6 ζ(k1)ζ (k2)ζ(k )ζ (k ) = ζ (k1)ζ(k2)ζ (k )ζ(k ) = (2π) Pζ (k1)Pζ (k2) h 1 2 i h 1 2 i 3 0 3 0 3 0 3 0 δ (k1 + k )δ (k2 + k ) + δ (k1 k )δ (k2 k ) . × 1 2 − 2 − 1 (A.42)

Thus we obtained the I integrals and expanded the 4-point functions in terms of the primordial curvature power spectrum. Now, we can ﬁnally integrate out the perpendicular component of k2 with the help of the 2D δ-functions to obtain the eight J integrals. The ﬁrst four of these are

¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k 2 E 2 J1 = (2π) δ (k⊥ k ) F (k1, k1,k) [T (c ηLSS, k2)] 4 − ⊥ (2π)2 2π 2π | | 20 ∗ ˆ ˆ ˆ ∗ ˆ Y (k2)Y22(k1)Y22(k2)Y (k1)Pζ (k1)Pζ (k2), (A.43) × 22 22 ¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k J2 = (2π) δ (k⊥ k ) F (k1, k1,k)F (k2, k2,k) 4 − ⊥ (2π)2 2π 2π E E ∗ ˆ ˆ ∗ ˆ ˆ T (ηLSS, k2) T (ηLSS, k1)Y (k2)Y22(k1)Y (k2)Y22(k1)Pζ (k1)Pζ (k2), (A.44) × 20 20 22 22 ¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k 2 E 2 J3 = (2π) δ (k⊥ k ) F (k1, k1,k) [T (ηLSS, k2)] − 4 − ⊥ (2π)2 2π 2π | | 20 ∗ ˆ ˆ ∗ ˆ ˆ Y (k2)Y22(k1)Y (k2)Y22(k1)Pζ (k1)Pζ (k2), and (A.45) × 22 22 ¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k J4 = (2π) δ (k⊥ k ) F (k1, k1,k)F (k2, k2,k) − 4 − ⊥ (2π)2 2π 2π E E ∗ ˆ ˆ ∗ ˆ ˆ T (ηLSS, k2) T (ηLSS, k1) Y (k2)Y22(k1)Y (k1)Y22(k2)Pζ (k1)Pζ (k2), (A.46) × 20 20 22 22

164 which have k2,⊥ = k1,⊥ + k⊥. The remaining integrals are

¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k 2 E 2 J5 = (2π) δ (k⊥ k ) F (k1, k1,k) [T (ηLSS, k2)] − 4 − ⊥ (2π)2 2π 2π | | 20 ˆ ∗ ˆ ∗ ˆ ˆ Y22(k2)Y (k1)Y (k1)Y22(k2)Pζ (k1)Pζ (k2), (A.47) × 22 22 ¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k ∗ ∗ J6 = (2π) δ (k⊥ k ) F (k1, k1,k)F (k2, k2,k) − 4 − ⊥ (2π)2 2π 2π E E ˆ ∗ ˆ ˆ ∗ ˆ T (ηLSS, k2)T (ηLSS, k1)Y22(k2)Y (k1)Y22(k1)Y (k2)Pζ (k1)Pζ (k2), (A.48) × 20 20 22 22 ¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k 2 E 2 J7 = (2π) δ (k⊥ k ) F (k1, k1,k) [T (ηLSS, k2)] 4 − ⊥ (2π)2 2π 2π | | 20 ˆ ∗ ˆ ˆ ∗ ˆ Y22(k2)Y (k1)Y22(k1)Y (k2)Pζ (k1)Pζ (k2), and (A.49) × 22 22 ¯2 Z 2 Z Z 2 A 2 0 d k1,⊥ dk1,k dk2,k ∗ ∗ J8 = (2π) δ (k⊥ k ) F (k1, k1,k)F (k2, k2,k) 4 − ⊥ (2π)2 2π 2π E E ˆ ∗ ˆ ∗ ˆ ˆ T (ηLSS, k2)T (ηLSS, k1)Y22(k2)Y (k1)Y (k1)Y22(k2)Pζ (k1)Pζ (k2), (A.50) × 20 20 22 22 where k2,⊥ = k1,⊥ k⊥. − A.8 Plasma delay integrals

Here we present the integrals involved in the detailed calculation from section

2.7.2. Concretely, the integrals stem from Eqs. (2.108,2.109). §§

We start with IT , which we expanded as IT = I1 I2 I3 + I4 I5 + I6 I7 I8. − − − − − The integrals are

∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I1 = R , (A.51) 3 3 2 ~ω /kBT 2 R 2 R 4π c kBT ωA 0 (e B 1) (ω + i) (ω + ωA + i) − B B ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I2 = R , (A.52) 3 3 ~ω /kBT 2 R 2 R 2 4π c kBT ωA 0 (e B 1) (ω i) (ω ωA i) − B − B − − ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I3 = R , (A.53) 3 3 2 ~ω /kBT 2 R R 2 4π c kBT ωA 0 (e B 1) (ω + i)(ω ωA i) − B B − − ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I4 = R , (A.54) 3 3 ~ω /kBT 2 R 2 R 2 4π c kBT ωA 0 (e B 1) (ω + i) (ω + ωA + i) − B B ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I5 = R , (A.55) 3 3 2 ~ω /kBT 2 R R 2 4π c kBT ωA 0 (e B 1) (ω i)(ω + ωA + i) − B − B 165 ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I6 = R , (A.56) 3 3 2 ~ω /kBT 2 R 2 R 4π c kBT ωA 0 (e B 1) (ω + i) (ω ωA i) − B B − − ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I7 = R , and (A.57) 3 3 2 ~ω /kBT 2 R ( R 2 2π c kBT ωA 0 (e B 1) (ω + i) ω + ωA + i) − B B ∞ R 5 2 Z ~ωB /kBT R R ~ e ωB dωB I8 = R , (A.58) 3 3 2 ~ω /kBT 2 R R 2 2π c kBT ωA 0 (e B 1) (ω i)(ω ωA i) − B − B − − where the exponential factor comes from the black body spectrum.

Similarly, for the case of IB = IB IB + IB + IB we have T 1 − 2 3 4 ∞ R 4 2 Z ~ωB /kBT R R B ~ e ωB dωB I1 = R , (A.59) 3 3 3 ~ω /kBT 2 R 4π c kBT ωA 0 (e B 1) (ω + ωA + i) − B ∞ R 4 2 Z ~ωB /kBT R R B ~ e ωB dωB I2 = R , (A.60) 3 3 3 ~ω /kBT 2 R 4π c kBT ωA 0 (e B 1) (ω ωA i) − B − − ∞ R 4 2 Z ~ωB /kBT R R B ~ e ωB dωB I3 = R , and 3 3 ~ω /kBT 2 R R R 2 4π c kBT 0 (e B 1) (ω + i)(ω i)(ω + ωA + i) − B B − B (A.61) ∞ R 4 2 Z ~ωB /kBT R R B ~ e ωB dωB I4 = R . (A.62) 3 3 ~ω /kBT 2 R R R 2 4π c kBT 0 (e B 1) (ω i)(ω + i)(ω ωA i) − B − B B − − In general the strategy for handling such integrals involved a combination of integration by parts and principal value while being careful about the redshift dependence of both temperature and frequency of CMB photons. Finally, one should take the limit as 0. →

166 Appendix B: Impact of inhomogeneous reionization in the Lyman-α forest

This appendix presents results relevant to Chapter3.

B.1 Special function: Legendre polynomials

The Legendre polynomials are a set of complete orthogonal polynomials. Thus, one can expand a function as follows

∞ X f(x) = a`P`(x) . (B.1) `=0

They satisfy the orthonormal condition

Z 1 2 P`(x)P`0 (x)dx = δ``0 . (B.2) −1 2` + 1

Here I provide the expressions for the Legendre polynomials used in this work

P0(cos θ) = 1 (B.3)

1 2 P2(cos θ) = (3 cos θ 1) (B.4) 2 − 1 4 2 P4(cos θ) = (35 cos θ 30 cos θ + 3) . (B.5) 8 −

167 Appendix C: White Dwarf Survival: mapping of hydrodynamical code

This appendix presents results relevant to Chapter4.

C.1 Equation of state

This appendix describes the equation of state of shock-heated white dwarf material, P (ρ, ).

C.1.1 Physical description

We will consider the WD matter to be made of ultra-relativistic degenerate e− and e+, and of (ideal gas) ions with average atomic number Z¯ = 7 and mass A¯ = 14 amu, which is roughly the case for carbon/oxygen (C/O) WD cores. The thermodynamics of the WD matter can be described by the mass density ρ and the chemical potential

µ − for the electrons and µ + = µ − for positrons. We introduce a dimensionless e e − e chemical potential J µ − /(kBT ). The number densities of electrons and positrons ≡ e are then given by

Z 3 d p 1 2 3 ∓J ± ne = 2 3 [E(p)±µ ]/k T = 2 3 (kBT ) Li3 e , (C.1) (2π~) e e− B + 1 −π (~c) −

168 41 where Lin( ) is the n-th order polylogarithm function , and the second equality ··· has used the fact that E(p) = pc for ultra-relativistic particles. With the electrical ¯ neutrality, the number density of ions is then nion = (n − n + )/Z. Since the mass e − e is dominated by the ions, we obtain the density of the WD matter as

¯ 3 ρ = nionmnucA = Af µe− f(J) , (C.4)

−27 ¯ ¯ 2 3 where mnuc = 1.67 10 kg is the mass of the nucleon, Af = mnucA/[Zπ (~c) ], × J −J 3 and f(J) = 2 Li3 e Li3 e /J . Similarly, we ﬁnd the total pressure − − − − from electrons, positrons and ions is given by

4 µe− 4 Af f(J) P = Agg(J)µe− + nion = µe− Agg(J) + ¯ , (C.5) J mnucA J

2 3 J −J 4 where Ag = 1/[π (~c) ], and g(J) = 6/ Li4 e + Li4 e /J . Denoting the − − − total speciﬁc energy (internal + kinematic energy per mass) of electrons, positrons,

± ions, and their sum as e− , e+ , ion, , respectively, we can write e total energy density

Z 3 d p E(p) 6 4 ∓J ± ρe = 2 3 [E(p)±µ ]/k T = 2 3 (kBT ) Li4 e . (C.6) (2π~) e e− B + 1 −π (~c) −

Thus, the total energy density of the material is given by

4 3 ρ µe− ρ = ρ(e− + e+ + ion) = Agg(J)µe− + ¯ , (C.7) 2 mnucA J

41The polylogarithm function is mathematically related to the complete Fermi-Dirac integral (see Eq. 5.4.1 in F. W. J. Olver et al. (2018)), i.e.

Z s 1 ∞ t x Fs(x) = t x dt = Lis+1( e ) , (s > 1) . (C.2) Γ(s + 1) 0 e − + 1 − − − A useful limiting case is given in Wood(1992),

s x x lim Lis( e ) = , (s = 1, 2, 3, ) , (C.3) Re(x) − −Γ(s + 1) 6 − − − ··· →∞ which tells us that positrons are negligible at low temperatures.

169 where the last term comes from the ideal gas law. Combining Eqs. (C.4) and (C.7),

we ﬁnd

−1/3 ρ = Ahh(J) , (C.8)

with the coeﬃcient Ah and the dimensionless function h(J) are deﬁned as

1/3 2 1/3 1/3 3~cZ¯ π Z¯ g(J) 1 2 A , h(J) + . (C.9) h ¯ 4/3 ≡ (mnucA) 2 ≡ 3 f(J) 2J f(J)

This set of equations (Eqs. C.4–C.8) suggests that the thermodynamic state of the

material can be fully determined by only two variables, ρ and J. Since h(J) is a

monotonically decreasing function, given ρ and , one can determine J from ρ and

by inverting Eq. (C.8) numerically. Then, the electron chemical potential can be

1/3 calculated by inverting Eq. (C.4), i.e. , µ − = ρ/[Af f(J)] , and from there, e { } we immediately have access to the pressure, temperature, and other thermodynamic

quantities.

C.1.2 Conversion to code units

In the simulation, we adopt the unit system cs,∞ = ρ∞ = Rc = 1. In these units, we have initial pressure Pinit = 1/γ = 3/4, and speciﬁc internal energy in,init =

1/[γ(γ 1)] = 9/4. − In order to express all the quantities in the code units, we need to express our basic units ρ∞, cs,∞ in SI units. Deﬁning a dimensionless Fermi momentum x as

x pF/(mec), where pF is the Fermi momentum and me is the rest mass of electron, ≡

170 we have

¯ 3 Amnuc mecx ρ∞ = 1 (C.10) 3π2Z¯ ~ ≡ 4 ~c mecx P∞ = (C.11) 12π2 ~ ¯ 2 4P∞ Z~c mecx cs,∞ = = ¯ 1 . (C.12) 3ρ∞ 3Amnuc ~ ≡

1/3 ¯ By eliminating x, Ah is reduced to Ah = 9/(6 Z) in the code units. To further simplify the problem, we rescale the electron chemical potential µe− by the background

electron Fermi energy EF,∞, which is also equal to the background electron chemical potential, i.e. 1/3 ρ∞ EF,∞ = µe−,∞ = . (C.13) Af f( ) ∞ Note that when J , f( ) = 1/3, g( ) = 1/4, so we have → ∞ ∞ ∞ 1/3 1/3 µ − ρ/ρ code ρ e = ∞ . (C.14) EF,∞ 3f(J) −−→ 3f(J)

Similarly, the pressure is

4 P 3ρ∞cs,∞ f(J) µe− P = = 3ρ∞cs,∞ g(J) + ¯ P∞ 4 ZJ EF,∞ 4/3 code f(J) ρ 3 g(J) + . (C.15) −−→ ZJ¯ 3f(J)

Thus, coeﬃcients Af and Ag are eliminated from the problem.

C.2 Carbon dissociation

In this appendix, we consider dissociation of 12C following a PBH-induced shock

in a white dwarf. The very high temperatures make this reaction potentially more

important than in standard SN Ia ignition scenarios. This sets the limit of validity

of our estimation of ˙nuc, since once carbon dissociates one must follow a full reaction

171 network rather than simply thinking of “carbon burning.” We consider this both from

a thermodynamics and reaction kinetics point of view: in order to be signiﬁcant,

carbon dissociation must be not just thermodynamically favorable, but must occur

on a relevant timescale.

Let us ﬁrst consider the thermodynamics of the dissociation reaction, 12C 3α. ↔ In nuclear statistical equilibrium, the chemical potentials satisfy µ(12C) = 3µ(4He).

The chemical potential of a nuclei AZ is given by

3/2 A A A (2π) nbX( Z) µ( Z) = Zmp + (A Z)mn B( Z) + T ln , (C.16) − − A 5/2 3/2 3/2 [2I( Z) + 1]A mN T where nb is the baryon number density, X is the mass fraction, mp, mn, and mN are

proton, neutron, and nucleon masses, B is the binding energy, and I is the spin. The

3 2 3 baryon density is nb = x /(3π λ¯eYe), where x is the electron Fermi parameter. If we

set an transition point by taking X(12C) and X(4He) to be 0.5, we can rewrite the

equilibrium condition as

7.4 MeV 1.07 1013(T/MeV)3 = ln × . (C.17) T x6

With a typical value of x 3, the equilibrium temperature is T = 0.3 MeV, above ∼ which carbon dissociation is thermodynamically favored. Nevertheless, given the

speed of the shock propagation we are interested in, one must investigate the ki-

netics of this reaction in order to estimate the eﬀect of carbon dissociation in our

calculation.

To assess reaction kinetics, one must consider the process to dissociate 12C. Rel-

evant nuclear data is taken from IAEA - online live chart of nuclides. We consider

172 ﬁrst the radiative sequence:

12C + γ 12C∗, ↔ 12C∗ + γ 12C∗∗, ↔ 12C∗∗ 8Be + 2α, ↔ 8Be 2α. (C.18) ↔

12 12 ∗ + 12 ∗∗ Here the excited states of C are C (at E1 = 4.440 MeV; spin 2 ) and C (at

+ E2 = 7.654 MeV; spin 0 ). The ﬁrst two steps are inverse electric quadrupole decays,

13 −1 12 −1 42 12 ∗∗ with rates of A1 = 1.64 10 s and A2 = 5.87 10 s respectively. Since C × × rapidly dissociates with 100% branching fraction, the rate-limiting step will be one ≈ of the ﬁrst two processes.

Treating the excited states of 12C by the steady state approximation, we have

1 12 ∗ −E1/T 12 −(E2−E1)/T 12 ∗ 12 ∗∗ 0 =n ˙ ( C ) = 5A1e n( C) A1 + A2e n( C ) + A2n( C ) − 5 1 12 ∗∗ −(E2−E1)/T 12 ∗ 12 ∗∗ 0 =n ˙ ( C ) = A2e n( C ) Γ2n( C ). (C.19) 5 −

In the limit Γ2 A2, the timescale for carbon dissociation is then 12 E2/T (E1+E2)/T n( C) e e −13 7.654 MeV/T tdissoc = 12 ∗∗ = + 1.7 10 e s, (C.20) Γ2n( C ) A2 5A1 ≈ × where at the end we see that the ﬁrst term is dominant. We see that the typical carbon dissociation lifetime is 10−10 s when T = 1.4 1010 K. Hence, for a massive WD, ∼ × the passage of a PBH may produce 4He in the inner mass shells; we have not followed the resulting reaction sequence, and it is not obvious to us whether this results in ignition of the carbon or not. However, at larger radii carbon may be ignited without this complication and the runaway explosion may occur.

42 14 12 These can be computed from the half-lives of the two levels, 4.22 10− s for C∗ and 4.91 17 12 4 × 12 × 10− s for C∗∗, and the branching ratio of 4.16 10− for radiative decay of C∗∗. × 173 Bibliography

B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley,

C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al. Observation of

Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters,

116(6):061102, February 2016. doi: 10.1103/PhysRevLett.116.061102.

B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams,

P. Addesso, R. X. Adhikari, V. B. Adya, and et al. GW170608: Observation of

a 19 Solar-mass Binary Black Hole Coalescence. ApJL, 851:L35, December 2017a.

doi: 10.3847/2041-8213/aa9f0c.

B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams,

P. Addesso, R. X. Adhikari, V. B. Adya, and et al. GW170104: Observation of

a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2. Physical Review

Letters, 118(22):221101, June 2017b. doi: 10.1103/PhysRevLett.118.221101.

B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams,

P. Addesso, R. X. Adhikari, V. B. Adya, and et al. GW170814: A Three-Detector

Observation of Gravitational Waves from a Binary Black Hole Coalescence. Physical

Review Letters, 119(14):141101, October 2017c. doi: 10.1103/PhysRevLett.119.

141101.

174 Katsuya T. Abe, Hiroyuki Tashiro, and Toshiyuki Tanaka. Sunyaev-Zel’dovich

anisotropy due to Primordial black holes. arXiv e-prints, art. arXiv:1901.06809,

January 2019.

M. Ackermann, W. B. Atwood, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri,

R. Bellazzini, B. Berenji, E. Bissaldi, R. D. Blandford, E. D. Bloom, R. Bonino,

E. Bottacini, J. Bregeon, P. Bruel, R. Buehler, R. A. Cameron, R. Caputo, P. A.

Caraveo, E. Cavazzuti, E. Charles, A. Chekhtman, C. C. Cheung, G. Chiaro,

S. Ciprini, J. Cohen-Tanugi, J. Conrad, D. Costantin, F. D’Ammando, F. de Palma,

S. W. Digel, N. Di Lalla, M. Di Mauro, L. Di Venere, C. Favuzzi, S. J. Fegan, W. B.

Focke, A. Franckowiak, Y. Fukazawa, S. Funk, P. Fusco, F. Gargano, D. Gaspar-

rini, N. Giglietto, F. Giordano, M. Giroletti, D. Green, I. A. Grenier, L. Guillemot,

S. Guiriec, D. Horan, G. J´ohannesson, C. Johnson, S. Kensei, D. Kocevski, M. Kuss,

S. Larsson, L. Latronico, J. Li, F. Longo, F. Loparco, M. N. Lovellette, P. Lu-

brano, J. D. Magill, S. Maldera, D. Malyshev, A. Manfreda, M. N. Mazziotta, J. E.

McEnery, M. Meyer, P. F. Michelson, W. Mitthumsiri, T. Mizuno, M. E. Monzani,

E. Moretti, A. Morselli, I. V. Moskalenko, M. Negro, E. Nuss, R. Ojha, N. Omodei,

M. Orienti, E. Orlando, J. F. Ormes, M. Palatiello, V. S. Paliya, D. Paneque,

M. Persic, M. Pesce- Rollins, F. Piron, G. Principe, S. Rain`o, R. Rando, M. Raz-

zano, S. Razzaque, A. Reimer, O. Reimer, S. Ritz, M. S´anchez- Conde, C. Sgr`o,

E. J. Siskind, F. Spada, G. Spandre, P. Spinelli, D. J. Suson, H. Tajima, J. G.

Thayer, J. B. Thayer, D. F. Torres, G. Tosti, E. Troja, J. Valverde, G. Vianello,

K. Wood, M. Wood, and G. Zaharijas. Search for Gamma-Ray Emission from Local

Primordial Black Holes with the Fermi Large Area Telescope. ApJ, 857:49, April

2018. doi: 10.3847/1538-4357/aaac7b.

175 F. Ajzenberg-Selove. Energy levels of light nuclei A = 11-12. Nuclear Physics A, 506:

1–158, January 1990. doi: 10.1016/0375-9474(90)90271-M.

A. Alastuey and B. Jancovici. Nuclear reaction rate enhancement in dense stellar

matter. ApJ, 226:1034–1040, December 1978. doi: 10.1086/156681.

C. Alcock, R. A. Allsman, D. R. Alves, T. S. Axelrod, A. C. Becker, D. P. Bennett,

K. H. Cook, N. Dalal, A. J. Drake, K. C. Freeman, M. Geha, K. Griest, M. J.

Lehner, S. L. Marshall, D. Minniti, C. A. Nelson, B. A. Peterson, P. Popowski, M. R.

Pratt, P. J. Quinn, C. W. Stubbs, W. Sutherland, A. B. Tomaney, T. Vandehei,

and D. Welch. The MACHO Project: Microlensing Results from 5.7 Years of Large

Magellanic Cloud Observations. ApJ, 542:281–307, October 2000. doi: 10.1086/

309512.

C. Alcock, R. A. Allsman, D. R. Alves, T. S. Axelrod, A. C. Becker, D. P. Bennett,

K. H. Cook, N. Dalal, A. J. Drake, K. C. Freeman, M. Geha, K. Griest, M. J.

Lehner, S. L. Marshall, D. Minniti, C. A. Nelson, B. A. Peterson, P. Popowski, M. R.

Pratt, P. J. Quinn, C. W. Stubbs, W. Sutherland, A. B. Tomaney, T. Vandehei,

and D. L. Welch. MACHO Project Limits on Black Hole Dark Matter in the 1-30

Msolar Range. ApJL, 550:L169–L172, April 2001. doi: 10.1086/319636.

Y. Ali-Ha¨ımoud and M. Kamionkowski. Cosmic microwave background limits on

accreting primordial black holes. Phys. Rev. D, 95(4):043534, February 2017. doi:

10.1103/PhysRevD.95.043534.

Yacine Ali-Ha¨ımoud. Correlation Function of High-Threshold Regions and Ap-

plication to the Initial Small-Scale Clustering of Primordial Black Holes.

176 Phys. Rev. Lett., 121(8):081304, Aug 2018. doi: 10.1103/PhysRevLett.121.081304.

Yacine Ali-Ha¨ımoud and Christopher M Hirata. Hyrec: A fast and highly accurate

primordial hydrogen and helium recombination code. Physical Review D, 83(4):

043513, 2011.

Yacine Ali-Ha¨ımoud, Ely D. Kovetz, and Marc Kamionkowski. Merger rate of

primordial black-hole binaries. Phys. Rev. D, 96(12):123523, Dec 2017. doi:

10.1103/PhysRevD.96.123523.

Elham Alipour, Kris Sigurdson, and Christopher M Hirata. Eﬀects of rayleigh scat-

tering on the cmb and cosmic structure. Physical Review D, 91(8):083520, 2015.

Philippe Andr´e, Carlo Baccigalupi, Anthony Banday, Domingos Barbosa, Belen Bar-

reiro, James Bartlett, Nicola Bartolo, Elia Battistelli, Richard Battye, George

Bendo, Alain Benoˆıt, Jean-Philippe Bernard, Marco Bersanelli, Matthieu B´ether-

min, Pawel Bielewicz, Anna Bonaldi, Fran¸cois Bouchet, Fran¸cois Boulanger, Jan

Brand, Martin Bucher, Carlo Burigana, Zhen-Yi Cai, Philippe Camus, Francisco

Casas, Viviana Casasola, Guillaume Castex, Anthony Challinor, Jens Chluba, Gay-

oung Chon, Sergio Colafrancesco, Barbara Comis, Francesco Cuttaia, Giuseppe

D’Alessandro, Antonio Da Silva, Richard Davis, Miguel de Avillez, Paolo de

Bernardis, Marco de Petris, Adriano de Rosa, Gianfranco de Zotti, Jacques De-

labrouille, Fran¸cois-Xavier D´esert, Clive Dickinson, Jose Maria Diego, Joanna

177 Dunkley, Torsten Enßlin, Josquin Errard, Edith Falgarone, Pedro Ferreira, Ka- tia Ferri`ere, Fabio Finelli, Andrew Fletcher, Pablo Fosalba, Gary Fuller, Sil- via Galli, Ken Ganga, Juan Garc´ıa-Bellido, Adnan Ghribi, Martin Giard, Yan- nick Giraud-H´eraud, Joaquin Gonzalez-Nuevo, Keith Grainge, Alessand ro Grup- puso, Alex Hall, Jean-Christophe Hamilton, Marijke Haverkorn, Carlos Hernandez-

Monteagudo, Diego Herranz, Mark Jackson, Andrew Jaﬀe, Rishi Khatri, Martin

Kunz, Luca Lamagna, Massimiliano Lattanzi, Paddy Leahy, Julien Lesgourgues,

Michele Liguori, Elisabetta Liuzzo, Marcos Lopez-Caniego, Juan Macias-Perez,

Bruno Maﬀei, Davide Maino, Anna Mangilli, Enrique Martinez-Gonzalez, Carlos

J. A. P. Martins, Silvia Masi, Marcella Massardi, Sabino Matarrese, Alessandro

Melchiorri, Jean-Baptiste Melin, Aniello Mennella, Arturo Mignano, Marc-Antoine

Miville-Deschˆenes, Alessandro Monfardini, Anthony Murphy, Pavel Naselsky, Fed- erico Nati, Paolo Natoli, Mattia Negrello, Fabio Noviello, Cr´eidhe O’Sullivan,

Francesco Paci, Luca Pagano, Rosita Paladino, Nathalie Palanque-Delabrouille,

Daniela Paoletti, Hiranya Peiris, Francesca Perrotta, Francesco Piacentini, Michel

Piat, Lucio Piccirillo, Giampaolo Pisano, Gianluca Polenta, Agnieszka Pollo, Nico- las Ponthieu, Mathieu Remazeilles, Sara Ricciardi, Matthieu Roman, Cyrille Ros- set, Jose-Alberto Rubino-Martin, Maria Salatino, Alessandro Schillaci, Paul Shel- lard, Joseph Silk, Alexei Starobinsky, Radek Stompor, Rashid Sunyaev, Andrea

Tartari, Luca Terenzi, Luigi Toﬀolatti, Maurizio Tomasi, Neil Trappe, Matthieu

Tristram, Tiziana Trombetti, Marco Tucci, Rien Van de Weijgaert, Bartjan Van

Tent, Licia Verde, Patricio Vielva, Ben Wand elt, Robert Watson, and Staﬀord

Withington. PRISM (Polarized Radiation Imaging and Spectroscopy Mission): an extended white paper. J. Cosmo. Astropart. Phys., 2014(2):006, Feb 2014. doi:

178 10.1088/1475-7516/2014/02/006.

G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo,

and F. S. Queiroz. The Waning of the WIMP? A Review of Models, Searches, and

Constraints. ArXiv e-prints, March 2017.

A. Arinyo-i-Prats, J. Miralda-Escud´e,M. Viel, and R. Cen. The non-linear power

spectrum of the Lyman alpha forest. J. Cosmo. Astropart. Phys., 12:017, December

2015. doi: 10.1088/1475-7516/2015/12/017.

Paolo Aschieri, B Jurˇco, Peter Schupp, and Julius Wess. Noncommutative guts,

standard model and c, p, t. Nuclear Physics B, 651(1-2):45–70, 2003.

R. Barkana and A. Loeb. A Method for Separating the Physics from the Astrophysics

of High-Redshift 21 Centimeter Fluctuations. ApJL, 624:L65–L68, May 2005. doi:

10.1086/430599.

A. Barnacka, J.-F. Glicenstein, and R. Moderski. New constraints on primordial

black holes abundance from femtolensing of gamma-ray bursts. Phys. Rev. D, 86

(4):043001, August 2012. doi: 10.1103/PhysRevD.86.043001.

L. Baudis. Dark matter detection. Journal of Physics G Nuclear Physics, 43(4):

044001, August 2016. doi: 10.1088/0954-3899/43/4/044001.

J. E. Bautista, N. G. Busca, J. Guy, J. Rich, M. Blomqvist, H. du Mas des Bour-

boux, M. M. Pieri, A. Font-Ribera, S. Bailey, T. Delubac, D. Kirkby, J.-M. Le

Goﬀ, D. Margala, A. Slosar, J. A. Vazquez, J. R. Brownstein, K. S. Dawson, D. J.

Eisenstein, J. Miralda-Escud´e, P. Noterdaeme, N. Palanque-Delabrouille, I. Pˆaris,

179 P. Petitjean, N. P. Ross, D. P. Schneider, D. H. Weinberg, and C. Y`eche. Mea-

surement of baryon acoustic oscillation correlations at z = 2.3 with SDSS DR12

Lyα-Forests. A&A, 603:A12, June 2017. doi: 10.1051/0004-6361/201730533.

J. Bechtold. The Lyman-Alpha forest near 34 quasi-stellar objects with Z greater

than 2.6. Astrophys. J. Supp., 91:1–78, March 1994. doi: 10.1086/191937.

George D. Becker, James S. Bolton, Martin G. Haehnelt, and Wallace L. W. Sargent.

Detection of extended He II reionization in the temperature evolution of the inter-

galactic medium. Mon. Not. R. Astron. Soc., 410:1096–1112, January 2011. doi:

10.1111/j.1365-2966.2010.17507.x.

Thomas Beckert and Heino Falcke. Circular polarization of radio emission from rela-

tivistic jets. Astronomy & Astrophysics, 388(3):1106–1119, 2002.

D. Benson. Computational methods in Lagrangian and Eulerian hydrocodes. Com-

puter Methods in Applied Mechanics and Engineering, 99(2-3):235–394, Sep 1992.

doi: 10.1016/0045-7825(92)90042-I.

G. Bertone, D. Hooper, and J. Silk. Particle dark matter: evidence, candidates and

constraints. Phys. Rep., 405:279–390, January 2005. doi: 10.1016/j.physrep.2004.

08.031.

S. Bird, I. Cholis, J. B. Mu˜noz, Y. Ali-Ha¨ımoud, M. Kamionkowski, E. D. Kovetz,

A. Raccanelli, and A. G. Riess. Did LIGO Detect Dark Matter? Physical Review

Letters, 116(20):201301, May 2016. doi: 10.1103/PhysRevLett.116.201301.

180 Diego Blas, Julien Lesgourgues, and Thomas Tram. The cosmic linear anisotropy

solving system (class). part ii: approximation schemes. Journal of Cosmology and

Astroparticle Physics, 2011(07):034, 2011.

M. Blomqvist, D. Kirkby, J. E. Bautista, A. Arinyo-i-Prats, N. G. Busca, J. Miralda-

Escud´e,A. Slosar, A. Font-Ribera, D. Margala, D. P. Schneider, and J. A. Vazquez.

Broadband distortion modeling in Lyman-α forest BAO ﬁtting. J. Cosmo. As-

tropart. Phys., 11:034, November 2015. doi: 10.1088/1475-7516/2015/11/034.

H. Bondi and F. Hoyle. On the mechanism of accretion by stars. Mon. Not. R.

Astron. Soc., 104:273, 1944. doi: 10.1093/mnras/104.5.273.

R. J. Bouwens, G. D. Illingworth, P. A. Oesch, J. Caruana, B. Holwerda, R. Smit, and

S. Wilkins. Reionization After Planck: The Derived Growth of the Cosmic Ionizing

Emissivity Now Matches the Growth of the Galaxy UV Luminosity Density. ApJ,

811:140, October 2015. doi: 10.1088/0004-637X/811/2/140.

J. D. Bowman, A. E. E. Rogers, R. A. Monsalve, T. J. Mozdzen, and N. Mahesh. An

absorption proﬁle centred at 78 megahertz in the sky-averaged spectrum. Nature,

555:67–70, March 2018. doi: 10.1038/nature25792.

T. D. Brandt. Constraints on MACHO Dark Matter from Compact Stellar Systems

in Ultra-faint Dwarf Galaxies. ApJL, 824:L31, June 2016. doi: 10.3847/2041-8205/

824/2/L31.

A. Brunthaler, G. C. Bower, H. Falcke, and R. R. Mellon. Detection of circular

polarization in m81*. The Astrophysical Journal Letters, 560(2):L123, 2001.

181 M. R. Buckley and A. H. G. Peter. Gravitational probes of dark matter physics.

ArXiv e-prints, December 2017.

Matthew R. Buckley and Annika H. G. Peter. Gravitational probes of dark matter

physics. Phys. Rep., 761:1–60, Oct 2018. doi: 10.1016/j.physrep.2018.07.003.

F. Capela, M. Pshirkov, and P. Tinyakov. Constraints on primordial black holes as

dark matter candidates from star formation. Phys. Rev. D, 87(2):023507, January

2013a. doi: 10.1103/PhysRevD.87.023507.

F. Capela, M. Pshirkov, and P. Tinyakov. Constraints on primordial black holes

as dark matter candidates from capture by neutron stars. Phys. Rev. D, 87(12):

123524, June 2013b. doi: 10.1103/PhysRevD.87.123524.

F. Capela, M. Pshirkov, and P. Tinyakov. Adiabatic contraction revisited: Implica-

tions for primordial black holes. Phys. Rev. D, 90(8):083507, October 2014. doi:

10.1103/PhysRevD.90.083507.

B. J. Carr. The primordial black hole mass spectrum. ApJ, 201:1–19, October 1975.

doi: 10.1086/153853.

B. J. Carr. Primordial Black Holes: Do They Exist and Are They Useful? ArXiv

Astrophysics e-prints, November 2005.

B. J. Carr and S. W. Hawking. Black holes in the early Universe. Mon. Not. R.

Astron. Soc., 168:399–416, August 1974. doi: 10.1093/mnras/168.2.399.

B. J. Carr, Kazunori Kohri, Yuuiti Sendouda, and Jun’Ichi Yokoyama. New cosmo-

logical constraints on primordial black holes. Phys. Rev. D, 81(10):104019, May

2010. doi: 10.1103/PhysRevD.81.104019.

182 Bernard Carr, Martti Raidal, Tommi Tenkanen, Ville Vaskonen, and Hardi Veerm¨ae.

Primordial black hole constraints for extended mass functions. Phys. Rev. D, 96

(2):023514, Jul 2017. doi: 10.1103/PhysRevD.96.023514.

G. R. Caughlan and W. A. Fowler. Thermonuclear Reaction Rates V. Atomic Data

and Nuclear Data Tables, 40:283, 1988. doi: 10.1016/0092-640X(88)90009-5.

R. Cen, J. Miralda-Escud´e,J. P. Ostriker, and M. Rauch. Gravitational collapse of

small-scale structure as the origin of the Lyman-alpha forest. ApJL, 437:L9–L12,

December 1994. doi: 10.1086/187670.

R. Cen, P. McDonald, H. Trac, and A. Loeb. Probing the Epoch of Reionization

with the Lyα Forest at z ˜ 4-5. ApJL, 706:L164–L167, November 2009. doi:

10.1088/0004-637X/706/1/L164.

Sol`ene Chabanier, Nathalie Palanque-Delabrouille, Christophe Y`eche, Jean-Marc Le

Goﬀ, Eric Armengaud, Julian Bautista, Michael Blomqvist, Nicolas Busca, Kyle

Dawson, Thomas Etourneau, Andreu Font-Ribera, Youngbae Lee, H´elion du Mas

des Bourboux, Matthew Pieri, James Rich, Graziano Rossi, Donald Schneider,

and AnˇzeSlosar. The one-dimensional power spectrum from the SDSS DR14 Lyα

forests. arXiv e-prints, art. arXiv:1812.03554, December 2018.

G. F. Chapline. Cosmological eﬀects of primordial black holes. Nature, 253:251,

January 1975. doi: 10.1038/253251a0.

Jens Chluba, Adrienne L Erickcek, and Ido Ben-Dayan. Probing the inﬂaton: Small-

scale power spectrum constraints from measurements of the cosmic microwave back-

ground energy spectrum. The Astrophysical Journal, 758(2):76, 2012.

183 T. R. Choudhury, E. Puchwein, M. G. Haehnelt, and J. S. Bolton. Lyman α emitters

gone missing: evidence for late reionization? Mon. Not. R. Astron. Soc., 452:

261–277, September 2015. doi: 10.1093/mnras/stv1250.

S. Clesse and J. Garc´ıa-Bellido. The clustering of massive Primordial Black Holes as

Dark Matter: Measuring their mass distribution with advanced LIGO. Physics of

the Dark Universe, 15:142–147, March 2017. doi: 10.1016/j.dark.2016.10.002.

Don Colladay and V Alan Kosteleck`y. Lorentz-violating extension of the standard

model. Physical Review D, 58(11):116002, 1998.

Michele Compostella, Sebastiano Cantalupo, and Cristiano Porciani. The imprint of

inhomogeneous He II reionization on the H I and He II Lyα forest. Mon. Not. R.

Astron. Soc., 435:3169–3190, November 2013. doi: 10.1093/mnras/stt1510.

Asantha Cooray, Alessandro Melchiorri, and Joseph Silk. Is the cosmic microwave

background circularly polarized? Physics Letters B, 554(1):1–6, 2003.

R. A. C. Croft, D. H. Weinberg, N. Katz, and L. Hernquist. Recovery of the Power

Spectrum of Mass Fluctuations from Observations of the Lyα Forest. ApJ, 495:

44–62, March 1998. doi: 10.1086/305289.

R. A. C. Croft, D. H. Weinberg, M. Pettini, L. Hernquist, and N. Katz. The Power

Spectrum of Mass Fluctuations Measured from the Lyα Forest at Redshift z = 2.5.

ApJ, 520:1–23, July 1999. doi: 10.1086/307438.

R. A. C. Croft, D. H. Weinberg, M. Bolte, S. Burles, L. Hernquist, N. Katz, D. Kirk-

man, and D. Tytler. Toward a Precise Measurement of Matter Clustering: Lyα

Forest Data at Redshifts 2-4. ApJ, 581:20–52, December 2002. doi: 10.1086/344099.

184 R. A. C. Croft, A. J. Banday, and L. Hernquist. Lyman α forest-CMB cross-correlation

and the search for the ionized baryons at high redshift. Mon. Not. R. Astron. Soc.,

369:1090–1102, July 2006. doi: 10.1111/j.1365-2966.2006.10292.x.

A. D’Aloisio, M. McQuinn, and H. Trac. Large Opacity Variations in the High-

redshift Lyα Forest: The Signature of Relic Temperature Fluctuations from Patchy

Reionization. ApJL, 813:L38, November 2015. doi: 10.1088/2041-8205/813/2/L38.

A. D’Aloisio, M. McQuinn, O. Maupin, F. B. Davies, H. Trac, S. Fuller, and P. R.

Upton Sanderbeck. Heating of the Intergalactic Medium by Hydrogen Reionization.

ArXiv e-prints, July 2018.

S. Das, T. Louis, M. R. Nolta, G. E. Addison, E. S. Battistelli, J. R. Bond, E. Cal-

abrese, D. Crichton, M. J. Devlin, S. Dicker, J. Dunkley, R. Dunner,¨ J. W. Fowler,

M. Gralla, A. Hajian, M. Halpern, M. Hasselﬁeld, M. Hilton, A. D. Hincks,

R. Hlozek, K. M. Huﬀenberger, J. P. Hughes, K. D. Irwin, A. Kosowsky, R. H.

Lupton, T. A. Marriage, D. Marsden, F. Menanteau, K. Moodley, M. D. Niemack,

L. A. Page, B. Partridge, E. D. Reese, B. L. Schmitt, N. Sehgal, B. D. Sherwin,

J. L. Sievers, D. N. Spergel, S. T. Staggs, D. S. Swetz, E. R. Switzer, R. Thornton,

H. Trac, and E. Wollack. The Atacama Cosmology Telescope: temperature and

gravitational lensing power spectrum measurements from three seasons of data. J.

Cosmo. Astropart. Phys., 4:014, April 2014. doi: 10.1088/1475-7516/2014/04/014.

M. Davis, G. Efstathiou, C. S. Frenk, and S. D. M. White. The evolution of large-

scale structure in a universe dominated by cold dark matter. ApJ, 292:371–394,

May 1985. doi: 10.1086/163168.

185 K. S. Dawson, D. J. Schlegel, C. P. Ahn, S. F. Anderson, E.´ Aubourg, S. Bailey, R. H.

Barkhouser, J. E. Bautista, A. Beiﬁori, A. A. Berlind, V. Bhardwaj, D. Bizyaev,

C. H. Blake, M. R. Blanton, M. Blomqvist, A. S. Bolton, A. Borde, J. Bovy, W. N.

Brandt, H. Brewington, J. Brinkmann, P. J. Brown, J. R. Brownstein, K. Bundy,

N. G. Busca, W. Carithers, A. R. Carnero, M. A. Carr, Y. Chen, J. Comparat,

N. Connolly, F. Cope, R. A. C. Croft, A. J. Cuesta, L. N. da Costa, J. R. A.

Davenport, T. Delubac, R. de Putter, S. Dhital, A. Ealet, G. L. Ebelke, D. J.

Eisenstein, S. Escoﬃer, X. Fan, N. Filiz Ak, H. Finley, A. Font-Ribera, R. G´enova-

Santos, J. E. Gunn, H. Guo, D. Haggard, P. B. Hall, J.-C. Hamilton, B. Harris,

D. W. Harris, S. Ho, D. W. Hogg, D. Holder, K. Honscheid, J. Huehnerhoﬀ, B. Jor-

dan, W. P. Jordan, G. Kauﬀmann, E. A. Kazin, D. Kirkby, M. A. Klaene, J.-P.

Kneib, J.-M. Le Goﬀ, K.-G. Lee, D. C. Long, C. P. Loomis, B. Lundgren, R. H.

Lupton, M. A. G. Maia, M. Makler, E. Malanushenko, V. Malanushenko, R. Man-

delbaum, M. Manera, C. Maraston, D. Margala, K. L. Masters, C. K. McBride,

P. McDonald, I. D. McGreer, R. G. McMahon, O. Mena, J. Miralda-Escud´e,A. D.

Montero-Dorta, F. Montesano, D. Muna, A. D. Myers, T. Naugle, R. C. Nichol,

P. Noterdaeme, S. E. Nuza, M. D. Olmstead, A. Oravetz, D. J. Oravetz, R. Owen,

N. Padmanabhan, N. Palanque-Delabrouille, K. Pan, J. K. Parejko, I. Pˆaris, W. J.

Percival, I. Pérez-Fournon, I. Pérez-Ràfols, P. Petitjean, R. Pfaffenberger, J. Pforr,

M. M. Pieri, F. Prada, A. M. Price-Whelan, M. J. Raddick, R. Rebolo, J. Rich,

G. T. Richards, C. M. Rockosi, N. A. Roe, A. J. Ross, N. P. Ross, G. Rossi, J. A.

Rubi˜no-Martin, L. Samushia, A. G. S´anchez, C. Sayres, S. J. Schmidt, D. P. Schnei-

der, C. G. Sc´occola, H.-J. Seo, A. Shelden, E. Sheldon, Y. Shen, Y. Shu, A. Slosar,

186 S. A. Smee, S. A. Snedden, F. Stauﬀer, O. Steele, M. A. Strauss, A. Streblyan-

ska, N. Suzuki, M. E. C. Swanson, T. Tal, M. Tanaka, D. Thomas, J. L. Tinker,

R. Tojeiro, C. A. Tremonti, M. Vargas Maga˜na, L. Verde, M. Viel, D. A. Wake,

M. Watson, B. A. Weaver, D. H. Weinberg, B. J. Weiner, A. A. West, M. White,

W. M. Wood-Vasey, C. Yeche, I. Zehavi, G.-B. Zhao, and Z. Zheng. The Baryon

Oscillation Spectroscopic Survey of SDSS-III. AJ, 145:10, January 2013. doi:

10.1088/0004-6256/145/1/10.

Pratika Dayal and Andrea Ferrara. Early galaxy formation and its large-scale eﬀects.

Phys. Rep., 780:1–64, Dec 2018. doi: 10.1016/j.physrep.2018.10.002.

Soma De and Hiroyuki Tashiro. Circular polarization of the cmb: A probe of the ﬁrst

stars. Physical Review D, 92(12):123506, 2015.

S Deguchi and WD Watson. Circular polarization of astrophysical masers due to

overlap of zeeman components. The Astrophysical Journal, 300:L15–L18, 1986.

T. Delubac, J. E. Bautista, N. G. Busca, J. Rich, D. Kirkby, S. Bailey, A. Font-Ribera,

A. Slosar, K.-G. Lee, M. M. Pieri, J.-C. Hamilton, E.´ Aubourg, M. Blomqvist,

J. Bovy, J. Brinkmann, W. Carithers, K. S. Dawson, D. J. Eisenstein, S. G. A.

Gontcho, J.-P. Kneib, J.-M. Le Goﬀ, D. Margala, J. Miralda-Escud´e,A. D. My-

ers, R. C. Nichol, P. Noterdaeme, R. O’Connell, M. D. Olmstead, N. Palanque-

Delabrouille, I. Pˆaris, P. Petitjean, N. P. Ross, G. Rossi, D. J. Schlegel, D. P.

Schneider, D. H. Weinberg, C. Y`eche, and D. G. York. Baryon acoustic oscillations

in the Lyα forest of BOSS DR11 quasars. A&A, 574:A59, February 2015. doi:

10.1051/0004-6361/201423969.

187 DES Collaboration, T. M. C. Abbott, F. B. Abdalla, A. Alarcon, J. Aleksi´c,S. Al-

lam, S. Allen, A. Amara, J. Annis, J. Asorey, S. Avila, D. Bacon, E. Balbinot,

M. Banerji, N. Banik, W. Barkhouse, M. Baumer, E. Baxter, K. Bechtol, M. R.

Becker, A. Benoit-L´evy, B. A. Benson, G. M. Bernstein, E. Bertin, J. Blazek,

S. L. Bridle, D. Brooks, D. Brout, E. Buckley-Geer, D. L. Burke, M. T. Busha,

D. Capozzi, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, F. J. Castander,

R. Cawthon, C. Chang, N. Chen, M. Childress, A. Choi, C. Conselice, R. Crit-

tenden, M. Crocce, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, R. Das, T. M.

Davis, C. Davis, J. De Vicente, D. L. DePoy, J. DeRose, S. Desai, H. T. Diehl,

J. P. Dietrich, S. Dodelson, P. Doel, A. Drlica-Wagner, T. F. Eiﬂer, A. E. El-

liott, F. Elsner, J. Elvin-Poole, J. Estrada, A. E. Evrard, Y. Fang, E. Fernandez,

A. Fert´e, D. A. Finley, B. Flaugher, P. Fosalba, O. Friedrich, J. Frieman, J. Garc´ıa-

Bellido, M. Garcia-Fernandez, M. Gatti, E. Gaztanaga, D. W. Gerdes, T. Gian-

nantonio, M. S. S. Gill, K. Glazebrook, D. A. Goldstein, D. Gruen, R. A. Gruendl,

J. Gschwend, G. Gutierrez, S. Hamilton, W. G. Hartley, S. R. Hinton, K. Hon-

scheid, B. Hoyle, D. Huterer, B. Jain, D. J. James, M. Jarvis, T. Jeltema, M. D.

Johnson, M. W. G. Johnson, T. Kacprzak, S. Kent, A. G. Kim, A. King, D. Kirk,

N. Kokron, A. Kovacs, E. Krause, C. Krawiec, A. Kremin, K. Kuehn, S. Kuhlmann,

N. Kuropatkin, F. Lacasa, O. Lahav, T. S. Li, A. R. Liddle, C. Lidman, M. Lima,

H. Lin, N. MacCrann, M. A. G. Maia, M. Makler, M. Manera, M. March, J. L.

Marshall, P. Martini, R. G. McMahon, P. Melchior, F. Menanteau, R. Miquel,

V. Miranda, D. Mudd, J. Muir, A. M¨oller, E. Neilsen, R. C. Nichol, B. Nord,

P. Nugent, R. L. C. Ogando, A. Palmese, J. Peacock, H. V. Peiris, J. Peoples,

W. J. Percival, D. Petravick, A. A. Plazas, A. Porredon, J. Prat, A. Pujol, M. M.

188 Rau, A. Refregier, P. M. Ricker, N. Roe, R. P. Rollins, A. K. Romer, A. Roodman,

R. Rosenfeld, A. J. Ross, E. Rozo, E. S. Rykoﬀ, M. Sako, A. I. Salvador, S. Samuroﬀ,

C. S´anchez, E. Sanchez, B. Santiago, V. Scarpine, R. Schindler, D. Scolnic, L. F.

Secco, S. Serrano, I. Sevilla-Noarbe, E. Sheldon, R. C. Smith, M. Smith, J. Smith,

M. Soares-Santos, F. Sobreira, E. Suchyta, G. Tarle, D. Thomas, M. A. Troxel,

D. L. Tucker, B. E. Tucker, S. A. Uddin, T. N. Varga, P. Vielzeuf, V. Vikram,

A. K. Vivas, A. R. Walker, M. Wang, R. H. Wechsler, J. Weller, W. Wester, R. C.

Wolf, B. Yanny, F. Yuan, A. Zenteno, B. Zhang, Y. Zhang, and J. Zuntz. Dark

Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering

and Weak Lensing. ArXiv e-prints, August 2017.

DESI Collaboration, Amir Aghamousa, Jessica Aguilar, Steve Ahlen, Shadab Alam,

Lori E. Allen, Carlos Allende Prieto, James Annis, Stephen Bailey, Christophe Bal-

land, Otger Ballester, Charles Baltay, Lucas Beaufore, Chris Bebek, Timothy C.

Beers, Eric F. Bell, Jos´eLuis Bernal, Robert Besuner, Florian Beutler, Chris Blake,

Hannes Bleuler, Michael Blomqvist, Robert Blum, Adam S. Bolton, Cesar Briceno,

David Brooks, Joel R. Brownstein, Elizabeth Buckley-Geer, Angela Burden, Eti-

enne Burtin, Nicolas G. Busca, Robert N. Cahn, Yan-Chuan Cai, Laia Cardiel-

Sas, Raymond G. Carlberg, Pierre-Henri Carton, Ricard Casas, Francisco J. Ca-

stand er, Jorge L. Cervantes-Cota, Todd M. Claybaugh, Madeline Close, Carl T.

Coker, Shaun Cole, Johan Comparat, Andrew P. Cooper, M. C. Cousinou, Mar-

tin Crocce, Jean-Gabriel Cuby, Daniel P. Cunningham, Tamara M. Davis, Kyle S.

Dawson, Axel de la Macorra, Juan De Vicente, Timoth´ee Delubac, Mark Der-

went, Arjun Dey, Govinda Dhungana, Zhejie Ding, Peter Doel, Yutong T. Duan,

189 Anne Ealet, Jerry Edelstein, Sarah Eftekharzadeh, Daniel J. Eisenstein, Ann El- liott, St´ephanie Escoﬃer, Matthew Evatt, Parker Fagrelius, Xiaohui Fan, Kevin

Fanning, Arya Farahi, Jay Farihi, Ginevra Favole, Yu Feng, Enrique Fernandez,

Joseph R. Findlay, Douglas P. Finkbeiner, Michael J. Fitzpatrick, Brenna Flaugher,

Samuel Flender, Andreu Font-Ribera, Jaime E. Forero-Romero, Pablo Fosalba,

Carlos S. Frenk, Michele Fumagalli, Boris T. Gaensicke, Giuseppe Gallo, Juan

Garcia-Bellido, Enrique Gaztanaga, Nicola Pietro Gentile Fusillo, Terry Gerard,

Irena Gershkovich, Tommaso Giannantonio, Denis Gillet, Guillermo Gonzalez-de-

Rivera, Violeta Gonzalez-Perez, Shelby Gott, Or Graur, Gaston Gutierrez, Julien

Guy, Salman Habib, Henry Heetderks, Ian Heetderks, Katrin Heitmann, Woj- ciech A. Hellwing, David A. Herrera, Shirley Ho, Stephen Holland, Klaus Hon- scheid, Eric Huﬀ, Timothy A. Hutchinson, Dragan Huterer, Ho Seong Hwang,

Joseph Maria Illa Laguna, Yuzo Ishikawa, Dianna Jacobs, Niall Jeﬀrey, Patrick

Jelinsky, Elise Jennings, Linhua Jiang, Jorge Jimenez, Jennifer Johnson, Richard

Joyce, Eric Jullo, St´ephanie Juneau, Sami Kama, Armin Karcher, Sonia Karkar,

Robert Kehoe, Noble Kennamer, Stephen Kent, Martin Kilbinger, Alex G. Kim,

David Kirkby, Theodore Kisner, Ellie Kitanidis, Jean-Paul Kneib, Sergey Ko- posov, Eve Kovacs, Kazuya Koyama, Anthony Kremin, Richard Kron, Luzius

Kronig, Andrea Kueter-Young, Cedric G. Lacey, Robin Lafever, Ofer Lahav, An- drew Lambert, Michael Lampton, Martin Land riau, Dustin Lang, Tod R. Lauer,

Jean-Marc Le Goﬀ, Laurent Le Guillou, Auguste Le Van Suu, Jae Hyeon Lee, Su-

Jeong Lee, Daniela Leitner, Michael Lesser, Michael E. Levi, Benjamin L’Huillier,

Baojiu Li, Ming Liang, Huan Lin, Eric Linder, Sarah R. Loebman, Zarija Luki´c,

Jun Ma, Niall MacCrann, Christophe Magneville, Laleh Makarem, Marc Manera,

190 Christopher J. Manser, Robert Marshall, Paul Martini, Richard Massey, Thomas

Matheson, Jeremy McCauley, Patrick McDonald, Ian D. McGreer, Aaron Meisner,

Nigel Metcalfe, Timothy N. Miller, Ramon Miquel, John Moustakas, Adam Myers,

Milind Naik, Jeﬀrey A. Newman, Robert C. Nichol, Andrina Nicola, Luiz Nicolati da Costa, Jundan Nie, Gustavo Niz, Peder Norberg, Brian Nord, Dara Norman,

Peter Nugent, Thomas O’Brien, Minji Oh, Knut A. G. Olsen, Cristobal Padilla,

Hamsa Padmanabhan, Nikhil Padmanabhan, Nathalie Palanque-Delabrouille, An- tonella Palmese, Daniel Pappalardo, Isabelle Pˆaris, Changbom Park, Anna Patej,

John A. Peacock, Hiranya V. Peiris, Xiyan Peng, Will J. Percival, Sandrine Per- ruchot, Matthew M. Pieri, Richard Pogge, Jennifer E. Pollack, Claire Poppett,

Francisco Prada, Abhishek Prakash, Ronald G. Probst, David Rabinowitz, Anand

Raichoor, Chang Hee Ree, Alexandre Refregier, Xavier Regal, Beth Reid, Kevin

Reil, Mehdi Rezaie, Constance M. Rockosi, Natalie Roe, Samuel Ronayette, Aaron

Roodman, Ashley J. Ross, Nicholas P. Ross, Graziano Rossi, Eduardo Rozo, Van- ina Ruhlmann-Kleider, Eli S. Rykoﬀ, Cristiano Sabiu, Lado Samushia, Eusebio

Sanchez, Javier Sanchez, David J. Schlegel, Michael Schneider, Michael Schubnell,

Aur´elia Secroun, Uros Seljak, Hee-Jong Seo, Santiago Serrano, Arman Shaﬁeloo,

Huanyuan Shan, Ray Sharples, Michael J. Sholl, William V. Shourt, Joseph H.

Silber, David R. Silva, Martin M. Sirk, Anze Slosar, Alex Smith, George F. Smoot,

Debopam Som, Yong-Seon Song, David Sprayberry, Ryan Staten, Andy Stefanik,

Gregory Tarle, Suk Sien Tie, Jeremy L. Tinker, Rita Tojeiro, Francisco Valdes,

Octavio Valenzuela, Monica Valluri, Mariana Vargas-Magana, Licia Verde, Alis- tair R. Walker, Jiali Wang, Yuting Wang, Benjamin A. Weaver, Curtis Weaverdyck,

Risa H. Wechsler, David H. Weinberg, Martin White, Qian Yang, Christophe Yeche,

191 Tianmeng Zhang, Gong-Bo Zhao, Yi Zheng, Xu Zhou, Zhimin Zhou, Yaling Zhu,

Hu Zou, and Ying Zu. The DESI Experiment Part I: Science,Targeting, and Survey

Design. ArXiv e-prints, October 2016.

R. H. Dicke, P. J. E. Peebles, P. G. Roll, and D. T. Wilkinson. Cosmic Black-Body

Radiation. ApJ, 142:414–419, July 1965. doi: 10.1086/148306.

S. Dodelson. Modern cosmology. 2003.

B. T. Draine and A. Lazarian. Magnetic Dipole Microwave Emission from Dust

Grains. Astrophys. J., 512:740–754, February 1999. doi: 10.1086/306809.

Tevian Dray. The relationship between monopole harmonics and spin-weighted spher-

ical harmonics. Journal of Mathematical Physics, 26(5):1030–1033, May 1985. doi:

10.1063/1.526533.

H. du Mas des Bourboux, J.-M. Le Goﬀ, M. Blomqvist, N. G. Busca, J. Guy, J. Rich,

C. Y`eche, J. E. Bautista, E.´ Burtin, K. S. Dawson, D. J. Eisenstein, A. Font-

Ribera, D. Kirkby, J. Miralda-Escud´e, P. Noterdaeme, N. Palanque-Delabrouille,

I. Pâris, P. Petitjean, I. Pérez-Ràfols, M. M. Pieri, N. P. Ross, D. J. Schlegel, D. P.

Schneider, A. Slosar, D. H. Weinberg, and P. Zarrouk. Baryon acoustic oscillations

from the complete SDSS-III Lyα-quasar cross-correlation function at z = 2.4. A&A,

608:A130, December 2017. doi: 10.1051/0004-6361/201731731.

Ruth Durrer. The cosmic microwave background, pages 222–224. Cambridge Univer-

sity Press Cambridge, 2008.

D. J. Eisenstein, D. H. Weinberg, E. Agol, H. Aihara, C. Allende Prieto, S. F. Ander-

son, J. A. Arns, E.´ Aubourg, S. Bailey, E. Balbinot, and et al. SDSS-III: Massive

192 Spectroscopic Surveys of the Distant Universe, the Milky Way, and Extra-Solar

Planetary Systems. AJ, 142:72, September 2011. doi: 10.1088/0004-6256/142/3/72.

Damian Ejlli. Magneto-optic eﬀects of the cosmic microwave background. arXiv

preprint arXiv:1607.02094, 2016.

Y. N. Eroshenko. Gravitational waves from primordial black holes collisions in binary

systems. ArXiv e-prints, April 2016.

F. W. J. Olver et al. (2018). NIST Digital Library of Mathematical Functions.

http://dlmf.nist.gov/, Release 1.0.18 of 2018-03-27. URL http://dlmf.nist.gov/.

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert,

C. W. Clark, B. R. Miller and B. V. Saunders, eds.

X. Fan, V. K. Narayanan, M. A. Strauss, R. L. White, R. H. Becker, L. Pentericci, and

H.-W. Rix. Evolution of the Ionizing Background and the Epoch of Reionization

from the Spectra of z˜6 Quasars. AJ, 123:1247–1257, March 2002. doi: 10.1086/

339030.

A. Font-Ribera, D. Kirkby, N. Busca, J. Miralda-Escud´e,N. P. Ross, A. Slosar,

J. Rich, E.´ Aubourg, S. Bailey, V. Bhardwaj, J. Bautista, F. Beutler, D. Bizyaev,

M. Blomqvist, H. Brewington, J. Brinkmann, J. R. Brownstein, B. Carithers, K. S.

Dawson, T. Delubac, G. Ebelke, D. J. Eisenstein, J. Ge, K. Kinemuchi, K.-G. Lee,

V. Malanushenko, E. Malanushenko, M. Marchante, D. Margala, D. Muna, A. D.

Myers, P. Noterdaeme, D. Oravetz, N. Palanque-Delabrouille, I. Pˆaris, P. Petitjean,

M. M. Pieri, G. Rossi, D. P. Schneider, A. Simmons, M. Viel, C. Yeche, and D. G.

York. Quasar-Lyman α forest cross-correlation from BOSS DR11: Baryon Acoustic

193 Oscillations. J. Cosmo. Astropart. Phys., 5:027, May 2014. doi: 10.1088/1475-7516/

2014/05/027.

A. Font-Ribera, P. McDonald, and A. Slosar. How to estimate the 3D power spectrum

of the Lyman-α forest. J. Cosmo. Astropart. Phys., 1:003, January 2018. doi:

10.1088/1475-7516/2018/01/003.

Andreu Font-Ribera, Jordi Miralda-Escud´e,Eduard Arnau, Bill Carithers, Khee-

Gan Lee, Pasquier Noterdaeme, Isabelle Pˆaris, Patrick Petitjean, James Rich, Em-

manuel Rollinde, Nicholas P. Ross, Donald P. Schneider, Martin White, and Don-

ald G. York. The large-scale cross-correlation of Damped Lyman alpha systems with

the Lyman alpha forest: ﬁrst measurements from BOSS. Journal of Cosmology and

Astro-Particle Physics, 2012:059, Nov 2012. doi: 10.1088/1475-7516/2012/11/059.

A. Friedmann. Uber¨ die M¨oglichkeit einer Welt mit konstanter negativer Krum-¨

mung des Raumes. Zeitschrift fur Physik, 21(1):326–332, Dec 1924. doi: 10.1007/

BF01328280.

L. Fu, M. Kilbinger, T. Erben, C. Heymans, H. Hildebrandt, H. Hoekstra, T. D. Kitch-

ing, Y. Mellier, L. Miller, E. Semboloni, P. Simon, L. Van Waerbeke, J. Coupon,

J. Harnois-D´eraps, M. J. Hudson, K. Kuijken, B. Rowe, T. Schrabback, S. Vafaei,

and M. Velander. CFHTLenS: cosmological constraints from a combination of cos-

mic shear two-point and three-point correlations. Mon. Not. R. Astron. Soc., 441:

2725–2743, July 2014. doi: 10.1093/mnras/stu754.

S. R. Furlanetto and S. P. Oh. The Temperature-Density Relation of the Intergalactic

Medium after Hydrogen Reionization. ApJ, 701:94–104, August 2009. doi: 10.1088/

194 0004-637X/701/1/94.

Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Patrick

Alken, Michael Booth, and Fabrice Rossi. Gnu scientiﬁc library. Network Theory

Ltd, 3, 2002.

J.-O. Gong and N. Kitajima. Small-scale structure and 21cm ﬂuctuations by pri-

mordial black holes. J. Cosmo. Astropart. Phys., 8:017, August 2017. doi:

10.1088/1475-7516/2017/08/017.

S. Gontcho A Gontcho, J. Miralda-Escud´e,and N. G. Busca. On the eﬀect of the

ionizing background on the Lyα forest autocorrelation function. Mon. Not. R.

Astron. Soc., 442:187–195, July 2014. doi: 10.1093/mnras/stu860.

A. Gould. Femtolensing of gamma-ray bursters. ApJL, 386:L5–L7, February 1992.

doi: 10.1086/186279.

P. W. Graham, S. Rajendran, and J. Varela. Dark matter triggers of supernovae.

Phys. Rev. D, 92(6):063007, September 2015. doi: 10.1103/PhysRevD.92.063007.

Anne M. Green. Microlensing and dynamical constraints on primordial black hole

dark matter with an extended mass function. Phys. Rev. D, 94(6):063530, Sep

2016. doi: 10.1103/PhysRevD.94.063530.

B. Greig, J. S. Bolton, and J. S. B. Wyithe. The impact of temperature ﬂuctuations

on the large-scale clustering of the Lyα forest. Mon. Not. R. Astron. Soc., 447:

2503–2511, March 2015. doi: 10.1093/mnras/stu2624.

195 K. Griest, A. M. Cieplak, and M. J. Lehner. Experimental Limits on Primordial Black

Hole Dark Matter from the First 2 yr of Kepler Data. ApJ, 786:158, May 2014.

doi: 10.1088/0004-637X/786/2/158.

Alan H. Guth. Inﬂationary universe: A possible solution to the horizon and ﬂatness

problems. Phys. Rev. D, 23:347–356, Jan 1981. doi: 10.1103/PhysRevD.23.347.

URL https://link.aps.org/doi/10.1103/PhysRevD.23.347.

S. Hannestad, S. H. Hansen, F. L. Villante, and A. J. S. Hamilton. Constraints on

inﬂation from cosmic microwave background and Lyman-/α forest. Astroparticle

Physics, 17:375–382, June 2002. doi: 10.1016/S0927-6505(01)00160-8.

S. Hawking. Gravitationally collapsed objects of very low mass. Mon. Not. R. Astron.

Soc., 152:75, 1971. doi: 10.1093/mnras/152.1.75.

L. Hernquist, N. Katz, D. H. Weinberg, and J. Miralda-Escud´e. The Lyman-Alpha

Forest in the Cold Dark Matter Model. ApJL, 457:L51, February 1996. doi: 10.

1086/309899.

H. Hildebrandt, M. Viola, C. Heymans, S. Joudaki, K. Kuijken, C. Blake, T. Er-

ben, B. Joachimi, D. Klaes, L. Miller, C. B. Morrison, R. Nakajima, G. Verdoes

Kleijn, A. Amon, A. Choi, G. Covone, J. T. A. de Jong, A. Dvornik, I. Fenech

Conti, A. Grado, J. Harnois-D´eraps, R. Herbonnet, H. Hoekstra, F. K¨ohlinger,

J. McFarland, A. Mead, J. Merten, N. Napolitano, J. A. Peacock, M. Radovich,

P. Schneider, P. Simon, E. A. Valentijn, J. L. van den Busch, E. van Uitert, and

L. Van Waerbeke. KiDS-450: cosmological parameter constraints from tomographic

196 weak gravitational lensing. Mon. Not. R. Astron. Soc., 465:1454–1498, February

2017. doi: 10.1093/mnras/stw2805.

R. Hills, G. Kulkarni, P. D. Meerburg, and E. Puchwein. Concerns about Modelling

of Foregrounds and the 21-cm Signal in EDGES data. ArXiv e-prints, May 2018.

C. M. Hirata. Wouthuysen-Field coupling strength and application to high-redshift

21-cm radiation. Mon. Not. R. Astron. Soc., 367:259–274, March 2006.

C. M. Hirata. Small-scale structure and the Lyman-α forest baryon acoustic oscillation

feature. Mon. Not. R. Astron. Soc., 474:2173–2193, February 2018. doi: 10.1093/

mnras/stx2854.

C. M. Hirata, A. Mishra, and T. Venumadhav. Detecting primordial gravitational

waves with circular polarization of the redshifted 21 cm line: I. Formalism. ArXiv

e-prints, July 2017.

Austin Hoag, Maruˇsa Bradaˇc,Kuang-Han Huang, Charlotte Mason, Tommaso Treu,

Kasper B. Schmidt, Michele Trenti, Victoria Strait, Brian C. Lemaux, Emily

Finney, and Maia Paddock. Constraining the neutral fraction of hydrogen in the

IGM at redshift 7.5. arXiv e-prints, art. arXiv:1901.09001, January 2019.

D. C. Homan, M. L. Lister, H. D. Aller, M. F. Aller, and J. F. C. Wardle. Full

polarization spectra of 3c 279. The Astrophysical Journal, 696(1):328, 2009.

Daniel C. Homan, Joanne M. Attridge, and John F. C. Wardle. Parsec-scale circular

polarization observations of 40 blazars. The Astrophysical Journal, 556(1):113,

2001.

197 W. Hu and M. White. Acoustic Signatures in the Cosmic Microwave Background.

ApJ, 471:30, November 1996. doi: 10.1086/177951.

Wayne Hu and Martin White. A CMB polarization primer. New Astronomy, 2(4):

323–344, Oct 1997. doi: 10.1016/S1384-1076(97)00022-5.

E. Hubble. A Relation between Distance and Radial Velocity among Extra-Galactic

Nebulae. Proceedings of the National Academy of Science, 15:168–173, March 1929.

doi: 10.1073/pnas.15.3.168.

L. Hui and Z. Haiman. The Thermal Memory of Reionization History. ApJ, 596:

9–18, October 2003. doi: 10.1086/377229.

L. Hui, N. Y. Gnedin, and Y. Zhang. The Statistics of Density Peaks and the Column

Density Distribution of the Lyα Forest. ApJ, 486:599–622, September 1997. doi:

10.1086/304539.

R. Hunt. A ﬂuid dynamical study of the accretion process. Mon. Not. R. Astron.

Soc., 154:141, 1971. doi: 10.1093/mnras/154.2.141.

Nuclear data section IAEA. Live chart of nuclides. https://www-nds.iaea.org/

relnsd/vcharthtml/VChartHTML.html. Accessed: 05/20/2019.

F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P. D. Serpico. Primordial nucleosyn-

thesis: From precision cosmology to fundamental physics. Phys. Rep., 472:1–76,

March 2009. doi: 10.1016/j.physrep.2009.02.002.

V. Irˇsiˇc, M. Viel, T. A. M. Berg, V. D’Odorico, M. G. Haehnelt, S. Cristiani, G. Cu-

pani, T.-S. Kim, S. L´opez, S. Ellison, G. D. Becker, L. Christensen, K. D. Denney,

198 G. Worseck, and J. S. Bolton. The Lyman α forest power spectrum from the XQ-

100 Legacy Survey. Mon. Not. R. Astron. Soc., 466:4332–4345, April 2017. doi:

10.1093/mnras/stw3372.

N. Itoh, H. Totsuji, and S. Ichimaru. Enhancement of thermonuclear reaction rate

due to strong screening. ApJ, 218:477–483, December 1977. doi: 10.1086/155701.

N. Itoh, H. Totsuji, S. Ichimaru, and H. E. Dewitt. Enhancement of thermonuclear

reaction rate due to strong screening. II - Ionic mixtures. ApJ, 234:1079–1084,

December 1979. doi: 10.1086/157590.

John David Jackson. Classical electrodynamics, 1999.

B. Jancovici. Pair correlation function in a dense plasma and pycnonuclear reactions

in stars. Journal of Statistical Physics, 17:357–370, November 1977. doi: 10.1007/

BF01014403.

Aris Karastergiou, S Johnston, D Mitra, AGJ Van Leeuwen, and RT Edwards. v : | | new insight into the circular polarization of radio pulsars. Monthly Notices of the

Royal Astronomical Society, 344(4):L69–L73, 2003.

Andrey Katz, Joachim Kopp, Sergey Sibiryakov, and Wei Xue. Femtolensing by dark

matter revisited. Journal of Cosmology and Astro-Particle Physics, 2018(12):005,

Dec 2018. doi: 10.1088/1475-7516/2018/12/005.

S. D. Kawaler, I. Novikov, G. Srinivasan, G. Meynet, and D. Schaerer. Stellar Rem-

nants. 1996.

199 L. C. Keating, E. Puchwein, and M. G. Haehnelt. Spatial ﬂuctuations of the inter-

galactic temperature-density relation after hydrogen reionization. Mon. Not. R.

Astron. Soc., 477:5501–5516, July 2018. doi: 10.1093/mnras/sty968.

James C. Kemp, Ramon D. Wolstencroft, and John B. Swedlund. Circular polariza-

tion: Jupiter and other planets. Nature, 232(5307):165–168, 1971.

Soma King and Philip Lubin. Circular polarization of the cmb: Foregrounds and

detection prospects. Physical Review D, 94(2):023501, 2016.

T. D. Kitching, A. F. Heavens, J. Alsing, T. Erben, C. Heymans, H. Hildebrandt,

H. Hoekstra, A. Jaﬀe, A. Kiessling, Y. Mellier, L. Miller, L. van Waerbeke, J. Ben-

jamin, J. Coupon, L. Fu, M. J. Hudson, M. Kilbinger, K. Kuijken, B. T. P. Rowe,

T. Schrabback, E. Semboloni, and M. Velander. 3D cosmic shear: cosmology

from CFHTLenS. Mon. Not. R. Astron. Soc., 442:1326–1349, August 2014. doi:

10.1093/mnras/stu934.

M. Klasen, M. Pohl, and G. Sigl. Indirect and direct search for dark matter. Progress

in Particle and Nuclear Physics, 85:1–32, November 2015. doi: 10.1016/j.ppnp.

2015.07.001.

A. Kogut, D. J. Fixsen, D. T. Chuss, J. Dotson, E. Dwek, M. Halpern, G. F. Hinshaw,

S. M. Meyer, S. H. Moseley, M. D. Seiﬀert, D. N. Spergel, and E. J. Wollack. The

Primordial Inﬂation Explorer (PIXIE): a nulling polarimeter for cosmic microwave

background observations. J. Cosmo. Astropart. Phys., 2011(7):025, Jul 2011. doi:

10.1088/1475-7516/2011/07/025.

200 S. M. Koushiappas and A. Loeb. Dynamics of Dwarf Galaxies Disfavor Stellar-Mass

Black Holes as Dark Matter. Physical Review Letters, 119(4):041102, July 2017.

doi: 10.1103/PhysRevLett.119.041102.

Florian Kuhnel¨ and Katherine Freese. Constraints on primordial black holes with

extended mass functions. Phys. Rev. D, 95:083508, April 2017. doi: 10.1103/

PhysRevD.95.083508.

J. Kwon, M. Tamura, J. H. Hough, N. Kusakabe, T. Nagata, Y. Nakajima, P. W.

Lucas, T. Nagayama, and R. Kandori. Near-infrared Circular Polarization Survey

in Star-forming Regions: Correlations and Trends. Astrophys. J. Lett., 795:L16,

2014.

S. Lange and L. Page. Detecting the Expansion of the Universe through Changes

in the CMB Photosphere. Astrophys. J., 671:1075–1078, December 2007. doi:

10.1086/523097.

A. Lazarian and T. Hoang. Radiative torques: analytical model and basic properties.

Mon. Not. R. Astron. Soc., 378:910–946, July 2007. doi: 10.1111/j.1365-2966.2007.

11817.x.

K.-G. Lee, A. Krolewski, M. White, D. Schlegel, P. E. Nugent, J. F. Hennawi,

T. Muller,¨ R. Pan, J. X. Prochaska, A. Font-Ribera, N. Suzuki, K. Glazebrook,

G. G. Kacprzak, J. S. Kartaltepe, A. M. Koekemoer, O. Le F`evre, B. C. Lemaux,

C. Maier, T. Nanayakkara, R. M. Rich, D. B. Sanders, M. Salvato, L. Tasca, and

K.-V. H. Tran. First Data Release of the COSMOS Lyα Mapping and Tomography

201 Observations: 3D Lyα Forest Tomography at 2.05 < z < 2.55. Astrophys. J. Supp.,

237:31, August 2018. doi: 10.3847/1538-4365/aace58.

B. V. Lehmann, S. Profumo, and J. Yant. The maximal-density mass function for

primordial black hole dark matter. J. Cosmo. Astropart. Phys., 4:007, April 2018.

doi: 10.1088/1475-7516/2018/04/007.

G. Lemaˆıtre. Expansion of the universe, A homogeneous universe of constant mass

and increasing radius accounting for the radial velocity of extra-galactic nebulae.

Mon. Not. R. Astron. Soc., 91:483–490, March 1931. doi: 10.1093/mnras/91.5.483.

A. Lewis and A. Challinor. Weak gravitational lensing of the CMB. Phys. Rep., 429:

1–65, June 2006. doi: 10.1016/j.physrep.2006.03.002.

Antony Lewis. Rayleigh scattering: blue sky thinking for future cmb observations.

Journal of Cosmology and Astroparticle Physics, 2013(08):053, 2013.

A. Lidz and M. Malloy. On Modeling and Measuring the Temperature of the z ˜ 5

Intergalactic Medium. ApJ, 788:175, June 2014. doi: 10.1088/0004-637X/788/2/

175.

A. Lidz, C.-A. Faucher-Gigu`ere, A. Dall’Aglio, M. McQuinn, C. Fechner, M. Zal-

darriaga, L. Hernquist, and S. Dutta. A Measurement of Small-scale Struc-

ture in the 2.2 <= z <= 4.2 Lyα Forest. ApJ, 718:199–230, July 2010. doi:

10.1088/0004-637X/718/1/199.

Adam Lidz, Oliver Zahn, Matthew McQuinn, Matias Zaldarriaga, Suvendra Dutta,

and Lars Hernquist. Higher Order Contributions to the 21 cm Power Spectrum.

ApJ, 659:865–876, April 2007. doi: 10.1086/511670.

202 Z. Luki´c,C. W. Stark, P. Nugent, M. White, A. A. Meiksin, and A. Almgren. The

Lyman α forest in optically thin hydrodynamical simulations. Mon. Not. R. Astron.

Soc., 446:3697–3724, February 2015. doi: 10.1093/mnras/stu2377.

Chung-Pei Ma and Edmund Bertschinger. Cosmological perturbation theory in the

synchronous and conformal newtonian gauges. arXiv preprint astro-ph/9506072,

1995.

J. H. MacGibbon. Quark- and gluon-jet emission from primordial black holes. II. The

emission over the black-hole lifetime. Phys. Rev. D, 44:376–392, July 1991. doi:

10.1103/PhysRevD.44.376.

M. E. Machacek, G. L. Bryan, A. Meiksin, P. Anninos, D. Thayer, M. Norman, and

Y. Zhang. Hydrodynamical Simulations of the Lyα Forest: Model Comparisons.

ApJ, 532:118–135, March 2000. doi: 10.1086/308551.

R Mainini, D Minelli, M Gervasi, G Boella, G Sironi, A Ba´u, S Banﬁ, A Passerini,

A De Lucia, and F Cavaliere. An improved upper limit to the cmb circular po-

larization at large angular scales. Journal of Cosmology and Astroparticle Physics,

2013(08):033, 2013.

Y. Mao, P. R. Shapiro, G. Mellema, I. T. Iliev, J. Koda, and K. Ahn. Redshift-space

distortion of the 21-cm background from the epoch of reionization - I. Methodology

re-examined. Mon. Not. R. Astron. Soc., 422:926–954, May 2012. doi: 10.1111/j.

1365-2966.2012.20471.x.

203 G. F. Marani, R. J. Nemiroﬀ, J. P. Norris, K. Hurley, and J. T. Bonnell. Gravitation-

ally Lensed Gamma-Ray Bursts as Probes of Dark Compact Objects. ApJ, 512(1):

L13–L16, Feb 1999. doi: 10.1086/311868.

T. Marrod´anUndagoitia and L. Rauch. Dark matter direct-detection experiments.

Journal of Physics G Nuclear Physics, 43(1):013001, January 2016. doi: 10.1088/

0954-3899/43/1/013001.

P. G. Martin. Momentum exchange between small particles and light. Mon. Not. R.

Astron. Soc., 158:63, 1972. doi: 10.1093/mnras/158.1.63.

P. G. Martin. Interstellar circular polarization. Monthly Notices of the Royal Astro-

nomical Society, 159(2):179–190, 1972.

Charlotte A. Mason, Adriano Fontana, Tommaso Treu, Kasper B. Schmidt, Austin

Hoag, Louis Abramson, Ricardo Amorin, Marusa Bradac, Lucia Guaita, Tucker

Jones, Alaina Henry, Matthew A. Malkan, Laura Pentericci, Michele Trenti, and

Eros Vanzella. Inferences on the Timeline of Reionization at z˜8 From the KMOS

Lens-Ampliﬁed Spectroscopic Survey. arXiv e-prints, art. arXiv:1901.11045, Jan-

uary 2019.

Makoto Matsumiya and Kunihito Ioka. Circular polarization from gamma-ray burst

afterglows. The Astrophysical Journal Letters, 595(1):L25, 2003.

P. McDonald, J. Miralda-Escud´e,M. Rauch, W. L. W. Sargent, T. A. Barlow, R. Cen,

and J. P. Ostriker. The Observed Probability Distribution Function, Power Spec-

trum, and Correlation Function of the Transmitted Flux in the Lyα Forest. ApJ,

543:1–23, November 2000. doi: 10.1086/317079.

204 P. McDonald, U. Seljak, R. Cen, D. Shih, D. H. Weinberg, S. Burles, D. P. Schneider,

D. J. Schlegel, N. A. Bahcall, J. W. Briggs, J. Brinkmann, M. Fukugita, Z.ˇ Ivezi´c,

S. Kent, and D. E. Vanden Berk. The Linear Theory Power Spectrum from the

Lyα Forest in the Sloan Digital Sky Survey. ApJ, 635:761–783, December 2005.

doi: 10.1086/497563.

P. McDonald, U. Seljak, S. Burles, D. J. Schlegel, D. H. Weinberg, R. Cen, D. Shih,

J. Schaye, D. P. Schneider, N. A. Bahcall, J. W. Briggs, J. Brinkmann, R. J.

Brunner, M. Fukugita, J. E. Gunn, Z.ˇ Ivezi´c,S. Kent, R. H. Lupton, and D. E.

Vanden Berk. The Lyα Forest Power Spectrum from the Sloan Digital Sky Survey.

Astrophys. J. Supp., 163:80–109, March 2006. doi: 10.1086/444361.

I. D. McGreer, A. Mesinger, and V. D’Odorico. Model-independent evidence in favour

of an end to reionization by z 6. Mon. Not. R. Astron. Soc., 447:499–505, ≈ February 2015. doi: 10.1093/mnras/stu2449.

M. McQuinn. The Evolution of the Intergalactic Medium. Ann. Rev. Astron. Astro-

phys., 54:313–362, September 2016. doi: 10.1146/annurev-astro-082214-122355.

M. McQuinn and M. White. On estimating Lyα forest correlations between multiple

sightlines. Mon. Not. R. Astron. Soc., 415:2257–2269, August 2011. doi: 10.1111/

j.1365-2966.2011.18855.x.

M. McQuinn, A. Lidz, M. Zaldarriaga, L. Hernquist, P. F. Hopkins, S. Dutta, and C.-

A. Faucher-Gigu`ere. He II Reionization and its Eﬀect on the Intergalactic Medium.

ApJ, 694:842–866, April 2009. doi: 10.1088/0004-637X/694/2/842.

205 Matthew McQuinn and Phoebe R. Upton Sanderbeck. On the intergalactic

temperature-density relation. Mon. Not. R. Astron. Soc., 456:47–54, February

2016. doi: 10.1093/mnras/stv2675.

Don Melrose. What causes the circular polarization in pulsars? In Radio Pulsars,

volume 302, page 179, 2003.

A. Mesinger and S. Furlanetto. Eﬃcient Simulations of Early Structure Formation

and Reionization. ApJ, 669:663–675, November 2007. doi: 10.1086/521806.

A. Mesinger, S. Furlanetto, and R. Cen. 21CMFAST: a fast, seminumerical simulation

of the high-redshift 21-cm signal. Mon. Not. R. Astron. Soc., 411:955–972, February

2011. doi: 10.1111/j.1365-2966.2010.17731.x.

A. Mesinger, A. Aykutalp, E. Vanzella, L. Pentericci, A. Ferrara, and M. Dijkstra.

Can the intergalactic medium cause a rapid drop in Lyα emission at z > 6? Mon.

Not. R. Astron. Soc., 446:566–577, January 2015. doi: 10.1093/mnras/stu2089.

Andrei Mesinger and Steven Furlanetto. Eﬃcient simulations of early structure for-

mation and reionization. The Astrophysical Journal, 669(2):663, 2007.

Andrei Mesinger, Steven Furlanetto, and Renyue Cen. 21cmfast: a fast, seminu-

merical simulation of the high-redshift 21-cm signal. Monthly Notices of the Royal

Astronomical Society, 411(2):955–972, 2011.

J. Miralda-Escud´eand M. J. Rees. Reionization and thermal evolution of a photoion-

ized intergalactic medium. Mon. Not. R. Astron. Soc., 266:343–352, January 1994.

doi: 10.1093/mnras/266.2.343.

206 Dipanjan Mitra, Janusz Gil, and George I. Melikidze. Unraveling the nature of coher-

ent pulsar radio emission. The Astrophysical Journal Letters, 696(2):L141, 2009.

Rohollah Mohammadi. Evidence for cosmic neutrino background from cmb circular

polarization. The European Physical Journal C, 74(10):1–12, 2014.

M. A. Monroy-Rodr´ıguez and C. Allen. The End of the MACHO Era, Revisited: New

Limits on MACHO Masses from Halo Wide Binaries. ApJ, 790:159, August 2014.

doi: 10.1088/0004-637X/790/2/159.

R. A. Monsalve, A. E. E. Rogers, J. D. Bowman, and T. J. Mozdzen. Results from

EDGES High-band. I. Constraints on Phenomenological Models for the Global 21

cm Signal. ApJ, 847:64, September 2017. doi: 10.3847/1538-4357/aa88d1.

Paulo Montero-Camacho and Christopher M. Hirata. Exploring circular polarization

in the CMB due to conventional sources of cosmic birefringence. Journal of Cosmol-

ogy and Astro-Particle Physics, 2018(8):040, Aug 2018. doi: 10.1088/1475-7516/

2018/08/040.

Paulo Montero-Camacho, Xiao Fang, Gabriel Vasquez, Makana Silva, and Christo-

pher M. Hirata. Revisiting constraints on asteroid-mass primordial black holes as

dark matter candidates. arXiv e-prints, art. arXiv:1906.05950, Jun 2019a.

Paulo Montero-Camacho, Christopher M. Hirata, Paul Martini, and Klaus Honscheid.

Impact of inhomogeneous reionization on the Lyman-α forest. Mon. Not. R. Astron.

Soc., 487(1):1047–1056, Jul 2019b. doi: 10.1093/mnras/stz1388.

Iman Motie and She-Sheng Xue. Euler-heisenberg lagrangian and photon circular

polarization. EPL (Europhysics Letters), 100(1):17006, 2012.

207 R. Murgia, G. Scelfo, M. Viel, and A. Raccanelli. Lyman-α forest constraints on

Primordial Black Holes as Dark Matter. arXiv e-prints, March 2019.

J. M. Nagy, P. A. R. Ade, M. Amiri, S. J. Benton, A. S. Bergman, R. Bihary, J. J.

Bock, J. R. Bond, S. A. Bryan, H. C. Chiang, C. R. Contaldi, O. Dor´e,A. J.

Duivenvoorden, H. K. Eriksen, M. Farhang, J. P. Filippini, L. M. Fissel, A. A.

Fraisse, K. Freese, M. Galloway, A. E. Gambrel, N. N. Gandilo, K. Ganga, J. E.

Gudmundsson, M. Halpern, J. Hartley, M. Hasselﬁeld, G. Hilton, W. Holmes, V. V.

Hristov, Z. Huang, K. D. Irwin, W. C. Jones, C. L. Kuo, Z. D. Kermish, S. Li, P. V.

Mason, K. Megerian, L. Moncelsi, T. A. Morford, C. B. Netterﬁeld, M. Nolta, I. L.

Padilla, B. Racine, A. S. Rahlin, C. Reintsema, J. E. Ruhl, M. C. Runyan, T. M.

Ruud, J. A. Shariﬀ, J. D. Soler, X. Song, A. Trangsrud, C. Tucker, R. S. Tucker,

A. D. Turner, J. F. Van Der List, A. C. Weber, I. K. Wehus, D. V. Wiebe, and

E. Y. Young. A New Limit on CMB Circular Polarization from SPIDER. ApJ, 844

(2):151, Aug 2017. doi: 10.3847/1538-4357/aa7cfd.

T. Nakama, B. Carr, and J. Silk. Limits on primordial black holes from µ distortions

in cosmic microwave background. Phys. Rev. D, 97(4):043525, February 2018. doi:

10.1103/PhysRevD.97.043525.

F. Nasir, J. S. Bolton, and G. D. Becker. Inferring the IGM thermal history during

reionization with the Lyman α forest power spectrum at redshift z 5. Mon. Not. ' R. Astron. Soc., 463:2335–2347, December 2016. doi: 10.1093/mnras/stw2147.

Robert J. Nemiroﬀ and Andrew Gould. Probing for MACHOs of Mass 10 -15 Msun to

10 -7 Msun with Gamma-Ray Burst Parallax Spacecraft. ApJ, 452:L111, Oct 1995.

doi: 10.1086/309722.

208 Hiroko Niikura, Masahiro Takada, Naoki Yasuda, Robert H. Lupton, Takahiro Sumi,

Surhud More, Toshiki Kurita, Sunao Sugiyama, Anupreeta More, Masamune Oguri,

and Masashi Chiba. Microlensing constraints on primordial black holes with Sub-

aru/HSC Andromeda observations. Nature Astronomy, page 238, Apr 2019a. doi:

10.1038/s41550-019-0723-1.

Hiroko Niikura, Masahiro Takada, Shuichiro Yokoyama, Takahiro Sumi, and Shogo

Masaki. Earth-mass black holes? - Constraints on primordial black holes with

5-years OGLE microlensing events. arXiv e-prints, art. arXiv:1901.07120, January

2019b.

J. O˜norbe, J. F. Hennawi, Z. Luki´c,and M. Walther. Constraining Reionization with

the z `ICˇ 5-6 Lyα Forest Power Spectrum: The Outlook after Planck. ApJ, 847:63,

Sep 2017. doi: 10.3847/1538-4357/aa898d.

Jose O˜norbe, F. B. Davies, Z. Luki´c,J. F. Hennawi, and D. Sorini. Inhomogeneous

Reionization Models in Cosmological Hydrodynamical Simulations. arXiv e-prints,

art. arXiv:1810.11683, October 2018.

Ohio Supercomputer Center. Ruby supercomputer, 2015. URL http://osc.edu/

ark:/19495/hpc93fc8.

B. Paczynski. Gravitational Microlensing by the Galactic Halo. ApJ, 304:1, May

1986. doi: 10.1086/164140.

D. N. Page and S. W. Hawking. Gamma rays from primordial black holes. ApJ, 206:

1–7, May 1976. doi: 10.1086/154350.

209 N. Palanque-Delabrouille, C. Y`eche, A. Borde, J.-M. Le Goﬀ, G. Rossi, M. Viel,

E.´ Aubourg, S. Bailey, J. Bautista, M. Blomqvist, A. Bolton, J. S. Bolton, N. G.

Busca, B. Carithers, R. A. C. Croft, K. S. Dawson, T. Delubac, A. Font-Ribera,

S. Ho, D. Kirkby, K.-G. Lee, D. Margala, J. Miralda-Escud´e,D. Muna, A. D.

Myers, P. Noterdaeme, I. Pˆaris, P. Petitjean, M. M. Pieri, J. Rich, E. Rollinde,

N. P. Ross, D. J. Schlegel, D. P. Schneider, A. Slosar, and D. H. Weinberg. The

one-dimensional Lyα forest power spectrum from BOSS. A&A, 559:A85, November

2013. doi: 10.1051/0004-6361/201322130.

N. Palanque-Delabrouille, C. Y`eche, J. Baur, C. Magneville, G. Rossi, J. Lesgourgues,

A. Borde, E. Burtin, J.-M. LeGoﬀ, J. Rich, M. Viel, and D. Weinberg. Neutrino

masses and cosmology with Lyman-alpha forest power spectrum. J. Cosmo. As-

tropart. Phys., 11:011, November 2015. doi: 10.1088/1475-7516/2015/11/011.

P. Pani and A. Loeb. Constraining primordial black-hole bombs through spectral dis-

tortions of the cosmic microwave background. Phys. Rev. D, 88(4):041301, August

2013. doi: 10.1103/PhysRevD.88.041301.

P. Pani and A. Loeb. Tidal capture of a primordial black hole by a neutron star:

implications for constraints on dark matter. J. Cosmo. Astropart. Phys., 6:026,

June 2014. doi: 10.1088/1475-7516/2014/06/026.

P. J. E. Peebles. Recombination of the primeval plasma. The Astrophysical Journal,

153:1, 1968.

B. Penning. The Pursuit of Dark Matter at Colliders - An Overview. ArXiv e-prints,

December 2017.

210 A. A. Penzias and R. W. Wilson. A Measurement of Excess Antenna Temperature

at 4080 Mc/s. ApJ, 142:419–421, July 1965. doi: 10.1086/148307.

Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Au-

mont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, N. Bartolo,

E. Battaner, R. Battye, K. Benabed, A. Benoˆıt, A. Benoit-L´evy, J. P. Bernard,

M. Bersanelli, P. Bielewicz, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond,

J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler,

E. Calabrese, J. F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, R. R.

Chary, H. C. Chiang, J. Chluba, P. R. Christensen, S. Church, D. L. Clements,

S. Colombi, L. P. L. Colombo, C. Combet, A. Coulais, B. P. Crill, A. Curto, F. Cut-

taia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti,

J. Delabrouille, F. X. D´esert, E. Di Valentino, C. Dickinson, J. M. Diego, K. Dolag,

H. Dole, S. Donzelli, O. Dor´e,M. Douspis, A. Ducout, J. Dunkley, X. Dupac, G. Efs-

tathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, M. Farhang, J. Fergusson, F. Finelli,

O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli,

K. Ganga, C. Gauthier, M. Gerbino, T. Ghosh, M. Giard, Y. Giraud-H´eraud,

E. Giusarma, E. Gjerløw, J. Gonz´alez-Nuevo, K. M. G´orski, S. Gratton, A. Gre-

gorio, A. Gruppuso, J. E. Gudmundsson, J. Hamann, F. K. Hansen, D. Hanson,

D. L. Harrison, G. Helou, S. Henrot-Versill´e,C. Hern´andez-Monteagudo, D. Her-

ranz, S. R. Hildebrand t, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup,

W. Hovest, Z. Huang, K. M. Huffenberger, G. Hurier, A. H. Jaffe, T. R. Jaffe, W. C.

Jones, M. Juvela, E. Keih¨anen, R. Keskitalo, T. S. Kisner, R. Kneissl, J. Knoche,

L. Knox, M. Kunz, H. Kurki-Suonio, G. Lagache, A. L¨ahteenm¨aki, J. M. Lamarre,

211 A. Lasenby, M. Lattanzi, C. R. Lawrence, J. P. Leahy, R. Leonardi, J. Lesgour- gues, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. L´opez-

Caniego, P. M. Lubin, J. F. Mac´ıas-P´erez, G. Maggio, D. Maino, N. Mandolesi,

A. Mangilli, A. Marchini, M. Maris, P. G. Martin, M. Martinelli, E. Mart´ınez-

Gonz´alez, S. Masi, S. Matarrese, P. McGehee, P. R. Meinhold, A. Melchiorri, J. B.

Melin, L. Mendes, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M. A. Miville-

Deschˆenes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi,

J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterﬁeld, H. U. Nørgaard-

Nielsen, F. Noviello, D. Novikov, I. Novikov, C. A. Oxborrow, F. Paci, L. Pagano,

F. Pajot, R. Paladini, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, T. J.

Pearson, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat,

E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, L. Popa,

G. W. Pratt, G. Pr´ezeau, S. Prunet, J. L. Puget, J. P. Rachen, W. T. Reach, R. Re- bolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha,

C. Rosset, M. Rossetti, G. Roudier, B. Rouill´ed’Orfeuil, M. Rowan-Robinson, J. A.

Rubi˜no-Mart´ın, B. Rusholme, N. Said, V. Salvatelli, L. Salvati, M. Sandri, D. San- tos, M. Savelainen, G. Savini, D. Scott, M. D. Seiﬀert, P. Serra, E. P. S. Shellard,

L. D. Spencer, M. Spinelli, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev,

D. Sutton, A. S. Suur-Uski, J. F. Sygnet, J. A. Tauber, L. Terenzi, L. Toﬀo- latti, M. Tomasi, M. Tristram, T. Trombetti, M. Tucci, J. Tuovinen, M. Turler,¨

G. Umana, L. Valenziano, J. Valiviita, F. Van Tent, P. Vielva, F. Villa, L. A. Wade,

B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Wilkinson, D. Yvon,

A. Zacchei, and A. Zonca. Planck 2015 results. XIII. Cosmological parameters.

A&A, 594:A13, Sep 2016. doi: 10.1051/0004-6361/201525830.

212 Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Bacci-

galupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Bat-

tye, K. Benabed, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R.

Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. But-

ler, E. Calabrese, J.-F. Cardoso, J. Carron, A. Challinor, H. C. Chiang, J. Chluba,

L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, F. Cuttaia, P. de Bernardis,

G. de Zotti, J. Delabrouille, J.-M. Delouis, E. Di Valentino, J. M. Diego, O. Dor´e,

M. Douspis, A. Ducout, X. Dupac, S. Dusini, G. Efstathiou, F. Elsner, T. A.

Enßlin, H. K. Eriksen, Y. Fantaye, M. Farhang, J. Fergusson, R. Fernandez-Cobos,

F. Finelli, F. Forastieri, M. Frailis, E. Franceschi, A. Frolov, S. Galeotta, S. Galli,

K. Ganga, R. T. G´enova-Santos, M. Gerbino, T. Ghosh, J. Gonz´alez-Nuevo, K. M.

G´orski, S. Gratton, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley,

D. Herranz, E. Hivon, Z. Huang, A. H. Jaﬀe, W. C. Jones, A. Karakci, E. Keih¨a-

nen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmalnicoﬀ,

M. Kunz, H. Kurki-Suonio, G. Lagache, J.-M. Lamarre, A. Lasenby, M. Lattanzi,

C. R. Lawrence, M. Le Jeune, P. Lemos, J. Lesgourgues, F. Levrier, A. Lewis,

M. Liguori, P. B. Lilje, M. Lilley, V. Lindholm, M. L´opez-Caniego, P. M. Lubin,

Y.-Z. Ma, J. F. Mac´ıas-P´erez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli,

A. Marcos-Caballero, M. Maris, P. G. Martin, M. Martinelli, E. Mart´ınez-Gonz´alez,

S. Matarrese, N. Mauri, J. D. McEwen, P. R. Meinhold, A. Melchiorri, A. Men-

nella, M. Migliaccio, M. Millea, S. Mitra, M.-A. Miville-Deschˆenes, D. Molinari,

L. Montier, G. Morgante, A. Moss, P. Natoli, H. U. Nørgaard-Nielsen, L. Pagano,

D. Paoletti, B. Partridge, G. Patanchon, H. V. Peiris, F. Perrotta, V. Pettorino,

F. Piacentini, L. Polastri, G. Polenta, J.-L. Puget, J. P. Rachen, M. Reinecke,

213 M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubi˜no-Mart´ın,

B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, E. P. S. Shellard,

C. Sirignano, G. Sirri, L. D. Spencer, R. Sunyaev, A.-S. Suur-Uski, J. A. Tauber,

D. Tavagnacco, M. Tenti, L. Toﬀolatti, M. Tomasi, T. Trombetti, L. Valenziano,

J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt,

I. K. Wehus, M. White, S. D. M. White, A. Zacchei, and A. Zonca. Planck 2018

results. VI. Cosmological parameters. ArXiv e-prints, July 2018a.

Planck Collaboration, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Bacci-

galupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Bat-

tye, K. Benabed, J. P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R.

Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. But-

ler, E. Calabrese, J. F. Cardoso, J. Carron, B. Casaponsa, A. Challinor, H. C.

Chiang, L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, F. Cuttaia,

P. de Bernardis, G. de Zotti, J. Delabrouille, J. M. Delouis, F. X. D´esert, E. Di

Valentino, C. Dickinson, J. M. Diego, S. Donzelli, O. Dor´e,M. Douspis, A. Ducout,

X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, E. Falgarone,

Y. Fantaye, J. Fergusson, R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis,

E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. G´enova-Santos,

M. Gerbino, T. Ghosh, J. Gonz´alez-Nuevo, K. M. G´orski, S. Gratton, A. Grup-

puso, J. E. Gudmundsson, J. Hamann, W. Hand ley, F. K. Hansen, G. Helou,

D. Herranz, E. Hivon, Z. Huang, A. H. Jaﬀe, W. C. Jones, A. Karakci, E. Keih¨a-

nen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmalnicoﬀ,

M. Kunz, H. Kurki-Suonio, G. Lagache, J. M. Lamarre, M. Langer, A. Lasenby,

214 M. Lattanzi, C. R. Lawrence, M. Le Jeune, J. P. Leahy, J. Lesgourgues, F. Lev-

rier, A. Lewis, M. Liguori, P. B. Lilje, M. Lilley, V. Lindholm, M. L´opez-Caniego,

P. M. Lubin, Y. Z. Ma, J. F. Mac´ıas-P´erez, G. Maggio, D. Maino, N. Mand olesi,

A. Mangilli, A. Marcos-Caballero, M. Maris, P. G. Martin, E. Mart´ınez-Gonz´alez,

S. Matarrese, N. Mauri, J. D. McEwen, P. D. Meerburg, P. R. Meinhold, A. Mel-

chiorri, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M. A. Miville-Deschˆenes,

D. Molinari, A. Moneti, L. Montier, G. Morgante, A. Moss, S. Mottet, M. Munch-¨

meyer, P. Natoli, H. U. Nørgaard-Nielsen, C. A. Oxborrow, L. Pagano, D. Paoletti,

B. Partridge, G. Patanchon, T. J. Pearson, M. Peel, H. V. Peiris, F. Perrotta,

V. Pettorino, F. Piacentini, L. Polastri, G. Polenta, J. L. Puget, J. P. Rachen,

M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A.

Rubi˜no-Mart´ın, B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott,

E. P. S. Shellard, M. Shiraishi, C. Sirignano, G. Sirri, L. D. Spencer, R. Sunyaev,

A. S. Suur-Uski, J. A. Tauber, D. Tavagnacco, M. Tenti, L. Terenzi, L. Toﬀolatti,

M. Tomasi, T. Trombetti, J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, F. Villa,

N. Vittorio, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Zacchei,

and A. Zonca. Planck 2018 results. I. Overview and the cosmological legacy of

Planck. arXiv e-prints, art. arXiv:1807.06205, Jul 2018b.

Planck Collaboration, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi,

M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, K. Benabed,

J. P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R.

Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J. F.

Cardoso, J. Carron, A. Challinor, H. C. Chiang, L. P. L. Colombo, C. Combet,

D. Contreras, B. P. Crill, F. Cuttaia, P. de Bernardis, G. de Zotti, J. Delabrouille,

215 J. M. Delouis, E. Di Valentino, J. M. Diego, S. Donzelli, O. Dor´e,M. Douspis,

A. Ducout, X. Dupac, S. Dusini, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K.

Eriksen, Y. Fantaye, J. Fergusson, R. Fernandez-Cobos, F. Finelli, F. Forastieri,

M. Frailis, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, C. Gauthier,

R. T. Génova-Santos, M. Gerbino, T. Ghosh, J. González-Nuevo, K. M. Górski,

S. Gratton, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley, F. K.

Hansen, D. Herranz, E. Hivon, D. C. Hooper, Z. Huang, A. H. Jaﬀe, W. C. Jones,

E. Keih¨anen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, N. Krachmalnicoﬀ,

M. Kunz, H. Kurki-Suonio, G. Lagache, J. M. Lamarre, A. Lasenby, M. Lattanzi,

C. R. Lawrence, M. Le Jeune, J. Lesgourgues, F. Levrier, A. Lewis, M. Liguori,

P. B. Lilje, V. Lindholm, M. Lpez-Caniego, P. M. Lubin, Y. Z. Ma, J. F. Mac´ıas-

P´erez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero,

M. Maris, P. G. Martin, E. Mart´ınez-Gonz´alez, S. Matarrese, N. Mauri, J. D.

McEwen, P. D. Meerburg, P. R. Meinhold, A. Melchiorri, A. Mennella, M. Migliac- cio, S. Mitra, M. A. Miville-Deschˆenes, D. Molinari, A. Moneti, L. Montier, G. Mor- gante, A. Moss, M. Munchmeyer,¨ P. Natoli, H. U. Nørgaard-Nielsen, L. Pagano,

D. Paoletti, B. Partridge, G. Patanchon, H. V. Peiris, F. Perrotta, V. Pettorino,

F. Piacentini, L. Polastri, G. Polenta, J. L. Puget, J. P. Rachen, M. Reinecke,

M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubi˜no-Mart´ın,

B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, E. P. S. Shellard,

M. Shiraishi, C. Sirignano, G. Sirri, L. D. Spencer, R. Sunyaev, A. S. Suur-Uski,

J. A. Tauber, D. Tavagnacco, M. Tenti, L. Toﬀolatti, M. Tomasi, T. Trombetti,

J. Valiviita, B. Van Tent, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K.

Wehus, S. D. M. White, A. Zacchei, J. P. Zibin, and A. Zonca. Planck 2018 results.

216 X. Constraints on inﬂation. arXiv e-prints, art. arXiv:1807.06211, Jul 2018c.

Andrew Pontzen. Scale-dependent bias in the baryonic-acoustic-oscillation-scale in-

tergalactic neutral hydrogen. Phys. Rev. D, 89:083010, Apr 2014. doi: 10.1103/

PhysRevD.89.083010.

V. Poulin, P. D. Serpico, F. Calore, S. Clesse, and K. Kohri. CMB bounds on disk-

accreting massive primordial black holes. Phys. Rev. D, 96(8):083524, October

2017. doi: 10.1103/PhysRevD.96.083524.

J. R. Pritchard, A. Loeb, and J. S. B. Wyithe. Constraining reionization using 21-cm

observations in combination with CMB and Lyα forest data. Mon. Not. R. Astron.

Soc., 408:57–70, October 2010. doi: 10.1111/j.1365-2966.2010.17150.x.

Jonathan R. Pritchard and Abraham Loeb. 21 cm cosmology in the 21st century.

Reports on Progress in Physics, 75(8):086901, Aug 2012. doi: 10.1088/0034-4885/

75/8/086901.

A. Raccanelli, E. D. Kovetz, S. Bird, I. Cholis, and J. B. Mu˜noz. Determining the

progenitors of merging black-hole binaries. Phys. Rev. D, 94(2):023516, July 2016.

doi: 10.1103/PhysRevD.94.023516.

M. Rauch. The Lyman Alpha Forest in the Spectra of QSOs. Ann. Rev. Astron.

Astrophys., 36:267–316, 1998. doi: 10.1146/annurev.astro.36.1.267.

J. I. Read. The local dark matter density. Journal of Physics G Nuclear Physics, 41

(6):063101, Jun 2014. doi: 10.1088/0954-3899/41/6/063101.

A. Refregier. Weak Gravitational Lensing by Large-Scale Structure. Ann. Rev. As-

tron. Astrophys., 41:645–668, 2003. doi: 10.1146/annurev.astro.41.111302.102207.

217 A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich,

R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips,

D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs,

N. B. Suntzeﬀ, and J. Tonry. Observational Evidence from Supernovae for an

Accelerating Universe and a Cosmological Constant. AJ, 116:1009–1038, September

1998. doi: 10.1086/300499.

H. P. Robertson. Kinematics and World-Structure. ApJ, 82:284, November 1935. doi:

10.1086/143681.

L. Roszkowski, E. M. Sessolo, and S. Trojanowski. WIMP dark matter candidates

and searches - current issues and future prospects. ArXiv e-prints, July 2017.

Leszek Roszkowski, Enrico Maria Sessolo, and Sebastian Trojanowski. WIMP dark

matter candidates and searches—current status and future prospects. Reports on

Progress in Physics, 81(6):066201, Jun 2018. doi: 10.1088/1361-6633/aab913.

V. C. Rubin and W. K. Ford, Jr. Rotation of the Andromeda Nebula from a

Spectroscopic Survey of Emission Regions. ApJ, 159:379, February 1970. doi:

10.1086/150317.

M. Sadegh, R. Mohammadi, and I. Motie. Generation of circular polarization in CMB

radiation via nonlinear photon-photon interaction. ArXiv e-prints, November 2017.

E. E. Salpeter. Energy and Pressure of a Zero-Temperature Plasma. ApJ, 134:669,

November 1961. doi: 10.1086/147194.

218 M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama. Primordial Black Hole Scenario

for the Gravitational-Wave Event GW150914. Physical Review Letters, 117(6):

061101, August 2016. doi: 10.1103/PhysRevLett.117.061101.

R. F. Sawyer. Photon-photon interactions as a source of cosmic microwave background

circular polarization. Physical Review D, 91(2):021301, 2015.

U. Seljak, A. Makarov, P. McDonald, S. F. Anderson, N. A. Bahcall, J. Brinkmann,

S. Burles, R. Cen, M. Doi, J. E. Gunn, Z.ˇ Ivezi´c,S. Kent, J. Loveday, R. H.

Lupton, J. A. Munn, R. C. Nichol, J. P. Ostriker, D. J. Schlegel, D. P. Schneider,

M. Tegmark, D. E. Berk, D. H. Weinberg, and D. G. York. Cosmological parameter

analysis including SDSS Lyα forest and galaxy bias: Constraints on the primordial

spectrum of ﬂuctuations, neutrino mass, and dark energy. Phys. Rev. D, 71(10):

103515, May 2005. doi: 10.1103/PhysRevD.71.103515.

U. Seljak, A. Slosar, and P. McDonald. Cosmological parameters from combining

the Lyman-α forest with CMB, galaxy clustering and SN constraints. J. Cosmo.

Astropart. Phys., 10:014, October 2006. doi: 10.1088/1475-7516/2006/10/014.

G. L. Sewell and C. A. Coulson. Stark eﬀect for a hydrogen atom in its ground state.

Proceedings of the Cambridge Philosophical Society, 45:678, 1949.

S. L. Shapiro and S. A. Teukolsky. Black Holes, White Dwarfs and Neutron Stars:

The Physics of Compact Objects. June 1986.

S. Singh, R. Subrahmanyan, N. Udaya Shankar, M. Sathyanarayana Rao, A. Fialkov,

A. Cohen, R. Barkana, B. S. Girish, A. Raghunathan, R. Somashekar, and K. S.

219 Srivani. First Results on the Epoch of Reionization from First Light with SARAS

2. ApJL, 845:L12, August 2017. doi: 10.3847/2041-8213/aa831b.

S. Singh, R. Subrahmanyan, N. Udaya Shankar, M. Sathyanarayana Rao, A. Fialkov,

A. Cohen, R. Barkana, B. S. Girish, A. Raghunathan, R. Somashekar, and K. S.

Srivani. SARAS 2 Constraints on Global 21 cm Signals from the Epoch of Reion-

ization. ApJ, 858:54, May 2018. doi: 10.3847/1538-4357/aabae1.

A. Slosar, A. Font-Ribera, M. M. Pieri, J. Rich, J.-M. Le Goﬀ, E.´ Aubourg,

J. Brinkmann, N. Busca, B. Carithers, R. Charlassier, M. Cortˆes,R. Croft, K. S.

Dawson, D. Eisenstein, J.-C. Hamilton, S. Ho, K.-G. Lee, R. Lupton, P. McDonald,

B. Medolin, D. Muna, J. Miralda-Escud´e,A. D. Myers, R. C. Nichol, N. Palanque-

Delabrouille, I. Pˆaris, P. Petitjean, Y. Piˇskur, E. Rollinde, N. P. Ross, D. J. Schlegel,

D. P. Schneider, E. Sheldon, B. A. Weaver, D. H. Weinberg, C. Yeche, and D. G.

York. The Lyman-α forest in three dimensions: measurements of large scale ﬂux

correlations from BOSS 1st-year data. J. Cosmo. Astropart. Phys., 9:001, Septem-

ber 2011. doi: 10.1088/1475-7516/2011/09/001.

A. Slosar, V. Irˇsiˇc,D. Kirkby, S. Bailey, N. G. Busca, T. Delubac, J. Rich, E.´ Aubourg,

J. E. Bautista, V. Bhardwaj, M. Blomqvist, A. S. Bolton, J. Bovy, J. Brownstein,

B. Carithers, R. A. C. Croft, K. S. Dawson, A. Font-Ribera, J.-M. Le Goﬀ, S. Ho,

K. Honscheid, K.-G. Lee, D. Margala, P. McDonald, B. Medolin, J. Miralda-Escud´e,

A. D. Myers, R. C. Nichol, P. Noterdaeme, N. Palanque-Delabrouille, I. Pˆaris,

P. Petitjean, M. M. Pieri, Y. Piˇskur, N. A. Roe, N. P. Ross, G. Rossi, D. J. Schlegel,

D. P. Schneider, N. Suzuki, E. S. Sheldon, U. Seljak, M. Viel, D. H. Weinberg,

and C. Y`eche. Measurement of baryon acoustic oscillations in the Lyman-α forest

220 ﬂuctuations in BOSS data release 9. J. Cosmo. Astropart. Phys., 4:026, April 2013.

doi: 10.1088/1475-7516/2013/04/026.

V. V. Sobolev. Moving envelopes of stars. 1960.

V. Springel. The cosmological simulation code GADGET-2. Mon. Not. R. Astron.

Soc., 364:1105–1134, December 2005. doi: 10.1111/j.1365-2966.2005.09655.x.

V. Springel, N. Yoshida, and S. D. M. White. GADGET: a code for collisionless

and gasdynamical cosmological simulations. N. Astron., 6:79–117, April 2001. doi:

10.1016/S1384-1076(01)00042-2.

V. Springel, C. S. Frenk, and S. D. M. White. The large-scale structure of the

Universe. Nature, 440:1137–1144, April 2006. doi: 10.1038/nature04805.

Sunao Sugiyama, Toshiki Kurita, and Masahiro Takada. Revisiting the wave optics

eﬀect on primordial black hole constraints from optical microlensing search. arXiv

e-prints, art. arXiv:1905.06066, May 2019.

John B. Swedlund, James C. Kemp, and Ramon D. Wolstencroft. Circular polariza-

tion of saturn. The Astrophysical Journal, 178:257–266, 1972.

V. Takhistov. Positrons from primordial black hole microquasars and gamma-ray

bursts. Physics Letters B, 789:538–544, February 2019. doi: 10.1016/j.physletb.

2018.12.043.

Hiroyuki Tashiro and Naoshi Sugiyama. The eﬀect of primordial black holes on 21-

cm ﬂuctuations. Mon. Not. R. Astron. Soc., 435(4):3001–3008, Nov 2013. doi:

10.1093/mnras/stt1493.

221 A. N. Taylor, S. Dye, T. J. Broadhurst, N. Ben´ıtez, and E. van Kampen. Gravitational

Lens Magniﬁcation and the Mass of Abell 1689. ApJ, 501:539–553, July 1998. doi:

10.1086/305827.

F. X. Timmes and S. E. Woosley. The conductive propagation of nuclear ﬂames. I

- Degenerate C+O and O+Ne+Mg white dwarfs. ApJ, 396:649–667, September

1992. doi: 10.1086/171746.

P. Tisserand, L. Le Guillou, C. Afonso, J. N. Albert, J. Andersen, R. Ansari,

E.´ Aubourg, P. Bareyre, J. P. Beaulieu, X. Charlot, C. Coutures, R. Ferlet,

P. Fouqu´e,J. F. Glicenstein, B. Goldman, A. Gould, D. Graﬀ, M. Gros, J. Haissin-

ski, C. Hamadache, J. de Kat, T. Lasserre, E.´ Lesquoy, C. Loup, C. Magneville,

J. B. Marquette, E.´ Maurice, A. Maury, A. Milsztajn, M. Moniez, N. Palanque-

Delabrouille, O. Perdereau, Y. R. Rahal, J. Rich, M. Spiro, A. Vidal-Madjar, L. Vi-

groux, S. Zylberajch, and EROS-2 Collaboration. Limits on the Macho content of

the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds. A&A, 469:

387–404, July 2007. doi: 10.1051/0004-6361:20066017.

H. Trac, R. Cen, and A. Loeb. Imprint of Inhomogeneous Hydrogen Reionization

on the Temperature Distribution of the Intergalactic Medium. ApJL, 689:L81,

December 2008. doi: 10.1086/595678.

Thomas Tram and Julien Lesgourgues. Optimal polarisation equations in ﬂrw uni-

verses. Journal of Cosmology and Astroparticle Physics, 2013(10):002, 2013.

M. A. Troxel, N. MacCrann, J. Zuntz, T. F. Eiﬂer, E. Krause, S. Dodelson, D. Gruen,

J. Blazek, O. Friedrich, S. Samuroﬀ, J. Prat, L. F. Secco, C. Davis, A. Fert´e,

222 J. DeRose, A. Alarcon, A. Amara, E. Baxter, M. R. Becker, G. M. Bernstein,

S. L. Bridle, R. Cawthon, C. Chang, A. Choi, J. De Vicente, A. Drlica-Wagner,

J. Elvin-Poole, J. Frieman, M. Gatti, W. G. Hartley, K. Honscheid, B. Hoyle,

E. M. Huﬀ, D. Huterer, B. Jain, M. Jarvis, T. Kacprzak, D. Kirk, N. Kokron,

C. Krawiec, O. Lahav, A. R. Liddle, J. Peacock, M. M. Rau, A. Refregier, R. P.

Rollins, E. Rozo, E. S. Rykoﬀ, C. S´anchez, I. Sevilla-Noarbe, E. Sheldon, A. Steb-

bins, T. N. Varga, P. Vielzeuf, M. Wang, R. H. Wechsler, B. Yanny, T. M. C.

Abbott, F. B. Abdalla, S. Allam, J. Annis, K. Bechtol, A. Benoit-L´evy, E. Bertin,

D. Brooks, E. Buckley-Geer, D. L. Burke, A. Carnero Rosell, M. Carrasco Kind,

J. Carretero, F. J. Castander, M. Crocce, C. E. Cunha, C. B. D’Andrea, L. N. da

Costa, D. L. DePoy, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, E. Fernandez,

B. Flaugher, P. Fosalba, J. Garc´ıa-Bellido, E. Gaztanaga, D. W. Gerdes, T. Gian-

nantonio, D. A. Goldstein, R. A. Gruendl, J. Gschwend, G. Gutierrez, D. J. James,

T. Jeltema, M. W. G. Johnson, M. D. Johnson, S. Kent, K. Kuehn, S. Kuhlmann,

N. Kuropatkin, T. S. Li, M. Lima, H. Lin, M. A. G. Maia, M. March, J. L. Mar-

shall, P. Martini, P. Melchior, F. Menanteau, R. Miquel, J. J. Mohr, E. Neilsen,

R. C. Nichol, B. Nord, D. Petravick, A. A. Plazas, A. K. Romer, A. Roodman,

M. Sako, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, M. Smith, R. C.

Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, M. E. C. Swanson, G. Tarle,

D. Thomas, D. L. Tucker, V. Vikram, A. R. Walker, J. Weller, and Y. Zhang.

Dark Energy Survey Year 1 Results: Cosmological Constraints from Cosmic Shear.

ArXiv e-prints, August 2017.

M. A. Troxel, N. MacCrann, J. Zuntz, T. F. Eiﬂer, E. Krause, S. Dodelson, D. Gruen,

J. Blazek, O. Friedrich, S. Samuroﬀ, J. Prat, L. F. Secco, C. Davis, A. Fert´e,

223 J. DeRose, A. Alarcon, A. Amara, E. Baxter, M. R. Becker, G. M. Bernstein,

S. L. Bridle, R. Cawthon, C. Chang, A. Choi, J. De Vicente, A. Drlica-Wagner,

J. Elvin-Poole, J. Frieman, M. Gatti, W. G. Hartley, K. Honscheid, B. Hoyle, E. M.

Huﬀ, D. Huterer, B. Jain, M. Jarvis, T. Kacprzak, D. Kirk, N. Kokron, C. Krawiec,

O. Lahav, A. R. Liddle, J. Peacock, M. M. Rau, A. Refregier, R. P. Rollins, E. Rozo,

E. S. Rykoﬀ, C. S´anchez, I. Sevilla-Noarbe, E. Sheldon, A. Stebbins, T. N. Varga,

P. Vielzeuf, M. Wang, R. H. Wechsler, B. Yanny, T. M. C. Abbott, F. B. Abdalla,

S. Allam, J. Annis, K. Bechtol, A. Benoit-L´evy, E. Bertin, D. Brooks, E. Buckley-

Geer, D. L. Burke, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, F. J. Ca-

stander, M. Crocce, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, D. L. DePoy,

S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, E. Fernandez, B. Flaugher, P. Fosalba,

J. Garc´ıa-Bellido, E. Gaztanaga, D. W. Gerdes, T. Giannantonio, D. A. Goldstein,

R. A. Gruendl, J. Gschwend, G. Gutierrez, D. J. James, T. Jeltema, M. W. G. John-

son, M. D. Johnson, S. Kent, K. Kuehn, S. Kuhlmann, N. Kuropatkin, T. S. Li,

M. Lima, H. Lin, M. A. G. Maia, M. March, J. L. Marshall, P. Martini, P. Melchior,

F. Menanteau, R. Miquel, J. J. Mohr, E. Neilsen, R. C. Nichol, B. Nord, D. Petrav-

ick, A. A. Plazas, A. K. Romer, A. Roodman, M. Sako, E. Sanchez, V. Scarpine,

R. Schindler, M. Schubnell, M. Smith, R. C. Smith, M. Soares-Santos, F. Sobreira,

E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas, D. L. Tucker, V. Vikram,

A. R. Walker, J. Weller, Y. Zhang, and DES Collaboration. Dark Energy Survey

Year 1 results: Cosmological constraints from cosmic shear. Phys. Rev. D, 98(4):

043528, Aug 2018. doi: 10.1103/PhysRevD.98.043528.

D. Tseliakhovich and C. Hirata. Relative velocity of dark matter and baryonic ﬂuids

and the formation of the ﬁrst structures. Phys. Rev. D, 82(8):083520, October 2010.

224 doi: 10.1103/PhysRevD.82.083520.

A. van Engelen, R. Keisler, O. Zahn, K. A. Aird, B. A. Benson, L. E. Bleem, J. E.

Carlstrom, C. L. Chang, H. M. Cho, T. M. Crawford, A. T. Crites, T. de Haan,

M. A. Dobbs, J. Dudley, E. M. George, N. W. Halverson, G. P. Holder, W. L.

Holzapfel, S. Hoover, Z. Hou, J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M.

Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Mil-

lea, J. J. Mohr, T. E. Montroy, T. Natoli, S. Padin, T. Plagge, C. Pryke, C. L.

Reichardt, J. E. Ruhl, J. T. Sayre, K. K. Schaﬀer, L. Shaw, E. Shirokoﬀ, H. G.

Spieler, Z. Staniszewski, A. A. Stark, K. Story, K. Vanderlinde, J. D. Vieira, and

R. Williamson. A Measurement of Gravitational Lensing of the Microwave Back-

ground Using South Pole Telescope Data. ApJ, 756:142, September 2012. doi:

10.1088/0004-637X/756/2/142.

Tejaswi Venumadhav, Antonija Oklopcic, Vera Gluscevic, Abhilash Mishra, and

Christopher M Hirata. A new probe of magnetic ﬁelds in the pre-reionization

epoch: I. formalism. arXiv preprint arXiv:1410.2250, 2014.

M. Viel, G. D. Becker, J. S. Bolton, M. G. Haehnelt, M. Rauch, and W. L. W.

Sargent. How Cold Is Cold Dark Matter? Small-Scales Constraints from the Flux

Power Spectrum of the High-Redshift Lyman-α Forest. Physical Review Letters,

100(4):041304, February 2008. doi: 10.1103/PhysRevLett.100.041304.

A. G. Walker. On Milne’s Theory of World-Structure. Proceedings of the London

Mathematical Society, 42:90–127, Jan 1937. doi: 10.1112/plms/s2-42.1.90.

225 M. Walther, J. F. Hennawi, H. Hiss, J. O˜norbe, K.-G. Lee, A. Rorai, and J. O’Meara.

A New Precision Measurement of the Small-scale Line-of-sight Power Spectrum of

the Lyα Forest. ApJ, 852:22, January 2018a. doi: 10.3847/1538-4357/aa9c81.

Michael Walther, Jose O˜norbe, Joseph F. Hennawi, and Zarija Luki´c.New Constraints

on IGM Thermal Evolution from the Lyα Forest Power Spectrum. ArXiv e-prints,

art. arXiv:1808.04367, August 2018b.

W. D. Watson and H. W. Wyld. The relationship between the circular polarization

and the magnetic ﬁeld for astrophysical masers with weak zeeman splitting. The

Astrophysical Journal Letters, 558(1):L55, 2001.

T. A. Weaver, G. B. Zimmerman, and S. E. Woosley. Presupernova evolution of

massive stars. ApJ, 225:1021–1029, November 1978. doi: 10.1086/156569.

D. H. Weinberg, S. Burles, R. A. C. Croft, R. Dave’, G. Gomez, L. Hernquist, N. Katz,

D. Kirkman, S. Liu, J. Miralda-Escude’, M. Pettini, J. Phillips, D. Tytler, and

J. Wright. Cosmology with the Lyman-alpha Forest. ArXiv Astrophysics e-prints,

October 1998.

D. H. Weinberg, R. Dav´e,N. Katz, and J. A. Kollmeier. The Lyman-α Forest as a

Cosmological Tool. In S. H. Holt and C. S. Reynolds, editors, The Emergence of

Cosmic Structure, volume 666 of American Institute of Physics Conference Series,

pages 157–169, May 2003. doi: 10.1063/1.1581786.

K. Wiersema, S. Covino, K. Toma, A. J. van der Horst, K. Varela, M. Min, J. Greiner,

R. L. C. Starling, N. R. Tanvir, R. A. M. J. Wijers, S. Campana, P. A. Curran,

Y. Fan, J. P. U. Fynbo, J. Gorosabel, A. Gomboc, D. G¨otz, J. Hjorth, Z. P. Jin,

226 S. Kobayashi, C. Kouveliotou, C. Mundell, P. T. O’Brien, E. Pian, A. Rowlinson,

D. M. Russell, R. Salvaterra, S. di Serego Alighieri, G. Tagliaferri, S. D. Vergani,

J. Elliott, C. Fari˜na, O. E. Hartoog, R. Karjalainen, S. Klose, F. Knust, A. J.

Levan, P. Schady, V. Sudilovsky, and R. Willingale. Circular polarization in the

optical afterglow of GRB 121024A. Nature, 509:201–204, 2014.

Eugene Wigner. Group theory: and its application to the quantum mechanics of

atomic spectra, volume 5. Elsevier, 2012.

David Wood. The computation of polylogarithms. Technical Report 15-92*, Univer-

sity of Kent, Computing Laboratory, University of Kent, Canterbury, UK, June

1992. URL http://www.cs.kent.ac.uk/pubs/1992/110.

D. G. Yakovlev and V. A. Urpin. Thermal and Electrical Conductivity in White

Dwarfs and Neutron Stars. Sov. Astron., 24:303, June 1980.

D. G. Yakovlev, A. D. Kaminker, O. Y. Gnedin, and P. Haensel. Neutrino emis-

sion from neutron stars. Phys. Rep., 354(1-2):1–155, Nov 2001. doi: 10.1016/

S0370-1573(00)00131-9.

C. Y`eche, N. Palanque-Delabrouille, J. Baur, and H. du Mas des Bourboux. Con-

straints on neutrino masses from Lyman-alpha forest power spectrum with BOSS

and XQ-100. J. Cosmo. Astropart. Phys., 6:047, June 2017. doi: 10.1088/

1475-7516/2017/06/047.

Qingjuan Yu, David N Spergel, and Jeremiah P Ostriker. Rayleigh scattering and

microwave background ﬂuctuations. The Astrophysical Journal, 558(1):23, 2001.

227 M Zarei, E Bavarsad, M Haghighat, R Mohammadi, I Motie, and Z Rezaei. Genera-

tion of circular polarization of the cmb. Physical Review D, 81(8):084035, 2010.

Guido Zavattini, Ugo Gastaldi, Ruggero Pengo, Giuseppe Ruoso, F. Della Valle,

and Edoardo Milotti. Measuring the magnetic birefringence of vacuum: the pvlas

experiment. International Journal of Modern Physics A, 27(15):1260017, 2012.

M. Zumalac´arregui and U. Seljak. Limits on Stellar-Mass Compact Objects as Dark

Matter from Gravitational Lensing of Type Ia Supernovae. Physical Review Letters,

121(14):141101, October 2018. doi: 10.1103/PhysRevLett.121.141101.

F. Zwicky. On the Masses of Nebulae and of Clusters of Nebulae. ApJ, 86:217,

October 1937. doi: 10.1086/143864.

228