Radiating Macroscopic Dark Matter

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RADIATING MACROSCOPIC DARK MATTER:

SEARCHING FOR EFFECTS IN COSMIC

MICROWAVE BACKGROUND AND

RECOMBINATION HISTORY

SAURABH KUMAR

Submitted in partial fulﬁllment of the requirements

for the degree of Doctor of Philosophy

Department of Physics

CASE WESTERN RESERVE UNIVERSITY

January, 2021 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of Saurabh Kumar

candidate for the degree of Doctor of Philosophy*.

Committee Chair Dr. Glenn Starkman

Committee Member Dr. Idit Zehavi

Committee Member Dr. John Ruhl

Committee Member Dr. Stacy McGaugh

Date of Defense

November 24, 2020

*We also certify that written approval has been obtained

for any proprietary material contained therein. Contents

List of Tables iii

List of Figures iv

Acknowledgments viii

Abstract ix

1 Introduction 1 1.1 Evidence of Dark Matter ...... 3 1.2 Challenges to the CDM paradigm ...... 7 1.3 Candidates of Dark Matter ...... 8 1.4 Macroscopic dark matter or macros ...... 10 1.5 Constraints on parameter space of macros ...... 13

2 CMB Spectral Distortions from Cooling Macroscopic Dark Matter 17 2.1 Spectral Distortions in the CMB ...... 19 2.1.1 The Boltzmann equation ...... 21 2.1.2 Thermalization/Scattering mechanisms ...... 22 2.1.3 Types of Spectral Distortions ...... 26 2.2 Cooling processes for macros ...... 32 2.3 Conclusions ...... 44

i CONTENTS

3 Radiating Baryonic Macroscopic Dark Matter and the Recombina- tion history of the Universe 45 3.1 Physics of cosmic recombination ...... 46 3.2 Change in recombination history from macros ...... 59 3.3 Conclusions ...... 63

4 Conclusions 68

A Photon Luminosity and Internal Temperature of Macro 70

B Neutrino Emission Luminosity 73

ii List of Tables

i 2.1 Deﬁnitions of various temperatures, T9, that appear in equations (2.66) and (2.67). For i = MU, DU, CP, the temperature values were determined by comparing the luminosities of various processes given by Eq. (2.58)...... 40

2.2 Deﬁnitions of various times, ti, that appear in equations (2.66) and (2.67). These time values very obtained by solving Eq. (2.66) for the most dominant cooling process...... 42

iii List of Figures

1.1 Constraints on mass and cross-section of macros from cosmological and astrophysical phenomena. The white region in the lower right corner is ruled out because it is denser than black holes. The grey region is ruled out by CMB constraints [119]; the yellow from mica observation [29, 94]; the red from superbursts in NSs; the dark blue from white dwarfs becoming supernovae [44]; the purple from a lack of human injuries and deaths [107]; the green from a lack of fast-moving bolides [105]; the maroon from a lack of microlensing events toward the Large Magellanic Cloud and the Galactic center [9, 112, 19, 45], and in pink toward M31 [83]. Image credit: Figure 1 in arXiv:2006.01200v2, Journal reference [104]...... 15

2.1 Measurements of the CMB spectrum. Image source: COBE webpage . 21 2.2 Multiple layers of a neutron star. Image credit: Figure 1 from arXiv:astro- ph/9906456v1, Journal reference [123]...... 33

2.3 Interior temperature TX and surface temperature Ts of a macro for

MX = M , λ = 1 and ρX = ρN , plotted versus time t and redshift z. Cooling is without DUrca, and both without CP and with CP for

two values of Tc. The ambient photon temperature TCMB is shown for comparison, and the eras when µ and y distortions occur are indicated. 40

iv LIST OF FIGURES

2.4 Dominant cooling mechanisms in a neutron star at different temperatures. Bottom Panel: T = 109 K, Middle Panel: T = 3 × 108 K, Top Panel: T = 108 K. The figures on the left and right panel describe cooling for neutron stars that begin cooling via MURCA and DURCA respectively at high temperatures. In all figures, x-axis and the y-axis is the logarithm of the critical temperature of onset of superfluidity

in neutrons and protons denoted by log10(Tcn) and log10(Tcp) respec-

tively. In our work, we choose log10(Tcn) = log10(Tcp). We ignore all bremsstrahlung processes in our work since for our choice of macro parameters they are ineﬃcient. Macro cooling is very well described by MURCA, DURCA and Cooper pair cooling. Image credit: Figure 13 in arXiv:astro-ph/9906456v1, Journal reference [123]...... 41 2.5 Top panel: µ distortion as a function of macro surface composition factor λ for three diﬀerent cooling scenarios. On the left, no DUrca, no CP; in the middle no DUrca, with CP; on the right; with DUrca, with

CP. Green lines denote ρX = ρN , red lines 0.1ρN ; blue lines 10ρN . The

9 9 panels with CP show results for Tc = 10 K (dashed) and Tc = 4×10 K (solid). Bottom panel: as for top panel but for y distortion. The vertical dashed line stands for λ = 1 which is similar to a Neutron Star. 43

3.1 Helium I energy levels for n ≤ 4 and continuum. In practice, the three energy levels of the He I atom - ground state, ﬁrst excited states and the continuum - very accurately describe Helium recombination. The dotted line shows that the triplet states are metastable states, they tend to hold charge and get ionized rather than decay into the ground state as is the case with singlet states. Image credit: Figure 8 from arXiv:astro-ph/9912182v2, Journal reference [102]...... 54

v LIST OF FIGURES

3.2 Electron fraction as a function of redshift. Notice that at lower redshifts the electron fraction value plateaus to few ×10−4. This happens because the net recombination rate falls below Hubble expansion rate. This plot was produced by solving the RECFAST code [101]...... 57 3.3 Electron temperature as a function of redshift. Until redshift of z ∼ 200, the residual free electrons in the universe are in thermal equilib-

< rium with the photons. For z ∼ 200, Hubble cooling becomes more eﬃ- cient and the electron temperature drops below that of CMB. Coulomb scattering between electrons, and ionized and neutral hydrogen and helium always keeps baryons in thermal equilibrium with electrons. This plot was produced by solving the RECFAST code [101]...... 58 3.4 Intensity spectrum of the CMB and photons emitted from macros of

9 three different masses at z = 1300. We used λ = 1, and TC = 10 K. In this plot, the spectrum from the 100 g macro is indistinguishable from a universe with no macros. The dashed, and dotted vertical lines stand for Lyman alpha and n = 2 ionization frequencies respectively. As we can see, at the two frequencies we are interested in, 3.4 eV and 10.2 eV, there is no significant excess of photons emitted from macros. The surplus at the Wein tail does not contribute in the excitation, and ionization processes since the cross-section of the Lyman Alpha excitation is sharply peaked at 10.2 eV, whereas that for the ionization rate falls off as ν−3 below 3.4 eV...... 64

vi LIST OF FIGURES

3.5 The n = 2 ionization rates without macros, and solar mass macros compared to the Hubble rate, H(z). We can see that the excess photons from the macro surface enhance the rate of ionization only at lower

< redshifts, z ∼ 500. However, by this redshift the ionization rate is below the Hubble rate. Therefore, the surplus 3.4 eV photons do not lead to excess ionization of excited hydrogen...... 65 3.6 The ratio of Lyman alpha excitation rates in a universe with solar mass macros to that without any. We see that the largest enhancement in

< < the excitation rate occurs at much lower redshifts between 40 ∼ z ∼ 200. Therefore, the surplus Lyman alpha photons from macros do not lead to any excess excitation of the ground state hydrogen atoms...... 66 3.7 The EBL (Extragalactic Background Light) intensity compared to that of CMB and macro photons. The EBL background is plotted with permission from [41]...... 67

vii Acknowledgments

Firstly, I am immensely thankful to my advisor Prof. Glenn Starkman for his support, encouragement and feedback. I am indebted to him for introducing me to many great collaborators whom I worked with throughout my graduate program. I am also extremely thankful to Prof. Idit Zehavi. She was my co-advisor during the ﬁnal year of my program. I am grateful for her support and patience. I thank my Particle Astrophysics Theory group friends for setting very high stan- dards which kept me motivated throughout my program. I specially thank my family in India for supporting me to pursue the less lucrative career path of academia. Last but not the least, I thank the faculty, friends, colleagues, and department assistants in physics for their help and support.

viii Radiating Macroscopic Dark Matter: Searching for Eﬀects in Cosmic Microwave Background and Recombination History

Abstract by SAURABH KUMAR

One of the most outstanding questions in fundamental physics is the nature of dark matter (DM). From multiple independent sources of evidence we infer that DM exists and constitutes nearly 85% of all matter in the universe. However, it is still not known what comprises the DM, how it is formed, and if and how it interacts non- gravitationally with Standard Model particles. In this dissertation, we are by and large concerned with the final question among the aforementioned list of puzzles, i.e., what signatures can DM leave in the observable universe? We propose a new mechanism by which DM can affect the early and late universe. The hot interior of a macroscopic DM, or macro, can behave as a heat reservoir so that energetic photons and neutrinos are emitted from its surface and interior respectively. In Part 1 of this dissertation we focus on the spectral distortions (SDs) of the cosmic microwave background before recombination. The SDs depend on the density and the cooling processes of the interior, and the surface composition of the macros. We use neutron stars as a straw-man for nuclear-density macros and find that the spectral distortions are mass-independent for fixed density. In our work, we find that, for macros of this type that constitute 100% of the dark matter, the µ and y distortions can be near or above detection threshold for typical proposed next-generation experiments such as PIXIE. In Part 2, we study the change in recombination history of hydrogen in the universe caused by macros. We numerically solve the Boltzmann equation to calculate the photon distribution function, electron fraction, and baryonic matter temperature as a function of redshift. During recombination, the excess photons emitted

ix Abstract by macros have frequencies well above the excitation and ionization energies of the hydrogen atom. At such high frequencies, the excitation and ionization processes have extremely small cross-sections, thereby limiting the eﬃciency of the excess photons in ionizing the hydrogen. We therefore ﬁnd that the photons emitted by the macros, across their entire parameter space, do not cause any detectable change in the recombination history of the universe.

x Chapter 1

Introduction

In standard cosmology, the ﬂat Λ−Cold Dark Matter (ΛCDM) universe contains dark energy, ΩΛ ∼ 0.6889 ± 0.0056, followed by matter, ΩM ∼ 0.3111 ± 0.0056, and radiation mostly in the form of photons of the Cosmic Microwave Background (CMB) [7]. While dark energy is a mysterious, as-yet-unexplained component that leads to the accelerated expansion of the universe, matter and radiation cause decelerated expansion. The interactions between matter and radiation in an expanding universe form the basis of many precise observations shaping our understanding of cosmology. Not all of the matter in the universe is well understood. In fact, the known

1 4 forms of matter or baryons such as hydrogen ( H) and helium (2He) only form about 1/6-th of the total matter. The nature of the rest 5/6-th of the matter is another mystery. This dominant form of matter is known as the Dark Matter (DM) of the universe. DM plays a very important role in the evolution of the universe. During structure formation, it forms a scaﬀolding into which baryonic matter collapses and forms galaxies and clusters of galaxies. As will be described in the next section, the only way that we know DM interacts with baryons is through gravity. Cosmological observations also tell us that DM must be “cold” (i.e. non-relativistic), and “dark” (i.e. interact rarely with ordinary matter and radiation) – hence Cold Dark Matter

1 CHAPTER 1. INTRODUCTION

(CDM). It must also interact rarely with itself. Another possibility that can reproduce the same effects as DM is that the fundamental force of nature that governs the dynamics of the universe at large scales, i.e. Einstein’s theory of General Relativity (GR) might not be the correct theory of gravitation. There exist theories such as MOdified Newtonian Dynamics (MOND) proposed by Milgrom which explore this possibility [73, 74, 75]. However, MOND has only been known to explain galactic physics (except very recently by [108]). DM is known to exist outside galaxies to form clusters of galaxies. Moreover, DM played an extremely important role at high redshifts when the universe was a hot plasma of photons, electrons and ionized baryons. The temperature and polarization anisotropies that we observe in the CMB today are related to the DM density, and density and velocity fluctuations in the early universe. At these cosmological scales, DM together with GR describe the universe very accurately. In the next section we will describe some of the key evidence of DM. We will see that CDM solves many problems at large cosmic scales. However, CDM predictions from simulations do not quite agree with observations at small scales. In Section 1.2 we will discuss some challenges to the CDM paradigm. The DM problem has been known since the 1930’s. Since then, many theories of DM have been proposed in literature. We will discuss some popular models of DM in Section 1.3. The most popular DM candidates belong to Beyond Standard Model (BSM) physics. However, considering the tremendous success of known theories of nature, namely the Standard Model of particle physics or simply the Standard Model (SM), and GR, searching for candidates within these well-understood and accurately tested theories needs serious consideration as well. With that in mind, we will not discuss BSM candidates in great detail throughout this work. In Section 1.4 we will introduce macroscopic DM or macros which arise from SM. In Section 1.5, we will discuss the observational constraints on cross-section and mass of macros.

2 CHAPTER 1. INTRODUCTION

1.1 Evidence of Dark Matter

The ﬁrst evidence of “missing mass” in the universe came from the velocity measurements of galaxies in the Virgo and Coma clusters of galaxies. Zwicky in 1933 [128] and Smith in 1936 [109] estimated the mass within these clusters which resulted in the measured velocity dispersions of the galaxies they contained. They found that the cluster mass is about two orders of magnitude larger than the sum of the masses of the individual galaxies. Further evidence came in the 1970s pioneered by Ru- bin [100, 98, 99, 97] who measured the rotation curves of the disks of nearby spiral galaxies, from the Doppler shift of the 21-cm line in atomic hydrogen gas in the disks. Instead of a decrease in velocity predicted by Newtonian mechanics, v ∝ r−1/2, the velocity was found to be constant outside the visible disk of the galaxies. This requires that the mass density of a galaxy is proportional to 1/r2, which implies that most of the mass in a galaxy resides outside the central luminous region. In the case of galaxy rotation curves, MOND provides a suitable alternative (see [36] for a review on MOND). MOND is eﬀective only below a characteristic

−10 −2 acceleration scale, a0 = 1.2 × 10 m s . Above a0, Newtonian gravity works well, meaning GM a = g = b . (1.1) N r2

Below a0, if we assume √ a = a0gN, (1.2) then at a suﬃcient distance from the center of the galaxy,

r V 2 GM max = a = a b . (1.3) r 0 r2

This gives us

1/4 Vmax = (a0GMb) = constant, (1.4)

3 CHAPTER 1. INTRODUCTION

thereby giving a ﬂat rotation curve beyond a certain distance, r, from the center of the

galaxy or below a0. Incidentally, MOND also reproduces the baryonic Tully-Fisher relation (BTFR), given by Equation (1.4) ([72, 115, 13]). BTFR has been seen to be

6 12 remarkably satisfied by galaxies of masses between 10 M − 10 M . Arguably the strongest evidence of DM comes from the power spectrum of the CMB fluctuations. The CMB consists of photons that were once tightly coupled to the ionized baryons and electrons of the universe mainly through Compton scattering. At a redshift, z ' 1100, the ionized baryons became neutral, resulting in free streaming of the CMB photons. Today, we can observe these photons in the microwave regime. The CMB is a near-perfect blackbody radiation with a mean temperature of 2.725 K. Besides the isotropic constant temperature, the CMB temperature has tiny fluctuations of the order of micro-kelvins. The CMB also has fluctuations in its polarization. These temperature and polarization anisotropies originate from the perturbations in the space-time metric, energy densities, and velocities of baryons, photons, CDM, and neutrinos. One can use the Boltzmann equations to follow the evolution of these perturbations, and thereby the temperature and polarization anisotropies, as the universe cools down. From the two-point correlation function of the temperature and polarization anisotropies, we can calculate the power spectrum of the CMB temperature,

TT EE BB TE Cl and polarization, Cl ,Cl , and Cl . One of the hallmarks of the temperature power spectrum is a sequence of oscillations that decay with increasing multipoles, l. The shape of this power spectrum, for example the set of ratios of the peak po-

2 2 sitions, or of the peak heights depend on ΩMh , and baryon density ΩBh , where h is the Hubble constant in units of 100 km s−1Mpc−1. The observed power spectrum from experiments such as PLANCK [7] can only be obtained with a DM density of

2 2 ΩDMh = 0.11933 ± 0.00091, and baryon fraction of ΩBh = 0.02242 ± 0.00014. The evolution of small perturbations continues after the recombination of ionized baryons to form neutral atoms. The evolution of matter density ﬂuctuations

4 CHAPTER 1. INTRODUCTION also provide complementary observations to the CMB anisotropies. Similar to the power spectrum of the CMB calculated from the two point correlation function of the temperature and polarization anisotropies, the matter power spectrum, P (k) can be calculated from the two point correlation function of the matter density fluctuations. Measurement of the power spectral function of matter density fluctuations by the Sloan Digital Sky Survey or SDSS provides the same values of DM density as CMB experiments [91]. At large wave numbers k or small scales, the density and velocity fluctuations of the baryon and photons satisfy the wave equation. The acoustic waves traveling in the baryon-photon plasma cause oscillations in the matter density fluctuations, known as Baryon Acoustic Oscillations (BAO). The strength of these oscillations is proportional to the baryon-matter density ratio, ΩB/ΩM. The BAO has been observed by SDSS and the 2dF Galaxy Redshift Survey and independently confirms the presence of DM [34, 25]. The abundances of nuclei formed primordially through the process known as Big Bang Nucleosynthesis (BBN) can also be used to calculate the baryon fraction of the universe [88, 124]. This is an evidence of the non-baryonic nature of DM. However, since BBN occurred at extremely high temperatures, T ' 109 K, when the universe was just three minutes old, it confirms that DM must have been present at least before BBN, possibly at or even before QCD phase transition [120]. At high temperatures, above 1 MeV, the neutrons, protons, electrons, and the neutrinos are in thermodynamic equilibrium through weak interactions. Below 1 MeV, because of the disappearance of electron-positron pairs, and inefficiency of many weak interactions, only neutron decay to proton continues. The decay continues until neutrons are captured in nuclei through nuclear reactions. More specifically, the neutron decay stops only when a sufficient amount of deuterium is formed. Due to the small binding energy of deuterium, sufficient deuterium is formed at temperatures much below an MeV. This is known as the “deuterium bottleneck”. At about t ' 180 s, the neutron decay

5 CHAPTER 1. INTRODUCTION

4 1 stops, and neutrons form the 2He nuclei. Along with the most abundant elements H

4 and 2He, isotopes and heavy elements like deuterium, helium-3, and lithium-7 are also formed in trace amounts. It is customary in BBN calculations to express abundances of elements as a function of baryon to photon ratio, η. While primordial helium is not sensitive to the value of η, the trace isotope of hydrogen - deuterium (D or 2H) - diminishes rapidly with growing η. Using quasistellar objects as a light source, one can observe the spectrum of these objects as their light passes through intergalactic clouds of primordial hydrogen and deuterium. From the absorption lines, deuterium abundance can be inferred, thereby providing us η. From the CMB blackbody temperature, one can calculate the number density of photons, whereas the CMB power

2 2 spectrum provides us the baryon fraction ΩBh . This allows us to write ΩBh as a function of η. Using the η from spectroscopic observations [56], one can infer the

2 baryon fraction, ΩBh = 0.0214 ± 0.0020, which again points to existence of DM. In fact, it’s the predictions of the abundance of ordinary nuclei from BBN calculations that provided the original evidence for non-baryonic DM, where “non-baryonic” is in the sense that DM cannot be composed of protons and neutrons. The observation of a double galaxy cluster 1E0657-558, also known as the “bullet cluster” at z = 0.296, has provided a picturesque evidence of DM [24]. The cluster consists of two sub-clusters of galaxies and hot gas which is visible through X-ray emission. It is a remnant of a collision between two clusters, with gas within the clusters having interacted with each other, whereas the galaxies within one cluster has passed through the other. Since clusters are perfect candidates for strong gravitational lensing, light bending from galaxies distant than 1E0657-558 have been studied to trace the mass content of the double cluster. It was found that most of the mass in this system is concentrated in the sub-clusters rather than in the central region where the gas is. The fraction of mass contained in the gas is 1/6-th the total mass of the double cluster, quite close to the baryon fraction in the universe.

6 CHAPTER 1. INTRODUCTION

1.2 Challenges to the CDM paradigm

Despite the excellent agreement between DM predictions at cosmological scales and observations, CDM predictions fail to explain observations at scales below ∼ 1 Mpc. This was ﬁrst noticed in the 1990s when N−body simulations could resolve DM subhalos. In this section we will discuss some of the small scale issues. We start with a discussion on the missing satellites problem (MSP) [57, 77]. CDM predictions show that a ﬁeld galaxy such as the Milky Way (MW) ought to have

7 about ∼ 1000 DM subhalos of mass M > 10 M which might host visible galaxies. Observationally, we have only detected about 50 MW satellites with stellar mass

(mass of a galaxy contained in its stars), M∗ > 300M [33]. This number might grow due to the capability of future surveys to detect ultra-faint dwarf galaxies [113, 47]. A more eﬀective way to solve this problem is to postulate that low mass subhalos do not support star formation and hence avoid detection. There are indications supporting this hypothesis. The observed stellar mass function of ﬁeld galaxies above

9 M∗ > 10 M can be extrapolated to lower mass satellite galaxies. Within the limits of the slopes of the observed stellar mass functions, the number of satellites, N, predicted could vary from N ∼ 40 to N > 200. Another notable challenge to CDM is the core-cusp problem [39, 76]. CDM sim-

8 9 ulations predict that for low mass galaxies, M ∼ 10 − 10 M , the rotation curve in the central region should rise quickly like a cusp with a density profile ρ ∝ r−1 (in practice the index is between −1 to −1.4 due to baryonic effects [80]) as compared to the outer region with flat velocity with density ρ ∝ r−2. Observations have shown that the central region in such galaxies have a more slowly rising velocity curve, which is best explained by a constant density core [71, 59]. Besides the discrepancy in the shape of the density function or the ρ − r relation, there is also an issue of central density normalization between simulations and observations: CDM predicts a more dense core than observed [8, 85].

7 CHAPTER 1. INTRODUCTION

A related problem to the MSP and core-cusp problem is the too-big-to-fail problem [17]. As mentioned earlier, a solution to the MSP is that small subhalos do not form stars and hence cannot be observed. One can reverse this argument and ask a related question: how many of the heavier subhalos predicted by CDM simulations, in which case they are too big to have failed to form stars, do we observe? Simulations suggest that there should be more heavier satellites of MW than we observe [18]. This resembles the MSP. Another formulation of the same problem could be: why do the predicted heavier satellites have higher central densities? If they had a constant density core instead, they would be lighter. This is the core-cusp version of the too-big-to-fail problem. Current galaxy formation simulations suggest that baryonic feedback can redistribute DM in dwarf galaxies [69, 92, 67] and therefore offer solutions to the small scale problems. The resultant DM density profile depends on the stellar mass content of the galaxies [43, 30]. However, DM profiles from baryonic feedback simulations also depend on other galaxy formation modeling parameters. The difficulty in simulating baryonic physics makes the small scale problems of CDM subject to speculation.

1.3 Candidates of Dark Matter

We now discuss some popular models of DM. Because DM cannot be composed of protons and neutrons, it is generally assumed that it must be a Beyond Standard Model (BSM) particle. Within the BSM paradigm, the two most well motivated candidates for DM are Weakly Interacting Massive Particles, or WIMPs, and Axions. Assuming that DM is formed of thermal WIMPs - meaning that in the early universe, DM-anti DM annihilation was in thermodynamic equilibrium with light species of the universe such as photons or neutrinos - to explain the observed abundance of DM in the universe, one ﬁnds that the annihilation cross-section is of the order of

8 CHAPTER 1. INTRODUCTION

weak interaction cross-section. This is known as the WIMP “miracle” and the mass of the WIMP must be in the GeV scale. The original candidate for WIMP was thought to be a heavy sterile neutrino. Given that ordinary neutrinos are below 1 eV, these neutrinos could belong to a fourth generation of heavy leptons that interact rarely with the three known generations. Furthermore, the mass of this neutrino should be greater than 45 MeV or half the mass of the Z0 boson because otherwise the Z0 boson would decay into the sterile neutrino-anti neutrino pair. This would not be in agreement with the total decay rate of the Z0 boson both predicted theoretically and conﬁrmed experimentally. Taking into consideration the number of annihilation channels and details of the annihilation process, the neutrino energy density fraction

2 −3 comes out to be, Ωνh < 10 , and hence insuﬃcient by itself to constitute DM. The next plausible candidate for WIMP was proposed to be one of the new particles predicted by supersymmetry: gravitino, the superpartner of the graviton with spin 3/2, or sneutrino, the spin 0 superpartner of the neutrino. Direct detection techniques of WIMPs include elastic scattering with atomic nuclei. The recoil of the atomic nuclei may be observed through collisional ionization or excitation, or by detecting vibrations in a crystal lattice. The direct detection experiments can probe up to a certain limit of the cross-section and mass of the WIMPs. This limit is known as the neutrino ﬂoor below which it is impossible to distinguish between neutrinos and WIMPs. Several direct detection experiments have and are still searching for DM signals: DAMA, CDMS, LUX, XENON to name a few. WIMP-anti WIMP annihilation can also produce gamma rays which could be an in- direct signal of DM (see [42] for a review). Such signals in the GeV energies coming from our galactic center have been observed by the FERMI telescope. However, there is much controversy over the origin of the galactic center excess. While it is true that DM is most dense at the galactic center [6], astrophysical phenomena such as stellar over-density, and the presence of milli-second pulsars in the central region are also

9 CHAPTER 1. INTRODUCTION viable explanations [12, 66]. In addition to nuclei recoil and gamma ray emissions, particle colliders are also searching for new physics. Axion is a prediction of an elegant solution to the strong charge conjugation and parity (strong-CP) problem in QCD [87]. The QCD lagrangian has two terms, one is the perturbative part, and the second is topological. In QCD, the topological/non- perturbative part strongly violates CP. Unless it is suppressed, it would lead to an electric dipole moment of the neutron ten orders of magnitude than what is observed experimentally. The CP violation was shown to be suppressed dynamically by introducing a global symmetry, known as the Peccei-Quinn symmetry. The spontaneous breaking of this symmetry results in a Nambu-Goldstone boson, the axion [116, 118]. In the original axion model, the energy-scale at which the Peccei-Quinn symmetry gets broken is of the order of electroweak symmetry breaking scale, around 100 GeV. If this were true, axions must be so weakly interacting that they are emitted from nuclear reactor cores and stellar interiors without any absorption leading to enhanced cooling eﬀects, the absence of which ruled out the original axion. The cooling rates of red giant stars by axion emission rule out symmetry breaking energy scales below 107 GeV [31]. Similarly, supernova SN1987A rules out energy scales below 1010 GeV [114]. Since the mass of the axion is inversely proportional to the symmetry breaking scale, the two observations provide an upper limit on the axion mass. Experimentally, axions can be observed by their conversion into photons in intense electromagnetic ﬁelds. Current limits restrict axion mass to be below 10−6 eV.

1.4 Macroscopic dark matter or macros

The ongoing infertility of particle DM searches, whether for WIMPS or axions, sug- gests that candidates arising from SM and GR return to serious consideration. Pri- mordial black holes (PBH) are very promising candidates because black holes are

10 CHAPTER 1. INTRODUCTION

naturally formed in GR. Gravitational wave detections of binary black hole mergers have revived interest in PBHs as DM candidates [2, 5, 3, 4]. PBHs can form in the early universe through gravitational collapse of highly overdense regions [21, 79, 15]. Such overdense regions can be caused by large primordial curvature perturbations during inflation. However, standard inflationary mechanisms cannot produce such large perturbations since such inflation will have a large spectral index which is already ruled out by CMB observations. PBHs can be produced in exotic inflationary mechanisms such as inflation with a small “plateau” in its potential [35, 52, 40, 78], multi-field inflation [106, 96], and other mechanisms which enhance primordial curvature perturbations at small scales. One characteristic of PBHs which make them attractive candidates for DM is that properties of black holes depend on very few parameters, mass and spin. In fact observationally, only the mass of the PBH is a free parameter. Current observations rule out PBHs of masses below 1015 g, since they would have evaporated by now through Hawking radiation. This would create extra relativistic degrees of freedom which are constrained by BBN, while an additional background of photons can be ruled out by CMB, galactic and extragalactic γ-ray and cosmic ray backgrounds. Further astrophysical constraints such as microlensing, disruptions of stellar objects through passage of PBHs, accretion effects, seeding cosmic structures, and gravitational waves rule out a large mass range of PBHs allowing

3 only the following windows: intermediate mass black holes, 10M < M < 10 M , lunar mass 1020g < M < 1024g, asteroid mass 1016g < M < 1017g, and Planck-mass relics of Hawking evaporation of mass ∼ 10−5 g. See Figure 10 of the review article by Carr and others [20] for current constraints on PBH masses and the sources of the constraints. With this brief introduction to PBHs, we turn our attention to macroscopic dark matter or macros. These have the particular virtue that they may be purely SM objects built of quarks or baryons. In this case they must have been formed before

11 CHAPTER 1. INTRODUCTION the freezeout of weak interactions at t ' 1s and the subsequent onset of BBN, if the success of the standard theory of BBN in predicting light-element abundances is to be preserved (although see [32]). A question relevant to macros is what makes macros weakly interacting with baryons and electrons of the universe even though they are themselves composed of SM particles. This can be understood as follows. Consider the interaction frequency of DM and baryons per baryon, Γ = nX σX,bv, where nX is the DM number density,

σX,b is the DM-baryon interaction rate, and v is the relative velocity between DM and baryons. In the case of particles DM candidates such as WIMPs, the cross- section, σX,b is small, making DM interact weakly with baryons or SM particles. However, there is another alternative to make Γ small. Since DM is non-relativistic, nX = ρX /MX , where ρX is the energy density of DM in the universe, one can make

MX very large so that nX is very small. This would make the interaction frequency,

Γ, very small too. In addition to making MX very large, one also needs σX or the effective cross-section, σX /MX , to be small. Thus, in the case of macros and even PBHs, the rarity of interactions makes them inert to SM particles. Witten first suggested [120] that DM could be composites of up, down and strange quarks assembled during the QCD phase transition. Subsequent proposals have included purely SM objects made of quarks [37] or baryons [65, 64] of substantial average strangeness. A variety of BSM variations also exist (e.g., [126]). Several authors have focused on the observational consequences ([29, 95, 51]). A very important consideration for the various models of macros is the question of durability. If macros aren’t stable during the lifetime of the universe, they might decay into ordinary baryons or quarks. A related question is: What is the ground state of QCD with non-zero baryon number? In the case of baryons composed of up and down quarks, ordinary nuclei seems to be the most stable configuration. The question becomes wide open when strange quarks are present with up and down quarks.

12 CHAPTER 1. INTRODUCTION

Witten showed that the Fermi momentum of up and down quark matter is 300-350

+11 MeV, greater than the strange quark mass, 93−5 MeV [120]. Thus it is energetically favorable for some nonstrange quarks to become strange quarks. Subsequent models required some amount of strangeness in the presence of some external fields such as kaon condensate [65], or quarks trapped inside domain walls [63] to make macros stable. Another related question is the epoch of macro formation. Macroscopic bound states of fermions (e.g. quarks or baryons) cannot be formed by gravitational collapse of adiabatic fluctuations in the radiation dominated era. They would arise typically from non-adiabatic fluctuations [120, 81] or topological defects [63] (e.g. from phase transitions) that enhance the fermion abundance relative to that of the radiation. Although there are stringent constraints on kaon or pion condensates, hyperons, and strange quark matter inside observed 2-M or heavier neutron stars [60, 61], these states may (or may not) be found in lighter neutron stars. Moreover, these exotic hadronic or quark matter equations of state (EOS) are theoretically allowed; hence, one should not abandon the possibility of their playing a role in the structure of macros, which are certainly not the endpoints of ordinary stellar evolution. The mass functions of macros are model-dependent and therefore difficult to predict [81]. Throughout this work, we do not discuss the origin of the macros, but concern our- selves with their detection.

1.5 Constraints on parameter space of macros

The presence of dense assemblages of quarks or baryons from before BBN through today would undoubtedly have as-yet unexplored observational consequences, no matter the speciﬁc mechanism of their formation or stabilization. Novel physics peculiar to such macros, with potential observational consequences include:

13 CHAPTER 1. INTRODUCTION

1) slow pre-recombination cooling of the macro compared to the ambient plasma: a. distorting the spectra of the cosmic microwave and neutrino backgrounds (CMB and CNB), b. heating the post-recombination universe, or c. contributing to the cosmic infrared background; 2) production of nuclei (including heavy nuclei) through: a. ineﬃciency in macro assembly at formation, b. evaporation, sublimation or boiling, especially xxx soon after macro formation, c. macro -macro collisions; 3),formation of binary macros, with potential gravitational-wave and electromagnetic signals; 4) DM self-interactions, especially in high-density environments such as galactic cores; 5) enhanced thermal and dynamical coupling of dark-matter to baryons and photons. These primary processes could have important secondary consequences, including implications for early star formation, assembly of supermassive black holes, and 21-cm emissions.

We will now discuss the current observational constraints on the mass, MX ,

2 and cross-section, σX = πRX where RX is the radius, of the macros. The current constraints are provided in Figure 1.1. Notice that unlike PBHs, macros below

15 MX < 10 g do not evaporate and therefore there is no reason to rule out that part of the parameter space. Non-observation of approximately nuclear-density macros

> through the tracks they would have left in 500 Myr old mica [51] demands mX ∼ 55g. Limits on larger macro masses have been obtained from gravitational micro-lensing

< 20 17 < < 20 (mX ∼ 2 × 10 g) [127, 46, 82] and femto-lensing (excluding 10 ∼ mX ∼ 2 × 10 g). These limits as quoted assume that the DM consists of macros of a single mass – an unlikely situation for a composite object.

14 CHAPTER 1. INTRODUCTION

1025 Atomic density 1020 Nuclear density Black holes 1015

] 10

2 10

m 5

c 10 [

X 100

10 5

10 10

10 15 101 105 109 1013 1017 1021 1025 1029 MX [g]

Figure 1.1: Constraints on mass and cross-section of macros from cosmological and astrophysical phenomena. The white region in the lower right corner is ruled out because it is denser than black holes. The grey region is ruled out by CMB constraints [119]; the yellow from mica observation [29, 94]; the red from superbursts in NSs; the dark blue from white dwarfs becoming supernovae [44]; the purple from a lack of human injuries and deaths [107]; the green from a lack of fast-moving bolides [105]; the maroon from a lack of microlensing events toward the Large Magellanic Cloud and the Galactic center [9, 112, 19, 45], and in pink toward M31 [83]. Image credit: Figure 1 in arXiv:2006.01200v2, Journal reference [104].

Cosmological constraints on macros, whether from the CMB or large scale structure, do not yet impinge on generic nuclear-density objects. They do rule out atomic density (1 g/cc) and lower density objects. The grey region in Figure 1.1, which rules

−7 2 −1 out σX /MX > 4.5 × 10 cm g , comes from CMB constraints: more interaction between photons and macros would lead to collisional damping of the CMB, aﬀecting the CMB acoustic peaks [119]. With this introduction to DM and macros, we will study the eﬀects of macros on some of the observables of the universe. Through current and future cosmological

15 CHAPTER 1. INTRODUCTION observations, we will put constraints on the mass and size of the macros. By studying eﬀects other than gravitation and particle annihilation and decay, we claim that what follows below is a novel way to explore DM phenomena.

16 Chapter 2

CMB Spectral Distortions from Cooling Macroscopic Dark Matter

The work described in this chapter is largely based on the paper published in Phys. Rev. D 99, 023521 (2019)/arXiv:1804.08601.

In this chapter, we focus our attention on the effect on the CMB and CNB of macroscopic objects that generically cool by volume emission of neutrinos and surface emission of photons. (BSM candidates may have additional cooling mechanisms.) By considering a specific example of a baryonic macro – a neutron star (NS) – as a macro, we demonstrate that the weak coupling of neutrinos to baryons and the inefficiency of surface cooling by photons generically lead the macros to remain significantly hotter than the ambient plasma through the epoch of recombination. Both energy and entropy are therefore injected into the plasma in the form of photons and neutrinos well after the time when thermal or statistical equilibrium can be restored. The CMB and CNB spectra are thereby distorted. In this first work, we characterize the distortion in terms of the traditional µ-type (photon-number excess) and y-type (photon-energy excess) spectral distortions (SD)

17 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

of the CMB, and by ∆Neff , the increase in the effective number of neutrino species. However, because the temperature of the macros can remain far above that of the ambient plasma, and because the cooling is ongoing through and after recombination, neither µ nor y will fully capture the shape of the resulting distortion. This will be considered in future work, as will the angular fluctuations in the distortion, its correlation with other observables, and other potential consequences of baryonic macro DM, as listed above. The magnitude of SD caused by macros is controlled of course by their abundance, but also by their specific internal physics. For NS material this includes: the thickness and insulating properties of the non-degenerate crust; the equation of state of the neutrino-emitting core, in particular the presence/absence of a superconducting phase and its detailed properties. For a solar-mass NS, known or anticipated properties result in µ and y-type distortions of the CMB that are potentially above the threshold of detection by feasible

next-generation SD experiments, and ∆Neff that are not. These speciﬁc conclusions will change for other microphysical models of macros, but may be instructive of what to expect and why. To our knowledge, this is the ﬁrst study of the radiative cooling of DM and the CMB spectral distortions it may cause. The only macroscopic objects of nuclear density known to exist in nature are

NS formed as endpoints of stellar evolution of stars of mass M ≥ 8M through

gravitational collapse of their cores. Most ordinary NS have a mass of M ∼ M and radius R ∼ 10 km. The core consists of mostly neutrons, and some protons electrons, and possibly even hyperons or other particles. A NS is born at an initial temperature of T ∼ 1011 K, but cools down rapidly due to powerful neutrino emission from the

internal layers. These appear to have masses below 2.2M [68, 10], well below the total mass within the horizon at z ∼ 109 (or even 1012, the epoch of quark conﬁnement and chiral symmetry breaking). We therefore use an ordinary NS as a proxy for a

18 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

14 3 macro. We take the macro’s central density to be ρX ' ρN ≡ 2.8 × 10 g/cm , which we refer to as nuclear density. Although microlensing limits preclude all the DM being NS, the macro mass function could include a sizable contribution from them.

Neutron stars theoretically are stable down to (0.09 − 0.19)M [26, 16, 93, 14], but do not appear to arise as the endpoints of the evolution of main-sequence stars below ∼ 1.2M . If formed in the early universe, these would be larger and of lower average density than post-stellar neutron stars. This motivates us to consider NS-like macros of somewhat lower-than-nuclear density.

The discovery of a NS with MNS < 1.2M would be exciting evidence for early- universe macro formation. Smaller-still composite baryonic objects require non-gravitational stabilization, whether within the SM through the incorporation of strange quarks/baryons [120],[65],[64] or by more exotic BSM mechanisms. Such SM or BSM baryonic composites may also exist in the mass range that includes stable NS. This chapter is organized as follows. In the next section, we provide a brief introduction to the mechanisms and characterization of SDs. In Section 2.1 we provide an outline for the Boltzmann equation. We then describe the various scattering processes in Section 2.1.2. In Section 2.1.3 we characterize the SDs. In Section 2.2, we discuss the neutrino and photon emission processes that cool the macro and calculate the SDs created by photon emission from the surface of macros. In Section 2.3, we present our conclusions. We provide a derivation for the photon luminosity of the macro, and describe the neutrino cooling processes in more detail in the appendices.

2.1 Spectral Distortions in the CMB

The COsmic Background Explorer (COBE), Far InfraRed Absolute Spectrophotome- ter (FIRAS) showed that the CMB is a near-perfect blackbody (BB) of tempera-

19 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

ture 2.725 ± 0.001 K, and any deviations to this BB spectrum must be less than,

< −5 Iν/I ∼ 10 [38, 70]. This groundbreaking discovery was awarded the Nobel Prize in Physics in 2006 and already constrains cosmological models that allow long peri- ods of excessive energy release in the CMB. However, there exist mechanisms which must have created very tiny spectral distortions (SDs) in the CMB spectrum. Some of the known mechanisms which arise from well understood baryon-photon interactions include reionization [48] and structure formation [84] , dissipation of primordial density ﬂuctuations [11, 27], adiabatic cooling of matter [55], cosmological recombination [110] and many other processes that add/subtract photons or redistribute energy in the CMB. Such SDs can be measured by future CMB missions such as PIXIE (Primordial Inﬂation Explorer) and PRISM (Polarized Radiation Imaging and Spectroscopy Mission) and can constrain already known physics. In addition to these known mechanisms, exotic mechanisms can also cause SDs such as decaying or annihilating particles in the early universe, cosmic strings, and PBHs.

6 At very high redshifts, z > zµ ≡ 2×10 , distortions would be wiped out by eﬃcient photon number and energy-changing interactions such as Compton scattering [50], double Compton emission [28] and Bremsstrahlung. Below this redshift, the distortion in the CMB spectrum is mainly characterized as a µ or a y distortion [125, 111]. At

4 6 higher redshifts, 5 × 10 . z . 2 × 10 , photon number-changing mechanisms such as double Compton emission and Bremsstrahlung are ineﬃcient, and photon injection results in a ﬁnite chemical potential in the Bose-Einstein distribution of photons,

4 the µ distortion. At lower redshifts, z . 5 × 10 , energy redistribution by Compton scattering becomes ineﬃcient, leading to y-type distortions. The intermediate era, z ≈ 104 −105, is also characterized by i-type distortions [53]. The current upper limits on the µ and y distortions from COBE-FIRAS are |µ| < 9×10−5, and y < 1.5×10−5. PIXIE and PRISM promise to probe SDs of magnitudes µ, y ∼ 10−8.

20 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

Figure 2.1: Measurements of the CMB spectrum. Image source: COBE webpage [1].

2.1.1 The Boltzmann equation

We now describe how photon addition/absorption or energy redistribution processes distort the BB spectrum of the CMB. To understand that, we have to solve the Boltzmann equation. The evolution of the CMB distribution function, f(rµ, pµ), where rµ = (t, r) is the position four-vector, and pµ = (hν, p) is the momentum four-vector, is given as

df = C + C + C + C (2.1) dt K DC BR S

where CK,CDC,CBR are the Compton (Kompaneets equation), double Compton, and

bremsstrahlung collision terms respectively. The last term, CS, is the source term which can arise from DM annihilation or decays, or in our case the photons radiating from the surface of the macros. We can expand the derivative term on the left hand

21 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

side of the above equation

df ∂f drµ ∂f dpµ = + (2.2) dt ∂rµ dt ∂pµ dt ∂f ∂f dri ∂f dν ∂f dpi = + + + . ∂t ∂ri dt ∂ν dt ∂pi dt

Since the mean CMB distribution function is isotropic, we can write

df ∂f dν ∂f = + (2.3) dt ∂t dt ∂ν ∂f ∂f = − Hν ∂t ∂ν

Alternatively, one can express f as f = f(t, x) where x = hν/kBTγ so that

df ∂f = (2.4) dt ∂t

since dx/dt vanishes.

2.1.2 Thermalization/Scattering mechanisms

We now discuss the most important scattering mechanisms between photons and electrons - Compton and double Compton - and electrons and ions - Bremsstrahlung. These mechanisms thermalize the CMB at high redshifts. On the other hand, as the universe cools down, their ineﬃciency causes SDs.

2.1.2.1 Compton scattering

Compton scattering is an energy redistribution process, given by the interaction between an electron and a photon:

e(p) + γ(k) → e(p0) + γ(k0). (2.5)

22 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

At high redshifts, the energy of any photons injected in the CMB is redistributed in frequency by the Compton process, thereby erasing any distortion eﬀect created by the injection. At lower redshifts however, the Compton interaction of photons

−1 −2 (Tγ ∝ a ) with baryons and electrons which cool as Te ∝ a keeps them in thermal equilibrium. As a result, the photons lose energy and the Compton process then becomes a mechanism for SDs. The term CK is given by the Kompaneets equation

∂f θe ∂ 4 ∂f Tγ ≈ 2 x + f(1 + f) (2.6) ∂τ x ∂x ∂x Te

2 where τ is a unitless quantity such that dτ = cσTnedt, and θe = kBTe/mec . In the above expression, the ﬁrst term in the brackets describes the Doppler scattering which describes the case where the electrons are relativistic. The second term in the bracket describes the recoil dominated scattering where the electrons are non-relativistic. One can verify that the Kompaneets equation satisﬁes dNγ/dt = 0 where

Z 2 Nγ ∝ x fdx, (2.7)

and thereby conserves photon number, Nγ.

The energy density of photons, ργ is proportional to

dρ Z ∂f γ ∝ x3 dx. (2.8) dt ∂t

This gives us eq dργ Te ≈ 4θeργ 1 − (2.9) dt Te where R x4f(1 + f)dx T eq = T . (2.10) e γ 4 R x3fdx

eq This means that when Te < Te, the CMB absorbs energy from the hot electrons, a

23 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER process known as Comptonization. The time scale of energy transfer from electrons to photons is given as

−1 tT teγ = (4θecσTne) = (2.11) 4θe = 1.2 × 1029(1 + z)−4 s.

4 The time scale teγ becomes larger than the Hubble time at redshift z ∼ 5 × 10 below which the CMB starts heating up the electron plasma thereby losing energy. This process is known as the Compton cooling process. It is far more eﬃcient than Comptonization since for every electron there are ∼ few × 109 photons. The time scale of Compton cooling, tγe is given as

3n(1 + fHe + xe)kBTe tγe ≈ teγ (2.12) 2ργ = 7.3 × 1019(1 + z)−4 s.

where n is the baryon number density, fHe is the helium number density fraction which is about 0.08, and xe is the electron fraction. Compton cooling keeps the residual electrons (and baryons through Coulomb scattering) of the universe in thermal equilibrium with the CMB even after recombination, until z ∼ 200.

2.1.2.2 Double Compton emission

The double Compton (DC) emission process describes the scattering of electron and photon that releases two photons.

0 0 e(p) + γ(k) → e(p ) + γ(k ) + γ(k2) (2.13)

Because of the very small baryon to photon ratio, the DC process is quite eﬃcient at high redshifts. However for low photon energies, this process is suppressed and

24 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

ineﬃcient than the Compton process because of the additional electron propagator and the additional vertex factor, α = 2πe2/hc = 1/137. The collision term for DC is given as

∂f K e−2x ≈ DC [1 − f(exe − 1)] (2.14) ∂τ x3 where 4α K = θ2I g (T ,T , x), (2.15) DC 3π γ dc dc e γ

and the Gaunt factor, gdc is given as

3 29 2 11 3 5 4 1 + 2 x + 24 x + 16 x + 12 x gdc ≈ . (2.16) 1 + 19.739θγ − 5.5797θe

2 In the above expressions, θγ = kBTγ/mec . Furthermore, Idc is given by the integral

Z 4π4 I ≈ x4 f [1 + f ] dx = . (2.17) dc BB BB 15

The above expressions for DC have been calculated for the case where the second emitted photon is of the same energy as the incoming photon (beyond the soft limit approximation), and the electrons are fast moving. The Gaunt factor is a correction term which captures the deviation from the soft photon limit and the assumption of the electron being at rest initially.

2.1.2.3 Bremsstrahlung

In the early universe, thermal Bremsstrahlung (BR) process given by

e(p) + H+(h) ↔ e(p0) + H+(h0) + γ(k) (2.18)

25 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

were important. We only need to consider electron-ion processes since the e−e− processes are ineﬃcient. In the case of BR, both emission and absorption processes are important. It is a lowest order radiative correction to the Coulomb scattering process. The BR collision term is given as

−xe ∂f KBRe xe ≈ 3 [1 − f(e − 1)] (2.19) ∂τ xe where α λ3 X K = √ e Z2N g (Z ,T ,T , x ) (2.20) BR 2π 6πθ i i ﬀ i e γ e e i and the BR Gaunt factor is given as

 √  3 ln 2.25 for x ≤ 0.37,  π xe e gﬀ(xe) ≈ (2.21)  1 otherwise.

In the above expressions λe = h/mec, Zi,Ni is the charge and number density of the ion, and the summation i is over hydrogen and helium ions. The time scale of BR process is given as

3 xe tT tBR ≈ −x (2.22) KBR(1 − e e ) g −1 Ω h2 −1 x3 ≈ 1.6 × 1026 ﬀ b e (1 + z)−5/2 s. 3.0 0.022 1 − e−xe

< One can verify that for low frequencies, xe ∼ 0.01, BR is highly eﬃcient above redshift,

> 5 z ∼ 10 .

2.1.3 Types of Spectral Distortions

We now describe how to deﬁne the two most important types SDs: µ and y distortions. In what follows below, we utilize a number of simplifying assumptions which will give

26 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

us insights about the behavior of each of the two types of distortions. In practice though, these assumptions are rarely true meaning that solutions can only be arrived at numerically. However, such numerical solutions do not provide a clear view behind the physics of SDs.

2.1.3.1 µ distortions

> 4 µ distortions take place at higher redshifts, µ ∼ 5 × 10 . At such redshifts, photon- electron scatterings are very frequent and energy redistribution of photons is highly eﬃcient. The spectrum of the CMB reaches equilibrium through Compton scattering. Neglecting photon number violating processes, the Boltzmann equation can be written as θe ∂ 4 ∂ Tγ 0 ≈ 2 x f + f(1 + f) (2.23) x ∂x ∂x Te

The general solution to the above equation is the solution of the equation

∂ T f + γ f(1 + f) = 0, (2.24) ∂x Te

which is given by a Bose-Einstein distribution

1 feq = (2.25) exe+µ0 − 1

where xe = hν/kBTe and must not be confused with the electron fraction, and µ0 is

the integration constant. If µ0 < 0, then there are more photons than that given by

a Planck distribution at temperature Te, meaning that µ distortion source is photon

injecting, whereas µ0 > 0 is a photon absorption source. The case xe + µ0 = 0 isn’t realized in practice because BR and DC prevent this from happening. When BR and DC processes are also considered, the µ distortion is no longer constant in frequency, which makes µ = µ(x).

27 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

It is quite convenient to express the µ(x) distortion as a product of the magnitude

of the distortion, µ0, which is independent of frequency and the spectral function, M(x), which describes the change in shape of the distribution function as a function of frequency,

µ(x) = µ0M(x) (2.26)

Magnitude of µ distortion

Let us now calculate the magnitude of the µ distortion, µ0. Consider the CMB with

an initial blackbody distribution at temperature Ti. Since µ distortions take place at high redshifts, we can assume thermal equilibrium of electrons with the CMB, i.e.

Te = Tγ = Ti. After an energy injection, let the equilibrium electron temperature be

BE BE Tf . The equilibrium photon number density, Nγ , and the energy density, ργ , can be written as

3 Z 2 BE Pl 8π kBTf x Nγ = Nγ (Ti)(1 + N ) = dx (2.27) c3 h ex+µ0 − 1 Pl 2 N (Tf ) Z x = γ dx Pl x+µ0 G2 e − 1 and Pl Z 3 BE Pl ργ (Tf ) x ρ = ρ (Ti)(1 + ρ) = dx (2.28) γ γ Pl x+µ0 G3 e − 1

Pl respectively. We deﬁne the quantity, Gi , as

Z xi GPl = dx (2.29) i ex − 1

which are basically the Riemann zeta functions. N , and ρ are changes in the number density and energy density respectively in the photon spectrum at equilibrium after energy injection. All distortions are assumed to be small.

28 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

0 Using change of variables, x → x+µ0, and assuming µ0 1, we can approximate the integrals in (2.27) and (2.28) as

Z 2 x Pl Pl dx = G2 − 2µ0G1 , (2.30) ex+µ0 − 1 Z 3 x Pl Pl dx = G3 − 3µ0G2 . ex+µ0 − 1

c Pl Pl Deﬁning M2 = kGk−1/Gk and ∆T = Tf − Ti, we get

∆T c N = 3 − µ0M2 (2.31) Ti ∆T c ρ = 4 − µ0M3 . (2.32) Ti

This gives us the deﬁnition of the magnitude of µ distortion, µ0 as [49]

∆ργ 4∆Nγ µ0 = 1.401 − . (2.33) ργ 3Nγ

The relative change in temperature is given as

∆T ∆Nγ = 0.4561µ0 + . (2.34) Ti 3Nγ

In an adiabatic process where ∆ργ/ργ = (4/3)∆Nγ/Nγ, the distortion vanishes and

the resultant distribution is still a blackbody but with the new temperature, Tf =

Ti(1 + ∆Nγ/3Nγ). Spectral shape of µ distortion Let us now calculate M(x), the spectral shape of the µ distortion. There are a couple of ways to do it.

29 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

Firstly, using Taylor expansion, we can expand the Bose-Einstein distribution as

1 1 G(xe) fBE = x +µ ≈ x − µ0 (2.35) e e 0 − 1 e e − 1 xe

which gives us xe G(xe) −µ0 e ∆f = − µ0 = x 2 . (2.36) xe (e e − 1)

In this definition, the spectral function, M = −G(xe)/xe, is also a function of µ0 through xe. This definition has been used in literature. However, one can also define the spectral function in such a way that it is independent of µ0. Similar to Compton scattering where the number density of the photons is con- served, let us define M(x) as

1 M(x) = G(x) a − . (2.37) µ x

Pl Pl where aµ = 2G2 /3G3 . One can verify that

Z x2M(x)dx = 0. (2.38)

Using an additional normalization factor of 3/κ where

Pl Pl G1 G2 κ = 8 Pl − 9 Pl (2.39) G2 G3

so that ∆ρ/ρPl = 1, one arrives at the spectral function M(x) which is given as

3 1 M(x) = G(x) a − . (2.40) κ µ x

Thus, we have expressed the spectral function completely as a function of x or Tγ =

T0(1 + z). We will use this deﬁnition from this point on.

30 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

2.1.3.2 y distortion

At lower redshifts, z < 1.5 × 104, when DC and BR emissions are ineﬃcient, the Boltzmann equation is given by

∂f θe ∂ 4 ∂ Tγ = 2 x f + f(1 + f) . (2.41) ∂τ x ∂x ∂x Te

Let’s start with a blackbody spectrum, f = fbb, at τ = 0. If ∆τ 1, then it can be showed that the change in the occupation number is given as

xex ex + 1 ∆f ≈ ∆τ(θ − θ ) x − 4 (2.42) e γ (ex − 1)2 ex − 1

= ∆τ(θe − θγ)YSZ(x)

= y YSZ(x).

As one can see from the above equation, the change in occupation number, ∆f, has been expressed as a product of the magnitude of distortion, y, and the spectral

function, YSZ(x), similar to the case of µ distortion. Examples of y distortion mechanisms include CMB photons scattering from hot

7 clusters where Te ∼ 10 K, in which case y > 0 [125]. Adiabatic cooling of electrons create negative y distortions since non-relativistic electrons cool as T ∝ 1/a2 due to cosmic expansion, but are maintained in thermal equilibrium with the hotter CMB due to Compton scattering [23]. The change in energy density of the CMB is given as

Z 3 ∆ργ ∝ y x YSZ(x) dx (2.43)

Pl = 4 y ργ .

31 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

Thus, one can estimate the y distortion as

Pl y = ∆ργ/4ργ (2.44) which serves as the deﬁnition of the y distortion magnitude. Finally, we can write down the change in distribution function of the CMB due to µ and y distortions as

∆f = M(x) µ0 + YSZ(x) y. (2.45)

with YSZ given by xex ex + 1 Y = x − 4 (2.46) SZ (ex − 1)2 ex − 1 and y, M(x), µ0 given by (2.44), (2.40), and (2.33) respectively. How to distinguish between µ and y distortions? If we look at the shape function of the µ and y distortions at low frequencies, we get

3 1 −1.401 M(x) = G(x) a − ≈ for x 1, (2.47) κ µ x x2 xex ex + 1 −2 Y = x − 4 ≈ for x 1. (2.48) SZ (ex − 1)2 ex − 1 x

From the above equations we notice that the µ distortion shape decreases more sharply at lower frequencies than the y distortion shape. This is a tell-tale signature of a µ distortion.

2.2 Cooling processes for macros

In this section, we provide expressions for the neutrino and photon luminosities from the interior and surface of the macro respectively. We then arrive at the temperature dependence of the interior of the macro.

32 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

atmosphere outer crust e,Z inner crust e ρd ,Z,n 0.5 ρ 0 e, p,n m outer core k 6 -1 8 2 ρ 0 inner core

ρ hyperons? pions? quarks?

(1.5−15) ρ 0

Figure 2.2: Multiple layers of a neutron star. Image credit: Figure 1 from arXiv:astro- ph/9906456v1, Journal reference [123].

Except for the inner core, the composition of which is still under debate, a NS is composed of neutrons, protons, electrons and heavy ions. After formation, it cools down via neutrino emission from the interior and photon emission from the surface. An ordinary NS is multi-layered as shown in Figure 2.2. However for our purposes, a very simplistic, bi-layered model of the NS is suﬃcient. It exactly captures the phenomenon we are interested in: the radiative eﬀects of hot, but non-relativistic DM in the universe. The cooling of a NS undergoes a transition when the core of the NS becomes degenerate. The Fermi energy of neutrons follows the equation

2 ~ 2 2/3 EF,n = 3π nn (2.49) 2mn ρ 2/3 = 60 MeV ρN

where mn, nn is the mass and number density of neutrons respectively. ρ is the NS

33 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

14 core density and we remind the reader that ρN ∼ 2.8×10 g/cc is the nuclear density. Assuming that inside the core of the NS, the neutrons, protons, electrons, and neutrinos are in thermodynamic equilibrium

− e + p n + ν, (2.50)

the chemical potentials of the non-relativistic species should follow the relation

µe + µp = µn. (2.51)

The chemical potentials are to a very good approximation the Fermi energies of the particles. The Fermi energy of non-relativistic neutrons and protons is given as

2 pF (n, p) EF (n, p) = (2.52) 2mn,p

whereas for relativistic electrons is given as

EF (e) = pF (e)c. (2.53)

3 Since number density, n, is proportional to pF , charge equality, ne = np gives us

pF (e) = pF (p). (2.54)

Using (2.52), (2.53), and (2.54) in (2.51), one arrives at

2 pF (n) pF (e)c ≈ (2.55) 2mn

which gives us ρ 2/3 EF (e) ≈ EF (n) = 60 MeV (2.56) ρN

34 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER whereas for protons,

2 2 4/3 pF (p) EF (n) ρ EF (p) = = 2 = 1.9 MeV. (2.57) 2mp 2mpc ρN

Thus, we see that EF for nuclear density is about 60 MeV for electrons and neutrons, and about 1.9 MeV for protons. We can therefore safely set 109 K ≈ 0.86 MeV as the degeneracy temperature for the core of the NS. Neutrino cooling occurs mainly through three processes: 1) direct Urca (DUrca)

− − n → p + e +ν ¯e , e + p → n + νe takes place at high temperatures, when neutrons and electrons are non-degenerate, but may also be important below the degeneracy temperature; 2) modiﬁed Urca (MUrca) in the neutron and proton branches

− − n + n → n + p + e +ν ¯e, n + e + p → n + n + νe

− − n + p → p + p + e +ν ¯e, p + e + p → n + p + νe is dominant at T < 109 K, when neutrons and electrons are degenerate; 3) nucleon Cooper pair (CP) cooling

N˜ + N˜ → CP + ν +ν, ¯

˜ > > (where N is a quasi-nucleon) is most eﬃcient for 0.98Tc ∼ T ∼ 0.2Tc, with neutrons in the NS interior become superconducting at Tc [123]. The luminosity of these neutrino-cooling processes is

i 45 −1/3 Lν = 10 Ci (MX/M )(ρX/ρN ) erg/s (2.58)

35 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

for i = DU, MU, CP . MX is the mass of the macro. ρX is the density of the core, and it partly characterizes our ignorance of the precise properties of the macro. T dependence is encoded in

  X 6 D 5.2 (T9 ) R i = DU   Ci = (3.0RM + 2.4RM )10−6 (T X )8 α i = MU  n p 9   −6 −1/3 X 7 7.1 × 10 (ρX /ρN ) (T9 ) a F i = CP.

T X is the macro’s internal temperature; the subscript 9 will be used for a temperature

9 2 2 −2 2 2 −1 in units of 10 K. In the above equation, α ' 2 (1 + mπ/pF (n)) −0.3 (1 + mπ/pF (n)) +

1/3 D M 0.07 where pF (n) = 340(ρX/ρN) MeV/c is the neutrons’ Fermi momentum; R , Rn

M and Rp ≤ 1 are reduction factors due to superfluidity [62, 121], a [86], and F [122] are the dimensionless factor and the control function respectively both of which depend on the type of superfluidity. The factors in the above expressions which depend on superfluidity have been discussed briefly in Appendix (B). The macro photon luminosity is

2/3 45 MX /ρX s4 CMB4 Lγ = 10 T9 − T9 erg/s (2.59) M /ρN

where T s is the surface temperature of the macro, and T CMB is the temperature of the ambient plasma. We assume that the macros have coalesced, and we can begin following their cooling from when the temperature of the ambient plasma is 109 K, at z = 3.7 × 108. (This is after any electroweak and QCD-associated phase transitions [81].) We take the macro to be isothermal at that moment with temperature equal to that of the plasma. The interior electrons, neutrons and protons will be degenerate. The cooling of neutron stars below this temperature has been well explored, and we have veriﬁed that our conclusions are insensitive to the details of the macro cooling before this

36 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

epoch. We assume that the macro, like a NS, has a degenerate isothermal interior contain- ing neutrons, protons and electrons, and a non-degenerate “atmosphere” of electrons and heavy ions. This keeps the interior warm as the ambient plasma cools. For con-

stant atmospheric photon luminosity Lγ the atmospheric density and temperature are related by

1/2 10 µ(MX/M )erg/s 3.25 ρatm = 1.2 × 10 ρN T9 (2.60) Z(1 + W )Lγ where µ, Z and W are the mean molecular weight, metallicity or mass fraction of elements heavier than hydrogen and helium, and mass fraction of hydrogen. In case of a low metallicity atmosphere, the Kramer’s opacity due to bound-free transitions assumed above will be exceeded by the opacity due to free-free transitions, which does not vanish for low metallicity. This means Eq. (2.60) will not diverge for small Z.

Where the atmosphere meets the interior, the relation between density ρ∗ and temperature T ∗ can be found (see Chapter 4 of [103]) by equating the electron pressure of the degenerate interior and the non-degenerate atmosphere

5 ∗3/2 3 ρ∗ = 7.6 × 10 µeT9 g/cm . (2.61)

where µe is the mean molecular weight per electron. Equating (2.60) and (2.61),

36 ∗3.5 Lγ = 8.9 × 10 λ (MX/M ) T9 erg/s , (2.62)

2 where λ ≡ 2µ/µeZ(1+W ). In the case of a NS, λ ≈ 1 (see [103], Chapter 11). We take

λ to be a free parameter that along with ρX, represent the unknown characteristics of the macros. We provide a brief derivation of the photon luminosity in Appendix (A).

37 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

The macro interior is nearly isothermal, due to the thermal conductivity of the degenerate electrons. Since T X ' T ∗, equating the photon luminosity (2.59) at the

s surface to (2.62) yields the macro’s surface temperature T9 (t). Starting from its assumed initial isothermal condition at 109 K, the macro cools according to dU X = −(LDU + LMU + LCP + L ). (2.63) dt ν ν ν γ where the internal energy is ([103] Eq.(11.8.2))

47 −2/3 X 2 UX = 6.1 × 10 (MX/M )(ρX/ρN ) T9 erg. (2.64)

We refer the reader to Appendix (A) for a derivation of the above equation. The interior temperature of the macro therefore obeys

d 1/3 −1 X T X = −8.3 × 10−4s−1 (ρ /ρ ) T X C (2.65) dt 9 X N 9 i i where the sum over i now includes photons, and

−9 1/3 X7/2 Cγ ≡ 8.9 × 10 (ρX /ρN ) λ T9 . Neutrino emission via MUrca occurs from the onset, since we take the initial tem-

9 perature to be 10 K. Emission via CP begins below Tc9. We explore two possibilities: ﬁrst, no DUrca cooling RD = 0; second, a proton fraction suﬃcient to support DUrca, with RD given by Eq.19 in [62]. In practice, SDs are relatively insensitive to the exact values of these various numerical factors. In the case where there is negligible DUrca emission, cooling proceeds in three stages:

X MU CP Stage 1: MUrca-dominated cooling from T9 = T9 = 1, at time t9, to T9 = 0.98Tc9, at tCP ;

38 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

CP γ Stage 2: CP-dominated cooling from T9 to T9 ' 0.2Tc9 at tγ; γ Stage 3: photon cooling below T9 , i.e. after tγ.

CP (If T9 is high enough, the ﬁrst stage may be omitted.) The macro cooling can be followed numerically, but by assuming that the dominant cooling mechanism in

M M each stage is the only one (and taking Rn ,Rp = 1), we ﬁnd

X T9 (t) ' (2.66)  −1/6 6 1/3  MU −8 MU ρX t−t9 T9 1 + 2.7 × 10 αT9  ρN s    MU X CP  for 1 ≡ T ≥ T ≥ T ' 0.98Tc9 ,  9 9 9   h 5 i−1/5  CP −2 CP t−tCP T9 1 + 1.8 × 10 aF T9 s

 CP X γ  for 0.98Tc9 ' T9 ≥ T9 ≥ T9 ' 0.2Tc9 ,   h i−2/3 T γ 1.0 + 1.1 × 10−11λT γ3/2 (ρ /ρ )2/3 t−tγ  9 9 X N s    γ X  for 0.2Tc9 ' T9 ≥ T9 .

The above relations can also be expressed in terms of redshift, z, using the time- redshift relation z = 4.9 × 109(t/s)−1/2. The times (and thus redshifts) at which the interior temperature TX falls to Tγ depend on detailed properties of the macro, such as its central density ρX , and the composition parameter λ. In Figure 2.3 we plot the central and surface temperatures of the macro, as well as the CMB temperature as a function of time for a representative value of these parameters.

X In the presence of DUrca cooling, Stage 1 is DUrca dominated until T9 becomes

γ DU T9 = 0.1 Tc9 at tγ. In this case, during Stage 1 (now T9 ≡ 1)

" 1/3 #−1/4 X DU DU4 D ρX t − t9 T9 (t) = T9 1 + 0.017 T9 R . (2.67) ρN s

For convenience, we provide Table where we describe the various temperatures and

39 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

log(1+z) 8.7 3.0 0 10

log(T/K) 4

Interior (No CP) 3 Surface (No CP) 9 Interior (Tc = 10 K) 2 9 Surface (Tc = 10 K) 9 Interior (Tc = 4 x 10 K) 9 µ era i era y era 1 Surface (Tc = 4 x 10 K) CMB 0 2 Recombination 17 log(t/s)

Figure 2.3: Interior temperature TX and surface temperature Ts of a macro for MX = M , λ = 1 and ρX = ρN , plotted versus time t and redshift z. Cooling is without DUrca, and both without CP and with CP for two values of Tc. The ambient photon temperature TCMB is shown for comparison, and the eras when µ and y distortions occur are indicated. times that appear in equations (2.66) and (2.67). A schematic which depicts the dominant cooling mechanisms at diﬀerent temperatures is provided in Figure 2.4.

Description

X T9 Interior temperature of macro CMB T9 CMB temperature MU T9 Temperature of macro at onset of MURCA DU T9 Temperature of macro at onset of DURCA CP T9 Temperature of macro at onset of CP cooling

i Table 2.1: Deﬁnitions of various temperatures, T9, that appear in equations (2.66) and (2.67). For i = MU, DU, CP, the temperature values were determined by comparing the luminosities of various processes given by Eq. (2.58).

40 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

Figure 2.4: Dominant cooling mechanisms in a neutron star at different temperatures. Bottom Panel: T = 109 K, Middle Panel: T = 3 × 108 K, Top Panel: T = 108 K. The figures on the left and right panel describe cooling for neutron stars that begin cooling via MURCA and DURCA respectively at high temperatures. In all figures, x-axis and the y-axis is the logarithm of the critical temperature of onset of superfluidity in neutrons and protons denoted by log10(Tcn) and log10(Tcp) respectively. In our work, we choose log10(Tcn) = log10(Tcp). We ignore all bremsstrahlung processes in our work since for our choice of macro parameters they are inefficient. Macro cooling is very well described by MURCA, DURCA and Cooper pair cooling. Image credit: Figure 13 in arXiv:astro-ph/9906456v1, Journal reference [123].

If one has to accurately calculate the SDs then one has to numerically solve the Boltzmann equation (2.1). However, there also exist analytical approximations which

41 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

Description X t9 Cosmic time at which T9 = 1

tCP Cosmic time at which CP cooling dominates

tγ Cosmic time at which photon cooling dominates

Table 2.2: Deﬁnitions of various times, ti, that appear in equations (2.66) and (2.67). These time values very obtained by solving Eq. (2.66) for the most dominant cooling process. help simplify the numerical estimates of the SDs. The pre-recombination contributions to µ and y distortions of the CMB can be approximated by

     µ  Z  1.4J     µ  1 ˙ = dt Jbb 2 Q. (2.68)    1  c ργ  y   4 Jy 

The window functions given in [111], [28], [54] are

−13 2.58−1 Jy(z) = 1 − Jµ(z) ≈ 1 + 4.7 × 10 z ,

5/2 Jbb(z) ≈ exp −(z/zµ) . (2.69)

−34 4 3 The CMB energy density ργ ≈ 7.0×10 z(t) g/cm , while the rate at which energy

density is injected into the photon distribution by macros of density nX is

˙ Q = nXLγ. (2.70)

3 −29 3 It is useful to rewrite nX = ΩX0ρcz(t) /MX with ρc ' 10 g/cm , and macro DM

< fraction ΩX0 ∼ 0.24.

In Figure 2.5, we plot µ and y (obtained numerically) vs. λ for MX = M , macro

densities near the ﬁducial ρN , and a variety of cooling scenarios. We have also calculated the perturbation to the neutrino energy density, since

neutrinos are injected well after weak-interaction freezeout at TCMB9 ' 10, but the

42 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

10-7 10-7 NS 10-7 PIXIE limit PIXIE limit PIXIE limit 10-8 NS 10-8 10-8 NS 10-9

10-9 10-9 -10

µ µ µ 10

10-10 10-10 10-11

T = 10 9 K T = 4 x 10 9 K 9 9 c c -11 -11 Tc = 10 K Tc = 4 x 10 K 10 10 0.1 -12 0.1ρ 0.1ρN ρN 10 N ρ ρN ρN N 10 10ρ 10ρN ρN N 10-12 10-12 10-13 10-2 10-1 100 101 102 10-2 10-1 100 101 102 10-2 10-1 100 101 102 λ λ λ 10-7 10-6 10-7 NS NS

-7 10 -8 -8 NS 10 10 PIXIE limit PIXIE limit 10-8 PIXIE limit -9 -9 10 y 10 y y

10-9

10-10 -10 10 T = 10 9 K T = 4 x 10 9 K -10 c c 9 9 10 Tc = 10 K Tc = 4 x 10 K 0.1ρN 0.1ρN 0.1ρN ρN ρN ρN 10-11 10ρN 10ρN 10ρN 10-11 10-11 10-2 10-1 100 101 102 10-2 10-1 100 101 102 10-2 10-1 100 101 102 λ λ λ Figure 2.5: Top panel: µ distortion as a function of macro surface composition factor λ for three diﬀerent cooling scenarios. On the left, no DUrca, no CP; in the middle no DUrca, with CP; on the right; with DUrca, with CP. Green lines denote ρX = ρN , red 9 lines 0.1ρN ; blue lines 10ρN . The panels with CP show results for Tc = 10 K (dashed) 9 and Tc = 4 × 10 K (solid). Bottom panel: as for top panel but for y distortion. The vertical dashed line stands for λ = 1 which is similar to a Neutron Star.

change in Neff is negligible. This could change if the internal physics of the macro were radically diﬀerent. The predicted y distortion is comparable to the target sensitivities of anticipated next-generation spectral distortion satellite missions, and the predicted µ distortion is nearly so. We remind the reader that since macros are much hotter than the plasma through much of their history and stay hot well after recombination, µ and y do not adequately capture the detectability of the SD signal.

Although we have presented results for MX = M , the spectral distortions µ and y

−1 are mass independent, for ﬁxed ρX , since Lγ ∝ MX from (2.62), while nX ∝ MX , and ˙ 0 so Q ∝ MX . The distortions will however depend on ρX , as well as on the detailed physics of the surface layer (as parametrized by λ), and the cooling mechanisms operative in an actual macro.

43 CHAPTER 2. CMB SPECTRAL DISTORTIONS FROM COOLING MACROSCOPIC DARK MATTER

2.3 Conclusions

We have demonstrated that the presence of macroscopic DM in the early universe may lead to observable signatures in the CMB spectrum. To fully characterize these distortions, the full spectral distortion must be inferred numerically using the Boltz- mann equation – this includes so-called intermediate distortions, a more complete characterization of the distorted spectrum and continued contributions to the distortions post-recombination. Also, the temperature of macros post-recombination may stay much higher compared to CMB for an extended period, implying the presence of hot relics that could be visible as an associated background radiation, or could heat the post-recombination universe. Other signatures can also be anticipated, such as correlations between CMB temperature anisotropies and spectral distortion anisotropies, the presence of heavy elements in the pre-recombination universe, and the continued production of these elements post-recombination and outside stars. The unexplored possibilities for observable consequences of standard model DM are yet rich.

44 Chapter 3

Radiating Baryonic Macroscopic Dark Matter and the Recombination history of the Universe

Throughout their cooling history, the macros emit photons from their surface. In the previous chapter, we calculated the µ and y spectral distortions in the CMB from photons emitted before recombination. Intriguingly, we found that the magnitude of these distortions was independent of the macro mass for the toy model we used for the macros. We continue the work in this chapter by studying the change in recombination history of neutral hydrogen in the universe. While in Chapter 2 we calculated the luminosity of the photons emitted by the macros, in this chapter we do a more detailed analysis by solving the Boltzmann equation to calculate the distribution function of the emitted photons. Simultaneously, we evaluate the matter temperature and the free-electron fraction in the universe as a function of redshift. Unlike the case of spectral distortions, we expect the mass of the macros to aﬀect the magnitude of

45 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE the change in recombination history. The chapter is organized as follows. In Section 3.1, we describe the standard recombination process. We then provide expressions for the evolution of the photon distribution function, the electron fraction, and the matter temperature as a function of redshift. In Section 3.2, we calculate the change in recombination history of hydrogen due to macros. We find that macros do not affect the hydrogen recombination history. We verify this by calculating the changes in Lyman alpha excitation rate, and the rate of ionization of the first excited state caused by macros. We also comment on the change in recombination history, and matter temperature. In Section 3.3, we summarize our conclusions.

3.1 Physics of cosmic recombination

At temperature higher than 4500 K, the ground state, the ﬁrst excited state, and the ionized state of the hydrogen atom are in thermodynamic equilibrium with each other:

− + e + p H2s + γ, and H2s H1s + γ. The number densities, ni, of non-relativistic fermions at a temperature T is given as

q2 Z − mi+ /kBT −3 µi/kBT 3 2mi ni = gi (2π~) e d q e (3.1)

where gi is the number of spin states of the particle, mi is the mass and µi is the chemical potential of the particle i. Since thermodynamic equilibrium implies chemical equilibrium, the sum of the chemical potentials of the non-relativistic species must

vanish, implying µH1s = µH2s . This gives us

−hνα/kBTM n2s = n1se . (3.2)

46 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

where n1s, and n2s are the number densities of 1s and 2s states of hydrogen atom respectively, να is the Lyman alpha frequency, and TM is the baryonic-matter temperature. From the Boltzmann equation, we can write the equation for the evolution of

number density of protons or hydrogen ions, np, as a function of cosmic time as

d n + 3Hn = −α n n + β n . (3.3) dt p p B e p H 2s

where ne is the number density of electrons, H is the Hubble rate at time t, αB, is the recombination rate of the n = 2 state also known as the Case B recombination coeﬃcient, and βH is the ionization rate of the n = 2 state.

3 Multiplying both sides by a , and then dividing both sides by the constant nH =

np + n1s + n2s + n2p where nH is the number density of hydrogen including both ions and neutral atoms, and using (3.2) gives us the Saha rate for proton fraction in the universe as a function of redshift z:

dxp 1 = α x x n − β (1 − x )e−xαTγ /TM (3.4) dz B e p H H p H(z)(1 + z)

In the above equation, xp = np/nH, xe = ne/nH, xα = hνα/kBTγ, and hνα ' 10.2 eV. As the universe cools down below 4500 K, the ground state is no longer in thermodynamic equilibrium with the ionized state. This means one can no longer use the Saha equation. The free electron goes through a cascade of radiative transitions from the continuum to the n = 2 state, and then decays to the ground state emitting a Lyman alpha photon. Following standard recombination calculations, we reduce the hydrogen electron energy states to three: ionized state, or the continuum, ﬁrst excited state including both the 2s, and 2p states, and the 1s ground state. We can ignore direct ground state recombination because this releases a 13.6 eV photon which ionizes another ground state hydrogen, leading to no net change in the number

47 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE of electrons in the universe. Hence, a free electron reaches the ground state by ﬁrst getting captured to the 2s, or 2p state followed by a radiative decay to the ground state. Furthermore, a 2p to 1s Lyman Alpha decay process releases a photon of 10.2 eV or more which can excite another ground state hydrogen that can easily be ionized. For recombination to occur, the ﬁrst excited state must decay via the slow 2s to 1s transition that releases two photons, none of which have the energy to ionize another ground state atom. This slow 2s to 1s “bottleneck” delays the recombination process. Putting everything together, the electron fraction depends on

1. βH, or the ionization rate of the 2s, and 2p states,

2. αB, or the Case-B recombination coeﬃcient,

−1 3.Γ 2s, or the slow 2s to 1s decay rate which is equal to 8.22458 s ,

8 −1 4.Γ 2p, or the fast 2p to 1s decay rate which is equal to 4.699 ×10 s , and

5. ε, or the 1s to 2p excitation rate.

We assume that stimulated recombination to the n = 2 state, and the stimulated decay of the first excited state can be ignored. We will see in the next section that the change in recombination history by an additional source of photons such as macros would occur through its effects on the ionization rate, βH, and Lyman Alpha excitation rate, ε. We further assume that the net recombination to n = 2 level is slow enough so that it is balanced by the net decay of the n = 2 state to the ground state. We must also take into consideration the probability, P , of an emitted Lyman alpha photon escaping to infinity without exciting another ground state hydrogen atom. Thus, we can write

αBnenp − βHn2s = (Γ2s + 3P Γ2p)n2s − εn1s (3.5)

48 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

Thus we get the number density of n2s atoms as

αBnenp + εn1s n2s = (3.6) Γ2s + 3P Γ2p + βH

The above equation is not particularly helpful since n1s state undergoes radiative transitions. However, during temperatures of recombination,

nH = n1s + n2s + n2p = n1s + 4n2s (3.7)

giving us

αBnenp + εnH n2s = (3.8) Γ2s + 3P Γ2p + βH + 4ε

We must now calculate the probability P of the Lyman alpha photon to escape to inﬁnity without exciting a ground state hydrogen atom. For a photon emitted at time t of angular frequency ω, the rate of 1s to 2p transition at any given time t0 after emission is given by

0 0 0 Γ1s→2p(t ) = σ(ω(t)a(t)/a(t )) c n1s(t ). (3.9) where σ is the cross-section of Lyman alpha absorption.

The number of Lyman alpha excitations, N1s→2p(t, ω(t)), caused by the emitted photon with angular frequency ω(t) at the time of emission t is given by

Z ∞ 0 0 N1s→2p(t, ω(t)) = dt Γ1s→2p(t ). (3.10) t

Assuming that photon absorption is a Poisson process, we can write the probability, p(N1s→2p = 0), that the photon isn’t absorbed at all as

−N1s→2p p(N1s→2p = 0) = e . (3.11)

49 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

Integrating over all emitted frequencies, ω(t), we get

Z ∞ P (t) = dω P(ω) p(N1s→2p = 0). (3.12) −∞

where P(ω) is the line proﬁle function given as

Γ2p 1 P(ω) = 2 2 . (3.13) 2π (ω − ωα) + Γ2p/4

where ωα = 2πνα. The cross-section of the Lyman alpha absorption is given by the Breit-Wigner formula: 2 3 2π Γ2p σ(ω) = 2 P(ω) (3.14) 2 kα

where kα is the mean wave number of the Lyman alpha photon. Typically the survival probability P (t) is very small, P (t) 1. This means that a Lyman alpha photon will be absorbed by a ground state hydrogen the instant it is

0 emitted. Thus, we can use the approximation, n1s(t ) = n1s(t) in (3.9). Similarly, the Hubble rate, H(t), can be treated as approximately constant in (3.9) which gives us

a(t0) ≈ a(t) + H(t) ∆t a(t) (3.15)

a(t)/a(t0) = 1 − H(t)(t0 − t)

We can then re-write (3.9) as

2 Z ∞ 3π Γ2pn1s(t)c R ω 0 0 − 2 −∞ dω P(ω ) P (t) = dω P(ω) e ωH(t)kα (3.16) −∞

50 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

The integral over ω0 gives us

Z ω 0 0 1 hπ −1 i dω P(ω ) = + tan (2 (ω − ωα) /Γ2p) . (3.17) −∞ π 2

Finally, using a change of variables, y = 2 (ω − ωα) /Γ2p, one arrives at

Z ∞ −v tan−1(y) 1 −v/2 e P (t) = e dy 2 (3.18) π −∞ 1 + y 1 − e−v = v where 2 3π Γ2pn1s(t)c v = 2 . (3.19) ωαH(t)kα

In the redshift eras we are interested in, v > 10. Therefore the exponential term in P (t) can be dropped giving us

2 ωαH(t)kα 8πH P (t) = 2 = 3 (3.20) 3π Γ2pn1s(t)c 3λαΓ2p(1 − xp)nH

where λα is the Lyman alpha wavelength.

Finally, if we go back to equation (3.3), we can write the change in xp using (3.8) as

dx p = α x x n (3.21) dz B p e H αBxpxe nH + ε(1 − xp) − βH(1 − xp) −1 −1 (1 − xp)(Γ2s + βH) + KH nH 1 × H(z)(1 + z)

3 −1 where KH = λα/8πH(z) = (3Γ2p(1 − xp)nH ) is the cosmological redshift term of Lyman alpha photons. In the above equation, the rate of ionization from the n = 2

51 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

state, βH is given as Z ∞ 8π 3 q(x) βH = 4π 2 σLν2s dx (3.22) c x2s x

−17 2 where σL = 1.6 × 10 cm , hν2s = 3.4 eV, q(x) is the photon occupation number which is equal to 1/(ex − 1) for a perfect black body.

From [90], the Case-B recombination coeﬃcient, αB, is given as

a tb α = F m3s−1 (3.23) B 1 + c td

where (F, a, b, c, d) = (10−19, 4.309, −0.6166, 0.6703, 0.53), and

−4 t = 10 TM. (3.24)

ε in (3.21) is given by

n2s 1 ε = Γ2s + fenh (3.25) n1s KHnH(1 − xp)

where fenh is the enhancement factor due to a source and is equal to unity in standard recombination. When there are excess photons in the universe due to macros, this factor will be larger than unity. We will derive the precise form of this term in the next section.

In ε, the prefactor n2s/n1s is given by

n −10.2 eV 2s = Exp . (3.26) n1s TM( in eV)

One can verify that using (3.25), and (3.26) in (3.21), we get the more familiar

52 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

equation

dx 1 p = P α x x n (3.27) dz H H(z)(1 + z) B p e H −hνα/kTM − βH(1 − xp)fenh e .

The factor PH is given as

1 + KHΓ2snH(1 − xp) PH = (3.28) 1 + KH(Γ2s + βH)nH(1 − xp)

For a more detailed discussion of standard recombination of hydrogen, see Chapter 2.3 of [117]. The universe also contains about 8.9% helium in number density in the form

4 of 2He. Helium recombination is slightly more complicated by the presence of two electrons as shown in Figure 3.1. However, the recombination of He III (doubly ionized) to He II (singly ionized) and He II to He I (neutral) can be calculated in a similar way as hydrogen recombination. We first start with a brief discussion on He III to He II recombination. This typically occurs at very high redshifts, z ∼ 6 × 103 so that it does not affect CMB anisotropies power spectrum. In contrast to hydrogen and He I recombination as will be discussed below, He II recombination does not go through any bottleneck because the 2s to 1s two photon decay rate from the first excited state of singly ionized helium is much faster than the net recombination rate. This implies that as soon as a He III atom captures an electron into the first excited state, the electron decays quickly to the ground state before the He II excited atom gets reionized. In the case of He I recombination, the triplet states, n3s and n3p, are metastable. At lower temperatures, this would have lead to faster recombination since metastable states tend to hold charge. However, as seen from Figure 3.1, the excited triplet states

53 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

Figure 3.1: Helium I energy levels for n ≤ 4 and continuum. In practice, the three energy levels of the He I atom - ground state, ﬁrst excited states and the continuum - very accurately describe Helium recombination. The dotted line shows that the triplet states are metastable states, they tend to hold charge and get ionized rather than decay into the ground state as is the case with singlet states. Image credit: Figure 8 from arXiv:astro-ph/9912182v2, Journal reference [102]. are closer to continuum than the ground state. As for example, the 23s metastable ground state is 4.77 eV away from the continuum as compared to the energy level diﬀerence between the metastable ground state and the ground state, about 19.83 eV. This along with the fact that He I recombination starts occurring at redshifts

54 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE z ∼ 3000 when the CMB has enough energetic photons reionizing excited He I atoms, the recombination rate is much slower. In fact, the radiation field is so strong that the triplet state is virtually depopulated. Detailed calculations have also showed that the effect of any neutral hydrogen possibly ‘stealing’ the CMB photons capable of reionizing excited He I atoms is negligible. Therefore, the recombination of the complex two electron He I atom can be very well approximated by an effective 3 level He I atom: the continuum, the singlet 21s and 21p states, and the ground state. It has been seen that the two photon 21s − 11s decay rate controls He I recombination since the decay rate is faster than the net decay rate of the 21p − 11s process. This is in contrast to the case of H I recombination where the net Lyman alpha decay rate was faster than the 2s − 1s two photon decay rate. We use expression 2 in [101] for change in singly ionized helium (denoted by HeII) fraction which is given as

d 1 x = P (x x n α (3.29) dz HeII HeII H(z)(1 + z) HeII e H HeI

−hν 1 /kTM − βHeI(fHe − xHeII)e HeI2 s )

In the above expression, xHeII = nHeII/nH, αHeI is the Case B He I recombination coeﬃcient for singlets given by

r r !1−p TM TM αHeI = r 1 + × (3.30) T2 T2 !1+p rT −1 1 + M m3s−1, T1

−16.744 5.114 r = 10 , p = 0.711,T1 = 10 K,T2 = 3 K,

55 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

βHeI is the ionization rate from n = 2 state given as

3/2 2 −hν2s,He/kTM βHeI = αHeI 2πmekTM/h e , (3.31)

hν2s,He ≈ 3.4 eV, fHe is given by

Yp fHe = nHe/nH = , (3.32) (1 − Yp)(mHe/mH)

Yp is the primordial helium mass fraction, and hνHeI21s ≈ 21.15 eV is the diﬀerence

1 1 between energy levels of the 2 s, and 1 s states of the helium atom. The factor PHeII is given as

−hνps/kTM 1 + KHeIΛHenH(fHe − xHeII)e PHeII = −hν /kT (3.33) 1 + KHeI(ΛHe + βHeI)nH(fHe − xHeII)e ps M

3 1 1 where KHeI = λHeI21p/8πH(z) is the cosmological redshift term of the 2 p − 1 s 1 1 −1 photons, λHeI21p = 58.4334 nm is the He I 2 p - 1 s wavelength, ΛHe = 51.3 s is the

1 1 1 2 s − 1 s two photon decay rate, hνps is the energy diﬀerence between the 2 p, and 21s levels of He I. Notice that the Case-B recombination coeﬃcient for both hydrogen and helium,

αB and αHeI, are functions of the baryonic-matter temperature, TM. Therefore, we must evolve TM as a function of redshift. It is given as

4 d 8σTaRTγ xe 2TM TM = (TM − Teﬀ ) + (3.34) dz 3H(z)(1 + z)mec 1 + fHe + xe 1 + z

where aR is the radiation constant. The ﬁrst term on the right hand side comes from energy exchange between electrons, and photons due to Compton scattering, and the second term describes the adiabatic cooling of electrons because of cosmic expansion. Teﬀ is the CMB temperature which in standard recombination takes the

56 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

101 xe = ne/nH

100

10-1

10-2 Electron Fraction

10-3

10-4 101 102 103 Redshift (z) Figure 3.2: Electron fraction as a function of redshift. Notice that at lower redshifts the electron fraction value plateaus to few ×10−4. This happens because the net recombination rate falls below Hubble expansion rate. This plot was produced by solving the RECFAST code [101].

value Teﬀ = 2.725(1 + z). In the above equation for baryonic-matter temperature, the photons exchange energy with electrons. The energy from electrons is then transferred to the neutral and residual ionized atoms through Coulomb scattering. The ratio of Compton and Coulomb scattering rates is given as [89]

photons-electrons 2.3 × 10−3 T 5/2 = 4 (3.35) electrons-nucleons xe logΛ where log Λ ≈ 16 for temperatures of interest. One can verify that below 104 K, the ratio is always less than unity implying that if Compton scattering is eﬃcient in the universe, so will Coulomb scattering. Therefore baryons are always in thermal equilibrium with electrons during and well after recombination.

57 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

104 Baryon Temperature CMB Temperature Compton Heating 103

102

Hubble Cooling Temperature (K)

101

100 101 102 103 Redshift (z) Figure 3.3: Electron temperature as a function of redshift. Until redshift of z ∼ 200, the residual free electrons in the universe are in thermal equilibrium with the pho- < tons. For z ∼ 200, Hubble cooling becomes more eﬃcient and the electron temperature drops below that of CMB. Coulomb scattering between electrons, and ionized and neutral hydrogen and helium always keeps baryons in thermal equilibrium with electrons. This plot was produced by solving the RECFAST code [101].

We plot the recombination history as a function of redshift in Figure 3.2 and baryonic-matter temperature as a function of redshift in Figure 3.3. In both plots we stop at redshift z = 10 since starting slightly above this redshift, structure in the universe in the form of galaxies and clusters starts reionizing and heating the electrons and baryons in the universe. A few kew observations from both these plots are:

1. The electron fraction freezes out at few ×10−4 at low redshifts. This is because the net recombination rate becomes ineﬃcient after which electrons become dilute due to slow expansion of the universe.

58 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

2. Compton scattering keeps the electrons in thermal equilibrium with the CMB even after recombination, until redshift z ∼ 200.

We acknowledge the use of the RECFAST code [101] in producing the Figures 3.2, 3.3, and 3.34. RECFAST is a highly efficient code that numerically computes the stiff coupled set of differential equations (3.27), (3.29) and (3.34). We refer the reader to [101] for a discussion about overcoming stiffness in the differential equations. After laying the foundation to calculate electron fraction and baryon temperature, we now proceed to calculate the same quantities in the presence of additional photon sources that are macros.

3.2 Change in recombination history from macros

In order to study the change in recombination history of neutral hydrogen in the universe due to photon emission from the surface of the macros, we must evaluate the photon distribution function as a function of time. This is given as

d n N 1 1 q(z, x) = X − (3.36) dz H(z)(1 + z) ex − 1 exTγ /Ts − 1

where q(z, x) is the photon occupation number, Ts is the surface temperature of the

macro, nX is the number density of macros, N is a normalization constant given

2 as N = πRX c, where RX is the radius of the macro, and x = hν/kBTγ. We note here that in the above expression, it is convenient to use x as a variable instead of frequency, ν, since dx/dz = 0. One can also verify that on switching from x to ν, and integrating the above expression over hν d3p, where p = hν/c, one gets the correct expression for the rate of change in energy density of photons,

∂ ρ + 4Hρ = n L (3.37) ∂t γ γ X γ

59 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

where Lγ is given by (2.59). We must now compute how much the source of ionizing photons, i.e. the surface of the macros, aﬀects the recombination history of hydrogen, and the matter temperature. To do that, we must numerically solve the diﬀerential equations for the evolution

of proton fraction, xp, ionized helium fraction, xHeII, and matter temperature, TM, given by (3.27), (3.29) and (3.34) respectively. While solving these equations, we must also evaluate the integral (3.22) to calculate the ionization rate βH for which we need to use the occupation number of photons given by equation (3.36). Having an excess number of photons will also increase the Lyman alpha excitation rate, ε. In other words, the fenh factor in (3.25) will be larger than unity and will be given by R ∞ dx x2 q(x) P(x) f = x 0 . (3.38) enh α R ∞ x3 P(x) 0 dx ex−1 where P is the line proﬁle function given by (3.13).

To calculate the above expression for fenh, we will use the method of Einstein coeﬃcients assuming that stimulated decay of n = 2 state to the ground state is negligible. We start with the equation

¯ n2s(3Γ2p + Γ2s) = n1sB12J (3.39) where J¯ denotes the spectral energy density of the radiation ﬁeld with units of

−2 ¯ erg cm and B12 is the Einstein coeﬃcient for photon absorption. J is given by the integral Z ∞ ¯ J = dν Jνφ(ν) (3.40) 0

where φ(ν) is the line proﬁle function which describes the eﬀectiveness of a photon

with frequency ν to cause a Lyman alpha transition. It is sharply peaked at ν = να,

60 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

and normalized such that Z ∞ dν φ(ν) = 1. (3.41) 0

In thermodynamic equilibrium, the mean intensity Jν is equal to the Planck function,

Bν, giving us 8πh ν3 J = B = . (3.42) ν ν c3 ex − 1

The line proﬁle function is given as

Γ 1 φ(ν) = 2p (3.43) 2 2 4π 2 Γ2p (ν − να) + 4π

This gives us

8πh Γ Z ∞ ν3 J¯ = 2p dν (3.44) 3 2 2 c 4π Γ 0 hν/kBT 2 2p (e − 1) (ν − να) + 4π

The cross section of Lyman alpha transition, σ, is given as

e2 σ = f12 φ(ν) (3.45) 40mec

where 0 is permittivity of free space, e is the Coulomb charge of an electron, and f12 is the oscillator strength which expresses the probability of photon absorption. It is given as hν f = 4B m 21 . (3.46) 12 12 0 e e2

Using the equation for f12 and (3.39), we get

n 3Γ P + Γ hν σ = 2s 2p 2s 21 φ(ν) (3.47) n1s J¯ c

61 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

The rate of Lyman alpha absorption can be written as

Z ε = c dν σ(ν) nγ(ν) (3.48)

where the number density of photons per frequency, nγ(ν), is given as

8π n (ν) = ν2 q(ν). (3.49) γ c3

Thus, we finally arrive at (3.25) with fenh given by (3.38). In our current work, we only focus on the change in recombination of hydrogen. In principle any additional source of energetic photons should also affect helium recombination. However, as we saw in Section 3.1, apart from the difference between energy levels of hydrogen, and helium, the physics of change in helium recombination is the same as in hydrogen. Hence, we defer the calculation of changes to helium recombination as a follow up.

In the expression for baryonic-matter temperature (3.34), Teﬀ is given as

ργ Teﬀ = Tγ 1 + Pl (3.50) 4ργ

Pl where Tγ = T0(1 + z),T0 = 2.725 K, and ργ/ργ is given as

ρ R x3q(x)dx γ = . (3.51) ρPl R x3 γ ex−1 dx

After numerically solving (3.36), one can obtain the spectrum of the CMB photons and those emitted from macros. In Figure 3.4, we provide the spectrum at a redshift

15 of z = 1300 for different masses of macros, MX = 100 g, 10 g, and M . We notice two things: first, there is no significant excess of 3.4 eV, and 10.2 eV photons. Secondly, the only excess occurs at the Wein tail of the photon spectrum. However, the cross-

62 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

section of ionization of the ﬁrst excited state, σH, falls oﬀ sharply with frequency,

−3 σH ∝ ν below 3.4 eV. Similarly, the Lyman alpha excitation rate is sharply peaked at 10.2 eV. Therefore, the excess high frequency photons do not aﬀect the Lyman

alpha excitation rate βH, and n = 2 ionization rate, ε, as can be seen in Figures 3.5 and 3.6 respectively. These features in the spectrum have the following eﬀect on hydrogen recombination: we observe no change in the electron fraction history for any parameter of macros. Another interesting question relevant to macro DM photon emissions is: what is the cumulative intensity of the macros as compared to the total light emitted by galaxies and clusters and the CMB? We answer this question in Figure 3.7 and observe that the intensity of light from macros is very subdominant (four orders of magnitude lower) as compared to the EBL at higher frequencies. On top of that, at higher frequencies the spectrum is indistinguishable from that of the CMB.

3.3 Conclusions

In this chapter, we studied the eﬀect of macros on the recombination history of hydrogen in the universe. We numerically calculated the distribution function of the photons emitted from the surface of the macros. From this distribution function, we

calculated the rate of excitation of the ground state, ε, and the ionization rate, βH, of the first excited state. There is no observable change in the recombination history. We understand this by observing the high frequencies of the photons emitted from the macros around recombination. The ground state excitation cross-section peaks falls off sharply above 3.4 eV according to the Breit-Wigner formula. Similarly, the first excited state ionization rate decays as ν−3 below 10.2 eV. The macros do emit photons around the excitation and ionization energies, but by that redshift the rates cause no effects since they are much below the expansion rate. The total light intensity of

63 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

10-5 10 2 g 10 15 g 10-10 10.2 eV Solar Mass -1 3.4 eV 10-15 -1 m Hz )/J s ν

I( 10-20

10-25

106 108 1010 1012 1014 1016 1018 ν/Hz Figure 3.4: Intensity spectrum of the CMB and photons emitted from macros of 9 three different masses at z = 1300. We used λ = 1, and TC = 10 K. In this plot, the spectrum from the 100 g macro is indistinguishable from a universe with no macros. The dashed, and dotted vertical lines stand for Lyman alpha and n = 2 ionization frequencies respectively. As we can see, at the two frequencies we are interested in, 3.4 eV and 10.2 eV, there is no significant excess of photons emitted from macros. The surplus at the Wein tail does not contribute in the excitation, and ionization processes since the cross-section of the Lyman Alpha excitation is sharply peaked at 10.2 eV, whereas that for the ionization rate falls off as ν−3 below 3.4 eV. the macros is also subdominant as compared to the extragalactic background light. These results are consistent across the entire mass range of macros, from 55 g to 1024 g. We therefore conclude that recombination history does not rule out any parameter space of macros. An important point worth mentioning here is that throughout our work, we did not take Compton scattering between photons and free electrons into account. More specifically, the photon occupation evolution equation (3.36) should contain the Compton scattering term in addition to the source term on the right hand side.

64 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

10-2 No Macro -4 10 Solar Mass

10-6 H(z)

10-8 ) -1 10-10 Rate (s 10-12

10-14

10-16

10-18 101 102 103 Redshift (z) Figure 3.5: The n = 2 ionization rates without macros, and solar mass macros compared to the Hubble rate, H(z). We can see that the excess photons from the < macro surface enhance the rate of ionization only at lower redshifts, z ∼ 500. However, by this redshift the ionization rate is below the Hubble rate. Therefore, the surplus 3.4 eV photons do not lead to excess ionization of excited hydrogen.

While there exists an expression for the Compton term, the Kompaneets equation [58], solving it numerically isn’t straightforward because of the derivative terms. However, qualitatively we expect that the Compton term would result in the transfer of energy from the excess photons to the electrons. This would only lead to thermalization of these excess photons with the baryon-CMB plasma of the universe. We therefore conclude that our results are robust to the inclusion of Compton scattering.

65 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

4 ε ε X / CMB

3 Ratio

1 10 100 1000 Redshift (z) Figure 3.6: The ratio of Lyman alpha excitation rates in a universe with solar mass macros to that without any. We see that the largest enhancement in the excitation < < rate occurs at much lower redshifts between 40 ∼ z ∼ 200. Therefore, the surplus Lyman alpha photons from macros do not lead to any excess excitation of the ground state hydrogen atoms.

66 CHAPTER 3. RADIATING BARYONIC MACROSCOPIC DARK MATTER AND THE RECOMBINATION HISTORY OF THE UNIVERSE

102

100

10-2 -1 ) -2 sr 10-4 I (nW m ν 10-6 EBL (z = 0) 2 10-8 10 g 10 18 g Solar Mass 10-10 10-1 100 101 102 103 ν (THz) Figure 3.7: The EBL (Extragalactic Background Light) intensity compared to that of CMB and macro photons. The EBL background is plotted with permission from [41].

67 Chapter 4

Conclusions

We started with a brief discussion on the observational evidences of dark matter in Chapter 1. These included the galaxy rotation curves, power spectrum of CMB anisotropies, matter power spectrum, BAO, BBN abundances, and “bullet cluster” observation. We discussed challenges to the CDM paradigm, mainly from observations at galactic scales and below: missing satellites problem, core-cusp problem, and the too-big-to-fail problem. Next, we discussed some candidates of DM. We discussed BSM candidates such as WIMPs and Axions. as well as PBHs which arise from GR. We then discussed the virtues of DM existing as a composite object of baryons or quarks known as macroscopic DM or macros. We looked at some existing models of macros in literature. Most of these models are of nuclear density. We saw that a vast parameter space of macros is still permitted by observations. In Chapter 2, we proposed a new mechanism of energy release from DM in the early universe: photon emissions from the surface of the macros. We adopted a two layered model of macros based on a neutron star since neutron stars are the only known objects in the universe that are of macroscopic size and have nuclear density. The macros are assumed to have formed around QCD phase transition, the core of the macro has an initial temperature similar to the ambient temperature at the time

68 CHAPTER 4. CONCLUSIONS of formation. The hot core cools down via slow neutrino mechanisms and therefore acts as a heat reservoir. The atmosphere of the macro acts as an insulation layer which keeps the interior hot. As the macro cools down, the atmosphere radiates photons and therefore injects energy in the CMB distorting its blackbody spectrum. After characterizing these spectral distortions, we numerically calculated the distortions before recombination and found that the magnitudes could be detected by next generation CMB spectral distortion experiments. In Chapter 3, we continued our work forward by calculating the distribution function of the emitted photons and calculating its eﬀects on the recombination history of hydrogen in the universe. We started with an introduction to the physics of standard recombination of hydrogen and helium. Later, we calculated the change in recombination history of hydrogen from the photons from macros and concluded that since the photons emitted have frequencies much above 3.4 eV and 10.2 eV, they do not cause an appreciable change in the ionization rate of ﬁrst excited state and Lyman alpha excitation rate. Therefore, we do not detect any change in the recombination history.

69 Appendix A

Photon Luminosity and Internal Temperature of Macro

In this Appendix, we derive the temperature dependence of the Macro given by Eq. (2.65), following closely the treatment of Chapter 4 in [103]. We assume that below degeneracy temperature of 109 K, the core of the Macro is composed of degenerate neutron-proton-electron plasma, and is isothermal. The atmosphere is composed of a non-degenerate layer. The energy transfer due to photon diﬀusion from the hot interior to the ambient CMB through the atmosphere can be described by the radiative heat transfer equation assuming local thermal equilibrium and steady state. The photon luminosity, Lγ, is given by

2 c d 4 Lγ = −4πr (aT ), (A.1) 3κρatm dr

where a is the radiation constant, r is the radial distance from the center of the Macro,

κ, ρatm and T are the Rosseland mean opacity, density, and the temperature of the atmosphere respectively.

70 APPENDIX A. PHOTON LUMINOSITY AND INTERNAL TEMPERATURE OF MACRO

Opacity, κ, can be approximated as Kramer’s opacity

−3.5 κ = κ0ρatmT , (A.2) where

24 2 −1 κ0 = 4.34 × 10 Z(1 + X) cm g . (A.3)

Hydrostatic equilibrium requires that the pressure of the atmosphere depends on the radius as dP Gm(r)ρ = − atm , (A.4) dr r2 where m(r) is the mass of the Macro within the radius r. Since the atmosphere is much thinner than the radius of the core, we can set m(r) = MX . The pressure for a non-degenerate gas is also given by the ideal gas law:

ρatm P (r) = kBT, (A.5) µmu

where µmu is the mean molecular weight. (mu is the atomic mass unit.) Substituting (A.5) in (A.4), and using (A.2) and (A.1),

πGM k P dP = 5.33ac X B T 7.5dT. (A.6) κ0Lγ µmu

Assuming a constant luminosity throughout the thin atmosphere, we can integrate the above equation with the boundary condition, P = 0 when T = 0. Thus we arrive at the density of the atmosphere given by Eq. (2.60):

s πGM µmu 3.25 ρatm = 1.25ac T (A.7) κ0Lγ kB 1/2 10 µ(MX/M )erg/s 3.25 = 1.2 × 10 ρN T9 . Z(1 + W )Lγ

71 APPENDIX A. PHOTON LUMINOSITY AND INTERNAL TEMPERATURE OF MACRO

At the point where the atmosphere meets the core, the non-degenerate electron pressure of the atmosphere given by the ideal gas law is equal to the electron degeneracy pressure of the core,

ρ k T ρ 5/3 ∗ B ∗ = 1.0 × 1013 ∗ , (A.8) µemu µe

where µe is the mean molecular weight per electron, and ρ∗ and T∗ are the density and temperature at this transition point. Solving for ρ∗ in the above equation and equating it with (A.7), we get the luminosity of the Macro given by Eq. (2.62):

5 µ 1 MX 3.5 Lγ = 5.7 × 10 erg/s 2 T∗ µe Z(1 + W ) M 36 ∗3.5 = (8.9 × 10 erg/s) λ (MX/M ) T9 . (A.9)

From [22], the heat capacity of the Macro at temperature TX is

2 2 1/2 dUX π (x + 1) kBTX Cv = = 2 NkB 2 , (A.10) dTX N,V x mc

where UX, N, and V are the internal energy, total number of neutrons, and volume of

the Macro respectively. In the above equation, x = pf /mncis the relativity parameter,

where pf is the Fermi momentum, and mn is the mass of neutron. Integrating the

above equation over TX gives us Eq. (2.64) for the internal energy:

−2/3 47 MX ρX X 2 UX = (6.1 × 10 erg) T9 . (A.11) M ρN

We can use this expression for UX in the left hand side of Eq. (2.63). The right hand side of Eq. (2.63) is a sum of the photon luminosity A.9, and neutrino luminosities that we will brieﬂy describe in the appendix below.

72 Appendix B

Neutrino Emission Luminosity

In this appendix, we describe brieﬂy the neutrino emissions from the Macro as given by Eq. (2.58). A detailed derivation of (2.58) is beyond the scope of the paper. Moreover, the expressions for the luminosities are very well established, and have been studied in great detail [62, 121, 122]. The DURCA luminosity [62]

−1/3 DU 45 X 6 D MX ρX Lν = 5.2 × 10 (T9 ) R erg/s (B.1) M ρN where RD is the reduction factor in DURCA rate due to superﬂuidity. As an example, we considered the type-AA superﬂuidity of neutrons and protons. The RD is given by Eq.19 in [62]: u RD = S + D, u + 0.9163

1/2 √ 1 π 1/4 − pe S = (K0 + K1 + 0.42232K2) ps e , I0 2

6 I0 = 457π /5040,

73 APPENDIX B. NEUTRINO EMISSION LUMINOSITY

√ p − q 2 2 K0 = (6p + 83pq + 16q ) 120 √ √ √ q p + p − q − p (4p + 3q)ln √ , 8 q

√ √ √ π2 p − q π2 √ p + p − q K = (p + 2q) − q pln √ , 1 6 2 q

7π4 √ K = p − q, 2 60

√ 2p = u + 12.421 + w2 + 12.350u + 45.171,

√ 2q = u + 12.421 − w2 + 12.350u + 45.171,

√ 2 2ps = u + w + 5524.8u + 6.7737,

√ 2 2pe = u + 0.43847 + w + 8.3680u + 491.32,

3/2 2 2 −u1−u2 D = 1.52(u1u2) (u1 + u2)e ,

q 2 2 u1 = 1.8091 + v1 + (2.2476) ,

q 2 2 u2 = 1.8091 + v2 + (2.2476) ,

2 2 u = v1 + v2,

74 APPENDIX B. NEUTRINO EMISSION LUMINOSITY

2 2 w = v2 − v1,

√ 0.157 1.764 v = v = v = 1 − τ 1.456 − √ + , (B.2) 1 2 A τ τ

where T τ ≡ X . (B.3) Tc

The MURCA luminosity [121]

MU M M 39 X 8 Lν = (3.0Rn + 2.4Rp )10 (T9 ) α

−1/3 × (MX/M )(ρX/ρN ) erg/s (B.4)

For simplicity, we consider only singlet-state neutron superﬂuidity of Type-A. The

M M associated reduction factors Rn and Rp are given by Eq. 32 and Eq. 37 in [121]

7.5 5.5 √ a + b 2 2 RM = e3.4370− (3.4370) +v , n 2

a = 0.1477 + p(0.8523)2 + (0.1175v)2,

b = 0.1477 + p(0.8523)2 + (0.1297v)2,

√ 0.157 1.764 v = 1 − τ 1.456 − √ + , τ τ

75 APPENDIX B. NEUTRINO EMISSION LUMINOSITY and

7 M p 2 2 Rp = 0.2414 + (0.7586) + (0.1318v) √ e5.339− (5.339)2+(2v)2 . (B.5) respectively. The CP cooling luminosity [122]

CP 39 X 7 Lν = 7.1 × 10 erg/s (T9 ) a F (B.6)

−2/3 × (MX/M )(ρX/ρN ) erg/s

The function F controls the eﬃciency of the CP cooling process. We select F to be

FA given by

2 4 6 FA(v) = (0.602v + 0.5942v + 0.288v ) 1/2 × 0.5547 + p(0.4453)2 + 0.01130v2 √ ×e− 4v2+(2.245)2+2.245 (B.7)

Eq.(34) in [122]. The factor a in the CP luminosity is a constant that depends on nucleon species and superﬂuidity type. It has the maximum value of 4.17 and 3.18 for triplet states of neutrons and protons respectively.

76 Bibliography

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