The effects of non-zero neutrino masses on the CMB determination of the cosmological parameters
Thesis
Presented in Partial Fulfillment of the Requirements for the Degree of Master of Physics in the Graduate School of The Ohio State University
By Michael A. Obranovich, B.S. Graduate Program in Physics
The Ohio State University
2012
Thesis Committee: Dr. Gary Steigman, Advisor Dr. Todd Thompson Copyright by Michael A. Obranovich 2012 Abstract
Since neutrinos have been determined to exhibit flavor oscillations from neutrino experiments, this implies that at least two of the three neutrinos have mass. Neutrinos are also the second most abundant particle to survive the Big Bang and thus have a powerful influence on Cosmological evolution. This is true despite their tiny ( eV) masses, of which the best limits have been set through ∼ their influence on the formation of Large Scale Structure and the Cosmic Microwave Background.
ii Dedication
For Julia
iii Acknowledgments
I’d like to thank my Mother Kathy, Father Greg, Sister Jennifer, and Brother Chris for their continuing support and advice through my academic career and beyond.
iv Vita
June 2005...... Del Oro High June 2009...... B.S. Astrophysics, The University of California, Santa Cruz 2009 to 2011...... Graduate Teaching Associate, Department of Physics, The Ohio State University
2011 to present...... Graduate Research Associate, Department of Physics, The Ohio State University
Fields of Study
Major Field: Physics
v Contents
Abstract ii
Dedication iii
Acknowledgments iv
Vita v
List of Tables viii
List of Figures ix
1 Introduction 1 1.1 CosmologyandanexpandingUniverse ...... 1 1.2 Neutrinos in Particle/Astroparticle Physics ...... 4
2 Neutrinos and Cosmology 8 2.1 CνB ...... 8 2.2 NeutrinosandBBN ...... 9 2.3 Massive Neutrinos and Large Scale Structure ...... 10
3 Temperature Anisotropies in the Cosmic Microwave Background 15
3.1 Effects of massive neutrinos on the CMB TT Spectrum ...... 16 3.2 Effects of a non-thermal distribution on the CMB TT Spectrum ...... 20 3.3 CMBparameterDetermination ...... 21
4 Conclusion 26
References 26
A Appendix 31
A.1 ξ, neutrino asymmetries and Neff ...... 31
vi A.2 Presentneutrinomassdensity...... 32
A.3 znr ...... 33 A.4 Neutrino Matter Power Spectrum suppression scales ...... 33 A.5 NeutrinoeffectsonHorizonscaling ...... 34
vii List of Tables
3.1 Constraints on mν from the WMAP collaboration. Clearly when additional infor- mation from LSS is included the constraints are improved...... 22
viii List of Figures
1.1 Normal and Inverted Hierarchies. 1.1(a) shows mν (eV ) for the normal (blue), inverted (red), and degenerate (gold) hierarchies. 1.1(b), Different hierarchies with (∆m2 = ∆m2 ) solar neutrino data (∆m2 = ∆m2 ).[1] ...... 6 atm 31 ≫ sol 21
in 2.1 Left, the evolution of ηντ,µ (top curves) and ηνe (lower curves) for initial value ηνe = 0.82 and zero total asymmetry. The black line corresponds to sin2 θ = 0 while the − 13 2 red lines were calculated in the NH for values of sin θ13 =0.04, 0.02 from left to right respectively. The blue lines use the same values but for the IH. Right, the final value
in of Neff after the flavors have equilibrated with the same initial ηνe as a function of
the total asymmetry ην . From top to bottom the lines correspond to no oscillations,
2 θ13 = 0, and sin θ13 =0.04intheNHandIH.Bothfiguresfrom[2]...... 11 2.2 Suppression of LSS due to massive neutrinos. 2.2(a), The upper black line is the
neutrino free streaming scale, kfs for a model with mν =1.2eV . The lower black line is the comoving Hubble scale, aH. All modes in the shaded region can grow, whereas modes above the shaded region are subject to free streaming suppression. 2.2(b), the matter power spectrum divided by the ΛCDM matter power spectrum for
various different neutrino masses. The lines are for mν =0, 0.15, 0.6, and 1.2eV indescendingorder. Bothfiguresfrom[3]...... ...... 13
2.3 The Matter Power Spectrum made with CAMB in a ΛCDM + mν model with
mν and ΩΛ varied. The more massive the neutrino the greater the MPK is sup- pressed,especiallyonsmallerscales. ...... 14
ix 3.1 3.1(a), zeq is calculated via eq. 3.4. As anticipated, zeq is mass dependent. The behavior changes as the mass of a single neutrino species becomes non-relativistic around z & 1100 m & 0.5eV . Using a cutoff point of 3.15y shifts the extrema rec ⇒ ν to a mass of 0.2eV . Since this is close to most limits of m . 0.6eV , the analysis ∼ ν is unchanged. 3.1(b), the total energy density ρν as a function of redshift for different
mass scale mν ...... 17 3.2 3.2(a), phase shift from affecting the early ISW contributions and horizon sizes. 3.2(b),
the change in height of the first peak from changes in the ISW effect as dependent
2 on the neutrino energy density ων = Ων h . Both figures are from [4] and use a flat ΛCDM model...... 19 3.3 CMB TT anisotropy spectrum made with CAMB. All curves are for a flat ΛCDM
model with H0 fixed but mν and ΩΛ varied. When mν & 0.5eV the neutrinos are non-relativistic at recombination and thus behave more like matter and decrease the early ISW contribution, lowering the peak. In all cases however, there is a phase shift to larger scales from horizon scaling and metric perturbations outside the sound horizon...... 19 3.4 CMB TT anisotropy spectrum made with CLASS ( [5]). All curves are for a flat
ΛCDM model with H0 fixed but with the individual neutrino masses mν and ΩΛvaried. In this mass range differences of a few percent arise...... 21
3.5 CMB TT anisotropy spectrum made with CLASS ( [5]). All curves are for a flat
ΛCDM model with everything fixed including mν . The two curves compare the use of a pure Fermi-Dirac distribution to describe the neutrino population and the distorted function described by equations 3.9 and 3.10. The difference is on the sub- percent level, on the same order as the numerical accuracy of thecode...... 22
3.6 All figures are from [6] using a ΛCDM +Ωk + Neff + mν model with data sets
W MAP + BAO + H0 + SPT + ACT . Allowing for Neff to vary shifts the mass
constraint to larger values. However, allowing for Ωk different from zero doubles the
value of mν allowed [6]. Also, by allowing both mν and Neff to vary, the positive
correlations between those parameters but opposite correlations with Ωk destroys the
correlation between Ωk and Neff (3.6(c))...... 24
2 3.7 2D contours for Left ΩΛ , middle w and right Ωmh vs mν,1. All contours are for a
ΛCDM +Ωk + Neff + mν + w model from a CosmoMC run with three chains. Data
sets include W MAP 7yr + SPT + LRGDR7+ H0 + SNe...... 25
x A.1 A.1(a), the sound horizon (blue) decreases since more energy density hastens the time of recombination. The distance to last scattering (red) decreases at a larger rate due
to the decrease in ΩΛ . A.1(b), the same effect occurs but because increasing Ων leads to a closed universe. Scaling is in reference to the massless neutrinocase...... 35 A.2 The first peak is shifted to larger scales due to massive neutrino effects on the horizons.
A.2(a) uses a zero curvature model with fixed Ωm and h, thus ΩΛ is decreased to
compensate for an increase in neutrino mass. A.2(b), Ωm +ΩΛ = 1, thus the curvature depends on the neutrino mass density. The shift is more dramatic due to the strong scalingdependenceonthecurvature...... 35
xi Chapter 1
Introduction
Elusive for twenty years after their initial proposal, neutrinos are now routinely observed by terrestrial detectors. While they’ve been useful in probing Standard Model physics their massive properties remain mostly a mystery. Knowing the details of their mass (are they Majorana/Dirac, why are the masses so tiny, what is their rest mass?) could lead to physics beyond the Standard
Model. To date, the best constraints on the neutrino mass scale mν have been placed by cosmological probes like the Cosmic Microwave Background (CMB) and Large Scale Structure (LSS). These combined probes constrain mν < 0.58eV (95% CL, [7]) and will improve with the advent of better data. This paper aims to describe the effects massive neutrinos have on the CMB Temperature spectrum and how that leads to constraints on mν and other CMB parameters. The structure of this paper is as follows: Current bounds on neutrino properties from neutrino astronomy will be reviewed followed by neutrino effects on Big Bang Nucleosynthesis (BBN) and Large Scale Structure (LSS), and finally the effects of neutrinos on the CMB Temperature anisotropies and their relations with CMB parameters will be discussed.
Chapter 1.1
Cosmology and an expanding Universe
On Cosmological scales the Universe appears to be homogeneous and isotropic with its dynamics dominated by gravity. Thus, General Relativity provides the framework for describing the evolution of the Universe and its contents. The Einstein equation describes how space-time geometry (g ν ) evolves under the influence of energy (T ν ),
1 R g R =8πGT (1.1) ν − 2 ν ν
1 10 0 0 − 0 a2(t)0 0 where g ν = is the metric (in a homogeneous and isotropic universe) 2 0 0 a (t) 0 2 0 0 0 a (t) 2 and a (t) is the scaling factor that accounts for an expanding (or contracting) universe. R ν is the
αβ 2 Ricci tensor and R = gαβR the Ricci scalar. The line element ds associated with this metric in spherical coordinates becomes
sin2 χ closed
ν 2 2 2 2 2 2 2 2 g ν dx dx = ds = dt + a (t) dχ + χ dθ + sin θdφ flat (1.2) − 2 sinh χ open where χ is the comoving (fixed) radial coordinate, and θ, φ are the usual angular coordinates accompanied by different function of χ depending on the geometry of the universe. The energy content is assumed to be a perfect fluid with some energy density and pressure related via the equation of state, w = p/ρ. T ν then takes the form,
ρ 0 0 0 0 p 0 0 T ν = (1.3) 0 0 p 0 0 0 0 p where ρ is the energy density and p is the pressure. By working out the components of the Einstein equation, the Friedmann equation may be derived, which describes the expansion history of the Universe and evolution of the scale factor with time,
a˙ 2 8πG k H2(t)= = ρ(t) (1.4) a 3 − a2(t) where the k is the same as in eq. 1.2 and accounts for curvature. Using the first law of Thermo- dynamics, the evolution of the energy density may be found. A homogeneous Universe implies that the temperature varies only in time and not in space, thus there is not heat flow and the first law can then be written. d (E(t)) = pd (V (t)) (1.5) −
2 Writing ∆E(t)= ρ(t)∆V (t) and knowing that V (t) a3(t) ∼
d da3 ρa3 = p (1.6) dt − dt allowing the scaling of the energy density to be found if the equation of state (pressure) is known. The equation of state may be found using statistical mechanics but the most useful values are w =0 for collision-less dark matter and w = 1/3 for radiation. Using those values it can be found that ρ a−4 and ρ a−3. radiation ∼ dm ∼ The Friedmann equation may be simplified if the various types of energy densities (Dark Mat- ter, Radiation, Dark Energy etc) are found in terms of a(t), and further parametrized by defining 2 3H0 ρcritical = 8πG and Ωi = ρi/ρcritical, where the index i denotes the different types of energy, and H0 is the Hubble constant, equal to the expansion rate today. The Friedmann equation becomes,
Ω Ω Ω H2 = H2 m,0 + r,0 + k,0 +Ω (a)+Ω (a) (1.7) 0 a3 a4 a2 Λ ν where Ωm,0 and Ωr,0 are the present matter and radiation energy densities today, and ΩΛ(a) and
Ων (a) are the dark energy and neutrino energy densities with unknown or non-trivial dependencies on a. Lastly, Ω is the curvature “density”, usually defined as Ω =1 Ω where Ω is the sum k,0 k − tot tot of all components that contribute to the total energy density today. Curvature plays an important role in calculating distances since distance is sensitive to the geometry, as can be seen in the geodesic eq. 1.2. Since the Universe is not perfectly homogeneous and isotropic, any observations of this nature must be accounted for theoretically. This is done by perturbing the metric to account for primordial fluctuations in the gravitational potential and curvature Ψ( x, t) and Φ( x, t) respectively. The metric is then modified as follows.
g ( x, t)= 1 2Ψ( x, t) 00 − −
g0i( x, t)=0 (1.8)
2 gij = a δij (1 + 2Φ( x, t))
To understand how the potentials evolve, one must also understand how the energy densities of various forms evolve which are then connected via the Einstein equation (eq. 1.1). This is done by solving the Boltzmann equations,
3 df(t) = C(t) (1.9) dt
for perturbations to the distribution functions. A common method for solving these equations is by solving for Fourier modes which are independent in the linear regime. For example, the photon perturbations are represented by Θ( x), which becomes
3 d k Θ( x)= eik xΘ(˜ k) (1.10) (2π)3 where k is the wavenumber and Θ(˜ k) is the Fourier transform of Θ( x). k has some significance in that it can be associated with the comoving size of the horizon at a given redshift, k = aH(a)/c. In this way, once can establish a maximum wavelength that should be considered at a given time, since any larger scale would be outside the horizon and thus not able to evolve. Or in the case of neutrinos, free streaming effects have an associated size scale and thus a wavenumber kfs (A.4), which produces features in the evolution of perturbations.
Chapter 1.2
Neutrinos in Particle/Astroparticle Physics
Since it’s proposal in the 1930’s, the neutrino has been detected in a variety of ways by both astrophysical and terrestrial sources. The reactions in which these neutrinos have been detected are based on three different flavors, the electron, muon, and tau (νe, ν , ντ respectively) neutrinos and their anti-particles (¯νe, ν¯ , ν¯τ ). Each flavor has a charged lepton counter-part corresponding to the
flavor name from which it decayed (except of course for νe which is in the first generation). The number of flavors has been constrained by the decay width Γ, of Z (Z ν ν¯ ), since Γτ ~/2 by → x x ≥ the Uncertainty Principle. The more families of particles with neutrinos, the more channels Z can decay to, thus τ decreases and the decay width increases. Current results are consistent with three
flavors of neutrinos. Current solar models predict various ways of producing νe ’s. Ultimately they are observed from the reaction, 8B 8 Be + e+ + ν (1.11) → e which produces only 0.02% of the 7 1010 particles per cm2 per sec flux at earth, but with high × enough energies to be detected (as opposed to pp De+ν , which produces the most neutrinos → e in number but with lower energy). These neutrinos are then detected by the decay of 37Ar, which
4 is produced in a tank of chlorine by ν +37 Cl e− +37 Ar. The first experiment to do this e → took place at the Homestake Mine under Ray Davis. While the collaboration was successful in detecting νe , the observed flux was one third of what was expected [8]. This was known as the “Solar Neutrino Problem” and some solutions ranged from detector systematics to problems with the standard solar model. The solution, confirmed in 2001 by the SNO collaboration [9] was that the three flavors of neutrinos exhibited oscillations. That is, a particular flavor state is a superposition of mass eigenstates which allows the neutrino to transform into another flavor when it has traveled a sufficient distance. This effect had been observed first with K and B meson oscillations. If we consider two flavor neutrinos (νe , ν ) with two mass eigenstates, the flavor states can take the form,
ν = ν cos(θ)+ ν sin(θ), ν = ν sin(θ)+ ν cos(θ) (1.12) | e | 1 | 2 | − | 1 | 2 where the indices 1, 2 indicate the mass eigenstates which evolve as,
− − ν(t) = ν sin(θ)e iE1t + ν cos(θ)e iE2t (1.13) | | 1 | 2 and θ is the mixing angle. Assuming a relativistic neutrino, E p + m2/2 p , the probability of i ≈ | | i | | transformation can be calculated:
2 2 2 2 2 1.27∆m (eV )L(km) P → (L)= ν ν(t) = sin (2θ) sin (1.14) νµ νe | e | | E(GeV )
It is important to note that this probability is dependent on on the square mass difference ∆m2 = m2 m2, and not the absolute mass scale of the neutrinos. Thus if we want to learn something about 2 − 1 the absolute mass scale, an alternative method to oscillation experiments is needed. If you knew
2 2 one of the neutrino masses, then only ∆m21 and ∆m31 are needed to find the other two. However, 2 2 2 current results show that ∆m21 > 0 and that ∆m31 > 0; thus the sign ambiguity on ∆m31 leads to what are known as the Normal and Inverted Hierarchies (NH and IH respectively) [1]. Fig. 1.1 shows the effects of a hierarchy on the total neutrino mass mν . Atmospheric neutrinos are created by primary cosmic rays interacting in the atmosphere to form pions which then decay to muons and ν . The muons then decay further to νe andν ¯ (eq. 1.15), leading to an expected number ratio of N /N 2. However, a deficit of muon neutrinos is observed, e ≈ leading back to oscillations as the mechanism for the discrepancy.
5 + + − − π + ν ; π +ν ¯ → → (1.15) − − + e+ + ν +ν ¯ ; e +ν ¯ + ν → e → e
Nuclear reactors have also been a source of neutrinos, having been detected by the KamLAND collaboration ( [10]) along with neutrinos produced in particle accelerators ( [11]). All experiments thus far, including astrophysical and terrestrial have observed flavor deficits and have been able to constrain oscillation parameters. Recently, [12] and [13] have done a global analysis where they have included various data sets (Solar, Atmospheric, Reactor/Particle) to constrain oscillation parameters.
Below (eq. 1.16) presents the results quoted in [12].
2 −5 2 +0.20 2 −3 2 +0.10 ∆m12[10 eV ]=7.59−0.18; ∆m31[10 eV ]=2.45 0.09(NH), (2.34−0.09)(IH) ± − (1.16) 2 +0.017 2 2 +0.009 sin θ =0.312− ; sin θ =0.51 0.06; sin θ =0.012− , 12 0.015 23 ± 13 0.006
Úm Ν
2.00 Normal Hierarchy
1.00 Inverted Hierarchy
0.50 No Hierarchy
0.20
0.10
0.05
lightest m Ν 0.001 0.005 0.010 0.050 0.100 0.500 1.000
(a) P mν for IH and NH m m 3 2 ∆m 2 sun 1 ∆m 2 atm ∆m 2 atm 2 ∆m 2 sun 1 3
NORMAL INVERTED (b) Normal and Inverted Hierarchies
Figure 1.1: Normal and Inverted Hierarchies. 1.1(a) shows mν (eV ) for the normal (blue), inverted 2 2 (red), and degenerate (gold) hierarchies. 1.1(b), Different hierarchies with (∆matm = ∆m31) solar 2 2 ≫ neutrino data (∆msol = ∆m21). [1]
6 Finally, it is worth mentioning the neutrinos detected from Super Nova 1987 A (SN1987A) since it is most likely a once in a lifetime occurrence, but we were able to collect valuable data ( [14], [15]). SN1987A has also led to a limit on the neutrino rest mass. If two neutrinos of the same mass were emitted at the same time but with slightly different (measured) energies, then there would be a time delay of detection since they would propagate with different velocities. It can be shown that the rest mass m0, 2∆t E2E2 1/2 m = 1 2 (1.17) 0 rc3 E2 E2 2 − 1
where ∆t is the time delay, r is the distance to the super nova event, and E1 and E2 are the different energies. This calculation has given an upper limit of m 26eV [16]. νe ≤ Some recent oscillation experiments, in particular the LSND experiment at Los Alamos have claimed surprising results of a ∆m2 > 1eV 2 forν ¯ ν¯ [17]. Since for three active flavors of → e 2 2 2 neutrinos only two independent mass-squared differences are needed (∆m31 = ∆m21 + ∆m32, [18]), two of the terms must be of the same order of magnitude for the equality to hold. From eq. 1.16, we see two of the terms are already of different magnitude. Since LSND results give a ∆m2 of differing order, a new neutrino must exist for the others to mix with if oscillations are to be used to explain the results. Since Z decay constrains the number of active neutrinos to three, the new neutrinos would be sterile and wouldn’t couple via normal Standard Model interactions. Also, the MiniBooNE experiment [19] is in disagreement with LSND, and it has become tricky to find a neutrino mass scheme with more than three neutrinos that agree with all current data. There is also
Cosmological evidence for the existence of an extra neutrino from Big Bang Nucleosynthesis (BBN) and the Cosmic Microwave Background (CMB) which depend on the relativistic degrees of freedom
Neff , which will be discussed in more detail later on.
7 Chapter 2
Neutrinos and Cosmology
It is evident that neutrino experiments are sensitive to ∆m2 and not the absolute mass scale. One promising exception is neutrino-less double beta decay, which would give the mass scale, and determine if neutrinos have Majorana or Dirac mass. The ongoing Heidelberg-Moscow experiment has set an upper bound of mββ < 0.27eV (90% CL) [20]. Fortunately, Cosmology offers the ability to constrain the neutrino mass since a cosmic neutrino background (CνB ) is present, and its history has had distinct imprints on precision measurements. Most notably, the CνB has made imprints on BBN, the CMB, and large-scale structure (LSS). The most simple constraint comes from assuming the neutrinos make up all of the dark matter. The present energy density of the CνB is (Appendix
A.2), m (eV ) Ω h2 = ν (2.1) ν 93.1 Current data constrain the matter density to Ω 0.3, thus assuming all of the neutrinos contribute m ≃ to this density and h =0.7, it is implied that mν < 14eV . Using more detailed analysis involving LSS and the CMB it is possible to set a much better constraint.
Chapter 2.1
CνB
The Cosmic neutrinos were created very early on in the universe when the temperatures were high enough that weak interactions were frequent. This also coupled the neutrinos to the plasma such that the neutrinos could thermalize and share the same temperature with the plasma. This continued until the thermally averaged weak interaction rate fell below the expansion rate and the neutrinos decoupled. Γ= n σv . H (2.2)
2 2 2 3 T Where σv G T , n T , and H 1.66√g∗ . And g∗ is the effective number of degrees ≈ F ≈ ≈ mpl
8 of freedom. 7 g∗ = g + g (2.3) i 8 j i=bosons j=fermions The last reactions that can keep the neutrinos coupled to the plasma are scatterings with electrons and positrons. Electrons and positrons annihilate away at a temperature of T . 0.5MeV and at this temperature g∗ = 10.75, which puts T 1.5MeV . Since at these temperatures only dec ≈ electrons keep νe coupled, Tdec is actually flavor dependent as well. More careful calculations set T (ν ) 2.4MeV, T (ν ) 3.7MeV [3]. The temperature of the decoupled neutrinos then dec e ≃ dec ,τ ≃ decreases as a−1. It is useful to calculate the temperature of the neutrinos relative to the photons so that the expan- sion rate can be written in terms of the photon temperature and number of neutrinos. However, the photons are heated after the electron/positron pairs annihilate, leading to a different temperature between the neutrinos and photons. Through an argument using the conservation of co-moving en- tropy, T /T = (11/4)1/3 1.40102 is obtained. More detailed calculations with non-instantaneous γ ν ≈ decoupling have been done, which obtain T /T 1.3978 [21]. At this point the relativistic energy γ ν ≈ density can be written. 7 4 4/3 ρ = ρ 1+ N (2.4) r γ 8 11 eff
Where 7/8 comes from integrating the Fermi-Dirac distribution, and Neff is the relativistic degrees of freedom. For three neutrino types taking into account non-instantaneous decoupling, Neff =3.046.
Neff , however, is considered a flexible parameter that can vary to take into account changes to the radiation density due to non-trivial or unknown physics. This form only holds so long as the neutrinos are relativistic, and the 4/11 factor appears only after electron positron annihilation.
Chapter 2.2
Neutrinos and BBN
Now that we know how neutrinos affect the expansion rate in the radiation era, in particular around the time of BBN, we can see how neutrinos (and their mass) affect some BBN observables. The most
4mN (nn/2) sensitive observable to Neff is Yp , the primordial helium abundance, which depends ≃ mN (nn+np) on the neutron to proton ratio n/p. n/p is sensitive to Neff from the expansion rate, since the decay time of neutrons (τ 885sec) is on the same time-scale as BBN. Since the universe cools to n ≈ T 0.08MeV ( [22]) faster when the energy density is larger (larger N ) there will be more BBN ≈ eff
9 4 neutrons available to form He. A fitting function can be used to calculate Yp ( [23], [24]):
Y 0.2485 0.0006+ 0.0016 [(η 6) + 100(S 1)] , where p ≃ ± 10 − − 10 2 η10 = 10 nb/nγ = 273.9Ωbh and (2.5) 7∆N S2 =1+ eff 43
Therefore the faster expansion rate means more neutrons are available to form 4He. Current mea- surements of Y 0.257 0.006 favor a N 3.7 0.5 [3], providing cosmological motivation for a p ∼ ± eff ≈ ± sterile neutrino(s). Neutrino mass can have an effect in that oscillations could create an asymmetry between the flavors which may alter the rates of the weak processes that keep the neutrons and protons in equilibrium before BBN. Due to the nearly maximal oscillation angles, the flavors are equilibrated before BBN [25]. A lepton asymmetry (η = (n n )/n ) contributes very little to νi νi − ν¯i γ N since BBN and the CMB constrain ξ . 0.1 so that the contribution to N from ξ is [22] eff | | eff
30 ξ 2 15 ξ 4 ∆N (ξ )= i + i . 0.01 (2.6) eff i 7 π 7 π i=e, ,τ where ξi = i/T and i is the chemical potential (Appendix A.1). This picture can change if θ13 > 0, which affects oscillations involving νe , leading to a different temperature when all the chemical potentials have equilibrated. If oscillation efficiency is delayed (θ13 = 0), then an asymmetry near neutrino decoupling would lead to a large Neff (see fig. 2.1) [2].
Chapter 2.3
Massive Neutrinos and Large Scale Structure
Around recombination (z 1100), in the cold (non-relativistic) dark matter scenario structures ≈ begin to form from the bottom up (galaxies clusters super-clusters). Cold dark matter is → → important to the formation of structures for two reasons. Being Dark, it is weakly coupled to the photons and baryons while the acoustic oscillations are taking place, which allows matter to begin clustering shortly after matter/radiation equality. And being Cold, it is already non-relativistic and so doesn’t need to fight pressure due to a high velocity. However, the massive neutrinos are not cold (Appendix A.3), since m z 1.99 103 ν (2.7) nr ≈ × 1eV
10 (a) Evolution of ην
(b) Final Neff
in Figure 2.1: Left, the evolution of ηντ,µ (top curves) and ηνe (lower curves) for initial value ηνe = 0.82 2 − and zero total asymmetry. The black line corresponds to sin θ13 = 0 while the red lines were 2 calculated in the NH for values of sin θ13 =0.04, 0.02 from left to right respectively. The blue lines use the same values but for the IH. Right, the final value of Neff after the flavors have equilibrated in with the same initial ηνe as a function of the total asymmetry ην . From top to bottom the lines 2 correspond to no oscillations, θ13 = 0, and sin θ13 =0.04 in the NH and IH. Both figures from [2]. the neutrinos may still be mildly relativistic down to a redshifts of a few hundred for sub-eV neutrino masses. Therefore, there is a suppression in the Matter Power Spectrum (MPK) beginning on the scale that enters the horizon when the neutrinos become non-relativistic (Appendix A.4).
m 1/2 − k 0.015 ν Ω1/2hMpc 1 (2.8) nr ≈ 1eV m with a suppression found in [26] and [1]:
∆P Ων mν 0.1Nν 8 0.8 2 (2.9) P ≈− Ωm ≈− 1eV Ωmh
11 Since knr is redshift independent and this is where suppression begins, it is also the maximum size scale that can be affected. Thus both the amount of suppression and the largest scale affected depend on the neutrino mass. This is due to the free-streaming characteristic of light neutrinos, which can move out of a region very quickly and carry away their own mass. There is a free streaming scale
(kfs) corresponding to the distance a neutrino can travel in one expansion time. In the matter dominated era it scales as Ω m 2 − (k /h)2 m ν Mpc 2 (2.10) fs ∼ (1 + z) 1eV On scales smaller than this the neutrinos cannot cluster, thus they don’t contribute to the matter perturbations on these scales. Figure 2.2(a) shows how the evolution of kfs determines which modes are allowed to grow and which free stream and therefore cannot grow. From 0.001 Ωk ) can weaken this constraint to mν . 1.2eV [27], [7], [28]. Thus, current LSS constraints are sensitive to mν only, and cannot determine the hierarchy. 12 (a) Growth (b) Matter Power Spectrum Suppression Figure 2.2: Suppression of LSS due to massive neutrinos. 2.2(a), The upper black line is the neutrino free streaming scale, kfs for a model with mν = 1.2eV . The lower black line is the comoving Hubble scale, aH. All modes in the shaded region can grow, whereas modes above the shaded region are subject to free streaming suppression. 2.2(b), the matter power spectrum divided by the ΛCDM matter power spectrum for various different neutrino masses. The lines are for mν =0, 0.15, 0.6, and 1.2eV in descending order. Both figures from [3]. 13 100000 Ω h2 = 0 2 ν Ωνh = 0.005 Ω h2 = 0.01 ν2 Ωνh = 0.025 10000 2 /k 1000 CDM ∆ 100 10 0.0001 0.001 0.01 0.1 1 k/h Mpc-1 Figure 2.3: The Matter Power Spectrum made with CAMB in a ΛCDM +mν model with mν and ΩΛ varied. The more massive the neutrino the greater the MPK is suppressed, especially on smaller scales. 14 Chapter 3 Temperature Anisotropies in the Cosmic Microwave Background Neutrino mass has a rather subtle and non-trivial affect on features of the CMB temperature anisotropies, thus it is important to understand how some of these features are formed. The acoustic oscillations start well before matter/radiation equality when the baryons are tightly coupled to the photons. As each mode of a photon perturbation Θ(k) enters the horizon it evolves like a standing wave oscillating at the speed of sound cs with gravity and radiation pressure acting as driving and restoring forces. In the radiation era, when a mode of the gravitational potential Ψ(k) enters the horizon, the rapid background expansion due to the radiation density causes Ψ(k) to decay very rapidly. Well into the matter dominated era, the collapse of matter balances the expansion rate allowing the potential to remain constant. The decay of the potential is such that at maximal compression the potential is at a minimum, giving way to a larger restoring force from the radiation (this is known as Radiation Driving). If the time of matter radiation equality is delayed then radiation driving can persist more efficiently until recombination, which will increase the heights of the peaks. After recombination, any residual radiation contributes toward increasing the expansion rate which causes the potentials to continue changing. Photons that would otherwise have experienced no net gain of temperature from falling in and climbing out of a potential do experience a change. This is known as the early Integrated Saches Wolfe (ISW) effect. These oscillations continue until the baryon-photon fluid decouples when the electrons combine with the protons to form neutral hydrogen (z 1100,T 0.26eV ) allowing the photons to rec ∼ rec ≈ freestream. These perturbations are frozen in at this time on a scale on the order of the size of the sound horizon at recombination rs(ηrec)= csdη. The photons then pick up temperature fluc- tuations depending on where they are in the compression/rar efaction phase and from gravitational redshifting, Θ + Ψ. The fluctuations are sampled from the variance of the coefficients of a spherical harmonic expansion of Θ: ∞ l Θ( x, p, η)= alm( x, η)Ylm( p) (3.1) − l=0 m = l 15 ∗ 2 2 The variance a a ′ ′ = δ ′ δ ′ C leads to the C s which are plotted as ∆ = l(l + 1)/2πC T lm l m ll mm l l T l against l. The cosmological parameters effectively alter the peak locations (l nπ/θ nπη /r ) n ∼ ∼ 0 s and heights in both unique and degenerate ways. Chapter 3.1 Effects of massive neutrinos on the CMB TT Spectrum Neutrino mass can affect the CMB Temperature spectrum in several different ways. First, neu- trinos can alter the redshift of matter/radiation equality z ( 3000) (and correspondingly the eq ≈ matter/radiation energy ratio) which can change the contribution from the early ISW effect. Sec- ond, mν changes the expansion rate by changing the energy density which can change the size of the sound horizon at recombination and/or the distance to last scattering. Changes in the expansion rate can also affect the damping scale, which instead of shifting to smaller scales, shifts to larger scales when the acoustic angular scale is kept fixed [29]. Weak lensing from LSS after recombination can also affect the CMB power spectrum on small scales and can mix E and B modes of the polar- ization spectrum ( [30]). It was also shown in [31] that neutrino perturbations outside the sound horizon could cause a small phase shift to larger scales. However, since most of the correlations with mν and other parameters can be understood through large scales (l < 1000), this discussion will focus on the location and height of the first acoustic peak. Neutrinos can affect the redshift of matter/radiation equality zeq via their contribution to ρm and ρr which depend on their mass. Thus, one must be careful as to how neutrinos get counted into each form of energy density. If they are relativistic at zrec (mν . 0.5eV , A.3) then it is reasonable to assume they contribute only to ρr, with the corollary partly true (if non-relativistic at zrec , they could be mildly relativistic near zeq ). One method to account for this is to split the neutrino distribution up into two parts that divide the contribution between the radiation and matter energy 2 densities. Adopting the method from [4], Ων h can be written, 4/3 ∞ 2 45 4 2 −4 2 2 2 x −1 Ων h = 4 Ωγ,0h a x + y x (e + 1) dx (3.2) π 11 0 where m a(11/4)1/3 y = ν (3.3) T γ,0 and x is the normalized momentum variable, Tγ,0 is the present CMB photon temperature, and three mass degenerate flavors of neutrinos are assumed. For x>y a neutrino is relativistic at a given 16 redshift and non-relativistic if x Since zrec is more or less fixed, changing zeq alters how long radiation can influence the evolution of the perturbations. If there is more residual radiation near the time of recombination then the potentials will be changing more rapidly and thus there is a larger contribution from the early ISW effect. Thus for small masses (mν . 0.5eV ) neutrinos enhance the early ISW effect and for larger masses they suppress it. The early ISW effect contributes to scales between the acoustic scale (l 300) and the scale at last scattering (l 150). An enhancement on these scales means l is A ∼ eq ∼ A effectively shifted to smaller multipoles and the height of the peak H1 is increased. Figure 3.1 shows how zeq can depend on mν . z ΡΝHm ΝLΡΝH0L 3350 mΝ 0.2 eV 3.0 mΝ 0.4 eV 3300 2.0 mΝ 0.8 eV 3250 1.5 3200 m z 0.0 0.2 0.4 0.6 0.8 1.0 Ν 500 1000 1500 2000 2500 3000 (a) zeq vs mν (b) ρν vs z Figure 3.1: 3.1(a), zeq is calculated via eq. 3.4. As anticipated, zeq is mass dependent. The behavior changes as the mass of a single neutrino species becomes non-relativistic around zrec & 1100 m & 0.5eV . Using a cutoff point of 3.15y shifts the extrema to a mass of 0.2eV . Since this⇒ is ν ∼ close to most limits of mν . 0.6eV , the analysis is unchanged. 3.1(b), the total energy density ρν as a function of redshift for different mass scale mν . The background expansion from the neutrino density can also change the size of the horizons. The sound horizon and comoving horizon at recombination ηrec are affected at early times before recombination. And contributions to the current size of the horizon η0 come mostly from late-time evolution and geometry. Thus the locations of the peaks are very sensitive to Ωk , shifting to larger (smaller) scales for Ωk < 0 (Ωk > 0). ΩΛ and the dark energy equation of state w have similar effects 17 (in a flat universe) since they contribute to the late time expansion which can change the distance to last scattering. Hence these parameters lead to the geometric degeneracy. By increasing mν in a flat universe the angular diameter distance decreases at a larger rate than the sound horizon, which projects the peaks onto larger scales (fig. A.1). These effects can be calculated by considering that the mth peak can be found near l = l (m φ ) (3.5) m A − m where lA is the acoustic scale, dA(ηrec) lA = π (3.6) rs(ηrec) and dA is the angular diameter distance to the last scattering surface (occurring at ηrec) and rs is the sound horizon at last scattering. dA depends on the expansion history and geometry and rs depends also on the expansion history but also the sound speed cs. In a flat universe, ΩΛ is decreased when Ων is increased, which decreases the distance to last scattering. In a non-flat universe, increasing Ων leads to a closed universe. Hence either way, lA, and thus l1 is shifted to smaller multipoles (Appendix A.5). The phase shift φm due to the Early ISW effect can be parametrized by the matter to radiation ratio in a fitting formula used by [4] from [32] r 0.1 φ 0.267 rec (3.7) 1 ≈ 0.3 where rrec is the radiation to matter ratio rrec = ρr/ρm at the time of recombination. This quantity is related to zeq , thus φ1 exhibits similar behavior to that of fig. 3.1. For mν . 0.5eV the effect is to enhance the early ISW contributions and shift the spectrum to larger scales. Larger masses still cause a shift to larger scales due to changes in the distance to last scattering. (Appendix A.5 2 ). The height of the first peak H1 = (∆Tl1 /∆Tl=10) increases for masses . 0.5eV from the early ISW effect. It can also be understood that in a flat universe, ΩΛ may decrease if Ων is increased. A decrease of ΩΛ can reduce the late time ISW effect which enhances the plateau region and by normalization raises the peaks. When mν is large enough, the neutrinos behave more like matter around recombination thus reducing the peak height. Figure 3.2 shows how the location and height 2 of the first peak depends on ων =Ων h in a flat ΛCDM universe. To summarize, the neutrinos affect the CMB temperature spectrum in a very non-trivial manner. Their mass contribution to the energy density alters zeq , affecting the early ISW effect which changes the heights and locations of the first few peaks. Changing mν can also change the size of the horizons 18 (a) (b) Figure 3.2: 3.2(a), phase shift from affecting the early ISW contributions and horizon sizes. 3.2(b), the change in height of the first peak from changes in the ISW effect as dependent on the neutrino 2 energy density ων =Ων h . Both figures are from [4] and use a flat ΛCDM model. by altering the expansion rate and/or other cosmological parameters. The effects just mentioned can be seen explicitly on the CMB Temperature spectrum in fig. 3.3. 7000 Ω h2 = 0 2 ν Ωνh = 0.005 Ω h2 = 0.01 ν2 Ωνh = 0.025 6000 WMAP 7 yr data 5000 2 µΚ 4000 π l(l+1)/2 3000 l C 2000 1000 0 10 100 1000 log l Figure 3.3: CMB TT anisotropy spectrum made with CAMB. All curves are for a flat ΛCDM model with H0 fixed but mν and ΩΛ varied. When mν & 0.5eV the neutrinos are non-relativistic at recombination and thus behave more like matter and decrease the early ISW contribution, lowering the peak. In all cases however, there is a phase shift to larger scales from horizon scaling and metric perturbations outside the sound horizon. 19 Chapter 3.2 Effects of a non-thermal distribution on the CMB TT Spectrum As mentioned in 2.1, the entire neutrino population does not decouple instantaneously from the plasma. Neutrinos in the tail end of the distribution still have enough energy to interact via elec- tron/positron scatterings as the electron/positron pairs annihilate transferring their energy to the photon plasma. This imposes flavor dependent, non-thermal distortions on the neutrino’s spectra. This can be accounted for in the neutrino energy density by using Neff =3.046 instead of 3. Since the anisotropies in the CMB are calculated by solving the Boltzmann Equation, df(t) = C(t) (3.8) dt where f is the distribution function and C(t) is the collision term, corrections to the distribution function must be accounted for. These corrections were calculated in [21] and are as follows: fν1 (y)=0.7fνe (y)+0.3fνx (y) fν2 (y)=0.3fνe (y)+0.7fνx (y) (3.9) fν3 (y)= fνx (y) −4 2 3 fνe (y)= feq(y) 1+10 1 2.2y +4.1y 0.047y − − (3.10) − f (y)= f (y ) 1+10 4 4+2.1y +2.4y2 0.019 y3 νµ,τ eq − − where y = pa is the comoving momentum, fνx = fνµ = fντ and feq is the Fermi-Dirac distribution. These equations are valid for θ13 = 0. It turns out that the corrections are at most on the sub- percent level, on order with the numerical error in the code. Figures 3.4 and 3.5 show how including non-thermal corrections to the distribution function are small, and less than the change to the CMB power spectrum for different neutrino masses. 20 6000 mν = 0.005 eV mν = 0.15 eV 5000 2 4000 µΚ π 3000 l(l+1)/2 l C 2000 1000 10 0 8 10 100 1000 6 log l 4 2 0 -2 Percentage difference -4 10 100 1000 log l Figure 3.4: CMB TT anisotropy spectrum made with CLASS ( [5]). All curves are for a flat ΛCDM model with H0 fixed but with the individual neutrino masses mν and ΩΛ varied. In this mass range differences of a few percent arise. Chapter 3.3 CMB parameter Determination Since there are many parameters that can be used to calculate the CMB power spectrum, and each one can have a unique and/or degenerate effect with another, a large sample of parameter space is needed to determine a best fit value for each one. This is where Monte Carlo simulations become useful, in particular CosmoMC ( [33]) which is the most widely used and has settled into being the standard. CAMB ( [34]) is used to calculate the CMB anisotropies and matter power spectrum and it has been shown that it is accurate to sub percent precision and consistent with other codes ( [5], [35], [36]). CosmoMC can then use CAMB in conjunction with other theoretical calculations like BBN and/or LSS as a theoretical standpoint that can be compared with real data to constrain cosmological parameters. Experimental collaborations usually produce their own analysis using their data such as WMAP ( [7]), the South Pole Telescope (SPT, [37]) and the Atacama Cosmology Telescope (ACT, [38]). The WMAP results for mν are listed below in table 3.3. 21 6000 mν = 0.15 eV, fid mν = 0.15 eV, mang 5000 2 4000 µΚ π 3000 l(l+1)/2 l C 2000 1000 0.4 0 0.3 10 100 1000 0.2 log l 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 Percentage difference -0.6 10 100 1000 log l Figure 3.5: CMB TT anisotropy spectrum made with CLASS ( [5]). All curves are for a flat ΛCDM model with everything fixed including mν . The two curves compare the use of a pure Fermi-Dirac distribution to describe the neutrino population and the distorted function described by equations 3.9 and 3.10. The difference is on the sub-percent level, on the same order as the numerical accuracy of the code. Data Set mν (ΛCDM) mν (wCDM) WMAP7yr < 1.3eV (95%CL) < 1.4eV (95%CL) WMAP7yr + LRG + H0 + SNe < 0.39eV (95%CL) < 0.53eV (95%CL) Table 3.1: Constraints on mν from the WMAP collaboration. Clearly when additional information from LSS is included the constraints are improved. Since the WMAP analysis is constrained to conservative models the constraints on mν can change when other parameters are allowed to vary or new data are added. In the case of mν , a parameter’s central value can change along with a relaxation of the error bars if there is a correlation to mν . Some of the most recent analysis by [27], [28] and [6] have been thorough in exploring all types of models and data sets. For example, in [6], when Ωk is allowed to vary but Neff is not, they quote a constraint of mν < 0.95eV which relaxes to mν < 1.12eV when Neff (∆Neff > 0) is allowed to vary. This can be understood by considering that increasing mν shifts the power spectrum to smaller multipoles. Neff can counter this effect since increasing Neff can decrease the size of the sound horizon at recombination which pushes the spectrum to larger multipoles. This is most easily compensated by allowing Ωk > 0, which counters the effect of neutrino mass, while Ωk < 0 can counter the effect of a larger Neff . Therefore there is a positive correlation between 22 mν and Ωk , and a negative correlation between Neff and Ωk . Since these effects are small, there is little to no correlation of mν and Neff from the CMB ( [27], [28]). Figure 3.6 shows these correlations. There are several other (CMB) parameters which exhibit correlations with mν , namely w, 2 ΩΛ, and Ωch . As discussed in detail, neutrino mass can have an effect on the size of the horizons. Therefore changing the dark energy equation of state or ΩΛ leads to a degeneracy with mν . In models with curvature, decreasing ΩΛ can help keep Ωk > 0, which as we’ve seen counters the effect 2 of mν on the CMB. In a flat universe decreasing ΩΛ can increase Ωm (but decrease h to keep Ωch , and thus the sound horizon, fixed) which then increases the distance to last scattering. This will project the multipoles to smaller scales, countering the effects of massive neutrinos. Hence there is a negative correlation between mν and ΩΛ from the CMB data. When information from LSS is included there is also a similar degeneracy since decreasing ΩΛ can reduce a similar suppression to the MPK caused by massive neutrinos. Decreasing w can also increase the distance to last scattering through slowing the expansion (increasing the age of the universe) by delaying when dark energy becomes dominant. When LSS data is considered then Ωm must increase to counter the suppression to the matter power spectrum from Ων . However, an increasing Ωm becomes incompatible with the supernova data unless w . 1 [39]. Thus there is still a negative degeneracy between m and w. − ν 2 Lastly, there is a correlation between Ωch and mν . Since the data seem to favor smaller masses (individual masses that are non-relativistic at recombination) increasing mν raises the first peak by 2 changing zeq and thus the contribution from the early ISW effect. A larger Ωch can counter this, 2 2 2 leading to a positive correlation with mν . In addition, CosmoMC calculates Ωdmh =Ωch +Ων h , 2 so of course increasing Ων h contributes to the positive correlation. All of these correlations can be seen clearly in fig. 3.7. 23 (a) (b) (c) Figure 3.6: All figures are from [6] using a ΛCDM + Ωk + Neff + mν model with data sets W MAP + BAO + H0 + SPT + ACT . Allowing for Neff to vary shifts the mass constraint to larger values. However, allowing for Ωk different from zero doubles the value of mν allowed [6]. Also, by allowing both mν and Neff to vary, the positive correlations between those parameters but opposite correlations with Ωk destroys the correlation between Ωk and Neff (3.6(c)). 24 0.76 −0.4 −0.6 0.74 −0.8 0.72 −1 Λ 0.7 w Ω −1.2 0.68 −1.4 0.66 −1.6 0.64 −1.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 m m ν1 ν1 (a) (b) 0.17 0.16 0.15 2 0.14 h DM Ω 0.13 0.12 0.11 0.1 0 0.1 0.2 0.3 0.4 m ν1 (c) 2 Figure 3.7: 2D contours for Left ΩΛ , middle w and right Ωmh vs mν,1. All contours are for a ΛCDM +Ωk + Neff + mν + w model from a CosmoMC run with three chains. Data sets include W MAP 7yr + SPT + LRGDR7+ H0 + SNe. 25 Chapter 4 Conclusion The neutrino background, although yet to be directly detected, has proven to play a critical role in the full history of the Universe. Neutrino oscillations and the number of species can affect flavor asymmetries before the BBN epoch and the expansion rate, which in turn can change the values of primordial element abundances. Their affect on the background expansion rate produces phase shifts in the CMB Temperature spectrum and can alter the heights of the peaks. In more recent times, their free streaming abilities affect the formation of structures. When lensing information from more precise data is available, m should be constrained to 0.1 0.15eV [3], [40]. And ν ∼ − with experiments sensitive to B-mode polarization such as CMBPol [41] mν could be constrained to a sensitivity of 0.04eV . For now, Cosmology has placed the most stringent bound on the ∼ neutrino rest mass at a few tenths of an eV. These current bounds may lead to interesting questions raised and/or answered within the Standard Model when experimental particle physics is able to place similar bounds. 26 Bibliography [1] Julien Lesgourgues and Sergio Pastor. Massive neutrinos and cosmology. arXiv:astro- ph/0603494v2, 2006. [2] Gianpiero Mangano, Gennaro Miele, Sergio Pastor, Ofelia Pisanti, and Srdjan Sarikas. 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[41] Daniel Baumann et al. Cmbpol mission concept study probing inflation with cmb polarization. arXiv:0811.3919, 2009. 30 Appendix A Appendix Appendix A.1 ξ, neutrino asymmetries and Neff Since in the radiation era the neutrino energy density can be written as ρ 7 T 4/3 ρ = ρ ν = ρ 1+ ν N (A.1) ν γ ρ γ 8 T eff γ γ then adding a neutrino chemical potential can change Neff since the distribution function will be modified. ∞ 3 4π p dp 4 4π i/T ρν,i = − = 6Tν,i Li4( e ) (A.2) (2π)3 e(p i)/T +1 − (2π)3 − 0 ∞ zk where Lis(z) = k=1 ks is a special function known as the polylogarithm or Jonquiere’s function. Another term for the anti-neutrinos will be dealt with later. There are only four terms, and if we assume the chemical potential to be close to zero, we can expand the exponentials. Expanding and defining ξi = i/T then leads to: 1 7π4 9 π2ξ2 ξ4 ρ T 4 + ζ(3)ξ + i + ln(2)ξ3 + i + O ξ5 (A.3) ν,i ≈ 2π2 ν,i 120 2 i 4 i 8 i The photon contribution from integrating over the Bose-Einstein distribution gives a factor of π2/30 (per one degree of freedom) that will be pulled out. Also, since the total contribution to the energy density will be a sum over all of the flavors and their anti-neutrino counterpart with = , ν,i − ν,i¯ all odd order terms will cancel. ρ 7 T 4/3 30 ξ 2 15 ξ 4 ν = ν,i 1+ i + i (A.4) ρ 8 T 7 π 7 π γ i=e, ,τ γ 31 Now we can identify the correction terms to Neff , such that; 30 ξ 2 15 ξ 4 ∆N = i + i (A.5) eff 7 π 7 π i=e, ,τ Appendix A.2 Present neutrino mass density The present energy density of neutrinos today, assuming they are non-relativistic is, ρν = nν mν (eV ) (A.6) To get the number density, we can utilize relations between the CMB photons and the CνBneutrinos. Specifically, − − n p2(ep/T + 1) 1dp q2(eq/Tν,0 + 1) 1dq ν = = (A.7) 2 p/T −1 2 q/T 0 −1 nγ p (e 1) dp q (e γ, 1) dq − − where q = pa is the co-moving momentum and I’ve used the fact that T a−1. ∼ n (3/2)T 3 ζ(3) 3 T 3 ν = ν,0 = ν,0 (A.8) n 2T 3 ζ(3) 4 T γ γ,0 γ,0 By definition of Ωi, 3 ρν 3 Tν,0 1 Ων = = nγ mν (eV ) (A.9) ρcr 4 Tγ,0 ρcr and we can calculate nγ like above, putting back in constants and numerical factors. 2ζ(3) − n = T 3 = 410.5cm 3 (A.10) γ π2(~c)3 γ,0 Using the temperature ratio T /T = 1/1.4010, ρ = 8.098 10−11h2(eV 4)/(~c)3 and putting ν,0 γ,0 cr × everything together we finally get 410.5 3 1 3 m (eV ) Ω h2 = m = ν (A.11) ν 8.098 10−11/(~c)3 4 1.4010 ν 94.1 × A more careful calculation using the non-thermal distributions (eq. 3.9, 3.10) and temperature difference from [21], Tν,0/Tγ,0 =1/1.3978, leads to a slightly different result: m (eV ) Ω h2 = ν (A.12) ν 93.1 32 Appendix A.3 znr The neutrinos transition to being non-relativistic when E =3.15T m (A.13) ≈ νi where the index i indicates a different neutrino flavor, the factor 3.15 coming from the average energy of a thermalized population of fermions, and T =(1+ z)T . With k T =1.68 10−4eV . ν,0 B ν,0 × m z 1990 νi (A.14) nr,i ≈ 1eV Appendix A.4 Neutrino Matter Power Spectrum suppression scales Only modes larger than the mode that enters the horizon as neutrinos become non-relativistic (knr) may be affected by the free-streaming effects of neutrinos. Where, knr = anrH(anr)/c (A.15) and H(a) H (1 + z)3/2√Ω in the matter dominated era. Using z from above, we find ≈ 0 m nr m 1/2 − k /h 0.015 ν Ω1/2Mpc 1 (A.16) nr ≈ 1eV m The free streaming scale is related to the distance a neutrino can travel in a Hubble time. (aH)2 d3pf v2 k2 ; c2 = o (A.17) fs c2 d d3pf ∼ d 0 Assuming the neutrinos are non-relativistic, then −4 p 3T (eV/c) 1 3 c 1.68 10 eV 1eV − c = ν,0 × × × 150(1 + z) kms 1 (A.18) d m ≃ m (eV/c2) a ≃ m (eV )a ≃ m ν ν ν ν 33 and in the matter/lambda dominated epoch, 2 2 3 H (z)= H0 Ωm(1 + z) +ΩΛ (A.19) so we can write the free streaming scale. 3 Ω (1 + z) +Ω m 2 − (k /h)2 m Λ ν Mpc 2 (A.20) fs ∼ (1 + z)4 1eV Appendix A.5 Neutrino effects on Horizon scaling Since l = π dA(ηrec) , changes in the expansion history or geometry can affect these quantities. A rs(ηrec) Neutrinos can change these quantities directly through their contribution to the energy density, or by changing other quantities like Ωm or Ωk . The horizons are calculated by the following equations: a ′ csda 2 1 rs = ′2 ′ ; cs = a H(a ) 3(1+3ρb/4ρr) 0 a rs dA = sin ΩkH0χ(a∗) (A.21) H0√Ωk − ≈ θs ′ ′ ′ t0 dt t dt a da ∗ χ(a )= ′ ; η(a)= ′ = ′2 ′ ∗ a(t ) a(t ) a H(a ) t 0 0 where rs is the sound horizon, dA is the angular diameter distance, χ is the comoving distance and η is the conformal time. Figures A.1 and A.2 show how changing the neutrino mass can affect these quantities. 34 1.00 1.00 0.98 rsrsHmΝ 0L 0.95 ΗΗHmΝ 0L 0.96 rsrsHmΝ 0L 0.90 ΗΗHm 0L 0.94 Ν m m 0.2 0.4 0.6 0.8 1.0 Ν 0.2 0.4 0.6 0.8 1.0 Ν (a) Ωk = 0 (b) Ωk =6 0 Figure A.1: A.1(a), the sound horizon (blue) decreases since more energy density hastens the time of recombination. The distance to last scattering (red) decreases at a larger rate due to the decrease in ΩΛ . A.1(b), the same effect occurs but because increasing Ων leads to a closed universe. Scaling is in reference to the massless neutrino case. l1 220 l1 220 218 215 216 210 214 205 200 212 mΝ m 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Ν (a) Ωk = 0 (b) Ωk =6 0 Figure A.2: The first peak is shifted to larger scales due to massive neutrino effects on the horizons. A.2(a) uses a zero curvature model with fixed Ωm and h, thus ΩΛ is decreased to compensate for an increase in neutrino mass. A.2(b), Ωm +ΩΛ = 1, thus the curvature depends on the neutrino mass density. The shift is more dramatic due to the strong scaling dependence on the curvature. 35