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 (gµν ) evolves under the influence of energy (Tµν ), 1 R g R =8πGT (1.1) µν − 2 µν µν 1 10 0 0 − 0 a2(t)0 0 where gµν = 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. Rµν 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 µ ν 2 2 2 2 2 2 2 2 gµν 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.