Massive neutrinos and cosmology

Massive neutrinos and cosmology Yvonne Y. Y. Wong RWTH Aachen Theory colloquium, Padova, November 18, 2009

Relic neutrino background: Temperature: 4 T,0 = 11 Origin of density perturbations? 1 /3 T CMB, 0=1.95 K Number density per flavour: n, 0= 6 3 3 3 T, 0= 112 cm 2 4 Nucleosynthesis Last scattering surface (CMB) Structure formation If mν > 1 mev Energy density per flavour: m 2 h = 93 ev Neutrino dark matter

Neutrino dark matter... Evidence for neutrino masses from neutrino oscillations experiments : 2 3 2 m ~2 10 ev atm m2sun ~8 10 5 ev 2 Large mixing between flavours: [ ][ e 0.82 0.54 = 0.31 0.47 0.47 0.71 ][ ] 0.21 1 0.69 2 0.69 3

Oscillation data allow for two mass hierarchies: m2atm ~2 10 3 ev2 2 msun ~8 10 5 ev 2 Normal hierarchy min m ~0.05 ev Mininum amount of neutrino dark matter Inverted hierarchy min m ~0.05 ev min ~0.1 %

Upper limit on neutrino masses from tritium β-decay: Large mixing means 2 U ei ~O 0.1 1 me U m 2 i ei 2 i 1/2 2.2 ev Electron energy (kev) max m ~7 ev max ~12 % Light neutrinos cannot be the only dark matter component

Neutrino dark matter is hot... Large velocity dispersion: v thermal 81 1 z ev m km s 1 A dwarf galaxy has a velocity dispersion of 10 km s-1 or less, a galaxy about 100 km s-1. Sub-eV neutrinos have too much thermal energy to be packed into galaxy-size self-gravitating systems. Neutrinos cannot be the dominant Galactic dark matter.

Why study neutrinos in cosmology... Hot dark matter leaves a distinctive imprint on the large-scale structure distribution. We can learn about neutrino properties from cosmology. Cosmological probes are getting ever more precise: Even a small neutrino mass can bias the inference of other cosmological parameters.

The concordance framework... We work within the ΛCDM framework extended with a subdominant component of massive neutrino dark matter. Flat geometry. Main dark matter is cold. Initial conditions from single-field slow-roll inflation. Baryons

Plan... What we can do now What we can do in the future The nonlinear matter power spectrum

1. What we can do now...

Two effects of massive neutrinos... On the background: Shift in time of matter radiation equality. On the perturbations: Suppression of growth.

Background... Sub-eV neutrinos become nonrelativistic at z<1000: Radiation at early times. Matter at late times. Comoving matter density today Comoving matter density before recombination Shift in matter-radiation equality relative to model with zero neutrino mass. mν = 1 ev mν = 0 ev

Two effects of massive neutrinos... On the background: Shift in time of matter radiation equality. On the perturbations: Suppression of growth.

Perturbations... At low redshifts, neutrinos become nonrelativistic:. ev But still have large thermal speed: c 81 1 z m hinder ν clustering on small scales. ν km s 1 ν c c Gravitational potential wells Free-streaming length scale & wavenumber: 8 2 c 2 1 z ev FS 4.2 m,0 m 3 m H 2 2 k FS FS h 1 Mpc FS Clustering k k FS FS Non-clustering k k FS

In turn, free-streaming (non-clustering) neutrinos slow down the growth of gravitational potential wells on scales λ << λfs or wavenumbers k >> kfs. Clustering potential wells become deeper ν c ν c Some time later... ν Both CDM and neutrinos cluster c ν c ν c ν c c ν Only CDM clusters

Depth of pot. well Initial time... δcdm δcdm Some time later... CDM-only CHDM k k Wide pot. well kfs Narrow pot. well The presence of HDM slows down the growth of CDM perturbations at large wavenumbers k. The density perturbation spectrum acquires a step-like feature.

Describing perturbations: CDM... Cold dark matter = collisionless, pressureless fluid: Density perturbations Continuity eqn Euler eqn Gravitational source c c =0 2 c H c =0 Velocity divergence Poisson eqn Expansion 3 2 = H m [ f c c f ] 2 2 f Neutrino m fraction

Describing perturbations: Neutrinos... Free-streaming neutrinos cannot be described by a perfect fluid. Must solve (linearised) collisionless Boltzmann equation: Nonrelativistic neutrinos f0 f p f a m =0 m a p f x, p, = f 0 f Momentum moments: Phase space density 1 3 Density d p f perturbation pi 1 3 d p i f Velocity divergence a m p p 1 ij d 3 p 2i 2j f Pressure and anisotropic stress a m

Describing perturbations: Neutrinos... Free-streaming neutrinos cannot be described by a perfect fluid. Must solve (linearised) collisionless Boltzmann equation: Nonrelativistic neutrinos f0 f p f a m =0 m a p f x, p, = f 0 f Momentum moments: Give rise to free-streaming behaviour Phase space density 1 3 Density d p f perturbation pi 1 3 d p i f Velocity divergence a m p p 1 ij d 3 p 2i 2j f Pressure and anisotropic stress a m

Clustering regime Massive neutrinos, mν=1 ev k =10 2 h Mpc 1 k FS kfs >> ~a δcdm δb δν during matter domination δγ ψ, φ Lesgourgues and Pastor 2006

Non-clustering regime Massive neutrinos, mν=1 ev 1 k =1 h Mpc k FS kfs << a δcdm δb a1-3/5fν δν ψ, φ (fν = Ων/Ωm) δγ Lesgourgues and Pastor 2006

Large scale matter power spectrum, P(k) CMB Galaxy clustering surveys Lyman-α fν = Neutrino fraction P 8 f 8 m P h = 2 m 93 ev

Large scale matter power spectrum, P(k) CMB Galaxy clustering surveys Linear k 3 P k 1 2 2 Lyman-α fν = Neutrino fraction P 8 f 8 m P h = 2 m 93 ev

Present status... WMAP5 only Dunkley et al. 2008 95% C.L. upper limit + Galaxy clustering Reid et al. 2009 + Galaxy + SN + HST Reid et al. 2009 Break degenercies + Weak lensing Tereno et al. 2008 Ichiki et al. 2008... and many more.

2. What we can do in the future...

WFMOS Planck High-z spectroscopic galaxy surveys, z>2 Photometric galaxy surveys with lensing capacity, zmax~3 Radio arrays, 5 < z < 15 MW A SNAP

Possible new techniques... Weak lensing of galaxies of the CMB Song & Knox 2004 Hannestad, Tu & Y3W 2006 Kitching et al. 2008 Lesgourgues et al. 2006 Perotto, Lesgourgues, Hannestad, Tu & Y3W, 2006 21 cm emission Mao et al. 2008 Pritchard & Pierpaoli 2008 Metcalf 2009 ISW effect Ichikawa & Takahashi 2005 Lesgourgues, Valkenburg & Gaztañaga 2007 Cluster abundance Wang et al. 2005

Weak lensing of galaxies/cosmic shear... Distortion (magnification or stretching) of distant galaxy images by foreground matter. Unlensed Distortions probe both luminous and dark matter (no galaxy bias problem!) Lensed

Unlensed Galaxies are randomly oriented, i.e., no preferred direction. Average galaxy shapes over cell Lensed Lensing leads to a preferred direction.

Shear map Weak lensing theory predicts: Cluster Void

Shear map Convergence map (projected mass) Weak lensing theory predicts: Cluster Void

Convergence power spectrum C l d a 0 2 [ ] 2 D ' d ' ngal ' D ' P k =l /D

Tomography = bin galaxy images by redshift z Tomography probes spectrum evolution and the growth function.

Convergence power spectra for 2 tomography bins Bin 2 autocorrelation 1 & 2 crosscorrelation Bin 2: 1.5 < z < 3 Bin 1 autocorrelation Bin 1: 0 < z < 1.5 Hannestad, Tu & Y3W 2006

Future surveys with lensing capacity 2008 Space-based 2010 2015 Ground-based 201X Time SNAP

Projected sensitivities... Currently disfavoured at 95% C.L. Planck (1 year) Lesgourgues et al. 2006 Perotto, Lesgouruges, Hannestad, Tu & Y3W 2006 95% sensitivities ~ 2015-2020 + Weak lensing with LSST (tomography) Hannestad, Tu & Y3W 2006 Kitching et al. 2008

Projected sensitivities... Currently disfavoured at 95% C.L. Planck (1 year) WARNING! Lesgourgues et al. 2006 Perotto, Lesgouruges, Hannestad, Tu & Y3W 2006 These sensitvities can be achieved only if theoretical predictions of the matter power spectrum are accurate to ~ 1%. 95% sensitivities ~ 2015-2020 + Weak lensing with LSST (tomography) Hannestad, Tu & Y3W 2006 Kitching et al. 2008

3. The nonlinear matter power spectrum...

N-body Total matter power spectrum, P(k) Nonlinearities... Semi-analyatic (PT, RG) Galaxies Weak lensing <1% CMB Linear 3 k P k 1 2 2

N-body Semi-analyatic (PT, RG) Galaxies Weak lensing <1% CMB Linear 3 k P k 1 2 2 What are the signatures of massive neutrinos in the nonlinear regime?? Total matter power spectrum, P(k) Nonlinearities... P lin ~8 P lin m?

Cosmological N-body simulations... = CDM Represent fluid O(100) to O(1000) Mpc

Cosmological N-body simulations... = CDM Represent fluid by point particles. O(100) to O(1000) Mpc

Cosmological N-body simulations... = CDM Represent fluid by point particles. Initial conditions from the Zel'dovich approximation: x= x i D i x i Initial positions dd & velocities v= x i d ψ = random displacement vector (from linear power spectrum) D = linear growth function O(100) to O(1000) Mpc

Simulating massive neutrinos... Neutrinos have thermal motion. Unperturbed phase space distribution: f p = Present day neutrino temperature 1 exp p/ T 0 1 T 0=1.95 K Average velocity: v = q ev 81 1 z m a m km s 1 For mν = 1 ev, <v> is 10% the speed of light at z = 100.

Same equations of motion as for CDM. Neutrino thermal motion is factored into the initial conditions. One possible implementation: Give neutrino particles an initial velocity: v= dd x i v thermal r d r = random direction vector v thermal = thermal velocity drawn randomly from a Fermi-Dirac distribution Brandbyge, Hannestad, Haugbølle & Thomsen 2008

Simulating CDM+massive neutrinos... Particle representation for both CDM and neutrinos. Modified GADGET-2. Neutrino particles drawn from Fermi-Dirac distribution. Brandbyge, Hannestad, Haugbølle & Thomsen 2008 Brandbyge and Hannestad 2008, 2009 z=4 CDM density z=0 512 h-1 Mpc density mν=0.6 ev

Change in the total matter power spectrum relative to the fν = 0 case: Pm P f Pm 0 k P f =0 k P f =0 k Linear perturbation theory: 0.15 ev 0.3 ev Pm ~8 m Pm 0.45 ev With nonlinear corrections: 0.6 ev Linear theory N-body Pm ~9.8 m Pm Power suppression due to neutrino free-streaming is larger than predicted by linear perturbation theory.

N-body Total matter power spectrum, P(k) Nonlinearities... Semi-analyatic (PT, RG) Galaxies Weak lensing <1% CMB Linear 3 k P k 1 2 2

Semi-analytic techniques... Nonlinear correction CDM Going beyond linear perturbation theory? c c =0 2 c H c =0 No nonlinear correction Neutrinos f0 f p f a m =0 m a p But see Shoji & Komatsu 2009 Y3W in prep Linearised continuity eqn Linearised Euler eqn 3 2 = H m [ f c c f ] 2 2 Linearised Vlasov eqn Poisson eqn

Corrections to the CDM component... Fluid description (linear): Continuity eqn c k, c k, = 0 Euler eqn c k, H c k, k 2 k, = Poisson eqn 3 k 2 = H 2 m [ f c c f ] 2 δc = CDM density perturbations δν = ν density perturbations θc = Divergence of velocity field 0

Corrections to the CDM component... Fluid description (incl. some nonlinear terms): Continuity eqn Vertex q q 121 k, q1, q 2 D k q12 12 2 1 q1 c k, c k, = d 3 q 1 d 3 q2 121 k, q1, q 2 c q 1, c q 2, Mode coupling Euler eqn c k, H c k, k 2 k, = d q 1 d q2 222 k, q 1, q 2 c q1, c q 2, 3 Poisson eqn 3 k = H 2 m [ f c c f ] 2 2 δc = CDM density perturbations δν = ν density perturbations θc = Divergence of velocity field 3 Vertex 222 k, q 1, q 2 D k q 12 q 212 q1 q2 2 q21 q 22 Starting point of most semi-analytic calculations in the literature.

Standard perturbation theory... Solve by perturbative expansion: c k, k, c k, / H n Juszkiewisz 1981, Vishniac 1983, Fry 1984, Goroff et al. 1986 k, = k, n m=1 nth order solution: n a k, =g ab, 0 b k, 0 n 1 d 3 q 1 d 3 q 2 0 d ' g ab, ' bcd k, q1, q 2 cn m q 1, ' dm q 2, ' m=1

Crocce & Scoccimarro 2006 Matarrese & Pietroni 2007 1 2 3 k =... Density/ Velocity = b1 0 a1 g ab, 0 + q1 ' q2 + +... time Linear 1 a 1 b k, =g ab, 0 k,0 Linear propagator Linear growth function 2nd order a2 k, = d q1 d q 2 0 d ' g ab, ' bcd k, q1, q2 3 3 1 1 g ce ', 0 e q 1, 0 g df ', 0 f q 2, 0

Crocce & Scoccimarro 2006 Matarrese & Pietroni 2007 1 2 3 k =... Density/ Velocity = a1 b1 0 g ab, 0 + q1 ' q2 + +... time 1 1 2 2 1 3 P k D k k ' k k ' = [ 2 ]... 22 11 Power = spectrum + Linear 13 +2 One-loop correction +...

Free-streaming suppression: One-loop corrected... Thin lines = linear Thick lines = 1-loop corrected fν~0.05 fν~0.1 fν~0.01 Change in power spectrum relative to the fν = 0 case: P Pf P Y3W 2008 also Saito et al. 2008, 2009 0 k P f Pf =0 k =0 k

Free-streaming suppression: One-loop corrected... Thin lines = linear Thick lines = 1-loop corrected fν~0.05 fν~0.01 fν~0.1 N-body simulations, Brandbyge et al. 2008 Y3W 2008 also Saito et al. 2008, 2009

Resummation and renormalisation group techniques... Many schemes have been proposed that go beyond standard perturbation theory: Crocce & Scoccimarro 2006, 2008 Taruya & Hiramatsu 2007 McDonald 2007 Matarresse & Pietroni 2007, 2008 Matsubara 2008 Valageas 2007 Pietroni 2008 Hiramatsu & Taruya 2009 etc..

Applied to massive neutrino cosmologies: 3.0 3.0 z = 2.33 mν = 0.3 ev 2.5 PNL(k)/Plin(k) = One-loop = Time-RG flow = N-body 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.0 0.2 0.4 0.6 z = 2.33 mν = 0.6 ev 0.8 1.0 1.2 1.4 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.8 z=1 mν = 0.3 ev 1.6 1.6 1.4 1.4 1.2 1.2 1.0 1.0 0.0 0.1 0.2 z=1 mν = 0.6 ev 0.3 0.4 0.5 0.0 k (h/mpc) 0.1 0.2 0.3 0.4 0.5 Lesgourgues, Matarrese, Pietroni & Riotto 2009

= One-loop = Time-RG flow = N-body = Linear Applied to massive neutrino cosmologies: 0.9 0.9 0.8 0.8 mν = 0.6 ev0.7 0.7 P(m )/P(m =0) mν = 0.3 ev 0.6 0.6 z=4 z= 2.33 0.5 0.5 1.0 0.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 z=1 0.5 0.0 0.5 0.5 1.0 0.0 z=0 0.5 0.0 k (h/mpc) 0.5 Lesgourgues, Matarrese, Pietroni & Riotto 2009

Summary... Using the large-scale structure distribution to probe neutrino physics is still fun. We can do even better in the future with forthcoming probes/new techniques. But we must make sure our theoretical predictions are reliable (1% accurate) at the (nonlinear) scales of interest. N-body simulations are the definitive way to go. Semi-analytic PT & RG techniques are also of some (limited) use.