Measuring neutrino masses with a photometric redshift survey
Yvonne Y. Y. Wong RWTH Aachen
[J. Hamann, S. Hannestad & Y3W, JCAP 1211 (2012) 052]
Theory seminar, Université Libre de Bruxelles, February 1, 2013
The cosmic pie(s)...
The ΛCDM model is the simplest model consistent with present observations.
Cosmological constant Massless Neutrinos (3 families)
Composition today 13.4 billion years ago (at photon decoupling) Plus flat spatial geometry+initial conditions from single-field inflation The cosmic neutrino background...
Embedding the standard model in FLRW cosmology necessarily leads to a thermal neutrino background (decoupling at T ~ 1 MeV). Fixed by weak interactions
Present temperature: 4 1/3 T = T =1.95K ν (11 ) γ Number density per flavour:
6 ζ(3) 3 −3 nν= T ν= 112 cm 4 π2
The cosmic neutrino background: energy density...
The present-day neutrino energy density depends on whether the neutrinos are relativistic or nonrelativistic.
● Relativistic (m << T): Photon energy density
2 4/3 7 4 7 4 π 3ρν ρν= T ν= ργ ∼0.68 8 15 8 ( 11) ργ
-4 ● Nonrelativistic (m >> T ~ 10 eV):
2 mν ρν=mν n ν Ω h = ν 93 eV
Neutrino dark matter (not part of the vanilla ΛCDM)
Flavour oscillations imply nonzero neutrino mass...
Atmospheric & solar neutrino oscillations indicate that at least one neutrino mass eigenstate has a mass of > 0.05 eV.
Cosmological implications:
● Relic neutrinos are nonrelativistic today.
● Present-day energy density: m Ω = ν >0.1 % ν 93 h2 eV
Tritium β-decay limit on neutrino mass...
Tritium β-decay end-point spectrum measurements impose an upper limit on the effective mass of the electron neutrino.
Current upper limit:
2 2 1/2 me≡ ∑∣U ei∣ mi <2.2 eV i ( ) max∑ mν∼7 eV→ maxΩν∼15 % Lobashev [Troitsk] 2003; Krauss et al. [Mainz] 2005 Neutrino dark matter is hot...
Neutrino dark matter come with large thermal motion. vthermal = 0 “Cold dark matter”
● Characteristic thermal speed:
T ν eV −1 vthermal = ≃ 50.4(1+z) km s mν ( mν )
finite v Structures on scales below the (maximum) thermal free-streaming scale could not have been formed via gravitational instability:
1/2 −1/2 eV −1 λFS,max=31.8Ωm , 0 h Mpc (mν ) http://www.itp.uzh.ch/ cf galaxies (~100 kpc), galaxy clusters (~ 1 Mpc) Why bother with neutrino dark matter then?
Because it's there.
● Neutrino dark matter = culmination of FLRW cosmology + SM + terrestrial experiment, a prediction as fundamental as the prediction of the cosmic microwave background.
– Confirmation of FLRW cosmology
– Determination of neutrino mass from cosmology
● You have to deal with it even if you don't care about it.
– When performing parameter inference, the presence of neutrino dark matter may shift the values of those cosmological parameters you care about, e.g., inflation parameters, dark energy properties, etc.
Plan...
Looking for neutrino dark matter
● Direct detection
● Indirect detection
Euclid sensitivity forecast
● Why Euclid is so good for the determination Cold dark matter of the neutrino mass from cosmology.
Disclaimer: We are dealing with subdominant, sub-eV to eV-mass neutrino dark matter here! The bulk of the DM content is explained by something else.
Neutrino DM 1. Looking for neutrino dark matter...
Direct detection of neutrino dark matter...
cf WIMP detection, … is a difficult business. ~ 10-46 cm2 G 2 m2 2 F ν −56 mν 2 ● Small interaction cross-section: σ ∼ ≃10 cm ν N π ( eV ) ● Neutrino energy too small to cross −3 most detection thresholds. p~m vearth ≃10 m
– Conventional WIMP detection techniques don't work here. Speed of Earth with respect to the CMB, ~ 370 km s-1
A zero threshold process?
● One unique candidate here...
Weinberg 1962 Cocco, Mangano & Messina 2007 Direct detection by neutrino capture... Lazauskas, Vogel & Volpe 2007 Blennow 2008 Lusignoli & Vignati 2010
β-decay end-point spectrum
- 1. N → N '+e +̄νe Cν B - 2. νe +N → N ' +e Always allowed if mν+mN >mN ' +me m u
r Local neutrino overdensity t c
e nν p Monochromatic signal from Rate 7.5 year 100 g tritium
s ∼ / / ( )
t relic neutrino capture (̄nν ) n i o
p KATRIN - ∼0.1 mg tritium d n e
y a c e
d 2m - ν β Electron energy
Qβ−mν Qβ≡mN −m N ' Qβ +mν Neutrino capture with KATRIN...
Requires a 109 local overdensity of neutrinos in a 3-year run for a 90% C.L. detection.
106 CνB events/yr 0 CνB events/yr
Neutrino clustering onto the Galactic halo
Expected background ~ 10 mHz
Kaboth, Formaggio & Monreal (2010) Indirect detection...
Exploit the effects of (subdominant) neutrino dark matter on precision cosmological observables:
● Cosmic microwave background anisotropies – Effects on the evolution of the homogeneous background
● Large-scale matter distribution – Effects on the level of the inhomegeneities
Neutrino dark matter and the CMB anisotropies...
Background effect 1 eV mass neutrino Relativistic Nonrelativistic ● eV-mass neutrinos become nonrelativistic close to equality and photon decoupling.
● A non-trivial transition from radiation to matter domination. → Affects sound horizon. → A nonstandard early Integrated Sachs-Wolfe effect.
Sachs-Wolfe effect:
Gravitational potential Redshift Blueshift Observer Ψ=0
Ψ CMB photon Δ T Δ T Observed CMB = +Ψ temperature fluctuation T observed T intrinsic
Integrated Sachs-Wolfe (ISW) effect:
Gravitational potential Redshift Observer Ψ=0
CMB photon
● Except during matter domination, gravitational potentials decay with time.
Observer – CMB photons suffer less redshift than in the case of a constant Ψ. – Larger observed temperature fluctuations: time τ T 0 Δ −κ( τ) ˙ ˙ (n̂ )=∫ d τ e [ Ψ(τ ,n̂ (τ 0−τ))+Φ(τ ,n̂ ( τ0−τ))] T ISW 0 Optical depth Early ISW effect
m =1×1.2 eV ∑ ν Current constraints from m 3 0.4 eV ∑ ν= × CMB alone: ∑ mν=0 eV ∑ mν<1.2 eV (95 %C.L.) ΛCDM + neutrino mass (7 parameters) Komatsu et al. [WMAP7] 2010
Fixed total matter density
∑ mν<1.3 eV (95 %C.L.) Hinshaw et al. [WMAP9] 2012
Sound horizon shift Indirect detection...
Exploit the effects of (subdominant) neutrino dark matter on precision cosmological observables:
● Cosmic microwave background anisotropies – Effects on the evolution of the homogeneous background
● Large-scale matter distribution – Effects on the level of the inhomegeneities
Subdominant neutrino DM and large-scale structure...
The presence of CDM acts as a source of density perturbations.
● No complete erasure of perturbations on small scales.
● But thermal motion of the relic neutrinos still makes neutrino clustering difficult.
ν ν c c
Gravitational potential wells
Free-streaming scale: ≫ FS Clustering 2 2 k ≪k FS 8 π vthermal 1+z eV −1 2 π λFS≡ 2 ≃4.2 h Mpc ; k FS≡ Ω ≪ FS 3Ω H m ,0 ( mν ) λFS m √ Non-clustering √ k ≫k FS Consider a neutrino and a cold dark matter particle encountering two gravitational potential wells of different sizes:
ν Clustering → potential ν wells become deeper c c
Some time later... λ≫λ FS λ≪λ FS ν
Both CDM and neutrinos cluster Only CDM c clusters c ν c ν c ν c ν
→ Free-streaming (non-clustering) neutrinos slows down the growth of density perturbations on scales λ<< λFS. δρ δρ ∣δ∣≡ ∣ ̄ρ ∣ Initial time... ∣ ρ̄ ∣ Some time later...
Perturbation spectrum (depth of “potential wells”)
CDM-only universe
A cold+neutrino DM Large length scales Small length scales universe k k Perturbation wavenumber kFS(znr)
znr = Redshift at which neutrinos become nonrelativistic The presence of neutrino dark matter induces a step-like feature in the spectrum of density perturbations.
) 2 k P (k)=〈∣δ(k)∣ 〉 ( P
, m CMB Galaxy u r redshift t c surveys e p s r e w o Replace some CDM p with neutrinos r Lyman-α e t f = Neutrino t ν a fraction m e l a Δ P Ω ν c ∝8 f ν≡8 s
P Ω m e g r 2 mν a Ων h = L ∑ 93eV
) 2 k P (k)=〈∣δ(k)∣ 〉 ( P
,
m CMB Galaxy u r redshift t c surveys e p s r e w o p
r Lyman-α e t f = Neutrino t ν a fraction m e l a Δ P Ων c ∝8 f ν≡8 s
P Ωm e g r 2 mν a Ων h = L ∑ 93eV
) 2 k P (k)=〈∣δ(k)∣ 〉 ( P
,
m CMB Galaxy u r redshift t c surveys e p s r e w
o “Linear” 3 p
k Pk r ≡ 2 ≪1 Lyman-α e 2 t f = Neutrino t ν a fraction m e l a Δ P Ων c ∝8 f ν≡8 s
P Ωm e g r 2 mν a Ων h = L ∑ 93eV
Change in the total matter power P P f 0k−P f 0k m ≡ ≠ = spectrum relative to the massless case: P P k m f =0
0.15 eV Linear perturbation theory:
0.3 eV P m ~8 P 0.45 eV m m
Linear 0.6 eV With nonlinear corrections: Simulations (particle representation for P m both CDM and neutrinos) ~9.8 P m m
Brandbyge, Hannestad, Haugbolle & Thomsen 2008; Viel, Haehnelt & Springel 2010; Bird, Viel & Haehnelt 2012 Approximation scheme Brandbyge & Hannestad 2009, 2010;
(linear neutrino evolution) Ali-Haimoud & Bird 2012 3 Semi-analytical: Loops and beyond... Saito et al. 2008; Y W 2008; Shoji & Komatsu 2009 Lesgourgues, Matarrese, Pietroni & Riotto 2009
0.9 0.9 ∑ mν=0.3 eV 0.8 0.8
∑ mν=0.6 eV 0.7 0.7
) z = 4 z= 2.33 0 0.6 0.6 = ν m
( 0.5 0.5
P 0.0 1.0 0.0 1.0 / 0.0 ) ν 0.9 0.9 m ( P 0.8 0.8
0.7 0.7
0.6 z = 1 0.6 z = 0 = One-loop 0.5 0.5 = RG Power = + + 2 + ... 0.5 = N-body spectrum = Linear Present constraints...
Hannestad, Mirizzi, Raffelt & Y3W 2010 ΛCDM+neutrino mass (7 parameters) 95% C.L. upper limits CMB only (WMAP7+ACBAR+BICEP+QuaD)
+ LSS power spectrum (SDSS7 LRG)
+ LSS + HST determination of H0
WMAP9 numbers are very similar Hinshaw et al. [WMAP9] 2012
Present constraints...
Hannestad, Mirizzi, Raffelt & Y3W 2010 ΛCDM+neutrino mass (7 parameters) 95% C.L. upper limits CMB only (WMAP7+ACBAR+BICEP+QuaD)
+ LSS power spectrum (SDSS7 LRG)
+ LSS + HST determination of H0
More complex parameter space
∑ mν<0.36→0.76 eV (95 %C.L.)
CMB+LSS+H0+SNIa Gonzalez-Garcia et al. 2010 Includes uncertainties in ● Number of neutrino species ● Dark energy equation of state ● Inflation physics (tensors, running) ● Spatial curvature
Future sensitivities...
95% C.L. upper limits Minimal nonlinear physics Planck alone (1 year) 2013
∑ mν<0.34→0.84 eV (95 %C.L.) Perotto et al. 2006
Present constraints
∑ mν<0.36→0.76 eV (95 %C.L.) 2020+ Nonlinear physics Planck+Euclid photo-z galaxy survey involved
∑ mν<0.022→0.044 (95 %C.L.) 2-5σ detection possible! (depending on treatment of galaxy bias) Hamann, Hannestad & Y3W 2012 2. Euclid sensitivity forecast...
ESA Euclid mission selected for implementation...
Launch planned for 2019.
● 6-year lifetime
● 15000 deg2 (>1/3 of the sky)
● Galaxies and clusters out to z~2
– Photo-z for 1 billion galaxies
– Spectro-z for 50 million galaxies
● Optimised for weak gravitational lensing (cosmic shear)
ESA Euclid mission selected for implementation...
Galaxy clusters Type Ia (cluster mass function) supernovae
Baryon acoustic oscillations (BAO)
Galaxy distribution (photo-z and spectro-z) Cosmic shear (weak gravitational lensing of galaxies) But everything I am about to say applies also to similar surveys such as LSST.
Weak lensing of galaxies/Cosmic shear...
Distortion (magnification or stretching) of distant galaxy images by foreground matter.
● Sensitive to both luminous and dark matter (no bias problem). Unlensed
Lensed Galaxies are randomly Unlensed 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 (or shear) power spectrum:
Shot noise dominates at large ℓ
2 〈γ 〉 Cosmic variance Δ Cℓ= 2 dominates uncertainties n at small ℓ γ = Intrinsic ellipticity n = source galaxy surface density
multipole ℓ ~ (angular separation)-1 ∞ ∞ 2 2 D(χ '−χ) C ∝ d χ a− d χ ' n (χ ') P (k=l /D (χ)) l ∫ ∫ gal D ' matter 0 [ χ (χ ) ] (Limber limit) Tomography = bin galaxy images by redshift
● Photometric redshifts for ~ 1 billion galaxies in Euclid survey.
z
Tomography probes the evolution of the matter distribution. Shear power auto & cross-spectra for 3 redshift bins:
● High redshift images are lensed more.
● Large cross-correlation signal.
Signal dominated measurement expected up to ℓ ~ 2000.
● Nonlinear corrections are important for the cosmic shear signal.
Solid = nonlinearities included Dots = linear evolution only
Our cosmic shear forecast...
We assume nonlinear corrections will be known to <1% accuracy up to ℓ = 2000.
● Mock nonlinear corrections from HaloFit = A fitting function calibrated HaloFit. Smith et al. 2003 against a set of N-body simulations
Realistic assumption? 1σ sensitivity to the ● Collisionless simulations are relatively neutrino mass sum easy to do. from cosmic shear
● But beware of dissipative baryon physics at ℓ > 1000 (believed to be a 1% effect, but who knows).
Tomography: 2 redshift bins (no gain with more bins).
Photo-z galaxy survey...
The same galaxy images used to determine the shear power spectra can be analysed for their own clustering properties.
● Photometric-z uncertainty: dz = 0.03 (1 + z) → Loss of small-scale information in the line-of-sight direction.
We account for this loss by considering 2D power angular spectra integrated over in a broad (> dz) redshift range: W = window function of the redshift bin 2 ∞ W (χ) (Limber limit) C l ∝∫ d χ 2 Pmatter (k=l /χ) 0 χ
● For neutrino masses, going 3D with spectro-z does not improve the sensitivity. Audren et al. 2012 ● But useful for dark energy/modified gravity/BAO.
Galaxy power auto & cross-spectra for 3 redshift bins:
● Appreciable cross-correlation signal between adjacent bins because of photometric redshift uncertainties.
● Otherwise, no intrinsic correlation
Nonlinear corrections
● Only nonlinearities in the matter density fluctuations has been accounted for in this figure.
● Other (not shown) sources include scale- dependent galaxy Solid = nonlinearities included bias. Dots = linear evolution only
Linear galaxy bias:
Luminous Red Galaxies
Regular Galaxies
On large scales, we expect the clustering of galaxies to trace the underlying matter density field up to a constant factor:
2 Pgalaxy (k)=b Pmatter (k )
Depends on the tracer; not predictable from first principles * This is really old data, but illustrates the point anyway... Scale-dependent galaxy bias:
Renormalised... Luminous Red Galaxies
Regular Galaxies
In reality, the bias factor is k- dependent:
2 Pgalaxy (k)=b (k) Pmatter(k)
Also tracer-dependent; even less predictable than * This is really old data, but illustrates the point anyway... the linear bias factor... Our galaxy clustering forecast...
Modelling galaxy bias is the main problem here.
Scale-dependent bias:
● We discard mock data where departure from linearity exceeds cosmic variance:
→ Redshift-dependent ℓmax.
● Tolerance factor εnl = 1 by default (also considered ultra-conservative
case εnl = 0.1).
Our galaxy clustering forecast...
Modelling galaxy bias is the main problem here.
Linear bias: two limiting cases
● Optimistic: Linear bias is exactly known. The reality is somewhere ● Pessimistic: No information at all. in between.
– N redshift bins → N linear bias parameters to marginalise.
Our galaxy clustering forecast...
Modelling galaxy bias is the main problem here.
Linear bias: two limiting cases
● Optimistic: Linear bias is exactly known. The reality is somewhere ● Pessimistic: No information at all. in between.
– N redshift bins → N linear bias parameters to marginalise. Pessimistic linear bias
Tomography: 8 redshift bins (not much improvement with more bins).
1σ sensitivity to the neutrino mass sum from galaxy clustering (optimistic linear bias) Redshift binning...
We choose our redshift bins by demanding the surface density of galaxies to be the same in all bins. Height of box =
corresponding ℓmax Differential galaxy in the redshift bin surface density (for galaxy clustering)
Black = 8 bins Red = 2 bins
Shear-galaxy cross correlation...
We assume the shear and the galaxy samples cover the exact same patch of the sky.
● Significant shear-galaxy cross-correlation signal.
● Turns out to be not very useful for neutrino mass measurement.
● But can improve dark energy constraints somewhat (more later).
Shear vs galaxy clustering...
Besides the galaxy bias, there is one more subtle difference between the two probes:
● Cosmic shear = bending of light rays → sensitive to the metric perturbations Φ.
● Galaxy clustering (if bias is known) probes the matter density perturbations δm.
In ΛCDM-type cosmologies (in the subhorizon limit):
ωm = physical 3 matter density k 2 Φ∝− ω δ Poisson equation 2 m m
● The extra ωm factor makes the parameter degeneracy directions of cosmic shear and galaxy clustering very different! Shear vs galaxy clustering: degeneracy directions...
ωm vs h degeneracy in a restricted 2-parameter analysis.
Black = galaxy only (optimistic bias) Red = cosmic shear only Gold = Planck CMB only Grey = galaxy only (pessimistic bias)
● The ωm-h degeneracy is completely broken when cosmic shear and galaxy clustering are used in combination.
→ Will also help to tighten the neutrino mass constraint.
Add neutrino mass: three-way degeneracy...
Including neutrino mass in the analysis Hubble creates a three-way degeneracy parameter between (ωm, h, Σmν). s e s s ● Different orientations of ellipsoids a Euclid galaxy m for cosmic shear and galaxy power spectrum o n i
clustering. r t u e n
● When combined, the three-way f o
degeneracy can be broken very m u
effectively. S Planck+Euclid cosmic shear
This is our central message! Physical matter density
ν m Σ
, s Euclid photo-z e s
s galaxy angular a power spectrum m
o n i r t u e n
f o
m u S
Planck+Euclid cosmic shear Hubble parameter, h (pointing out of the page)
Physical matter density, ωm Some numbers...
A 7-parameter forecast: matter density Initial conditions
Neutrino mass sum Θ=(ωb , ωm , h , As ,ns , τ ,∑ mν) (modelled as 1 massive + 2 massless neutrinos) baryon density Hubble parameter Optical depth (affects CMB only)
● Dramatic improvement in
sensitivities to (ωm, h, Σmν) when shear and galaxy clustering are combined.
● Shear-galxay cross- correlation is not so important.
c = CMB (Planck); g = galaxy clustering s = cosmic shear; x = shear-galaxy cross Some numbers...
A 7-parameter forecast: matter density Initial conditions
Neutrino mass sum Θ=(ωb , ωm , h , As ,ns , τ ,∑ mν) (modelled as 1 massive + 2 massless neutrinos) baryon density Hubble parameter Optical depth (affects CMB only)
● Galaxy clustering adds some marginal improvement on the constraint from CMB+cosmic shear even if the linear galaxy bias is completely unknown.
c = CMB (Planck); g = galaxy clustering s = cosmic shear; x = shear-galaxy cross Black = CMB+galaxy (optim. bias) Red = CMB+shear Green = CMB+galaxy+shear
Good enough to measure the mass spectrum?
Unfortunately not.
● In fact we can model the mass spectrum any way we like and the data cannot tell.
Fiducial value 1+2 = 1 massive + 2 massless 2+1 = 2 (equally) massive + 1 massless We always recover the 3+0 = 3 (equally) massive + 0 massless fiducial value within 1σ. Extended models...
Besides the 7 vanilla parameters, some other cosmological parameters are known to be degenerate with the neutrino mass. Extra radiation Dark energy EoS
c=CMB (Planck); g=galaxy power spectrum; s=cosmic shear; x=shear-galaxy cross-correlation
● Optimistic linear bias: No deterioration of sensitivity to Σmν; still 5σ+.
● Pessimistic linear bias: Factor of 2 deterioration in sensitivity.
Summary...
● The existence of neutrino dark matter is a fundamental prediction of SM+FLRW cosmology+neutrino oscillation experiments.
● Existing precision cosmological data already place strong constraints on the abundance of neutrino dark matter and hence the neutrino mass.
● Future observations with Planck+Euclid will do much better, especially when cosmic shear and galaxy clustering data are analysed in combination.
– A 2-5σ detection is possible even if the neutrino mass is the smallest allowed by oscillation experiments.
● The challenge (for theorists): get the nonlinear corrections right!