Probing the Largest Cosmological Scales with the CMB-Velocity Correlation

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by  Pablo Fosalba

Probing the largest cosmological scales with the CMB-Velocity correlation
Pablo Fosalba1,∗ and Olivier Dor´e2,†
1Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB,
Facultat de Ci`encies, Torre C5 par-2, Barcelona 08193, Spain
2CITA, University of Toronto, 60 St George Street, Toronto, ON M5S 3H8, Canada
(Received February 5, 2008)
Cross-correlation between the CMB and large-scale structure is a powerful probe of dark-energy
and gravity on the largest physical scales. We introduce a novel estimator, the CMB-velocity
correlation,thathasmostofhispoweronlargescalesandthat,atlowredshift,deliversuptofactor
of two highersignal-to-noise ratio than therecently detected CMB-dark matter density correlation
expected from the Integrated Sachs-Wolfe effect. We propose to use a combination of peculiar
velocitiesmeasuredfromsupernovaetypeIaandkineticSunyaev-Zeldovichclustersurveystoreveal
8 thissignalandforecast dark-energyconstraintsthatcanbeachievedwithfuturesurveys. Westress
0 that low redshift peculiar velocity measurements should be exploited with complementary deeper
0 large-scale structure surveysfor precision cosmology.
2
PACSnumbers: 04.40.Dg,04.25.Dm,97.10.Kc,04.30.Db
n
a
J
INTRODUCTION affected by shot-noise unlike galaxy densities (specially
4
in deep samples). Peculiar velocities have more power
1
on the largest (linear) scales than the dark matter den-
The Integrated Sachs-Wolfe (ISW) effect carries infor-
2 mation onthe evolutionof the universe at low and mod- sity, what follows from the continuity equation (see be-
v erate redshifts (z < 3) and thus is a powerful probe of low), and thus provides a natural counterpart to the
2 density field in reconstructing the gravitational poten-
dark-energy (DE)∼properties [1–3]. Since the ISW effect
8
tial. However probing large-scale velocity flows is not
7 is most sensitive to gravity on the largest scales (hun-
an easy endeavor. Although galaxy surveys now sam-
1 dreds of Mpc), it also offers a novel way to distinguish
0 dark-energy from modified gravity theories [4–6]. Mea- ple large volumes, the intrinsic inaccuracy plaguing dis-
7 tance estimations (20-25%) limits the volume available
suringthe ISW signalthatcontributesto the totalCMB
0 for such measurements [16–18]. Nonetheless better dis-
temperature power spectrum is mainly limited by sam-
/
h pling variance errors on large scales, that has tradition- tanceindicatorsandotherprobesofthevelocityfieldsare
p already used in cosmology but their power to reveal the
ally challenged its detection. An alternative path to ex-
- ISW effect and the physics behind it has been neglected
o tractingtheISWeffectisbycross-correlatingCMBmaps
r withtracersofthelarge-scalestructure,suchasgalaxies. so far: SNe currently yield distance measurements with
t a 5%-10% error [19, 20] and kinetic Sunyaev-Zeldovich
s
Recently, first detections of the ISW effect from cross-
a (kSZ,[21])couldinprinciple deliverredshiftindependent
v: correlationanalysesofWMAPdatawithdifferentgalaxy velocity errors around 100 km.s−1. These surveys al-
surveys,andx-raymapshavebeenobtained[7–13]. Re-
i ready allow to map the large-scalevelocity field at low z
X ported detections are at the 2 4 σ level and favor a
− with a good agreement with other probes [19, 22, 23].
r DEdominateduniverse,independentofSupernovaetype
a
Ia (SNe hereafter). Future galaxy surveys(such as DES,
Pan-STARRS, WFMOS) hold the promise of raising the In this paper we propose to use the large-scale veloc-
significanceofthesedetectionsbygoingdeeperandwider ity flows from low-redshift SNe and kSZ cluster surveys
to optimally sample the ISW signal. Primary anisot- to measure the ISW effect from the CMB-velocity corre-
ropies of the CMB are the dominant source of noise in lation. We propose to include velocity measurements at
such correlationanalysesthat seriously limits the ability lowredshiftincombinationwithdeeperlarge-scalestruc-
with which this probe can constrain cosmology [14, 15]. ture probes in tomographic analyses to better probe DE
Inaddition,CMB-galaxycorrelationsareaffectedbysys- andgravityonthelargestscales. Thecomplementarityof
tematics such as galaxy bias and shot noise. Moreover velocitieswith respecttodensities comesfromthe larger
reconstructing the gravitational potential (and its time signal-to-noise of the CMB-velocity correlation estima-
evolution) from the density field is a noisy process in- tor at low redshift (up to a factor of 2 gain depending
volving second order spatial derivatives. on cosmology), as we will show in below. Unless oth-
§
Alternatively, velocity flow measurements from esti- erwise stated, we will focus on flat ΛCDM models with
mates of the luminosity distance and redshift are unbi- Ω =0.75, Ω = 0.05,n =1,h =0.7,σ =0.9, and as-
DE b s 8
asedtracersofthe gravitationalpotential,andthe latter sumelarge-scalestructuresourcesfollowadensitydistri-
can be simply reconstructed through first spatial deriva- bution dn/dz z2exp (z/(√2z ))1.5 , with a width,
m
∝ −
tives of the previous. Also, these measurements are not σ z /2, where z is(cid:2)the survey medi(cid:3)an redshift.
z m m
≃
2
field spreads in comoving scales are given by ([25])
2
WT(k) = dηF˙(η)j (kη) (2)
ℓ c2 Z ℓ
3k2H2Ω dn(η)
Wδ(k) = 0 0 dηF(η) j (kη) (3)
ℓ 2 Z dη ℓ
k dn(η)
Wv(k) = dη x(η)j′(kη), (4)
ℓ H Z dη ℓ
0
being dn/dη the space density of sources and we denote
the radial velocity v = v(r) ˆr v. On sufficiently
r
| · | ≡
large (linear) scales, gravitational instability generates
irrotational flows, thus we expect vanishing tangential
component v = 0, (only the line-of-sight modes k = k
⊥ k
are non-zero,[26]) and we define j′(x) = dj(x)/dx with
j (x) being the spherical Bessel function of order ℓ. In
ℓ
ourcomputations,weshallusevelocitiesindimensionless
units, by normalizing to the speed of light.
FIG. 1: Cross power spectra for the CMB temperature-
velocity,Tv(top),andthetemperature-density,Tδ(bottom), Fig.1showstheCMBtemperature-velocity,Tv,power
for different mean survey depths, zm. Most of the signal in is about 2 orders of magnitude smaller than that of the
theTvcorrelationcomesfromlargerscales(lowermultipoles) temperature-density,Tδ,estimator. However,asweshall
thantheTδ. Weamplifythevelocitycorrelationsignalby100 see below, the noise contribution is more than 100 times
to match the ISW amplitudefrom thedensity correlation.
smaller thanfor the Tδ,what makesthis signaleasierto
extract in cross-correlationsanalyses. Note that most of
the signal in the Tv correlation comes from larger scales
MODELING THE OBSERVABLES (lower multipoles) than the Tδ and therefore very wide
surveysarerequiredtomeasuretheexpectedcorrelation.
On the other hand, as illustrated by Fig. 2, the peak
Dynamics ofthe fluctuations ofthe cosmologicalfields
amplitude of the Tv kernels (i.e, integrand in eq. (1)),
on the largest scales can be accurately described by lin-
for relatively shallow surveys, z < 0.5, peaks at ℓ = 2,
ear perturbation theory. In this regime, the Poisson and
whereas for deeper surveys the s∼ignal is dominated by
continuity equation in Fourier space are given by (Pee-
bles 1980, 2) k2/a2Φ(k,t) = 3/2H2ΩDδ(k,t ), where higher ℓ’s. Note that the CMB-velocity is mainly sensi-
we have us§ed t−hat H2 = (8πG/3)ρ¯/Ω, with ρ¯b0eing the tive to modes at k 0.01 0.001Mpc−1, i.e, almost one
∼ −
order of magnitude larger comoving scales than for Tδ.
meanmatterdensity,a(t)istheFRWscalefactor,kisthe
comoving wavenumber and D(t) accounts for the linear
growthof dark-matterdensity perturbations. The linear
OBSERVATIONAL PROSPECTS
continuity equation reads, v(k,t) = iD˙aδ(k,t )/kkˆ
0
where kˆ = k/k and denotes derivat−ive with respect
· Current probes of the velocity flow on large scales in-
to comoving time t. Combining both Fourier equa-
clude galaxy peculiar velocities, SNe galaxy host pecu-
tions we get the linear evolution of velocity modes as
liar velocities and the kinetic Sunyaev-Zel’dovich effect
a function of the present day gravitational potential
(e.g. [24]), v(k,t)=ik/H Φ(k,t )g(t)kˆ where Φ(k,t)= (kSZ). The radial component of peculiar velocities of
0 0
galaxies and SNe, v is determined by inferring their dis-
D/aΦ(k,t )isthetimeevolutionofgravitationalpoten-
0
tance, d, and then subtracting off the Hubble flow con-
tial modes and g(t)=2a2/3Ω Ω /a3+Ω dD/da.
0 m Λ
tribution to the measured redshift, z, as v = cz H d
Angular cross-power spectrapbetween the CMB tem- − 0
[34]. The empirical correlation –between light-curve
perature anisotropy and the dark-matter density δ and
shapeandluminosity,andbetweencolor-curveshapeand
peculiar velocity v fields can be derived from the cor-
extinction–usedtoinfertheSNeluminositydistancecur-
responding angular 2-point correlation function w =
XY rently yield a dispersion in apparent magnitude, m, of
X(ˆr)Y(ˆr′) = (2ℓ+1)/4π XYP (ˆr ˆr′), and yields,
h i Cℓ ℓ · σ(m) = 0.1 0.15. Since m is related to the luminosity
P −
distance in Mpc, d , andthe absolute magnitude, M, as
L
CℓXY = π2 Z k2dk PΦΦ(k)WℓX(k)WℓY(k). (1) ma r=ed5shloigft10inddLe+penMd,enatddiissptaernscieonmoefasσu(rmem)e=nt0s.1erreonrtsaiolsf
5% when ignoring the effect of the marginalization over
where the 3D power spectra is defined as, the constantM. Thereforemeasuringthe velocityofthe
(2π)3 (k)δ(k k′) = X˜(k)Y˜(k′) , and the host galaxy of one SNe can be as accurate as the veloc-
XY
P − h i
kernels WX(k) that define how the signal from the X ity obtained from 25 galaxies using the D σ relation.
ℓ −
3
FIG. 2: Contribution to the ISW effect power spectra from FIG.3: Contributionofmultipolestothesignal-to-noiseratio
different 3D wavenumbers, k, and thin redshift z shells, for S/N for Tv (solid) and Tδ (dashed) for full-sky surveys with
various multipoles ℓ. As in Fig. 1, top panels display the differentdepths,forourbaselineΛCDMmodel. Total(S/N)2
Tv kernel, amplified by a factor of 100, whereas the bottom isgivenbytheareaunderthecurves. VelocitiesgiveaS/Nin
panels show the Tδ kernel. Kernels are given by ℓ(ℓ+1)/2π cross-correlation with the temperature that is systematically
times the integrand in eq.(1). largerthanthatfordensities,Tδ,thegainfactordependingon
surveydepthandcosmology. NotethatS/Nscalesaspfsky,
where f is thefraction of sky covered bythesurvey.
sky
In turn, the redshift for nearby galaxies (SNe host or
not) is currently measured using narrow lines from the
hostgalaxieswithanuncertaintyσ(cz)=30km.s−1 that and DE constraints as compared to the Tδ correlation.
Inparticular,weproposetocombinefull-skypeculiarve-
is dominated by statistical errors. As such for a Hubble
locitysurveysfromSNeatz <0.1and,complementarily,
constantofh=0.72,errorsinredshiftmeasurementswill
kSZ clustersurveysforsourc∼esatz >0.1. We define the
besubdominantascomparedtoerrorsinH dforz >0.02
0
cross-correlationsignal-to-noise ratio∼S/N as
(0.004)ifweuseSNe(galaxy)baseddistanceestimators,
respectively. Since galaxy peculiar radial velocities typ-
(CTX)2
ically have a rms of 300 km.s−1 we reach a signal-to- (S/N)2 = (2ℓ+1)f ℓ (5)
sky (CTX)2+CTTCXX
noise ratio S/N 1 per source at z 0.02(0.004). If Xℓ ℓ ℓ ℓ
≃ ≃
we were to volume-average the distance measurements
whereX =vorδ. Fig.3showsthecontributionofdiffer-
coming from SNe observations over samples at fixed z,
entmultipolestotheS/Ndependingonsurveydepth,for
the systematic error limit would be σ(m) = 0.02 [27].
a ΛCDM model with Ω = 0.75 that will be our base-
Assuming that we had enough SNe to reach this limit DE
within boxes of depth σ(cz)=30km.s−1, then one could line. Theareaunderthecurvesgivethetotal(S/N)2. We
find that for low z, Tv peaks at similar (albeit slightly
get a 1% distance measurement and the S/N per source
larger)ℓ’sthanTδ,i.e,ℓ 10 20. Forourbaselinecos-
would reach 1 at z 0.1.
∼ −
≃ mology one gets, for the z = 0.3 survey, (S/N) = 5,
On the other hand, the kSZ determined peculiar ve- m Tv
that is 25% larger than the corresponding significance
locities of galaxy clusters do not rely on a distance
fortheISWdetectionfromtheTδ correlation. Theshal-
determination, and associated z-independent errors of
100kms−1 may be achievable [28, 29]. However the lower survey, zm = 0.15, leads to more moderate sig-
∼signal is limited to z > 0.1 in order for the primordial nificance, (S/N)Tv = 3.2. However, this is about factor
CMBnottobeadomin∼antsourceofconfusion. Itspower of 2 larger than that for Tδ. This comes from the fact
that the lowest redshift sources give the dominant con-
as a tracer of the large-scale gravitational potential has
tributiontotheoverallS/N.Inotherwords,asshownby
already been studied in [24] and as a DE probe in con-
Fig. 2, progressively deeper velocity surveys are less op-
junction with other density tracers in [30, 31].
timal for ISW measurements since the Tv signal quickly
The nature of the signalwe areinvestigating demands
drops with redshift unlike Tδ. In general, the relative
wide but rather shallow surveys in comparison to other
gain in S/N when using velocities as compared to densi-
probes. To illustrate the detectability of this new corre-
tiescanbeunderstoodbycomputingtheratioofS/Nfor
lation, we will considertwo ideal full-sky surveys,with a
Tv withrespecttoTδ foragivenmultipoleℓ,thatscales
distributionofsourceswithmedianredshiftz =0.15or
m
zm =0.3, and study their performance in ISW detection as (CℓTv/CℓTδ)qCℓδδ/Cℓvv. Although at low multipoles
4
ror σ(Ω ) = 0.065 for our baseline ΛCDM cosmology,
DE
that is almost a factor of 2 better accuracy than what
densities can deliver (see e.g, lower left panel in Fig 4).
These constraints are comparable to what is expected
for σ(Ω ) from PLANCK[35] (see comparisonbetween
DE
thinandthicksolidlinesintheleftpanels),althoughsuch
an outstanding CMB dataset will largely help breaking
the degeneracywith w. Forthe deeper survey,z =0.3,
m
velocities yield a factor of 2 better measurement of the
DE density σ(Ω ) = 0.03 than the shallower survey
DE
(solidcontoursin rightpanels ascomparedto samelines
in left panels), but this is comparable to what a density
tracer would deliver (dashed contours in corresponding
panels). Velocities can also help breaking the degener-
acy with w, but, as expected from a low redshift survey,
to a much lower extent than PLANCK. Note however
how the DE errors depend on the baseline cosmology
used: Tvmeasurementscanprovidesignificanltysharper
constraints on quintessence models than on ΛCDM and
FIG. 4: Constraints on the DE parameters from the cross- yield better measurements of Ω than PLANCK. This
DE
correlation of the CMB with shallow full-sky surveys. Con-
is shown in the lower panels of Fig 4 where errors from
tours show 68% (inner line) and 95% (outer line) confidence
theTv correlationalone(thinsolidlines)donotimprove
levels for 1 DOF. Solid (dashed) lines show parameter errors
when adding PLANCK priors (thick solid lines).
from the Tv (Tδ) correlation, whereas thick (thin) lines cor-
respond to constraints with (without) PLANCK priors. Left In practice several factors could degrade these opti-
(right) panel shows the case for surveys with median red- mal expectations. In cross-correlations S/N σ and
8
tswhioftdzimffer=en0t.1c5ho(izcmes=of0.t3h)eaDnEd teoqpua(tbiootntoomf)stpaatne:elswd=isp−lay1 thus current estimates of σ8 ≃ 0.8 would imp∝ly a 10%
weaker detection than we estimated from our baseline
(ΛCDM) and −2/3 (quintessence).
model. Lensing could introduce noise in Tv as we go
deeper in redshift although its large scale contribution
is small. More importantly, we neglected so far the fact
the signal is about two orders of magnitude smaller for
that we deal with a discrete sampling of the underly-
velocities,(CTv/CTδ) 0.01(seeFig.1),thecorrespond-
ℓ ℓ ∼ ing velocity field. Note that for velocities no shot noise
ingratioofnoisetermsarisingfromtheauto-correlations
affects their measurement, unlike the case for densities.
Cδδ/Cvv > 104, what makes the relative S/N exceed
ℓ ℓ However, if we consider the case for a z = 0.3 survey,
unity. We∼note that this Tv significance gain over Tδ m
90% of sources lie below z =0.8. Since there are around
increases for more strongly DE dominated cosmologies.
105 clusters with masses greater than 1014 h−1 M in
Inparticular,forashallowsurvey(z =0.15),velocities ⊙
m this volume, a full-sky cluster kSZ survey up to z = 0.8
canmeasurethecosmiclowmultipoleswithasignificance
would provide a good sampling of the field for modes
> 2 times larger than dark-matter density tracers, such
k < 0.1 Mpc−1 which is sufficient for the CMB-velocity
∼as galaxies, in cross-correlationwith the CMB.
sig∼nal according to Fig. 2.
Inconclusion,theuseofwideandshallowvelocitysur-
veys can substantially increase our ability to extract the
COSMOLOGICAL IMPACT AND CONCLUSIONS
ISW signal from a cross-correlation with the CMB and
thus help other cosmological probes to distinguish grav-
InthecontextofflatCDMmodels,anincreasedsignif- ity from alternatives. Peculiar velocities are highly cor-
icance in the ISW measurement from the Tv correlation related with other large-scale structure tracers such as
directly translates into tighter DE constraints. Fig. 4 the galaxy density distribution but probe larger physi-
displays the expected parameter constraints that can cal scales. In addition, the newly proposed correlation
be achieved for two shallow full-sky surveys. For com- is a priori less affected by systematics that plague the
parison purposes, we show constraints from Tv (solid temperature-galaxy correlation such as galaxy bias and
lines) and the usual Tδ (dashed) correlations. In gen- shotnoise. Inparticular,combiningshallowvelocitysur-
eral, the Tv estimator yields better DE constraints than veys with complementary data from planned deep sur-
theTδirrespectiveofthebaselinecosmologyusedorsur- veys (such as LSST, DES, Pan-STARRS, WFMOS) can
vey depth, provided z < 0.3, beyond which the usual fullyexploittheabilityoftomographicanalysesoffuture
m
Tδ cross-correlationoutp∼erforms Tv. A shallow velocity surveys to reveal the nature of dark-energy and grav-
survey with z = 0.15 can deliver a (symmetrized) er- ity on the largest physical scales [32, 33]. Furthermore,
m
5
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