Direct Determination of Expansion History Using Redshift Distortions

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by  Yong-Seon Song

Direct Determination of Expansion History Using Redshift Distortions
Yong-Seon Song
Korea Astronomy and Space Science Institute, Daejeon 305-348, R. Korea
(Dated: October25, 2012)
Weinvestigate direct determination of expansion history usingredshift distortions without plug-
gingintodetailedcosmologicalparameters. Theobservedspectrainredshiftspaceincludeamixture
ofinformation: fluctuationsofdensity–densityandvelocity–velocityspectra,anddistancemeasures
ofperpendicularandparallelcomponentstothelineofsight. Unfortunatelyitishardtomeasureall
thecomponents simultaneously without anyspecific priorassumption. Common prior assumptions
include a linear/quasi-linear model of redshift distortions or a model for the shape of the power
2
spectra, which eventually breaks down on small scales at later epochs where nonlinear structure
1
formation disturbs coherent growth. The degeneracy breaking between the effect of cosmic dis-
0
tances and redshift distortions for example depends on the prior we assume. As an alternative
2
approach is to utilize the cosmological principle inscribed in the heart of the Friedmann–Lematre–

t Robertson–Walker (hereafter FLRW) universe, that is, the specific relation between the angular
c
diameter distance and the Hubble parameter, in this degeneracy breaking. We show that utilizing
O
thisFLRWpriorearlyinthestepofdistinguishingthedistanceeffectfromredshiftdistortionshelp
4 usimprovethedetectabilityofpowerspectraanddistancemeasureswithnoleaningoncombination
2 of other experiments.

] PACSnumbers: draft
O
C
I. INTRODUCTION fundamental cosmological observables without plugging
.
h into theoretical models described by detailed cosmologi-
p cal parameters.
- Fundamental cosmological observables of the universe The full history of cosmic expansion can be recon-
o
arecosmicexpansion,largescalestructureformationand
r structed using galaxy redshift surveys. Despite the en-
t curvatureofspace. Thosearekeytracersofpast,present richednonlinearstructures,the zero-thorderdescription
s
and future of the universe. Cosmic expansion is defined
a ofourcurrentuniverseishomogeneousandisotropicover
[ bythe expansionofthe distanceamongcosmologicalob- sufficiently largescales[6–16]. The measuredspatialdis-

1 jects of the universe with time. History of the universe tribution of galaxies is deteremined by the density fluc-
is known by this expansion rate at each epoch [1]. The
v tuations and the coherent peculiar velocities of galaxies.
universe at large scale appears as a compilation of swiss
6 Even though we expect the clustering of galaxies in real
9 cheese–like bubbles bordered by filaments of galaxies. space to have no preferred direction, galaxy maps pro-
5 This contemporary spatial distribution is clearly observ-
ducedby estimating distancesfromredshifts obtainedin
6 able in a three dimensional reconstruction of the spec-
spectroscopicsurveysrevealananisotropicgalaxydistri-
0. troscopy wide–deep survey [2]. The fate of the universe bution. The anisotropies arise because galaxy recession
1 isdeterminedbythe globalshapeofspace. Spatialman- velocities,fromwhichdistancesareinferred,includecom-
2 ifold of the universe appears compact or non–compact
ponentsfromboththeHubbleflowandpeculiarvelocities
1 without boundary [3]. Consistency of those key observ-
driven by the clustering of matter. Measurements of the
:
v ables with predictions of known physical sciences on the anisotropiesallowconstraintsto be placedonthe rateof
i earth will be an evidence of existence of the Friedmann–
X growth of clustering and Hubble flow along the line of
Lematre–Robertson–Walker(hereafter FLRW) universe.
sight [17–38].
r
a Discovery of metric expansion revolutionized our un- In practice, ‘referenced’ spectra of redshift distortions
derstandingoftheuniverse. Themetricexpansionissuc- areobservedinfiducialcoordinates,insteadoftruespec-
cessfully modeled by FLRW model based upon the cos- tra. Cosmological parameters are commonly applied to
mologicalprinciple. Thecosmologicalprincipleisaphilo- projectmeasured‘referenced’spectrainto ‘bestfit’spec-
sophicalstatementthat allpropertiesofthe universeare tra. Those ‘best fit’ spectra will be genuine only if un-
viewed the same for all observers on a sufficiently large derlying theoretical models are true. But cosmological
scale. However, the first evidence of cosmic acceleration observables should be provided a priori without theoret-
in 1998 [4, 5] brought an issue of inconsistency between icalmodels tobe tested. Tobeginwith,we wouldliketo
observables of FLRW universe and predictions of physi- answerbasicquestionsinFLRWuniverse;whatisexpan-
calsciences. Aprimegoalofprecisioncosmologyinnext sion rate at each epoch, what are spectra of large scale
decades is to provide cosmological observables in a the- structure, and what is globalcurvature of the space. We
oretical model independent way for a fair judgement of would not jump into theoretical issues of cosmic accel-
confirmation or exclusion. In order words, cosmic ob- eration; what is nature of dark energy, and what is a
servationshould be unplugged fromour prior knowledge cosmologicalmeasure of Einstein’s gravity.
of underlying sciences. We investigate methods to probe Theobservedspectrainredshiftspacenotonlydepend
2
on fluctuations of density & velocity fields but also de- in detail in this section. In the following subsections, we
pend on distance measures of perpendicular & parallel present various methods to probe distance measures out
components to the line of sight [39–44]. Unfortunately, of the observed spectra. The future tomographic wide–
those are not simultaneously decomposed out of redshift deep survey from z =0 to 2 is assumed through this pa-
distortions in precision due to high degeneracy. We pro- per. Fiducial cosmology model is ΛCDM universe with
pose a couple of approaches to break the degeneracy cosmological parameters of (Ω = 0.24, Ω = 0, h =
m k
among observables. The first approach is an easier way 0.73, A2 =2.41×10−9, n =0.96).
S S
but less model–independent, and the second approach is
a harder way but more model–independent.
First approach,we can, in principle, resolvethis prob- A. Components of observed galaxy power spectra
lem if we understand the shape of the power spectrum
precisely. In the context of standard cosmology, the An observed galaxy power spectrum P˜(k,µ,z) in red-
shape of spectra is determined before the last scatter- shift space is commonly modeled as [49] over k <
ing surface, and in linear theory, it evolves coherently 0.1hMpc−1,
through all scales. In this case, the shape of spectra is
determined by CMB experiments, both coherent growth P˜(k,µ,z)=P (k,z)+2µ2P (k,z)+µ4P (k,z),
gg gΘ ΘΘ
functions of density & velocity and distance measures
canbedeterminedseparatelyinprecisionusingthestan- where the subscripts g and Θ denote the inhomogeneity
dard ruler test or Alcock-Paczynski test [45]. Unfortu- of galaxy number density and the divergence of peculiar
nately, this ansatz is not applicable for a specific theo- velocity measured in the unit of aH. Powerspectra P ,
gg
retical model of cosmic acceleration in which structure P andP correspondtocomponentsoffluctuationsof
gΘ ΘΘ
formation does not grow coherently at later epochs, i.e. density and velocity fields. The subscript g denotes per-
the determined shape of spectra has been altered since turbations of galaxy distribution. Galaxy number over-
last scattering surface [46–48]. density δ (k,z) is in generalbiased relativeto mass fluc-
g
Second approach, we propose an alternative method tuations and assumed to be δ (k,z) = b(k,z)δ (k,z).
g m
to make measurements with assuming theoretical prior The fiducial value of b is assumed to be b = 1 in the
as minimal as possible. In FLRW universe, the perpen- manuscript.
dicularandtheparallelcomponentscanbeunifiedbased The observed spectra are transformed by variation
upon the spartial homogeneity. Instead of making addi- of distance measures through correspondent coordinate
tional theoretical antsatz, we propose the configuration components in Fourier space. With the plane wave ap-
ofspectroscopic wide–deepfield surveyto be fully tomo- proximation, k and µ are given by k and k as,
⊥ k
graphic,i.e., composedof a series of redshift bins, overa
certain range of redshift. If this observational constraint k = k2 +k2
issatisfied,there isnofurther needforusto assumethe- ⊥ k
oretical prior more than FLRW universe. We find that µ = qkk/k. (1)
bothpowerspectraanddistancemeasuresaresimultane-
ouslymeasurablewithout anyassistanceofotherexperi- Given the observational quantities, such as k⊥DA (i.e.,
ments. Remarkably,theexpansionhistoryofHubbleflow anangularscale)andkkH−1 (i.e.,ascalealongredshift),
is measured well within a couple of percentage precision different distance measures (DA and H−1) result in a
at each redshift. feature of power spectra in a different wavenumber k⊥
We summarize the layout of this paper. In section II, andkk. Whenwehaveapriorinformationonthelocation
probing distance measures are presented. In section III, ofafeatureink⊥ andkk,wethencandetermineDA and
model independent observation of spectra is presented. H−1.
In section IV, remaining issues are discussed.
B. Dependence of P˜ on distance measures
II. DETERMINATION OF DISTANCE
MEASURES Anisotropyofredshift distortionsgivesus anopportu-
nity to probe D and H−1 separately. The mechanism
A
The observed galaxy–galaxy correlation in redshift of simultaneous determination is explained in this sub-
spacedependsonfluctuationsofdensity&velocityfields section.
and distance measures of perpendicular & parallel com- Decomposition of Pgg and PΘΘ out of P˜ was studied
ponents to the line of sight. The perpendicular compo- previously, when DA and H−1 are assumed to be well–
nent of distance measures is represented by the angu- known[23]. Pgg spectraarepreciselydecomposedatµ→
lar diameter distance D , and the parallel component is 0, and P spectra are decomposed by running of P˜ on
A ΘΘ
represented by the inverse of Hubble flow H−1 at each µ coordinate. Here we are interested in the reverse case.
givenredshift. Thelinearresponseoftheobservedpower We study decompositionofD andH−1 outofP˜, when
A
spectra to the variation of distance measures is studied spectra of P and P are fixed.
gg ΘΘ
3
of dlnP˜(k,µ)/dlnD . Dash curve in the top panel of
A
Fig. 1 represents the derivative in terms of H−1.
WeobserveorthogonalbehaviorofP˜tothevariationof
D andH−1. IfspectraofP andP areknown,then
A gg ΘΘ
distance measures will be determined precisely through
thisorthogonality. Itisveryinterestingtonotethatthere
is indeed a way to probe perpendicular and parallel dis-
tance measures separately.
Inpractice,P˜ isnotobservedintruecoordinateframe
but in reference frame. The following transformation is
obtained,
P˜(k ,µ ;z)/V =P˜(k,µ;z)/V (4)
ref ref ref
where V is a volume factor given by V =D2H−1. If we
A
define projected spectra P as,
P(k,µ,z)≡P˜(k,µ,z)[V (z)/V(z)] , (5)
ref
then logarithmic derivatives of P in terms of D and
A
H−1 are given by,
dlnP(k,µ) dlnP˜(k,µ)
FIG.1: (Toppanel)DerivativesofP˜intermsofdistancemeasures = −2 (6)
dlnD dlnD
of DA and H−1 are presented with given k = 0.05hMpc−1 and A A
z = 1. Solid and dash curves represent derivatives of P˜ in terms
otefrDmAs oafnddisHta−n1ceremspeaecsutirveeslyo.f(DBAottaonmd pHa−ne1l)arDeeprirveasteinvteesdo.fSPoliidn dlnP(k,µ) = dlnP˜(k,µ) −1. (7)
anddashcurvesrepresentderivativesofPintermsofDAandH−1 dlnH−1 dlnH−1
respectively.
Constants in the above equations represent the volume
factor effects. If those effects dominate, then orthogonal
LogarithmicderivativeofP˜ intermsofD isgivenby, featureofD andH−1 willbewipedout. Wepresentre-
A A
sults inthe bottom panelofFig.1. Althoughit becomes
dlnP˜(k,µ) dlnP˜(k,µ)
= −(1−µ2) (2) degenerate at µ →0, distinct feature remains at µ → 1,
dlnDA dlnk whichenablesustodecomposeDA andH−1 instillgood
P (k)+µ2P (k) precision. Hereafter,thisvolumefactorisincludedinthe
+ 4µ2(1−µ2) gΘ P˜(k,µ)ΘΘ , derivatives of P˜.
We investigate detectability of D and H−1 in this
A
where dlnP˜/dlnk is, subsection, when P and P are fixed. In general,
gg ΘΘ
those spectra are unknown. If P and P float as
dlnP˜ dlnP dlnP dlnP gg ΘΘ
= gg +2µ2 gΘ +µ4 ΘΘ . well, additional degeneracies arise between DA and Pgg
dlnk dlnk dlnk dlnk at µ → 0, and between H−1 and P at µ → 1. In the
ΘΘ
Derivative of dlnP˜(k,µ)/dlnD maximizes at µ → 0, following subsections, we introduce approaches to break
A
full degeneracies among all observables.
and vanishes at µ → 1. We present it graphically in
Fig. 1 with given scale of k =0.05hMpc−1 and redshift
of z = 1. Solid curve in the top panel of Fig. 1 repre-
C. Estimation to derive detectability
sentsthe derivativeinterms ofD . Itis reasonablethat
A
correlationpairsalongthelineofsightareunchangedby
Weestimateerrorstodecomposethoseallcomponents
variation of perpendicular component of distance mea-
introducedintheprevioussubsectionusingthefollowing
sures.
Logarithmic derivative of P˜ in terms of H−1 is given Fishermatrixanalysis[23]. Withtheconsiderationofthe
non-perturbativecontributionandtheperfectcorrelation
by,
between g and Θ at linear regime, the power spectra in
dlnP˜(k,µ) dlnP˜(k,µ) redshift space, P˜ in Eq. 1, are rewritten as [51],
= −µ2 (3)
dlnH−1 dlnk
− 4µ2(1−µ2)PgΘ(k)+µ2PΘΘ(k). P˜(k,µ,z) = Pgg(k,z)+2µ2r(k)[Pgg(k,z)PΘΘ(k,z)]1/2
P˜(k,µ) + µ(cid:8)4PΘΘ(k,z) GFoG(k,µ,σz). (8)
Derivative of dlnP˜(k,µ)/dlnH−1 vanishes at µ → 0, Thecross-correlationcoeffi(cid:9)cientr(k) isdefinedasr(k)≡
and maximizes at µ→1. It is opposite to the derivative P / P P . The density–velocity fields are highly
gΘ gg ΘΘ
p
4
correlated for k < 0.1hMpc−1 thus we assume that the and radiation energy densities and sets the location of
density and velocities are perfectly correlated, r(k) ∼ 1. thematter-radiationequalityinthetimecoordinate[54].
The density-velocity cross-spectra become the geomet- Therefore, the broadband shape is well determined, in a
ric mean of the two auto-spectra to leave only two free model–independent way, when the shape parameters of
functions,P andP . Uncertaintyintheobservedred- n , ω and ω are given [42].
gg ΘΘ S m b
shifts is modeled by a line-of-sight smearing effect of the Assuming that linear theory holds, the parameter set
structure using fitting function GFoG = e−k2σz2µ2 where pα is given by (Gg(zj), GΘ(zj), DA(zj), H−1(zj), ωm,
σz denotes one–dimensional velocity dispersion (FoG: ωb, nS). Here GX denotes the coherent growth factor
Finger-of-God effect) [30, 50–53]. For simplicity, we set defined by,
G =1 in this work.
FEoGrrors of determining parameter p out of P˜ can be PΦΦ(k,a) = QΦ(k)G2Φ(a),
α
estimated using Fisher matrix analysis determining the Pgg(k,a) = Qm(k)G2g(a),
sensitivity of a particular measurement. Fisher matrix P (k,a) = Q (k)G2(a), (12)
for this decomposition, Fdec, is written as, ΘΘ m Θ
αβ
wherethesubscriptΦdenotesthecurvatureperturbation
Fdec = ∂P˜(~k)Veff(P˜)∂P˜(~k) d3k . (9) in the Newtonian gauge,
αβ Z ∂pα P˜(~k)2 ∂pβ 2(2π)3 ds2 =−(1+2Ψ)dt2+a2(1+2Φ)dx2. (13)
The effective volume V (P˜) is given by,
eff
The growth function G is defined as G ≡ bg where
nP˜ 2 b is the standardlineargbias parameter bgetweenδmthe den-
Veff(P˜)="nP˜+1# Vsurvey, (10) switeyfoolflogwalathxeiepsoasnitdivtehesigunndceornlyvienrgsiodna,rkGmaistttehre, δgmro.wAths
Θ
where n denotes galaxy number density, here n = 5× function of −Θ. We further assume the scale indepen-
10−3(h−1Mpc)−3. Comoving volume, V , given by dent bias at scales k <0.1hMpc−1. The shape factor of
survey
eachredshiftshellfromz =0to2with∆z =0.2(fsky = the perturbed metric power spectra QΦ(k) is defined as
1/2) is written as,
2π2 9
V =f 4π(D3 −D3 ), (11) QΦ(k)= k3 25∆2ζ0(k)TΦ2(k), (14)
survey sky 3 outer inner
whichisadimensionlessmetricpowerspectraata (the
eq
where D and D denote comoving distances of
outer inner matter–radiation equilibrium epoch), and ∆2 (k) is the
outerandinnershellofthegivenredshiftbinrespectively. ζ0
initial fluctuations in the comoving gauge and T (k) is
Theoretical estimation of density–density and velocity– Φ
the transfer function normalised at T (k →0)=1. The
velocity spectra using this method is studied in [23]. Φ
primordialshape∆2 (k)depends onn (the slopeofthe
ζ0 S
primordialpowerspectrum),as∆2ζ0(k)=A2S(k/kp)nS−1,
D. Approach I: shape of spectra prior where A2 is the amplitude of the initial comoving fluc-
S
tuations at the pivot scale, k = 0.002 Mpc−1. The in-
p
In this subsection, we present an easier approach to termediate shape factor TΦ(k) depends on ωm.
determine distance measures. Although full information The shape factor for the matter fluctuations, Qm(k),
ofspectraareunknown,D andH−1areobservablewith which is important for both the galaxy–galaxy and
A
given shape of spectra. velocity–velocitypowerspectra in Eq. 12 above,is given
Theshapeofthepowerspectraofdensity–densityand by the conversionfrom QΦ(k) of,
velocity–velocity correlations critically depends on the
4 k4
epoch of matter–radiationequality. Under the paradigm Q (k)≡ Q (k), (15)
of inflationary theory, initial fluctuations are stretched m 9H04Ω2m Φ
outside the horizon at different epochs which generates
where, assuming c = 1, we can write H ≡
0
thetiltinthepowerspectrum. Thepredictedinitialtilt- 1/2997hMpc−1.
ing is parameterised as a spectral index (n ) which is
S Derivatives in Fisher matrix in terms of (G (z ),
g j
just the shape dependence due to the initial conditions.
G (z ) are given by,
Θ j
Whentheinitialfluctuationsreachthecoherentevolution
epoch after matter-radiation equality, they experience a ∂lnP˜(k,µ,z ) 2Q (k)
shcoarliez-odnepteontdheenteqshuiafltitfyroempotchhe.mGormavenittatthioenyarlei-nesnttaebriltihtye ∂Gg(zj) j = P˜(k,mµ,zj) Gg(zj)+µ2GΘ(zj)
is governed by the interplay between radiative pressure ∂lnP˜(k,µ,z ) 2µ2Q (k) (cid:2) (cid:3)
j = m G (z )+µ2G (z )(1,6)
resistance and gravitational infall. The different dura- ∂GΘ(zj) P˜(k,µ,zj) g j Θ j
tion of modes during this period result in a secondary (cid:2) (cid:3)
shape dependence in the power spectrum. This shape and derivatives in terms of (D (z ), H−1(z )) are given
A j j
dependence is determined by the ratio between matter by,
5
FIG. 2: (Left panel) The fiducial DA(z) is presented as black curve in the top panel. Cases for the fractional errors are presented in
the bottom panel; dotted curve (fixed Gg & GΘ, fixed shape of spectra, no Planck prior),dashcurve (floating Gg & GΘ,fixed shape of
spectra, noPlanckprior),longdashcurve(fixedGg &GΘ,floatingshapeofspectra,noPlanckprior),andthinsolidcurve(floating Gg
&GΘ,floatingshapeofspectra,applyingPlanckprior). (Right panel)Thesamecaptions forH−1 asintheleftpanel.
∂lnP˜(k,µ,z ) 1 ∂P˜(k,µ,z ) ∂lnk ∂P˜(k,µ,z ) ∂lnµ
j j j
= +
∂lnDA(zj) P˜(k,µ,zj)" ∂lnk ∂lnDA(zj) ∂lnµ ∂lnDA(zj)#
∂lnP˜(k,µ,z ) 1 ∂P˜(k,µ,z ) ∂lnk ∂P˜(k,µ,z ) ∂lnµ
j j j
= + , (17)
∂lnH−1(zj) P˜(k,µ,zj)" ∂lnk ∂lnH−1(zj) ∂lnµ ∂lnH−1(zj)#
where, functionswiththefixedshapeofspectra. Itshowsweak-
enedconstraintsonD andH−1 byafactorof3and1.5
A
∂lnk = −(1−µ2) respectively. Next, we float the shape spectra with the
∂lnD fixedgrowthfunctions. LongdashcurvesinFig.2repre-
A
∂lnµ = (1−µ2) sentconstraintsonDA andH−1 weakenedbyafactorof
3 and 2 respectively. Finally, when we float both growth
∂lnD
A
functions and shape of spectra, constraints on D and
∂lnk A
= −µ2 H−1 are blown by a factor of 10. In reality, we do not
∂lnH−1
know either growth functions nor shape of spectra.
∂lnµ
= −(1−µ2). (18)
∂lnH−1 This blown–offdetectability is commonlyimprovedby
a combination with other experiments such as CMB. In
Fisher matrixcomponents with the variationof(ωm, ωb, practice,wearenotabletofixtheshapeparameters. We
nS)arecalculatedcomputationallyusingCAMBoutput. floatthoseshapeparameters,andmarginalizewithCMB
We present the detectability of distance measures. experiments. For the CMB power spectra marginaliza-
Dotted curves in the bottom panels of Fig. 2 represent tion, we include in our analysis the unlensed C spectra
l
errors when fixing both growth functions and shape of of temperature–temperature, temperature–polarization
spectra. Both D and H−1 are well measured at a sub– and polarization–polarization and use the expected er-
A
percentage accuracy. rors for Planck survey. This treatment is nearly equiv-
We investigate the effects of the unknown overall am- alent to using distance information measured by CMB.
plitudes of growth functions and shape of spectra. Dash WelistdiagonalelementsofCMBpriorinformationused
curves in Fig. 2 represent detectability to float growth for prior; σ(ω ) = 4.9 × 10−4, σ(ω ) = 3.7 × 10−5,
m b
6
σ(n ) = 1.7 × 10−3. Estimated errors are shown as
S
thin blue solid curves in the bottoms panels of Fig. 2.
Thinbluesolidcurvesapproachtodashcurveswhichare
the case of floating distance measures and fixed shape of
spectra. Planck prior resolves uncertainties caused by
unknown shape of spectra. The error bars shown in the
top panels of Fig. 2 are estimated using Planck prior.
E. Approach II: FLRW prior
In this subsection, we investigate an alternative
method to probe distance measures using redshift sur-
vey alone without an assistance of other experiments
like Planck in the previous subsection. Mathematical
description of our universe is possible by assuming the
cosmologicalprinciple. Ourunderstandingofthepertur-
bation theory is based upon the spartial homogeneity of
the universe. In homogenous FLRW universe, the per-
pendicularcomponentofdistancemeasuresareexpressed
by parallel component of Hubble flow. The angular di- FIG. 3: Black solid curve in the top panel represents fiducial
ameterdistancerepresentstheperpendicularcomponent, H−1(z)witherrorbarsestimatedusingFLRWpriorinsmallcur-
vatureapproximation. Bluesolidcurveinthebottompanelrepre-
and it is expressedby the integratedsum of Hubble flow
sentsfractionalerrors.
from the current to the targeted epochs. Therefore, if
the wide–deep field survey is designed to be fully tomo-
graphic through all redshift spaces, then both different where
components of distance measures can be unified.
∂lnk ∂lnk ∂lnD (z )
With suggested tomographical wide–deep field survey ∂lnH−1(zj′) = ∂lnDA(zj)∂lnH−A1(zjj′) −µ2δjj′ (20)
design, the angular diameter distance of D is approxi-
A ∂lnµ ∂lnµ ∂lnD (z )
amtaetaeclyherxepdrsehsisftedbibnyadsi,screte sum of Hubble flow ofH−1 ∂lnH−1(zj′) = ∂lnDA(zj)∂lnH−A1(zjj′) −(1−µ2)δjj′,
where zj′ < zj and δjj′ = 1 at j = j′ otherwise 0. The
D (z ) = 1 zjdz′H−1(z′) reduction of parameter space for distance measures en-
A j 1+z hances the detectability of p parameter.
j Z0 α
In Fig. 3, we present the detectability of Hubble flow
1 Nj
∼ H−1(zj′)∆zj′, (19) from z = 0.1 to z = 1.9. Solid curve in the bottom
1+zj j′=1 panel represents the achievable precision level. Due to
X volume limit, the detectability is slightly over 1% accu-
racyat z <0.5. But at the redshift bins of z >0.5, H−1
where N represents the total number of redshift bins
j is measured much better than 1% accuracy. There are
up to the targeted redshift of z . Then p parameter
j α many observationalmethods to probe Hubble flow using
space for distance measures is unified into a single pa-
cosmological parameters, but none of those probe Hub-
rameter of Hubble flow H−1. Most future spectroscopy
bleflowintheoreticalmodelindependentway. Although
wide–deep field redshift survey programs such as SUB-
theHubbleflowH atcurrentepochcanbemeasureddi-
0
ARU PFS, BigBOSS and EUCLID are planned to make
rectly using distance ladder, the full evolution of H has
full tomographic scanning [55–57]. The parameter set of
yet been attempted to be measured in this precision.
p becomes(G (z ), G (z ), H−1(z′),ω ,ω , n ). The
α g j Θ j j m b S In addition to this, it is interesting to observe the
derivativesintermsof(P (k ,z ),P (k ,z ))aregiven
gg i j ΘΘ i j self–determinationof shape factors with Friedman prior.
inEq.24,andthederivativeintermsofH−1(z )isgiven
j EvenwithoutPlanckcombined,we achievethe following
by, constraints on shape parameters; σ(ω ) = 1.1×10−3,
m
σ(ω )=5.8×10−4, σ(n )=5.9×10−3. Itsuggeststhat
b S
∂lnP˜(k,µ,z ) 1 ∂P˜(k,µ,z ) ∂lnk full non–parameteric reconstruction of spectra is possi-
j j
=
∂lnH−1(zj′) P˜(k,µ,zj) ∂lnk ∂lnH−1(zj′) ble, which is discussed in next section in detail.
The expressionin Eq. 19 is not applicable for the case
1 ∂P˜(k,µ,zj) ∂lnµ of curved space–time. This implicit flat universe prior
+
P˜(k,µ,zj) ∂lnµ ∂lnH−1(zj′) canbe liftedoffeasily. We define curvatureparameterK
7
as,
K= |Ω |h2/2997.2hMpc−1. (21)
k
p
Then parameter space of p is extended into (G (z ),
α g j
G (z ),H−1(z′),ω ,ω ,n ,K). Theestimatedangular
Θ j j m b S
diameter distance D (z ) is given with open curvature
A j
as,
1 Nj
DA(zj)= sinh K H−1(zj′)∆zj′ , (22)
K(1+z )  
j j′=1
X
 
and with closed curvature as,
1 Nj
DA(zj)= sin K H−1(zj′)∆zj′ . (23)
K(1+z )  
j j′=1
X
 
We investigate impact on the detectability from added
curvaturedegreeoffreedom. Wefindthatconstraintson
H−1(z )becomespoorerbyafactorof1.5withcurvature
j
marginalization. FIG. 4: (Top panel)Casesforthefractionalerrorsarepresented;
However, constraints on curvature of the space have dotted curve (fixed DA &H−1,fixed shapeof spectra, noPlanck
been nailed down to the vicinity about flatness. Even prior),dashcurve(floatingDA &H−1,fixedshapeofspectra,no
though the space is curved, it is likely to be a couple Planck prior), long dash curve (fixed DA & H−1, floating shape
ofpercentagedeviationfromflatness,whichisconsistent oHf−s1p,ecfltoraat,inngo Pshlaapncekopfrsipoer)c,traan,dapthpilnyinsogliPdlacunrcvkep(rflioora)t.in(gBDotAtom&
with prediction of inflationary paradigm as well. In this panel)ThesamecaptionsforGΘ asinthetookpanel.
small curvature approximation of K ≪ 1, D can be
A
approximately given by Eq. 19 in our interest redshift
range of z < 2. From now, when we mention FLRW space of pα becomes (Pgg(ki,zj), PΘΘ(ki,zj), DA(zj),
prior, this small curvature approximation is applied. H−1(zj)). The subscript “i” denotes 20 k bins, and the
subscript“j” denotes 10 z bins from0 to 2. Variationof
(P (k ,z ), P (k ,z )) represents simultaneous floata-
gg i j ΘΘ i j
III. CHALLENGE TO MEASURE SPECTRA IN tion of growth functions and shape of spectra. This ap-
MODEL INDEPENDENT WAY proach has advantage for an exotic dark energy or mod-
ified gravity model in which growth functions become
We presentthedetectability ofgrowthfunctions ofG scale dependent [46, 47].
g
and GΘ in Fig. 4. The detectability of Gg and GΘ is Thederivativesintermsof(Pgg(ki,zj),PΘΘ(ki,zj))in
studied in detail [23], when both distance measures and Fisher matrix are given by,
shape of spectra are fixed. Dotted curves in the bottom
∂lnP˜(k,µ,z ) Θ (k) P (k ,z )
panelsofFig.4reproducewellthepreviousresults. First, j = i 1+µ2 ΘΘ i j
we test the impact of floating distance measures on er- ∂Pgdgec(ki,zj) P˜(k,µ,zj)" s Pgg(ki,zj) #
rors represented by dotted curves. Dash curves in Fig. 4
represent fractional errors when floating DA and H−1. ∂lnP˜(ki,µ,zj) = µ2Θi(k) Pgg(ki,zj) +µ2 ,
The detectability of Gg and GΘ is weakened by a factor ∂PΘdeΘc(ki,zj) P˜(ki,µ,zj)"sPΘΘ(ki,zj) #
of 3 and 1.5 respectively. Second, we test the impact of
(24)
floating shape parameters on errors represented by dot-
ted curves. Long dash curves in Fig. 4 represent those whereΘ (k)denotes stepfunction inwhichΘ (k)=1at
i i
results. If distance measures are fixed, growth functions k belong to k bin, otherwise Θ (k) = 0. The derivative
i i
and shape parameters are determined in good precision terms of distance measures are the same as in Eq. 17.
evenwithoutcombinationofCMB experiments. Finally, The shape factor floats intrinsically in this general pa-
we float both distance measures and shape of spectra. rameterization. If distance measures float as well, none
None of G and G are measured in precision. Con- of spectra are measured precisely. If one wants to mea-
g Θ
straints on G and G become much worse. sure fluctuations of (P , P ), then distance measures
g Θ gg ΘΘ
We generalize parameterization of growth functions should be provided by other experiments such as super-
and shape of spectra with Fourier modes of power spec- novae.
tra. Fourier space is binned into 20 bins from 0.005 In this section, we investigate a new method to probe
h−1Mpc to 0.105 h−1Mpc. The full set of parameter spectrausingredshiftsurveyalonewithoutanassistance
8
FIG. 5: (Left panel) Spectra at z = 1.1 are presented. Solid curve in the top panel represents fiducial Pgg(k), and solid curve in the
bottom panel represents PΘΘ(k). Error bars on the curves represent theoretical estimation using FLRW prior. (Right panel) Summed
fractionalerrorsofspectraateachgivenredshiftarepresentedwithapplyingFLRWprior.
of other experiments. With FLRW prior, the set of pa- P (k ,z ) are uniquely determined at µ bins approach-
gg i j
rameter space of p becomes (P (k ,z ), P (k ,z ), ing µ→0, while P (k ,z ) aremainly determined atµ
α gg i j ΘΘ i j ΘΘ i j
H−1(z )). Constraints on power spectra of P (k ,z ) bins approachingµ→1. However,the impactof param-
j gg i j
andP (k ,z )withfixeddistancemeasuresarestudied eterizedH doesnotpivotatµ→0limitnorµ→1limit,
ΘΘ i j
in detail [23]. The previous results are updated by float- but prevails through µ bins in coherent way. It breaks
ing distance measures with applying Friedman universe the degeneracy among parameters.
prior. In the left panel of Fig. 5, solid curves in the top
and bottom panels represent fiducial P (k ,z = 1.1)
gg i j
and PΘΘ(ki,zj = 1.1), and error bars on the curves are IV. DISCUSSION
estimated using FLRW prior. When FLRW prior is ap-
plied, the measuredpowerspectra with floating distance
If we float freely both the spectra ofP andP and
gg ΘΘ
measures are probed as precise as the measured power the distancemeasuresofD andH−1,then weareinca-
A
spectra with fixed distance measures.
pable ofprovidinganyobservableout ofgalaxydistribu-
We continue our test at the other redshift bins. Here
tions in redshift space. It has been underestimated the
we present the results using cumulated error of σtot(P)
significance of this degeneracy between fluctuations and
which is given by,
distance measures. Unless distance measures of D and
A
H−1 are given by other experiments, there is no obser-
ikmax
σtot(P) −2 = P(k )C−1[P(k ),P(k )]P(k )(,25) vation of fluctuations. Unless any piece of information
i i j j
about fluctuations is known, there is no probe of dis-
(cid:2) (cid:3) i,j=Xikmin tancemeasures. Specific conditionsoneithercomponent
where C−1 is the inverseof covariancematrix calculated should be imposed to make targeted observation.
using Fisher matrix. In the top right panel of Fig. 5, We investigate the detectability of parameter p with
α
σtot(P ) is presented from z =0.1 to z =1.9. The pre- a given broadband shape of spectra. In this manuscript,
gg
cisionto measureP (k ,z )is notweakenedthroughall the broadband shape is given by a few shape cosmologi-
gg i j
redshifts. In the bottom right panel of Fig. 5, summed calparametersof(ω ,ω ,n ),insteadofphenomenolog-
m b S
errors for P (k ,z ) are presented. Again, power spec- ical parameters. The primordial shape during inflation
ΘΘ i j
tra of coherent motions are well measured even after is determined by n , and the transferred shape before
S
marginalizing distance measures. matter–radiation equality epoch is described by ω and
m
We have to explain the reason why the degeneracy ω . The scale dependence of spectra of P and P is
b gg ΘΘ
amongthe parametersarebroken,eventhoughprincipal dropped off to leave only the time dependent factors pa-
componentsapproachisusedforspectra. Thespectraare rameterized by growth functions of G and G . Tight
g Θ
not principally parameterizedalongµ direction. Spectra constraints on (G , G , D , H−1) are expected when
g Θ A
9
theshapeparametersaremarginalizedwithCMBexper- distanceladdertechnique,butonlyatthelocaluniverse.
iments. However,ifCMBexperimentsmarginalizationis There are other geometric probes developed, but indi-
lifted off, the detectability gets worse by a factor of 10. rectly measured by luminosity or angular diameter dis-
Therearesomeclassesoftheoreticalmodelsinwhichthe tances. ButourmethodtoprobeHubbleflowisnotonly
shape of spectra is not fully determined at last scatter- providingH measurementsthroughallredshifts,butalso
ing surface. Then there will be no precise observables independent of specific cosmological parameters. Red-
despite the given broadband shape of spectra, as far as shift survey will reveal tomographic evolution of Hubble
no external priors on the shape of spectra are provided. flowfromthecurrenttoanytargetedepochasfarasitis
The degeneracy between fluctuations and distance mea- reached by spectroscopic survey. We also show that de-
suresstillprevailswiththe conditionofgivenbroadband tectability of P and H−1 is influenced by uncertainty
ΘΘ
shape of spectra. ofcurvature,butitisnotmuchsevereinsmallcurvature
Weproposenewtreatmenttoextractobservableswith approximation.
assuming a minimal theoretical ansatz. Our standard However, we have to mention caveats. As it is well
modelofperturbationtheoryisbaseduponthecosmolog- known, redshift distortion map is severely contaminated
ical principle implying spartial homogeneity of the uni- by non–linear smearing effect. It is indeed troublesome
verse. It also leads us to unify two distinct components todecomposelinearinformationoutofthismixture. But
ofdistance measuresinto a singleone with no additional therearemanyreportshavingbeenmadeforthepossible
theoretical prior. The perpendicular component of dis- precise decompositionof fluctuations with fixed distance
tance measures is represented by angular diameter dis- measures. The contaminated maps can be decomposed
tance which is an integrated sum of parallel component intolineartheoryofspectraandnon–smearingeffectcon-
represented by Hubble flow. This reduction is possible tributions up to limited scales of k <0.1hMpc−1. Then
by demanding anexperimental designof wide–deep field the similar control of non–linear effect might be avail-
redshift survey to be fully tomographic through all red- able for the case of relaxing distance measure prior. In
shifts. We find that Pgg, PΘΘ and H−1 are measured in near future, we will be back to the following workto full
precision about a sub–percentage level using this reduc- reconstruction of fluctuations and distance measures.
tion of distance measure parameters only assuming the
FLRW universe available.
It is remarkable that Hubble flow is measured in high
Acknowledgments
precisionatallredshiftsshownintherightpanelofFig.5.
ThedirectmeasurementofHubbleflowisakeyinforma-
tion to understand the history of the universe. The dis- We thank Hee-Jong Seo for assistance to initiate this
covery of cosmic accelerationdemands us to probe Hub- work and for substantial inputs in this manuscript. We
ble flow in a model independent way, because we lose thankEricLinderforhelpfuldiscussionsduringaKITPC
confidence of the detailed matter–energy components of workshop. Numericalcalculationswereperformedbyus-
theuniverseorthe validationofgeneralrelativityatcos- ing a high performance computing cluster in the Korea
mologicalscales. DirectmeasurementofH ispossibleby Astronomy and Space Science Institute.
[1] A.Friedman, Zeitschrift furPhysik 10, 377-386 (1922). Roy.Astron. Soc. 353, 1201 (2004) [astro-ph/0406513].
[2] E. R.Harrison, Phys. Rev.D 1, 2726 (1970). [10] I.Zehaviet al.[SDSSCollaboration], Astrophys.J.630,
[3] A. Einstein and W. de Sitter, Proc. Nat. Acad. Sci. 18, 1 (2005) [astro-ph/0408569].
213 (1932). [11] M. Tegmark et al. [SDSS Collaboration], Phys. Rev. D
[4] A. G. Riess et al. [Supernova Search Team Collabora- 74, 123507 (2006) [astro-ph/0608632].
tion], Astron. J. 116, 1009 (1998) [astro-ph/9805201]. [12] L. Guzzo, M. Pierleoni, B. Meneux, E. Branchini,
[5] S. Perlmutter et al. [Supernova Cosmology Project O. L. Fevre, C. Marinoni, B. Garilli and J. Blaizot et
Collaboration], Astrophys. J. 517, 565 (1999) al.,Nature451, 541 (2008) [arXiv:0802.1944 [astro-ph]].
[astro-ph/9812133]. [13] M. J. Drinkwater, R. J. Jurek, C. Blake, D. Woods,
[6] D. G. York et al. [SDSS Collaboration], Astron. J. 120, K. A. Pimbblet, K. Glazebrook, R. Sharp and
1579 (2000) [astro-ph/0006396]. M. B. Pracy et al., Mon. Not. Roy. Astron. Soc. 401,
[7] J. A. Peacock, S. Cole, P. Norberg, C. M. Baugh, 1429 (2010) [arXiv:0911.4246 [astro-ph.CO]].
J. Bland-Hawthorn, T. Bridges, R. D. Cannon [14] E. A. Kazin et al. [SDSS Collaboration], Astrophys. J.
and M. Colless et al., Nature 410, 169 (2001) 710, 1444 (2010) [arXiv:0908.2598 [astro-ph.CO]].
[astro-ph/0103143]. [15] B. A. Reid, W. J. Percival, D. J. Eisenstein, L. Verde,
[8] E. Hawkins, S. Maddox, S. Cole, D. Madgwick, D. N. Spergel, R. A. Skibba, N. A. Bahcall and T. Bu-
P. Norberg, J. Peacock, I. Baldry and C. Baugh davarietal.,Mon.Not.Roy.Astron.Soc.404,60(2010)
et al., Mon. Not. Roy. Astron. Soc. 346, 78 (2003) [arXiv:0907.1659 [astro-ph.CO]].
[astro-ph/0212375]. [16] B. A. Reid, L. Samushia, M. White, W. J. Perci-
[9] W.J.Percivalet al.[2dFGRSCollaboration], Mon.Not. val, M. Manera, N. Padmanabhan, A. J. Ross and
10
A.G. Sanchez et al.,arXiv:1203.6641 [astro-ph.CO]. [37] P. Zhang, J. Pan and Y. Zheng, arXiv:1207.2722 [astro-
[17] B. Jain and P. Zhang, Phys. Rev. D 78, 063503 (2008) ph.CO].
[arXiv:0709.2375 [astro-ph]]. [38] E. Gaztanaga, A. Cabre and L. Hui, Mon. Not. Roy.
[18] S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 77, Astron. Soc. 399, 1663 (2009) [arXiv:0807.3551 [astro-
023504 (2008) [arXiv:0710.1092 [astro-ph]]. ph]].
[19] Y. -S. Song and K. Koyama, JCAP 0901, 048 (2009) [39] C. Blake and K. Glazebrook, Astrophys. J. 594, 665
[arXiv:0802.3897 [astro-ph]]. (2003) [astro-ph/0301632].
[20] Y.-S.Song and W. J. Percival, JCAP 0910, 004 (2009) [40] H. -J. Seo and D. J. Eisenstein, Astrophys. J. 598, 720
[arXiv:0807.0810 [astro-ph]]. (2003) [astro-ph/0307460].
[21] Y. Wang, JCAP 0805, 021 (2008) [arXiv:0710.3885 [41] Y. Wang, Astrophys. J. 647, 1 (2006)
[astro-ph]]. [astro-ph/0601163].
[22] W.J.PercivalandM.White,arXiv:0808.0003[astro-ph]. [42] Y.-S.Song,C.G.Sabiu,I.KayoandR.C.Nichol,JCAP
[23] M.White,Y.-S.SongandW.J.Percival,Mon.Not.Roy. 1105, 020 (2011) [arXiv:1006.4630 [astro-ph.CO]].
Astron. Soc. 397, 1348 (2008) [arXiv:0810.1518 [astro- [43] C. -H. Chuang and Y. Wang, arXiv:1205.5573 [astro-
ph]]. ph.CO].
[24] P. McDonald and U. Seljak, JCAP 0910, 007 (2009) [44] F.Beutler,C.Blake,M.Colless,D.H.Jones,L.Staveley-
[arXiv:0810.0323 [astro-ph]]. Smith, G. B. Poole, L. Campbell and Q. Parker et al.,
[25] Y. -S. Song and I. Kayo, Mon. Not. Roy. Astron. Soc. arXiv:1204.4725 [astro-ph.CO].
407, 1123 (2010) [arXiv:1003.2420 [astro-ph.CO]]. [45] C. Alcock and B. Paczynski, Nature 281, 358 (1979).
[26] J.Tang,I.KayoandM.Takada,arXiv:1103.3614 [astro- [46] S.M.Carroll,V.Duvvuri,M.TroddenandM.S.Turner,
ph.CO]. Phys. Rev.D 70, 043528 (2004) [astro-ph/0306438].
[27] D.JeongandE.Komatsu,Astrophys.J.651,619(2006) [47] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett.
[astro-ph/0604075]. B 484, 112 (2000) [hep-th/0002190].
[28] D.JeongandE.Komatsu,Astrophys.J.691,569(2009) [48] Y. -S. Song, W. Hu and I. Sawicki, Phys. Rev. D 75,
[arXiv:0805.2632 [astro-ph]]. 044004 (2007) [astro-ph/0610532].
[29] V.DesjacquesandR.K.Sheth,Phys.Rev.D81,023526 [49] N. Kaiser, Mon. Not. Roy.Astron. Soc. 227, 1 (1987).
(2010) [arXiv:0909.4544 [astro-ph.CO]]. [50] K. B. Fisher, Astrophys. J. 448, 494 (1995)
[30] A.Taruya,T.NishimichiandS.Saito,Phys.Rev.D82, [astro-ph/9412081].
063522 (2010) [arXiv:1006.0699 [astro-ph.CO]]. [51] R. Scoccimarro, Phys. Rev. D 70, 083007 (2004)
[31] E.Jennings,C.M.BaughandS.Pascoli,Mon.Not.Roy. [astro-ph/0407214].
Astron. Soc. 410, 2081 (2011) [arXiv:1003.4282 [astro- [52] T. Matsubara, Phys. Rev. D 77, 063530 (2008)
ph.CO]]. [arXiv:0711.2521 [astro-ph]].
[32] B. A. Reid and M. White, arXiv:1105.4165 [astro- [53] M.CrocceandR.Scoccimarro,Phys.Rev.D77,023533
ph.CO]. (2008) [arXiv:0704.2783 [astro-ph]].
[33] T.Okumura,U.Seljak,P.McDonaldandV.Desjacques, [54] D. J. Eisenstein and W. Hu, Astrophys. J. 496, 605
JCAP1202,010(2012)[arXiv:1109.1609[astro-ph.CO]]. (1998) [astro-ph/9709112].
[34] J. Kwan, G. F. Lewis and E. V. Linder, Astrophys. J. [55] R. Ellis, M. Takada, H. Aihara, N. Arimoto, K. Bundy,
748, 78 (2012) [arXiv:1105.1194 [astro-ph.CO]]. M. Chiba, J. Cohen and O. Dore et al.,arXiv:1206.0737
[35] L. Samushia, W. J. Percival and A. Raccanelli, Mon. [astro-ph.CO].
Not.Roy.Astron.Soc.420,2102(2012)[arXiv:1102.1014 [56] D. Schlegel et al. [BigBOSS Collaboration],
[astro-ph.CO]]. arXiv:1106.1706 [astro-ph.IM].
[36] C.Blake,S.Brough,M.Colless,C.Contreras,W.Couch, [57] A. Cimatti, R. Laureijs, B. Leibundgut, S. Lilly,
S. Croom, T. Davis and M. J. Drinkwater et al., Mon. R. Nichol, A. Reefregier, P. Rosati and M. Steinmetz
Not.Roy.Astron.Soc.415,2876(2011)[arXiv:1104.2948 et al.,arXiv:0912.0914 [astro-ph.CO].
[astro-ph.CO]].