Cold dark matter isocurvature perturbations: Cosmological constraints and applications

Hong Li,1,2,* Jie Liu,1,† Jun-Qing Xia,3 and Yi-Fu Cai4

1Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-4, Beijing 100049, Peoples Republic of China2Theoretical Physics Center for Science Facilities (TPCSF), Chinese Academy of Science, Beijing 100049, Peoples Republic of China

3Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, I-34136 Trieste, Italy4Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

(Received 17 January 2011; published 15 June 2011)

In this paper we present the constraints on cold dark matter (CDM) isocurvature contributions to the

cosmological perturbations. By employing Markov chain Monte Carlo method, we perform a global

analysis for cosmological parameters using the latest astronomical data, such as 7-year Wilkinson

Microwave Anisotropy Probe (WMAP7) observations, matter power spectrum from the Sloan Digital

Sky Survey, luminous red galaxies, and the Union2 type Ia supernovae sample. We find that the correlated

mixture of adiabatic and isocurvature modes is slightly favored by the current observational data. We also

obtain a tight limit on the fraction of the CDM isocurvature contributions, which should be less than

14.6% at 95% confidence level. With the presence of the isocurvature modes, the adiabatic spectral index

becomes slightly bigger, nadis ¼ 0:972 0:014 (1), and the tilt for isocurvature spectrum could be large,

namely, the best fit value is nisos ¼ 3:020. Finally, we discuss the effect on WMAP normalization priors,

shift parameter R, acoustic scale lA, and z, from the CDM isocurvature perturbation. By fitting the mixed

initial condition to the combined data, we find the mean values of R, lA, and z can be changed about 2:9,2:8, and 1:5 respectively, comparing with those obtained in the pure adiabatic condition.

DOI: 10.1103/PhysRevD.83.123517 PACS numbers: 98.80.Cq, 98.65.Dx, 98.70.Vc, 98.80.Es

I. INTRODUCTION

The accumulation of WMAP 7 yr measurements ofcosmic microwave background (CMB) radiation [1], asso-ciated with observations of the Sloan Digital Sky Survey(SDSS) [2], provide a wealth of information on the anisot-ropies and inhomogeneities of our Universe, in light ofwhich the perturbation theory has been tested at a certainlevel. Currently, various observational data favor the sim-plest concordance cosmological model, which has six freeparameters and the pure adiabatic initial condition [1,3–6].Although the concordance model fits the data quite well,it is always worthy to consider alternative candidates.Namely, it is important to study observational constraintson initial states of cosmological perturbations at a reheat-ing surface.

Generically, there exist two classes of modes of cosmo-logical perturbations, with one being adiabatic of which thetrajectory is parallel to the background evolution, while theother is isocurvature of which the trajectory is orthogonalto the background evolution [7–15]. However, a first lessonfrom observational data is that the primordial fluctuationsare nearly adiabatic; additionally, an isocurvature mode isalso expected to be negligible as is predicted by the sim-plest inflation model in terms of a single inflaton field anda rapidly reheating process [16]. Therefore, one usuallymakes the data fitting without considering the isocurvaturemode.

Accompanied with developments of the inflationarycosmology, models of multiple field inflation were exten-sively studied in the literature [17–27], which predicted anexistence of primordial isocurvature fluctuations. Theseprimordial isocurvature modes could be transferred intocosmological perturbations after reheating, such as baryon,cold dark matter (CDM), dark energy, and neutrino, re-spectively [28–31]. Consequently, the attention on isocur-vature modes has been awakened in recent years [32,33].One may notice that, although the pure isocurvature

primordial perturbation has been ruled out by theBoomerang and MAXIMA-1 data [34] already, a mixtureof adiabatic and isocurvature modes can be in agreementwith the current data, fortunately. In the literature [35–46],the studies on constraining the isocurvature fluctuationhave been performed extensively using various observa-tional data, such as cosmic microwave background (CMB),large scale structure theory (LSS), integrated Sachs-Wolfeeffect, or Lyman- forest data. With the WMAP7 data, thetotally uncorrelated and anticorrelated adiabatic, nonadia-batic perturbations are constrained [1], which are per-formed by fixing the correlation coefficients to be 0 or1, respectively.In this paper, we study the constraints on the isocurva-

ture modes of cosmological perturbations in light of thelatest observational data. We consider a generic scenariothat the adiabatic modes and isocurvature ones are allowedto be mixed through a correlation matrix which could bearising from a time-varying background trajectory. In thedetailed analysis, we treat the coefficients of the correla-tion matrix as a free parameter and make the data fitting.

*hongli@ihep.ac.cn†liujie@ihep.ac.cn

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Comparing with the previous results in the literature, ourresult shows a slight improvement on the final constraintsof the mixed initial condition parameters. Namely, CDMisocurvature components are stringently limited, the con-tribution from the isocurvature modes are only allowed insmall scales, and the error bars of initial condition parame-ters become smaller than the past works in the literature,which can be observed from the one-dimensional proba-bility distribution of the fraction parameter as analyzedin the main context.

TheWMAP normalization priors, R, lA, and z encodingthe information of background cosmic distances, can beapplied to greatly simplify the numerical calculations ofdetermining cosmological parameters, such as the equationof state of dark energy. It was found that these priors couldbe sensitive to the peak locations and local structures of theCMB temperature power spectrum [47]. However, it iswell-known that these quantities can be affected by CDMisocurvature perturbation. Therefore, we study the effectson the WMAP normalization priors given by the WMAPgroup from the isocurvature mode perturbation.

The outline of this paper is as follows. In Sec. II, wedescribe the parametrization of cosmological perturbationswith adiabatic and isocurvature modes mixed, and then weconsider a specific example to illustrate the effects ofisocurvature perturbation imprinted on the CMB tempera-ture power spectrum and LSS matter power spectrum,respectively. In Sec. III, we perform a global analysis andintroduce the data we applied and the parameters used inthe data fitting. The constraints on these parameters arepresented in detail in Sec. IV, and the corresponding effectson reduced distance parameters are discussed. Finally,Sec. V includes the conclusions and discussion.

II. PARAMETRIZATION OF CORRELATEDADIABATIC AND ISOCURVATURE

PERTURBATIONS

For model-independent consideration, the initial condi-tions for the correlated mixture of adiabatic and isocurva-ture modes can be written as

P ij ¼ Aijs

k

k0

nijs1

; (1)

where k0 is the pivot scale. Here, Aijs and n

ijs are 2 2

symmetric matrices which characterize the amplitude andpower spectrum index, respectively. We have

Aijs ¼ Aadi

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAadis Aiso

s

pcosffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Aadis Aiso

s

pcos Aiso

s

!; (2)

where cos ¼ Aadi;isos =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAadis Aiso

s

pdescribes the correlation

between the adiabatic mode and the isocurvature mode[20] and Aadi

s and Aisos are the amplitude of adiabatic and

isocurvature modes, respectively. The index matrix nijs isgiven by

nijs ¼ nadis ncors

ncors nisos

; (3)

where nadis and nisos are the spectrum indices for adiabaticand isocurvature modes. Following Ref. [39], we assumencors ¼ ðnadis þ nisos Þ=2 for simplicity.Both adiabatic and isocurvature perturbations seed the

structure formation of our Universe, and can be imprintedin CMB temperature, polarization power spectrum, as wellas matter power spectrum of galaxy surveys. With initialisocurvature modes, the CMB power spectrum is given by

CTT‘ ¼ Aadi

s Cadi‘ þ Aiso

s Ciso‘ þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAadis Aiso

s

qCadi;iso‘ cos; (4)

where the initial power spectrum can be obtained asfollows:

Cij‘ ¼ 4

2‘þ 1

Zd lnk

k

k0

nijs1

i‘ðkÞj

‘ðkÞ; (5)

with i, j standing for adiabatic or isocurvature and x‘ are

the transfer functions of the radiation at initial moment.There are similar formulas for CMB polarization powerspectra CEE

‘ , CBB‘ and temperature-polarization cross

correlation power spectrum CTE‘ .

Moreover, the matter power spectrum PðkÞ can be writ-ten as

PðkÞ ¼ Aadis PadiðkÞ þ Aiso

s PisoðkÞþ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAadis Aiso

s

qPadi;isoðkÞ cos; (6)

with

P ijðkÞ ¼k

k0

nijs1

TiðkÞTjðkÞ; (7)

where TxðkÞ are the transfer functions of matter componentat initial moment.In order to show the effect of isocurvature perturbations,

in Fig. 1 we plot the CMB temperature power spectra(upper panel) and LSS matter power spectra (lower panel).The red solid lines are given by the best fit model withmixed initial conditions by fitting the current observationaldata: !b ¼ 0:0226, !c ¼ 0:1144, Aadi

s ¼ 2:407 109,Aisos ¼ 3:885 1012, nadis ¼ 0:968, nisos ¼ 3:020,

cos ¼ 0:149. The blue dashed line and black dash-dotted lines are given by choosing Aiso

s ¼ 1 1010 andcos ¼ 1 instead, respectively. Since the value of nisos ismuch larger than unity, one can see that by taking intoaccount the CDM isocurvature modes and their correlationwith adiabatic perturbations, the height and location of thepeaks of the temperature power spectrum of CMB (TT)spectra will be modified, and the amplitudes of the matterpower spectrum on small scales are changed significantly.For the scale-invariant isocurvature spectrum, we find itseffects on TT spectra and matter power spectrum aremodest comparing with the blue-tilted one.

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III. OBSERVATIONAL CONSTRAINTS

A. Data

We perform a global analysis by employing a modifiedMarkov chain Monte Carlo (MCMC) package COSMOMC

[48–50], in which we have made an extension of the initialperturbations by including correlated adiabatic and CDMisocurvature modes. In computation of CMB we have in-cluded the WMAP7 temperature and polarization powerspectra with the routine for computing the likelihood sup-plied by the WMAP team. Since the CDM isocurvatureperturbations affect the power spectrum on small scales,we also include some small-scale temperature anisotropiesmeasurements, ACBAR [51], CBI [52], and Boomerang[53]. Furthermore, we use the matter power spectrum mea-sured by the observations of luminous red galaxies (LRG)from SDSS data release seven [2], and the Union2 super-novae data sets calibrated by the SALT2 template [54].

B. Parameters

As pointed out in Refs. [39,44], the constraints of thespectral index may depend on the pivot scale. In order to

avoid such dependence, we do not take the spectra indices

nijs as free parameters in our analysis. We take two different

amplitudes Asðk1Þ and Asðk2Þ at two pivot scales k1 ¼0:002 and k2 ¼ 0:05 to be free. And the spectra indicescan be derived by

nijs 1 ln½P ijðk2Þ=P ijðk1Þ

ln½k2=k1 ¼ ln½Aijs ðk2Þ=Aij

s ðk1Þln½k2=k1 : (8)

We assume that the overall amplitude is composed of twocomponents,

As Aadis þ Aiso

s ; (9)

and the fraction of the adiabatic part is defined by

Aadis =As: (10)

Thus, our most general parameter space is

P f!b;!c;s; ; cos; 1; 2; Asðk1Þ; Asðk2Þg; (11)

where !b bh2, !c bh

2 denotes physical baryondensity and cold dark matter density relative to criticaldensity, respectively; s 100 rs

dAis the ratio of sound

horizon to angular diameter distance at decoupling and characterizes the optical depth to reionization. Asðk1Þ,Asðk2Þ, 1, and 2 are the amplitudes and the fractionsof adiabatic mode at two scales, respectively. In our cal-culation, we assume the flat universe and dark energy is thecosmological constant.

IV. GLOBAL FITTING RESULTS

In this section, we present the main constraints on thecosmological parameters, as well as the parameters relatedto the CDM isocurvature perturbations. Then, we discussthe effects on theWMAP normalization priors when takinginto account CDM isocurvature perturbations.

A. Constraints on cosmological parameters

In Table I and II we list the constraints on the initialcondition associated parameters by fitting the latest obser-vational data. The parameters presented in Table I are thefree parameters during our calculation, while those inTable II are the derived parameters from those in Table Iaccording to Eqs. (8)–(10), and both results are given bythe statistics of the samples given by our MCMC approach.One-dimensional posterior probability distributions ofAadis , Aiso

s , nadis , and nisos , as well as cos and at the twodifferent pivot scales are plotted in Fig. 2.

101

102

103

104

0

1000

2000

3000

4000

5000

6000

7000

l

l(l+

1)C

lTT/2

π

10−3

10−2

10−1

100

101

102

103

104

105

k (h Mpc−1)

P(k

)

FIG. 1 (color online). The CMB temperature power spectraand LSS matter power spectra are plotted using different initialconditions in the upper and lower panels. The red solid lines aregiven by the best fit model with the mixed initial condition. Theblue dashed lines and the black dash-dotted lines are given bychoosing Aiso

s ¼ 1 1010 and cos ¼ 1, respectively.

TABLE I. The mean values and 1 errors of As given by Eq. (9) at two different pivot scales,2 and cos, while the limit of 1 is for 95% confidence level.

log½1010Asðk1Þ log½1010Asðk2Þ 1 2 cos

3:185þ0:0410:042 3:622þ0:242

0:247 >93:9% 0:594þ0:0560:083 0:094þ0:075

0:095

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From the mixed initial conditions, the constraints on theadiabatic perturbation are log½1010Aadi

s ¼ 3:165 0:044and nadis ¼ 0:972 0:014 at 68% confidence level. Theadiabatic spectrum index becomes larger, when comparingwith the constraint nadis ¼ 0:964 0:010ð1Þ from thepure adiabatic initial condition. For the CDM isocurvaturemode, the constraints are log½1010Aiso

s ¼ 1:375þ1:2331:306

and nisos ¼ 2:246þ0:4940:428 (68% C.L.), which indicate that

the current observational data favor a very blue-tilted iso-curvature spectrum. The coefficient of the correlation islimited as 0:113< cos< 0:473 at 95% C.L. We com-pare the fitting results between the two cases: the pureadiabatic model and the mixed model based on the mini-mal 2 analysis. We find that the minimal 2 is reduced by3.5 for the mixed model with three additional free parame-ters. We also find that the fraction of CDM isocurvaturemodes is tightly limited by the data, which cannot be largerthan 14.9% at 95% confidence level. The constrainingpower mainly comes from WMAP7 temperature and po-larization power spectra, since a too-large fraction of theCDM isocurvature mode will lead to the overstated modi-fication of the shape of acoustic peaks in TT and TE powerspectra, as illustrated in Fig. 1. Our results are consistentwith the previous results reported in Ref. [44].

The constraints on the other cosmological parametersare also affected by different initial conditions. We findthat, with the mixed initial conditions, the results favor a

slightly smaller !m and larger , and this is due to thecorrelation of the CDM isocurvature perturbations withother cosmological parameters when considering the non-adiabatic initial condition in our analysis.The correlations among the isocurvature mode parame-

ters and those of adiabatic modes inevitably existed. InFig. 3 we plot a two-dimensional cross correlation of theparameters, such as nisos Aiso

s , nisos nadis , Aisos Aadi

s , andso on. We find a strong correlation between nisos Aiso

s .Also, the coefficient cos is correlated withm, because apositive correlated adiabatic and isocurvature componentraise the peaks of the TT spectrum, which can be compen-sated by lower m.

B. Information on reduced distance parameters

The WMAP normalization priors given by WMAPgroup include the shift parameter R, the acoustic scale lA,and the photon decoupling epoch z. R and lA correspondto the ratio of angular diameter distance to the decouplingera over Hubble horizon and sound horizon at decoupling,respectively, given by

RðzÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffimH

20

qrðzÞ; (12)

lAðzÞ ¼ ðzÞ=rsðzÞ; (13)

TABLE II. The mean values and 1 errors of the parameters related to the mixed initial adiabatic and CDM isocurvatureperturbations.

log½1010Aadis nadis log½1010Aiso

s nisos cos 2min

Adiabatic 3:204 0:036 0:964 0:010 – – – 8217.0

Mixed 3:165 0:044 0:972 0:014 1:375þ1:2331:306 2:246þ0:494

0:428 0:094þ0:0750:095 8213.5

3.1 3.2 3.3

log[1010 Asadi]

−4 −2 0

log[1010 Asiso]

−1 0 1cos∆

0.94 0.96 0.98 1

nsadi

1 2 3 n

siso

0 0.5 1α

FIG. 2 (color online). The one-dimensional probability distributions of initial conditions associated parameters are plotted with redsolid lines. For log½1010Aadi

s and nadis , we specially make a comparison with the results obtained from the pure adiabatic initialconditions which are plotted with black dashed lines. We also plot the adiabatic perturbation modes fraction parameter 1 and 2 usingthe purple dash-dotted line and blue solid line, respectively.

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where rðzÞ and rsðzÞ denote the comoving distance to zand comoving sound horizon at z, respectively. Thedecoupling epoch z is given by [55]

z ¼ 1048½1þ 0:00124ðbh2Þ0:738½1þ g1ðmh

2Þg2;(14)

where

g1 ¼ 0:0783ðbh2Þ0:238

1þ 39:5ðbh2Þ0:763 ;

g2 ¼ 0:560

1þ 21:1ðbh2Þ1:81 :

(15)

The WMAP normalization priors encode in part of theCMB information and can be used to constrain cosmologi-cal parameters to some extent. They are derived parametersfrom the CMB power spectra based on the fiducial cosmo-logical model. Thus, they are model-dependent [47,56].By using the WMAP7 data, in Fig. 4 we show theone-dimensional probability distributions of the WMAPnormalization priors in two different initial perturbationconditions. The red solid lines are given by pure adiabaticinitial perturbations, while the black dash-dotted lines arethe mixed case. We find the obvious difference in theprobability distributions of R, lA, and z. As we know,the WMAP normalization priors mainly include the infor-mation on the oscillatory structures of the CMB powerspectrum. Based on the constraints on isocurvature modeswe get above, the current CMB data favor a very bluepower index and consequently this blue-tilted isocurvaturespectrum can significantly modify the locations and theheight of the peaks of the CMB TT power spectrum, whichis shown in Fig. 1. Thus, the information of WMAPnormalization priors can be different for the cases of thepure adiabatic model and the mixed model1

In Table III, we list 1 constraints on WMAP normal-ization priors from the full WMAP7 data for differentinitial conditions. One can see that, by taking into accountthe contribution from the initial CDM isocurvature modes,

cos∆A

adi

−1 0 1

3.1

3.2

3.3

Aadi

Ais

o

3.1 3.2 3.3

−4

−2

0

Aiso

n iso

−4 −2 0

1

2

3

cos∆

n iso

−1 0 1

1

2

3

ns

n iso

0.94 0.96 0.98 1

1

2

3

cos∆

Ωm

−1 0 10.22

0.24

0.26

0.28

0.3

0.32

FIG. 3 (color online). Two-dimensional contours among some cosmological parameters.

142 144 146 148 150 152rs

295 300 305lA

1.6 1.65 1.7 1.75 1.8R

1086 1088 1090 1092 1094z∗

FIG. 4 (color online). One-dimensional posterior distributionsof lA, R, z, and rs with WMAP7 data. The red solid lines aregiven by the mixed initial perturbations, while the black dashedlines are from pure adiabatic perturbations, and the blue dash-dotted lines are given by fitting with the data combination ofCMBþ LSSþ SN Ia.

1For the scale-invariant isocurvature spectrum, which onlyenhances the CMB TT power spectrum at large scale, theinfluence on the final results of WMAP normalization priorsare modest.

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the constraints of the WMAP normalization priors areobviously changed. The mean value of R is changed about1 when comparing with the pure adiabatic case, while forlA and z, the changes are about 1 and 1:4, respectively.

Furthermore, we also provide the 1 constraints on R,lA, z, and rs using the combined data sample of WMAP7,ACBAR, CBI, Boomerang, as well as SDSS, LRG, andsupernovae (SN) type Ia in Table III. As we expect, com-bining the data of SN and LSS can help in constraining theother background cosmological parameters and signifi-cantly shrink error bars of R, lA, z, and rs. When consid-ering the CDM isocurvature mode perturbation, we find thedifferences between the constraints from pure adiabaticperturbation and mixture of adiabatic and isocurvatureperturbations are clearly improved. In this case, the differ-ence of the mean values of R, lA, z, and rs become 2:9,2:8, 1:5, and 0:7, respectively.

V. SUMMARY

In this paper, we study the constraints onmixed adiabaticand isocurvature modes of cosmological perturbations.Using the current observational data, such as WMAP7CMB power spectrum, matter power spectrum of SDSSdata, release seven LRGdata, and the SN IaUnion2 sample,we find that an adiabatic initial condition with the presenceof certain isocurvature modes can explain the experimentsbetter than a pure adiabatic initial condition. Moreover, weobtained more stringent constraints on the parameters ofcosmological perturbations by virtue of the improvement ofaccuracy of observational data in recent years. Our resultshows that the spectral index of isocurvature perturbationhas a very blue tilt. This quantity may lead to either en-hancement or depression of the power spectrum on smallscales, which depends on the correlation angle of the iso-curvature and adiabatic modes.

Given that WMAP normalization priors are widely usedin performing testing of cosmological models, we providethe comparison on constraints of R, lA, and z from differ-ent initial conditions. Since WMAP normalization priorsare derived parameters from the CMB power spectra, theconstraints are changed when the isocurvature mode modi-fies the shape of the peaks and troughs in the CMB powerspectra.As an end, we would like to mention that, from the point

of view of information criteria, detailed constraints oncosmological parameters depend on the set of parametersof the model which is compared with observational data[57]. The simplest model with a purely adiabatic scale-invariant primordial power spectrum is able to capture themost relevant clues of early-universe physics, however,more information deserves to be explored along with theimprovement of observational data from forthcoming ex-periments; we expect the global analysis on both CMBpower spectrum and matter power spectrum on smallscales will be crucial to constrain the isocurvatureperturbation.

ACKNOWLEDGMENTS

It is a pleasure to thank Andrew R. Liddle for comments.The work of H. L. and J. L. is supported in part by theKnowledge Innovation Program of Chinese Academy ofScience, the National Natural Science Foundation of Chinaunder Grants No. 11033005 and 10803001, by the 973Program No. 2010CB833000, and by the YouthFoundation of the Institute of High Energy Physics underGrant No. H95461N. Y. F. C. thanks the Department ofPhysics of McGill University for the hospitality whenthis work was finalized. The research of Y. F. C. is sup-ported in part by the Arizona State University CosmologyInitiative.

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