Inflation predicts primordial scalar perturbations with a nearly
scale-invariant spectrum and a spectral index approximately unity (the
Harrison--Zel'dovich (HZ) spectrum). The first important step for inflationary
cosmology is to check the consistency of the HZ primordial spectrum with
current observations. Recent analyses have claimed that a HZ primordial
spectrum is excluded at more than 99% c.l.. Here we show that the HZ spectrum
is only marginally disfavored if one considers a more general reionization
scenario. Data from the Planck mission will settle the issue.Comment: 4 Pages, 2 Figure

A. A. Starobinsky

Alessandro Melchiorri

Asantha Cooray

Edward W. Kolb

Elena Giusarma

Olga Mena

Paolo Serra

Stefania Pandolfi

Y. B. Zel’dovich

English

arXiv

Inflation predicts primordial scalar perturbations with a nearly scale-invariant spectrum and a spectral index approximately unity (the Harrison--Zel'dovich (HZ) spectrum). The first important step for inflationary cosmology is to check the consistency of the HZ primordial spectrum with current observations. Recent analyses have claimed that a HZ primordial spectrum is excluded at more than 99% c.l.. Here we show that the HZ spectrum is only marginally disfavored if one considers a more general reionization scenario. Data from the Planck mission will settle the issue

Pandolfi, Stefania

Giusarma, Elena

Kolb, Edward W.

Lattanzi, Massimiliano

Melchiorri, Alessandro

Mena Requejo, Olga

Cooray, Asantha

Serra, Paolo

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:1003.4763v2  [astro-ph.CO]  26 Mar 2010Harrison-Zel’dovich primordial spectrum is consistent with observationsStefania Pandolfi,1 Asantha Cooray,2 Elena Giusarma,3Edward W. Kolb,4 Alessandro Melchiorri,5 Olga Mena3, and Paolo Serra21ICRA and INFN, Universita’ di Roma “La Sapienza”, Ple. Aldo Moro 2, 00185, Rome, Italy2Center for Cosmology, Department of Physics & Astronomy,University of California, Irvine, California 92697, USA3IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain4Department of Astronomy & Astrophysics, Enrico Fermi Institute,and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA5Physics Department and INFN, Universita’ di Roma “La Sapienza”, Ple. Aldo Moro 2, 00185, Rome, ItalyInflation predicts primordial scalar perturbations with a nearly scale-invariant spectrum and aspectral index approximately unity (the Harrison–Zel’dovich (HZ) spectrum). The first importantstep for inflationary cosmology is to check the consistency of the HZ primordial spectrum withcurrent observations. Recent analyses have claimed that a HZ primordial spectrum is excluded atmore than 99% c.l. Here we show that the HZ spectrum is only marginally disfavored if one considersa more general reionization scenario. Data from the Planck mission will settle the issue.PACS numbers: 98.80.-k 95.85.Sz, 98.70.Vc, 98.80.CqOne usually models the dynamics of primordial infla-tion by the dynamics of a scalar field, know as the in-flaton, evolving under the influence of a scalar potential.The scalar and tensor perturbation spectra produced byinflation depend upon the value of the inflaton potentialduring inflation and how rapidly the scalar field evolvesduring inflation. Observational information of the spec-tra can be translated into information about the inflatonpotential, thus about physics far beyond the scales of thestandard model of particle physics [1].Cosmic microwave background (CMB) observablesthat map onto the the potential and its slope are 1) n,the spectral index of comoving scalar perturbations PR:n−1 = d lnPR/d lnk, and 2) r, the ratio of tensor (grav-itational wave) perturbations to PR [2].CMB temperature and polarization anisotropies arethe most promising route to explore the physics behindinflation. Recent measurements by the Wilkinson Mi-crowave Anisotropy Probe (WMAP) satellite seven-yearmission [3], combined with the ground-based and balloon-borne experiments such as BOOMERanG [4], QUAD [5],ACBAR [6], and BICEP [7] have sharpened our knowl-edge of some key inflationary parameters.With regard to the dynamics of inflation, a hotly de-bated question is whether the case of n = 1 is significantlyexcluded by current observations (see e.g., Refs. [3]-[13]).The scalar spectral index, n, has been recently con-strained to the value n = 0.963 ± 0.014 at 68% c.l. bythe WMAP seven-year dataset (WMAP7) [3], disfavor-ing the value of n = 1 at about two standard deviations.A combined analysis with galaxy clustering data givesn = 0.963± 0.012 at 68% c.l., ruling out n = 1 at morethan three standard deviations ([3], [13]).Compelling evidence that n 6= 1 would be quite reveal-ing for two reasons. First, it would provide an indicationfor the dynamical evolution of the expansion rate of theuniverse as perturbations are being produced. Secondly,in many models of slow-roll inflation the amplitude of thetensor perturbations are proportional to |n−1|. Thus, thelarger the departure of n from unity, the more likely ten-sor perturbations would be within observational reach.As it is well known, a scalar spectral index with anexact value of n = 1 corresponds to the phenomenologi-cal model proposed by Harrison, Zel’dovich, and Peebles[14].1 While one can construct inflationary models thatgive n = 1 either approximately [15] or exactly [16], theyare less than compelling. In fact, observations pointingto n exactly unity may indicate that the origin of cosmicperturbations lies in some unknown fundamental processand may not arise from inflation.In this paper we show that the value of n, and its uncer-tainty, derived from CMB datasets are sensitive to howone models the reionization of the universe. In particu-lar, the statement that n = 1 is observationally excludedis very much weakened if one treats reionization in a gen-eral manner.It is well known from a large set of astrophysical ob-servables that after primordial recombination (which oc-curred at a redshift of z ∼ 1100) the universe “reionized”at a redshift z > 6. Current constraints from CMB dataassume a sudden and complete reionization at a redshiftzr. This parameter is usually assumed to be in the range4 < zr < 32, and the constraints on n are obtained aftermarginalization over zr. The electron ionization frac-tion xe(z) is parametrized by zr such that for z  zrxe(z) = 1 (xe(z) = 1.08 for z < 3 in order to take intoaccount Helium recombination) and xe(z) = 2×10−4 for1 For the purpose of our paper, we will assume that the tensormodes are small enough as to be irrelevant.2z > zr, i.e., joining the value after primordial recom-bination with a smooth interpolation. We refer to thiscommon procedure in what follows as “sudden” reioniza-tion.However, the precise details of the reionization processare not very well known, and therefore the reionizationhistory of the universe at those redshifts could have easilybeen very different.As already remarked in Refs. [17] and [18], the assump-tion of a more general reionization scheme could affect thecosmological constraints on n. It is therefore timely, es-pecially in view of the recent CMB and galaxy clusteringdata that have improved the bound on n, to investigatethe impact of the reionization history on the current con-straints on n in a more general reionization scenario.We adopt two methods for parametrizing the reioniza-tion history. The first method, developed in Ref. [17], isbased on principal components that provide a completebasis for describing the effects of reionization on large-scale E-mode polarization. Following Ref. [17], one canparametrize the reionization history as a free function ofredshift by decomposing xe(z) into its principal compo-nents:xe(z) = xfe (z) +∑µmµSµ(z), (1)where the principal components, Sµ(z), are the eigen-functions of the Fisher matrix that describes the depen-dence of the polarization spectra on xe(z) (again, seeRef. [17]), mµ are the amplitudes of the principal com-ponents for a particular reionization history, and xfe (z)is the WMAP fiducial model at which the Fisher matrixis computed and from which the principal componentsare obtained. In what follows we use the publicly avail-able Sµ(z) functions and varied the amplitudes mµ forµ = 1, ..., 5 for the first five eigenfunctions. Hereafter werefer to this method as the MH (Mortonson-Hu) case.In the second approach, we use a different parametriza-tion, sampling the evolution of the ionization fractionxe as a function of the redshift z at 7 points (z =9, 12, 15, 18, 21, 24, 27), and interpolating the value ofxe(z) between them with a cubic spline. For 30 < zwe fix xe = 2 × 10−4 as the value of xe expected beforereionization (and after primordial recombination), whilexe = 1 for 3 < z < 6 and xe = 1.08 for z < 3 in order tobe in agreement with both Helium ionization and Gunn-Peterson test observations. This approach is very similarto the one used in Ref. [18], and we will refer to it as theLWB (Lewis-Weller-Battye) case.Exploiting two different and more general reionizationschemes will provide us with not only a consistency checkof the results, but also a broader set of reionization his-tories. Finally, we also consider here the standard reion-ization model, i.e., sudden reionization.We have then modified the Boltzmann CAMB code[19] incorporating the two generalized reionization sce-narios and extracted cosmological parameters from cur-rent data using a Monte Carlo Markov Chain (MCMC)analysis based on the publicly available MCMC packagecosmomc [20].We consider here a flat ΛCDM universe described bya set of cosmological parameters{ωb, ωc,Θs, n, log[1010As]}, (2)where ωb ≡ Ωbh2 and ωc ≡ Ωch2 are the physical baryonand cold dark matter densities relative to the critical den-sity, Θs is the ratio of the sound horizon to the angulardiameter distance at decoupling, As is the amplitude ofthe primordial spectrum, and n is the scalar spectral in-dex. In one case we will also consider the possibility of aredshift-dependent dark energy component evolving withequation of state w(z) = w0 + wzz/(1 + z).The extra parameters needed to describe the reioniza-tion are the five amplitudes of the eigenfunctions for theMH case, or the five amplitudes in the five bins for theLWB case, and one single common parameter, the opticaldepth, τ , for the sudden reionization case.Our basic data set is the seven–year WMAP data[3] (temperature and polarization) with the routinefor computing the likelihood supplied by the WMAPteam. Together with the WMAP data, we also aug-ment the WMAP7 data with the CMB datasets fromBOOMERanG [4], QUAD [5], ACBAR [6], and BICEP[7]. For all these experiments we marginalize over a pos-sible contamination from the Sunyaev-Zeldovich compo-nent, rescaling the WMAP template at the correspond-ing experimental frequencies. We therefore consider twocases: we first analyze the WMAP data alone, referringto it as to the “WMAP7” case, and we then includethe remaining CMB experiments (“CMB All”). Finallywe also consider, separately, the galaxy clustering resultsfrom LRG-7 of Ref. [21] and the baryonic acoustic oscil-lation (BAO) data from the same dataset, see Ref. [22].Our main results are reported in Table I, where welist the constraints on n for the different reionization sce-narios using different datasets. Notice that using theMH parametrization drastically alters the constraints onn. For the WMAP7 case alone, the HZ spectrum is notonly in agreement with the data, but is also close to thebest-fit value. When the remaining CMB experimentsare included, the best-fit value of n shifts to lower val-ues, but it is still consistent at better than 95% c.l. withn = 1. When information from galaxy clustering is in-cluded, a value of n = 1 is excluded at about the 95%c.l., i.e., in a much less conclusive way than in the sud-den case where n = 1 is excluded at more than 99.7% c.l.(see e.g., Ref. [13]).Adopting the LWB parameterization of reionizationhas very similar effects. The HZ spectrum is onlymarginally excluded at a 95% confidence level if the com-plete CMB dataset is considered. Notice that in thecase of the LWB parametrization, the results are closer3nτ0.92 0.94 0.96 0.98 1 1.02 1.04 1.060.060.070.080.090.10.110.12FIG. 1: Contours of the 68% and 95% c.l. constraints in the nvs τ plane for different datasets. The contours regions comefrom a generalized reionization scenario using (from right toleft) the MH approach from WMAP-7 data (Yellow), “CMBall” (Green), All CMB+BAO (Blue).to those obtained in the standard sudden reionizationscheme. Therefore, the LWB parameterization is lessgeneral and samples fewer reionization scenarios than theMH method.Dataset Ionization n (68% c.l.) 95% c.l.WMAP7 sudden 0.965 ± 0.014 n ≤ 0.993CMB All sudden 0.959 ± 0.013 n ≤ 0.984WMAP7 MH 0.993 ± 0.023 n ≤ 1.042CMB All MH 0.975 ± 0.017 n ≤ 1.011CMB All+LRG-7 MH 0.966 ± 0.014 n ≤ 0.994CMB All+BAO MH 0.965 ± 0.014 n ≤ 0.995CMB All+BAO MH+w(z) 0.985 ± 0.018 n ≤ 1.025WMAP7 LWB 0.977 ± 0.018 n ≤ 1.01CMB All LWB 0.963 ± 0.015 n ≤ 0.998TABLE I: Constraints and upper limits on n from severaldatasets in different recombination scenarios.Considering the WMAP7 data alone, the HZ spec-trum is at ∆(−2 ln(L)) = 0.72 with respect to the bestfit model in the case of a MH recombination scenario(∆(−2 ln(L)) = 2.2 when all CMB experiments are con-sidered), compared with ∆(−2 ln(L)) = 3.3 when recom-bination is sudden (∆(−2 ln(L)) = 5.8 in case of CMBAll).It is interesting to consider the constraints on the op-tical depth τ , derived by integrating xe(z) up to z = 32.Figure 1 shows the 68% and 95% c.l. constraints in the nvs. τ plane arising from different datasets and assum-ing the MH reionization parametrization. As we cansee the optical depth is always in the range 0.08-0.11,slightly higher than in the standard analysis but consis-tent with several physical reionization models. Noticealso that the HZ spectrum (n = 1) is perfectly com-patible with WMAP7. Including more datasets shrinksand shifts the contours towards lower values of n, butthe HZ case is always reasonably consistent at the 95%c.l.. Including BAO data rules out the HZ spectrumat about 2 standard deviations that is considerably lessstringent than in the standard case. Moreover, consid-ering a redshift-dependent equation of state weakens thegeometrical probes and HZ is again inside the 68% con-fidence level for the CMB+BAO case.It is important to investigate if a more general reioniza-tion scenario with the assumption of n = 1 could providea viable cosmology, i.e., if the value of cosmological pa-rameters such as the baryon density or the age of theuniverse are compatible with complementary cosmolog-ical information from Big Bang Nucleosythesis and ageconstraints. It has been shown, for example, in Ref. [13]that the assumption of an HZ spectrum in the standardanalysis with sudden reionization gives a baryon abun-dance that is at odds with current bounds from BBN.We have therefore performed an analysis using WMAP7and assuming n = 1 for the MH reionization scenario,and the results for the different cosmological parametersare summarized in Table 2.Parameter Constraint (68% c.l.)Ωbh2 0.0234 ± 0.0004Ωch2 0.106 ± 0.005H0 (km s−1Mpc−1) 74.2± 2.1Age (Gyr) 13.6± 0.1ΩΛ 0.765 ± 0.022TABLE II: Constraints on cosmological parameters fromWMAP7 assuming a HZ primordial spectrum and a reion-ization scenario characterized by the MH parametrization.Notice from the results of Table 2 that the constraintson the remaining cosmological parameters are all viablewhen compared with independent cosmological observ-ables. Current Big Bang Nucleosynthesis constraintsfrom observations of Deuterium abundances provide aconstraint of Ωbh2 = 0.0213 ± 0.002 at 95% c.l. [23],which is compatible with the WMAP7 constraint assum-ing HZ. The constraints shown in Table 2 on H0 are alsowell in agreement with the recent HST determinationh = 0.748 ± 0.036 at 68% c.l. [24]. Figure 2 depicts theconstraints from WMAP7 in the Ωbh2 vs. H0 plane ina generalized recombination scenario, assuming both avarying n and a fixed n = 1 scenarios. Notice that fixingn = 1 implies larger values of both Ωbh2 and H0.In conclusion, the details of the reionization processesin the late universe are not very well known. In the ab-4Ωb h2H0 [Km s−1  Mpc−1 ]0.020 0.022 0.024 0.0266570758085FIG. 2: 68% and 95% c.l. WMAP7 constraints in the H0 vs.Ωbh2 plane in a generalized reionization scenario described bythe MH parametrization. The larger (blue) contours regionsarise when n is allowed to vary. The smaller (red) contourscorrespond to the HZ (n = 1) spectrum.sence of a precise, full redshift evolution description ofthe ionization fraction during the reionization period, asimple parametrization, with a single parameter zr, hasbecome the standard reionization scheme in numericalanalyses. However, more general reionization scenariosare certainly plausible and their impact on the cosmo-logical constraints should be carefully explored. For ex-ample, processes of dark matter annihilation as in Ref.[25] could easily be present.In this paper we have investigated the stability of theCMB constraints on n in generalized reionization scenar-ios. Our study is motivated by the recent claim thatan HZ spectrum is ruled out by current WMAP data atmore than 95% c.l. In agreement with previous studies,we have found that a more general treatment of reion-ization drastically weakens current CMB constraints onn. If only the WMAP 7-year data is considered, a HZn = 1 spectrum lies well within the 68% c.l. allowed re-gion. If all current CMB datasets are considered in theanalysis, the HZ spectrum still lies well within the 95%c.l. allowed region. If the assumption of constant darkenergy is relaxed, HZ is again inside the 68% c.l. There-fore, current data do not yet rule out an HZ spectrum ina conclusive way. It is also worth mentioning that nonstandard processes during recombination (see e.g., Ref.[26]) or extra relativistic particles (see e.g., Ref. [27])could be present and could also put n = 1 in agreementwith current observations.Near future data from the Planck satellite mission areclearly needed to solve this important question. In thisrespect, performing a parameter analysis with simulateddata with the experimental configuration described inRef. [28], and assuming a model with n = 0.965, wefound that Planck will be able to discriminate it froma HZ spectrum at more than five standard deviationseven in the generalized reionization cases describes here.[1] J. E. Lidsey, A. R. Liddle, E. W. Kolb, E. J. Copeland, T.Barreiro and M. Abney, Rev. Mod. Phys. 69, 373 (1997).[2] S. Dodelson, W. H. Kinney and E. W. Kolb, Phys. Rev.D 56, 3207 (1997).[3] E. Komatsu et al., arXiv:1001.4538 [astro-ph.CO]; D.Larson et al., arXiv:1001.4635 [astro-ph.CO].[4] W. C. Jones et al., Astrophys. J. 647, 823 (2006); F. Pi-acentini et al., Astrophys. J. 647, 813 (2006), Astrophys.J. 647, 833 (2006).[5] M. L. Brown et al. [QUaD collaboration], Astrophys. J.705, 978 (2009).[6] C. L. Reichardt et al., Astrophys. J. 694 (2009) 1200.[7] H. C. Chiang et al., arXiv:0906.1181 [astro-ph.CO].[8] W. H. Kinney, E. W. Kolb, A. Melchiorri, and A. Riotto,Phys. Rev. D 74, 023502 (2006); W. H. Kinney, E. W.Kolb, A. Melchiorri, and A. Riotto, Phys. Rev. D 78(2008) 087302.[9] D. Parkinson, P. Mukherjee, and A. R. Liddle, Phys. Rev.D 73, 123523 (2006).[10] J. Hamann, S. Hannestad, M. S. Sloth, and Y. Y. Y.Wong, Phys. Rev. D 75, 023522 (2007).[11] M. Bridges, A. N. Lasenby, and M. P. Hobson,arXiv:astro-ph/0607404.[12] J. Martin and C. Ringeval, JCAP 0608, 009 (2006).[13] F. Finelli, J. Hamann, S. M. Leach, and J. Lesgourgues,arXiv:0912.0522 [astro-ph.CO].[14] E. R. Harrison, Phys. Rev. D 1, 2726 (1970); Y. B.Zel’dovich, Mon. Not. Roy. Astron. Soc. 160, (1972), P.J. E. Peebles and J. T. Yu Astrophys. J. 162, 815 (1970).[15] A. Vallinotto, E. J. Copeland, E. W. Kolb, A. R. Liddle,and D. A. Steer, Phys. Rev. D 69, 103519 (2004).[16] A. A. Starobinsky, JETP Lett. 82, 169 (2005) [Pisma Zh.Eksp. Teor. Fiz. 82, 187 (2005)].[17] M. J. Mortonson and W. Hu, Astrophys. J. 686, L53(2008).[18] A. Lewis, J. Weller, and R. Battye, Mon. Not. Roy. As-tron. Soc. 373, 561 (2006).[19] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J.538 (2000) 473.[20] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002).[21] B. A. Reid et al., arXiv:0907.1659 [astro-ph.CO].[22] W. J. Percival et al., arXiv:0907.1660 [astro-ph.CO].[23] M. Pettini, B. J. Zych, M. T. Murphy, A. Lewis, and C.C. Steidel, Mon. Not. Roy. Astron. Soc. 391, 1499 (2008).[24] A. G. Riess et al., Astrophys. J. Suppl. 183, 109 (2009).[25] S. Galli, F. Iocco, G. Bertone and A. Melchiorri, Phys.Rev. D 80 (2009) 023505.[26] S. Galli, R. Bean, A. Melchiorri and J. Silk, Phys. Rev.D 78 (2008) 063532.[27] R. Bowen, S. H. Hansen, A. Melchiorri, J. Silk and R.Trotta, Mon. Not. Roy. Astron. Soc. 334, 760 (2002);J. Hamann, S. Hannestad, J. Lesgourgues, C. Rampf andY. Y. Y. Wong, arXiv:1003.3999 [astro-ph.CO].[28] [Planck Collaboration], arXiv:astro-ph/0604069.

Harrison-Zel'dovich primordial spectrum is consistent with observations

Crossref

Harrison-Zel’dovich primordial spectrum is consistent with observations

Inflation predicts primordial scalar perturbations with a nearly scale-invariant spectrum and a spectral index approximately unity [the Harrison-Zel'dovich (HZ) spectrum]. The first important step for inflationary cosmology is to check the consistency of the HZ primordial spectrum with current observations. Recent analyses have claimed that a HZ primordial spectrum is excluded at more than 99% c. l. Here we show that the HZ spectrum is only marginally disfavored if one considers a more general reionization scenario. Data from the Planck mission will settle the issue

MELCHIORRI, Alessandro

PANDOLFI, STEFANIA

Archivio della ricerca- Università di Roma La Sapienza

Inflation predicts primordial scalar perturbations with a nearly scale-invariant spectrum and a spectral index approximately unity [the Harrison-Zel'dovich (HZ) spectrum]. The first important step for inflationary cosmology is to check the consistency of the HZ primordial spectrum with current observations. Recent analyses have claimed that a HZ primordial spectrum is excluded at more than 99% c. l. Here we show that the HZ spectrum is only marginally disfavored if one considers a more general reionization scenario. Data from the Planck mission will settle the issue.Peer reviewe

Harrison-Z'eldovich primordial spectrum is consistent with observations

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

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Archivio della ricerca- Università di Roma La Sapienza

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