Preprintastro-ph Bumpy Power Spectra and ∆T/T Louise M. Griffiths1, Joseph Silk1 and Saleem Zaroubi2 1Astrophysics, Nuclear and Astrophysics Laboratory, KebleRoad, Oxford OX13RH, United Kingdom. 2Max Planck Institute for Astrophysics, Karl Schwarzschild Str. 1, 85748 Garching, Germany. 1February2008 1 0 ABSTRACT 0 Withtherecentpublicationofthe measurementsofthe radiationangularpowerspec- 2 trum from the BOOMERanG Antarctic flight (de Bernardis et al. 2000), it has be- n comeapparentthatthecurrentlyfavouredspatially-flatcolddarkmattermodel(mat- a ter density parameter Ωm = 0.3, flatness being restored by a cosmological constant J ΩΛ = 0.7, Hubble parameter h = 0.65, baryon density parameter Ωbh2 = 0.02) no 8 longer provides a good fit to the data. We describe a phenomenological approach to 1 resurrecting this paradigm. We consider a primordial power spectrum which incorpo- 2 rates a bump, arbitrarily placed at kb, and characterized by a Gaussian in log k of v standard deviation σb and amplitude Ab, that is superimposed onto a scale-invariant power spectrum. We generate a range of theoretical models that include a bump at 1 7 scales consistent with cosmic microwave background and large-scale structure obser- 5 vations, and perform a simple χ2 test to compare our models with the COBE DMR 0 data and the recently published BOOMERanG and MAXIMA data. Unlike models 1 that include a high baryoncontent, our models predict a low third acoustic peak. We 0 find that low ℓ observations (20 < ℓ < 200) are a critical discriminant of the bumps 0 becausethe transferfunctionhasa sharpcutoffonthe highℓ sideofthe firstacoustic / h peak. Current galaxy redshift survey data suggests that excess power is required at p a scale around 100 Mpc, corresponding to kb ∼ 0.05 h Mpc−1. For the concordance - model, use of a bump-like feature to accountfor this excess is not consistentwith the o constraints made from recentCMB data. We note that models with an appropriately r t chosen break in the power spectrum provide an alternative model that can give dis- s tortions similar to those reported in the APM survey as well as consistency with the a : CMB data (Atrio-Barandela et al. 2000; Barriga et al. 2000). We prefer however to v discounttheAPMdatainfavourofthelessbiaseddecorrelatedlinearpowerspectrum i X recently constructed from the PSCz redshift survey (Hamilton & Tegmark 2000). We show that the concordancecosmology can be resurrectedusing our phenomenological r a approach and our best-fitting model is in agreement with the PSCz observations. Key words: cosmology:theory — cosmic microwave background 1 INTRODUCTION erablespeculationastotheadditionalfreedomthatcouldbe added to the concordance cold dark matter (CDM) model The recent BOOMERanG and MAXIMA measurements of (matterdensityparameterΩm =0.3,flatnessbeingrestored an acoustic peak in theangular power spectrum of thecos- by a cosmological constant ΩΛ = 0.7, Hubble parameter mic microwave background (CMB) temperature at l ≈ 200 h = 0.65, baryon density parameter Ωbh2 = 0.02) to ac- (deBernardis et al. 2000; Hanany et al. 2000) has provided commodate such an effect. Ideas that have been proposed remarkable confirmation that the growth via gravitational include enhancement of the baryon fraction (Lange et al. instabilityofprimordialadiabaticdensityfluctuationsseeds 2000; White et al. 2000), a large neutrino asymmetry (Les- large-scalestructure.Oneconsequenceofthelocationofthis gourgues&Peloso2000),delayofrecombination(Hu&Pee- peak, due to the compression of an acoustic wave on first bles 2000), an admixture of a component of cosmological entering the horizon of last scattering, is that the spatial defects (Bouchet et al. 2000) and models employing double geometry of theuniverseis flat. inflation in supergravity (Kanazawa et al. 2000). However, the weakness of the second acoustic peak at l≈400,duetothesubsequentfirstrarefactionoftheacous- Here we suggest a more phenomenological solution, ticwaveonthelastscatteringhorizon,hasprovokedconsid- which is motivated by suggestive, although not overwhelm- (cid:13)c 0000RAS 2 Louise M. Griffiths, Joseph Silk and Saleem Zaroubi ing,evidencefromgalaxysurveysthatthereisexcesspower relative to the scale-invariant (n≈1) fluctuation spectrum Table 1. Thedatausedinthisstudy. of the conventional model near 100 h−1 Mpc. The case for excess power has not hitherto been completely convincing Experiment ℓeff δTℓdeaffta±σdata(µK2) because one is probing the limit of current surveys. Nev- ertheless, several independent data sets have provided such COBE 2.1 72.25−+57228.5.0 indications (See e.g. Broadhurst et al. 1990; Einasto et al. COBE 3.1 784.0−+447760..2751 1997; Landy et al. 1996). COBE 4.1 1156.0−+444347..076 In fact themultiple inflationary model of Adams, Ross COBE 5.6 630.01−+229847..1756 & Sarkar (1997) predicts the suppression of the second COBE 8 864.36−+222244..6247 acoustic peak through thegeneration of features in thepri- COBE 10.9 767.29−+223219..2075 mordial power spectrum from phase transitions that occur COBE 14.3 681.21−+224494..044 during inflation. More generally, there are strong theoret- COBE 19.4 1089.0−+332247..7264 BOOMERanG 50.5 1140±280 ical arguments which suggest that arbitrary features can BOOMERanG 100.5 3110±490 be dialled onto the primordial power spectrum predicted BOOMERanG 150.5 4160±540 by generic inflationary models (See e.g. Chung et al. 1999; BOOMERanG 200.5 4700±540 Garc´ıa-Bellido et al 1996; Lesgourges et al. 1998; Linde & BOOMERanG 250.5 4300±460 Mukhanov 1997; Martin et al. 1999; Randall et al 1996; BOOMERanG 300.5 2640±310 Starobinsky 1998). BOOMERanG 350.5 1550±220 We therefore consider a primordial power spectrum BOOMERanG 400.5 1310±220 which incorporates a phenomenological bump, arbitrarily BOOMERanG 450.5 1360±250 placed at kb, and characterized by a Gaussian in log k of BOOMERanG 500.5 1440±290 standard deviation σb and amplitude Ab, that is superim- BOOMERanG 550.5 1750±370 BOOMERanG 600.5 1540±430 posed onto a scale-invariant power spectrum as advocated by Silk & Gawiser (1998). We examine the constraint on MAXIMA 73 2000+−658100 the bump parameters for the ΛCDM concordance model MAXIMA 148 2960+−658500 0c(Oh.6os5itc,reiΩkobefrhb2&u=mS0pt.e0pi2na)rhapamrodseettde1rb9sy9t5ot)hteh(ΩeCmrMegB=iodn0a.ot3af,,pΩaanΛradm=reest0te.rr7ic,stphathc=ee MMMAAAXXXIIIMMMAAA 223297383 632072727000+−+−+−1963053029444000000 thatisconsistentwithobservationsoflarge-scalepowerand MAXIMA 448 1530+−321700 CMB anisotropies. MAXIMA 523 2340+−433800 MAXIMA 598 1530+−338400 MAXIMA 673 1830+−449400 2 THE THEORETICAL MODELS MAXIMA 748 2180+−760200 In our paper we consider one particular spatially-flat CDM model; the ΛCDM concordance model of Ostriker & Stein- inrelationtothefirstacousticpeak,itmaybethatadipin hardt (1995) (Ωm =0.3, ΩΛ =0.7, h=0.65, Ωbh2 =0.02). theprimordial power spectrum around the scale of the sec- WeassumeGaussianandadiabaticinitialconditionswitha ondpeakcouldalsoenabletheconcordancemodeltofitthe scale-invariant (n= 1) power law form as predicted by the data.Theoriesthatpredictabumpintheprimordialpower simplest inflationary models. The radiation angular power spectrumhavebeeninspiredbyhintsofsuchafeaturefrom spectrum is calculated using the cmbfast program (Seljak large-scale structure observations. We do not investigate a &Zaldarriaga 1996) oncethecodehasbeen modifiedtoin- dip in this paper because there is less theoretical motiva- corporate a bumpin theprimordial spectrum as advocated tion for this scenario. We attempt to increase the first to by Silk & Gawiser (1998). We model this bump as a Gaus- second peakratio withtheincorporation ofabumparound sian in log k with a central location in wavenumber kb, a the scale of the first peak, then renormalize the radiation standarddeviationσb,andanamplitudeAb,resultinginthe angular power spectrum to fit thedata. newprimordialpowerspectrumgivenbelow,whereP0(k)is thepower spectrum of the model without thefeature. 3 THE OBSERVATIONAL DATA P(k) = P0(k) (cid:18)1 + Ab exp(cid:18)−(log k 2−σl2og kb)2(cid:19)(cid:19) , (1) Ourdatasampleislisted inTable1.Itconsistsofthe8un- b correlated COBE DMR points from Tegmark & Hamilton We restrict the choice of bump parameters to the re- (1997), the 12 data points from the BOOMERanG Antarc- gion of parameter space that is consistent with large-scale ticflight(deBernardisetal.2000)andthe10recentlypub- structure and CMB observations. The parameters are var- lished MAXIMA data points(Hanany et al. 2000). ied as follows: 0.05 < σb < 2.0, 0.0 < Ab < 3.0, 0.001 < kbhMpc−1 < 0.140. Our focus is directed towards determining whether it 4 CONSTRAINING THE MODELS is possible to resurrect the concordance model without re- sortingtotheproposedideaslistedinourintroductionthat We use a simple χ2 goodness-of-fit analysis employing mayprovetocontradict observation.SincetheCMBobser- the data in Table 1 along with the corresponding win- vationsindicatethatthesecondacousticpeakissuppressed dow functions for the uncorrelated COBE DMR points (cid:13)c 0000RAS,MNRAS000,000–000 Bumpy Power Spectra and ∆T/T 3 appropriate distribution for the χ2 statistic has Ndata - 3 degreesoffreedom.Nothingfurtheristobesubtractedfrom this to allow for the number of parameters, as they are not beingvariedinthefit.Toassesswhetheramodelisagoodfit tothedata,weneedtheconfidencelevelsofthisdistribution. These are χ2 < 29.87 at the 68% confidence level, χ2 < 27 27 40.11 at the 95% confidence level and χ2 < 46.96 at the 27 99% confidence level. Models which fail these criteria are rejected at thegiven level. Although we are unable to give the overall best-fitting modelforcurrentlypermittedcosmologies, sincethiswould requirevarying each of thecosmological parameters as well asthosedescribingthebump,wefindthatfortheparadigm considered the best-fitting model is kb = 0.004hMpc−1, Ab = 0.9, σb = 1.05. This model has a χ2 of 22.0 which is in good agreement with expectations for a fit to 30 data pointswith6adjustableparameters(the3bumpparameters and the3 hidden parameters). Bymarginalizingoverthebumpparametersweareable Figure 1. The observational data set of Table 1. The crosses todeterminethe68%confidencelevellimitsoneachparam- indicate the eight uncorrelated COBE DMR points (Tegmark eter.Wefindthatthelowerlimitonkb extendsrighttothe &Hamilton1997),thecirclesindicatethetwelveBOOMERanG edge of the region of parameter space that we are investi- data points (de Bernardis et al.2000) and the triangles indicate gating.Wedonotfeelitnecessarytopushthislimitfurther the ten MAXIMA data points (Hanany et al. 2000). The solid curve shows the best-fitting model (kb = 0.004hMpc−1, σb = sinceitextendsintotheregionofgreatestobservationalun- 1.05, Ab = 0.9) normalized to the full observational data set. certainty due to cosmic variance. The upper limits in both The remaining curves show the same model with varying kb as σb andAb alsoreachtheedgesofourparameterspaceindi- indicated.Allmodelsarenormalizedtothebest-fittingmodel. cating that the CMB data allows a lot of freedom with the amplitude and standard deviation of a bump. We find that at the 68% confidence level kb ≤ 0.014hMpc−1, σb ≥ 0.15 (Tegmark & Hamilton 1997) and assuming a top hat win- and Ab ≥0.3. dow function over the BOOMERanG and MAXIMA bins. In Figure 1 we show the best-fittingmodel as well as a The window functions describe how the anisotropies at dif- rangeofkb modelswiththesameAb,σb andnormalization ferentℓcontributetotheobservedtemperatureanisotropies toillustratetheeffectofvaryingkb.InFigure2weplotthe (Lineweaveretal.1997).Foragiventheoreticalmodel,they best-fitting model together with models of varying σb and enableustoderiveaprediction for theδT that each exper- Ab. From these figures it can be seen that, unlike models iment would see, to be compared with the observations in incorporating a high baryon content, our model predicts a Table 1. lowthirdacousticpeak.Alsothesefigureshighlightthatlow It has been noted that the use of the χ2 test can give ℓ observations (20 <ℓ< 200) are a critical discriminant of a bias in parameter estimation in favour of permitting a thebumpsbecausebeyondthefirstacousticpeak themod- lower power spectrum amplitude because in reality there is els become less distinguishable. This because the transfer atailtohightemperaturefluctuations.Othermethodshave functionhasamuchsharpercutoffonthehighℓsideofthe been proposed (Bond, Jaffe & Knox 1998; Bartlett et al. first peak, relative to thecutoff on thelow ℓ side. 1999)whichgivegoodapproximationstothetruelikelihood, A confidencelevelcontour map of kb versusAb for the though they require extra information on each experiment cosmology of interest with σb fixed at 1.05 is shown in Fig- whichisnotyetreadilyavailable.Wedonotusethesemore ure3.Thisindicatesthat,forthechosencosmology,thescale sophisticated techniqueshere. at which a bump can appear in the primordial power spec- ThereareNdata =30datapoints.Ratherthanadopting trum is quiteconstrained by the CMB data sample. At the the COBE normalization, the theoretical models are nor- 68% confidencelevel weare limited models with a bumpat malized tothefullobservational dataset resultinginahid- scales kb ≤0.010hMpc−1 for thisvalueofσb,although we denparameter.WeusethemethodofLineweaver&Barbosa haverather more freedom with theamplitude of thebump. (1998) totreat thecorrelated calibration uncertaintyof the 12BOOMERanGdatapointsandthatofthe10MAXIMA data points as free parameters with Gaussian distributions 5 FROM CMB TO GALAXY SURVEYS about their nominal values of 10% for BOOMERanG and 4%forMAXIMA.Thisresultsintwofurtherhiddenparam- Sinceourincorporationofabumpintotheprimordialpower eters. We do not account for the 10% correlation between spectrum of density perturbations was inspired by large- theBOOMERanGbinsnorthatbetweentheMAXIMAbins scalestructureobservations,it isinterestingtoask howour whichwouldfurtherreducethedegreesoffreedom.Account- modifiedmodelcompareswiththedecorrelatedlinearpower ingforthecorrelations wouldprovidetighterconstraintson spectrum that was recently generated from the PSCz cata- themodels, so the constraints we make are conservative. logue (Hamilton & Tegmark 2000). We treat the 22 decor- Because we are measuring absolute goodness-of-fit on related PSCz data points as uncorrelated so that the the- a model-by-model basis, with three hidden parameters, the oretical model that we find to be the best-fit to the CMB (cid:13)c 0000RAS,MNRAS000,000–000 4 Louise M. Griffiths, Joseph Silk and Saleem Zaroubi Figure2. ThesamedatasampleasinFigure1.Thesolidcurve Figure4. ThePSCzdecorrelatedlinearprimordialpowerspec- shows the best-fitting model (kb = 0.004hMpc−1, σb = 1.05, trum (Hamilton & Tegmark 2000). The solid curve shows the Ab = 0.9) normalized to the full observational data set. The standard spatially-flatcosmological model with abump at kb = dotted curve shows the same model with σb = 2.0, the dotted– 0.004hMpc−1,normalizedtotheCMBdatasamplewithabias dashedcurvethesamemodelwithσb=0.5andthedashedcurves factorof1.07.Thedottedcurveshowsthestandardmodelwith- thesamemodelwithAb=0.5(smalldashes)and1.5.Allmodels out the bump, normalized to the CMB data sample with a bias arenormalizedtothebest-fittingmodel. factorof1.16. Figure 4 plots our best-fitting CMB normalized stan- dard low-density cosmological model with and without the bumpagainstthePSCzdecorrelatedlinearpowerspectrum. Our best-fitting model has been renormalized to the PSCz data set with a bias parameter of 1.07 and themodel with- out the bump takes a bias parameter of 1.16. Both models are a very good fit to these up-to-datelarge-scale structure observations (χ2best−fit = 17.20, χ2nobump = 16.35, χ221 < 23.46 at the 68% confidence level), but it is clear that the current data does not probe the scales that are critical to discriminating between the models. As stated in our introduction, several independent large-scale structure data sets provide suggestive, although not overwhelming, evidence that there is excess power rel- ative to the scale-invariant (n ≈ 1) fluctuation spectrum of the conventional model near 100 h−1 Mpc, correspond- ing to kb ∼ 0.05h Mpc−1 (See e.g. Broadhurst et al. 1990; Einasto et al. 1997; Landy et al. 1996). Figure 5 shows our best-fittingbumpmodeltogetherwiththestandardΛCDM Figure3. ConfidencelevelcontoursfortheconcordanceΛCDM model without a bump and the same model with a bump model as a function of kb and Ab, σb fixed at 1.05. The region at kb =0.05h Mpc−1, each model independently taking its withinthe68%contourlineisallowedatthe68%confidencelevel. optimal normalization to thefull CMB data set. It is inter- esting to note that a bump in the primordial power spec- observationaldatasetcanbecomparedwiththegalaxysur- trumofanyamplitudeorstandarddeviationatkb =0.05h vey observations using a χ2 test. The theoretical model is Mpc−1 isruledoutatthe95%confidencelevelbytheCMB renormalizedtotheobservationaldatasetresultinginahid- observational datafor theΛCDM concordance model. den parameter. This allows for a bias factor b where P(k)PSCz=b2P(k)CMB, (2) 6 SUMMARY Wenotethatnon-linearitycorrectionstothedataareomit- tedinourcomparison.InΛCDMmodelstheeffectsofnon- We describe a toy model for a bump to be included in the linearity in the matter power spectrum are partially can- primordialdensitypowerspectrumwiththehopeofreviving celled by galaxy-to-mass antibias, so that the PSCz power thestandardmodelwithoutresortingtorevisingfundamen- spectrum is close to linear all the way to k = 0.3h Mpc−1 tal cosmological theories. We have confronted our theoret- [Hamilton 2000]. ical models with the recent BOOMERanG and MAXIMA (cid:13)c 0000RAS,MNRAS000,000–000 Bumpy Power Spectra and ∆T/T 5 and SDSS should be able to probe thelarge-scale structure power spectrum at the depths required to further test our conjecture.Large-scale velocity field dataareusefulonly at higherkasadiscriminantofbump-likefeatures,andwewill address this issue in a later paper. ACKNOWLEDGEMENTS LMGwouldliketothankPedroFerreiraandAndrewLiddle for providing productive insights into the analysis and un- derstanding of this problem and Alessandro Melchiorri and Bill Ballinger for useful discussions. We acknowledge use of the Starlink computer system at the University of Oxford anduseofthecmbfastcodeofSeljak&Zaldarriaga(1996). REFERENCES AdamsJ.A.,RossG.G.,SarkarS.,1997,Nucl.Phys.B503,405 Figure5. ThesamedatasampleasinFigure1.Thesolidcurve shows the best-fitting model (kb = 0.004hMpc−1, σb = 1.05, Atrio-Barandela F., Einasto J., Mu¨ller V., Mu¨cket J. P., A. A. StarobinskyA.A.,2000,astro-ph/0012320 Ab = 0.9), the dotted curve shows the standard ΛCDM model Barriga J., Gaztanaga E., Santos M. G., Sarkar S., 2000, astro- without a bump and the dashed curve shows the same model withabumpatkb=0.052hMpc−1,σb=1.05,Ab=0.9.Each ph/0011398 BartlettJ.G.etal.,1999,astro-ph/9903045 modelisindependentlynormalizedtothefullCMBobservational deBernardisP.etal.,2000,Nature,404,955 dataset. BondJ.R.,JaffeA.H.,KnoxL.,1998,astro-ph/9808264 Bouchet F.R., Peter P., Riazuelo A., Sakellariadou M., 2000, astro-ph/0005022 data and have shown that it is indeed possible to resurrect BroadhurstT.J.etal.,1990,Nature,343,726 the standard model, although we are somewhat restricted Chung D.J.H., Kolb E.W., Riotto A., Tkachev, I.I., 1999, hep- with where we place our additional feature. We find that ph/9910437 our model, unlike models that include a high baryon con- EinastoJ.etal.,1997Nature,343,726 tent,predicts a low third acoustic peak. Garc´ıa-Bellido J., Linde A., Wands D., 1996, Phys. Rev. D, 54, There are two CMB measurements that will help to 6040 discriminate between such a bump and cosmological alter- HamiltonA.,2000,privatecommunication HamiltonA.,TegmarkM.,2000, astro-ph/0008392 natives for suppressing the second peak. One is of course HananyS.etal.,2000,astro-ph/0005123 thedetection of thethird peak. In addition, although low ℓ HuW.,Peebles P.J.E.,2000,astro-ph/0004389 (20-100)observationshavereceivedrelativelylittleattention Kanazawa T., Kawasaki M., Sugiyama N., Yanagida T., 2000, and hence are currently a poor constraint on cosmological astro-ph/0006445 parameters, we have found that the low ℓ power is poten- LangeA.E.etal.,2000,astro-ph/0005004 tially a critical discriminant for the possible bump feature. LandyS.D.etal.,1996,ApJ,456,L1-L4 Thisbecausethetransferfunctionhasamuchsharpercutoff LesgourguesJ.,PelosoM.,2000,astro-ph/0004412 on thehigh ℓ sideof thefirst peak,relative tothecutoffon Lesgourgues J., Polarski D., Starobinsky A.A., 1998, MNRAS, thelow ℓ side. 297,769 Current galaxy redshift survey data suggests that ex- LindeA.,MukhanovV.,1997,Phys.Rev.D,56,R535 LineweaverC.H.,BarbosaD.,1998,A&A,329,799 cess power is required at a scale around 100 Mpc, corre- sponding to kb ∼ 0.05 h Mpc−1 (See e.g. Broadhurst et LMinaretwineaJv.e,rRCia.zHu.eelotAal..,,S1a9k9e7l,laAri&adAo,u3M22.,,3169599,astro-ph/9904167 al. 1990; Einasto et al. 1997; Landy et al. 1996). For the OstrikerJ.P.,SteinhardtP.J.,1995,Nature,377,600 concordance paradigm, use of a bump-like feature to ac- RandallL.,Soljaˇci´cM.,GuthA.H.,1996,Nucl.Phys.B472,249 count for this excess is not consistent with the constraints SeljakU.,ZaldarriagaM.,1996,ApJ,469,437 from the CMB data. We note that models with an appro- Silk J., Gawiser E., 1999, in COSMO-98: Particle Physics and priately chosen break in thepower spectrum provide an al- the Early Universe, ed. D. O. Caldwell (American Institute ternative model that can give distortions similar to those ofPhysics,NewYork,)148-156 reportedintheAPMsurveyaswellasconsistency withthe StarobinskyA.A.,1998, Grav.Cosmol.,4,88 CMBdata(Atrio-Barandelaetal.2000;Barrigaetal.2000). TegmarkM.,HamiltonA.,1997, astro-ph/9702019 WhiteM.,ScottD.,PierpaoliE.,2000,astro-ph/0004385 We prefer however to discount the APM data in favour of the less biased decorrelated linear power spectrum recently ThispaperhasbeenproducedusingtheRoyalAstronomical constructed from the PSCz redshift survey (Hamilton & Tegmark 2000). Society/Blackwell ScienceLATEX style file. The incorporation of a bump in the primordial spec- trum at a scale of kb = 0.004h Mpc−1, as the CMB data prefers,isingoodagreementwiththePSCzpowerspectrum with a bias parameter of 1.07. Future surveys such as 2DF (cid:13)c 0000RAS,MNRAS000,000–000