Bumpy Power Spectra and dT/T

5 Pages·0.3 MB·English

by  Louise M. Griffiths

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