Non-parametric reconstruction of the primordial power spectrum at horizon scales from WMAP data

0.37 MB·English

by  Domenico Tocchini-Valentini

Mon.Not.R.Astron.Soc.000,1–9(2005) Printed5February2008 (MNLATEXstylefilev2.2)
Non-parametric reconstruction of the primordial power
spectrum at horizon scales from WMAP data
Domenico Tocchini-Valentini12⋆, Yehuda Hoffman3† and Joseph Silk1‡
1Department of Physics, University of Oxford, DenysWilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom
6 2Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
0 3Racah Institute of Physics, HebrewUniversity, Jerusalem 91904, Israel
0
2

n Accepted ...Received...;inoriginalform...
a
J

7 ABSTRACT
1 We extend to large scalesa method proposedin previousworkthat reconstructs non-
parametrically the primordial power spectrum from cosmic microwave background
2
data at high resolution. The improvement is necessary to account for the non-
v
gaussianity of the Wilkinson Microwave Anisotropy Probe (WMAP) likelihood due
8
primarilyto cosmic variance.We assume the concordanceΛCDMcosmology,utilise a
7
smoothing prior and perform Monte Carlo simulations around an initial power spec-
4
9 trum that is scale-free and with spectral index ns = 0.97, very close to the concor-
0 dance spectrum. The horizon scale for the model we are considering corresponds to
5 the wavenumber kh = 4.52×10−4Mpc−1. We find some evidence for the presence
0 of features and we quantify the probabilities of exceeding the observed deviations in
/ WMAP data with respect to the fiducial models. We detect the following marginal
h
p departures from a scale-free (spectral index ns = 0.97) initial spectrum: a cut-off at
- 0.0001 < k < 0.001Mpc−1 at 79.5% (92%), a dip at 0.001 < k < 0.003Mpc−1 at
o 87.2% (98%) and a bump at 0.003 < k < 0.004Mpc−1 at 90.3% (55.5%) confidence
r level.
t
s Thesefrequentistconfidencelevelsarecalculatedbyintegratingoverthe distribu-
a
tionoftheMonteCarloreconstructionsbuiltaroundthefiducialmodels.Thefrequen-
:
v tist analysis finds the low k cutoff of the estimated power spectrum to be about 2.5σ
Xi away from the ns = 0.97 model, while in the Bayesian analysis the model is about
1.5σ away from the estimated spectrum. (The σ’s are different for the two different
r
a methods.)
Key words: methods: data analysis – cosmic microwavebackground– cosmological
parameters – early Universe.
1 INTRODUCTION to the initial power spectrum reconstruction, due to its
potential importance. For example in Spergel et al. (2003);
The spectacular results provided by the Wilkinson Mi-
Peiris et al. (2003); Bridle et al. (2003); Barger et al.
crowave Anisotropy Probe (WMAP) experiment (Bennett
(2003); Mukherjee & Wang (2003); Mukherjee & Wang
et al. 2003) offer the possibility of testing the nature of the
(2005) a shape parametrized a priori is used. Model
initial conditions that seeded the structureof the Universe.
independent methods that use so-called non para-
The inflationary paradigm puts forward a convincing pic-
metric algorithms have also been proposed e.g. in
turethat explains the formation of thefirst seeds. However
Gawiser & Silk (1998); Tegmark & Zaldarriaga (2002);
inflation has not been yet incorporated successfully in the
Matsumiya et al. (2002, 2003); Kogo et al. (2004a,b, 2005);
frameworkoffundamentalphysics.Thereforehintsfromob-
Shafieloo & Souradeep (2004); Hu & Okamoto (2004);
servationsmayproveusefulindiscerningsuchanimportant
Hannestad (2004); Tocchini-Valentiniet al. (2005); Leach
connection.
(2005); Sealfon et al. (2005). In Tocchini-Valentini et al.
An increasing number of studies have been devoted
(2005) we proposed a method based on regularized least
squares to non-parametrically reconstruct under a general
smoothing assumption the primordial power spectrum
⋆ E-mail:[email protected], [email protected] from CMB temperature data in the wave number range
† E-mail:hoff[email protected] 0.01 < k < 0.1hMpc−1, with h standing for the Hubble
‡ E-mail:[email protected]
2 Domenico Tocchini-Valentini, Yehuda Hoffman and Joseph Silk
constant in units of 100Kmsec−1Mpc−1. Here we extend result.Weconcludewith a discussion of ourresultsin Sec-
the analysis to large scales by proposing the necessary tion IV.
modifications dictated by the non-gaussianity of the data
errorscausedbycosmicvariance.Whilewefixthelate-time
cosmologicalparameterstothebestfitvaluesobtainedfrom
2 ESTIMATION OF THE POWER SPECTRUM
thecombinedWMAPandSloanDigitalSkySurvey(SDSS)
data (Tegmark et al. 2004), our method has the virtue of 2.1 Introduction to the Method
allowing an estimate of the power spectrum, at the highest
The problem at hand is how to estimate the primordial
resolution to date, that permits a refined local search for
power spectrum P(k) from a CMB observational data base
deviations from a scale free power spectrum. The horizon
whichiscompressedintotheformoftheangularpowerspec-
scale for the model we are considering translates into the
wavenumber k ≃2π/η =4.52×10−4Mpc−1, where η is trum,Cl.
h 0 0
We start first with a pedagogical example where the
theconformaltimeatpresent.Ahighresolutionreconstruc-
cosmic variance is neglected, a situation applicable in the
tion at large scales (using the Richardson-Lucy algorithm)
large l case.
has been presented previously by Shafieloo & Souradeep
The temperature CMB angular power spectrum is re-
(2004), that used binned data after the hexadecapole.
lated to the primordial power spectrum, P(k), by a convo-
We propose here a different deconvolution method that
lution with a window function that depends on the cosmo-
startsfrom theunbinnedWMAPdataandprovidesfor the
logical parameters:
first time an estimate of the errors in the reconstruction.
Deconvolving WMAP data we find indications for the dk
C =4π ∆2(k)P0(k) ≃ W x , (1)
following three features that deviate from a fiducial scale ℓ Z ℓ k ℓi i
free (spectral index n = 0.97) initial spectrum: a cut-off Xi
s
at 0.0001 < k < 0.001Mpc−1 at 79.5% (92%), a dip at where the last term represents a numerical approximation
0.001 < k < 0.003Mpc−1 at 87.2% (98%) and a bump to the integral obtained through a fine discretization and
at 0.003 < k < 0.004Mpc−1 at 90.3% (55.5%). Similar xi ≡ P(ki)/ki. We fix the units in the transfer function
features were also detected in Shafieloo & Souradeep in order to express the primordial fluctuation spectrum x,
(2004) and suggestions of a similar cutoff were also pro- correspondingtothecomovingcurvatureperturbationspec-
vided by Mukherjee& Wang (2003); Bridle et al. (2003); trum ∆2R in Verdeet al. (2003),in unitsof 2.95×10−9.
Contaldi et al. (2003); Cline et al. (2003). The cutoff effect Let ussuppose that ourscope is to estimate a solution
isduemainlytothelowlargescalepoweratthequadrupole for thelinear problem
and to a minor extent to the octopole present in WMAP d=Wx+n, (2)
data (Bennett et al. 2003) and, since it is intriguingly
presentalso intheCOBEDMR data(Bennettet al. 1996), wheredistheCMBvectordata,nisthenoisevector,char-
certainly deserves some consideration. acterized by zero mean and covariance matrix
Wewishtoemphasizethatourapproachisverysimilar N= nnt , (3)
in spirit to traditional image reconstruction methods and D E
that our aim is not to search for the minimal parametric and x is the vector of the unknowns. The CMB measure-
description of the power spectrum. Given a dense enough mentsaregivenbytheangularpowerspectrumevaluatedat
discretization to closely approximate the integral in Eq.(1) the multipoles ℓ, W is the transfer function calculated nu-
below, the data automatically select the degree of smooth- merically with a Boltzmann code such as CMBFAST (Sel-
ing necessary to allow the highest resolution in the recon- jak & Zaldarriaga 1996), and x represents theinitial power
structedsignalcompatiblewiththeactualnoiselevel,being spectrum, as defined above. Cosmic variance is neglected
the result of a tradeoff between resolution and noise filter- hereand N is independentof x.
ing. Strictly speaking we are using a large number of band Consider a Gaussian likelihood,
amplitudes as parameters, however since they correspond
to a dense grid in wavenumber space that translates in a L∝exp −1(Wx−d)tN−1(Wx−d) ≡exp −1S′ , (4)
(cid:20) 2 (cid:21) (cid:20) 2 (cid:21)
good approximation of the continuum, we call our method
non-parametric. Aswillbecomeapparentlater, ourmethod whereweareignoringanoverallfactorthatdoesnotdepend
iscapableof estimating simultaneously thelarge numberof on the model and Eq.(4) is implicitly defining S′. Our pur-
amplitude parameters. Widely used multi-parameter esti- pose is to findan estimate for thevector x, given the data,
mationmethodssuchasMonteCarloMarkovChainwould, theircovariance matrix and thewindow function.
in their simplest implementations, find some difficulties in The solution tothe minimization problem
evaluating hundredsof parameters. After having tested the
reliability of the method through Monte Carlo simulations, ∂ S′[x] =0, (5)
∂x
thereconstructioncanbecomparedwithsimpleparametric (cid:8) (cid:9)
shapes to look for local deviations, that we estimated with maximizes thelikelihood and provides a minimum variance
two typesof errors, as detailed below. estimator that is unbiased. The solution is found to be
InSectionIIweintroduceindetailthemethod;inSec- x= WtN−1W −1WtN−1d, (6)
tion III we show the outcome of some tests on mock data (cid:2) (cid:3)
and its covariance matrix is given by
that allow us to proceed with confidence to the reconstruc-
tion from the actual WMAP temperature data, our main Σ= WtN−1W −1. (7)
(cid:2) (cid:3)
Non-parametricreconstruction ofthe primordialpowerspectrum athorizonscalesfrom WMAP data 3
Howeverinthiscontextitisnotpossibletocomputethein- Clearly, a compromise is needed when the parameter ǫ
verse matrix defined above, essentially because the transfer has to be chosen. A reasonable choice is to select an ǫ such
functionWdoesnotrepresentaone-to-onerelationconnect- that
ingthedataspacetothesolutionspace,sincetheprojection S′[x]=N (13)
onthelastscatteringsurfaceandsmearingeffects(including d
gravitational lensing, that is however not considered in the withN representingthenumberofdatapoints.Inessence,
d
present work) destroy part of the information (Tegmark & thiswas themethod adopted inourfirst reconstruction pa-
Zaldarriaga 2002). Note that the formal solution provided per, Tocchini-Valentini et al.(2005),where,forhighenough
by Eqs.(6) and (7) is valid only under the assumption that multipoles, cosmic variance could be considered negligible
N is independentof x. with respect to experimental noise and a Gaussian shape
Clearly to solve the problem, some external informa- for the likelihood of the data given a model that was a safe
tion must be injected in the form of a prior. In order to do choice, as will also beconfirmed below.
thisinameaningfulway,theproblemmustberephrasedin Clearly the algorithm can now be cast as a classical
Bayesian terms. We are interested in evaluating the proba- Lagrange multiplier problem. Namely the equations to be
bility P(x|d) of having a primordial power spectrum given solved are
thedata. Bayes theorem states that:
S′[x]=N (14)
d
P(d|x)P(x)
P(x|d)= P(d) , (8) and
where P(d|x) is the likelihood of the data given a model ∂ S′[x]+ǫR[x] =0. (15)
∂x
for the power spectrum, P(x) is the prior assumed on the (cid:8) (cid:9)
power spectrum and P(d) is the probability of having the
data. In what follows we will fix the late-time cosmological 2.2 Extension to Large Scales
parameters to the best fit values found by Tegmark et al.
Sinceweareinterestedinanestimateofthepowerspectrum
(2004) when they assumed a featurless power law for the
at largescales, correspondingtolarge angle CMBdata, the
initial power spectrum.
simple Gaussian likelihood of the previous section can no
We will consider a prior that favours smooth solutions
longer beused.The first non-Gaussian modifications to get
and a useful choice is to adopt a prior of theform
a reliable fit for the CMB likelihood and the need for them
P(x)∝exp −ǫxtLtLx ≡exp(−ǫR), (9) wereproposedinBond,Jaffe,Knox(1998and2000).Forthe
WMAPexperiment,thelikelihoodisapproximatedindetail
(cid:0) (cid:1)
with L representing an approximation to the first deriva- bytheproductofalognormal andaGaussian distributions
tive operator and ǫ is a constant parameter. Eq.(9) defines (Verdeet al. 2003):
R as the regularizing operator whose strength depends on
ǫ. In other words, the logarithm of the prior, namely the −2lnL = 2(zth−zd)tQ−1(zth−zd)+
3
regularization function, approximates a penalty term that
corresponds to the squared norm of the first derivative of 1(Wx−d)tN−1(Wx−d), (16)
3
the solution. After neglecting the probability of the data,
that does not depend on the initial power spectrum, the with zth =ln(Wx+N) and zd =ln(d+N). Here
minimization problem generalizes to Q−ℓℓ1′ =(Wx+N)ℓNℓ−ℓ′1(Wx+N)ℓ′. (17)
∂∂x S′[x]+ǫR[x] =0. (10) The noise vector N was computed by the WMAP team
(cid:8) (cid:9) (Verdeet al. (2003)) through calibration with Monte Carlo
The parameter ǫ weighs the amount of smoothing that the simulations. It is important to note that the matrix N de-
solutionisrequestedtohave.Inotherwords,itispossibleto pends on the angular power spectrum Wx, due to cosmic
finda solution, provided it is smoothed toa certain degree.
variance and in addition the galaxy cut provokes some cor-
Under the assumption that the covariance matrix N does
relation due tomode mixing.
notdependon thetheoretical powerspectrum,thesolution
Sincetheprimordialpowerspectrumreconstructionon
can bewritten as
large scales is very sensitive to the quadrupole likelihood
x= WtN−1W+ǫLtL −1WtN−1d, (11) function,itisrelevanttobesurethatforegroundcontamina-
(cid:2) (cid:3) tionhasbeenproperlytakenintoaccount.Therehavebeen
with thenow computable covariance matrix various post-WMAP papers in which alternative methods
Σ= WtN−1W+ǫLtL −1. (12) to clean foregrounds have been presented (e.g. in Tegmark
et al. (2003), Efstathiou (2004) and Slosar et al. (2004)).
(cid:2) (cid:3)
The dependence of the solution on the smoothness param- Wewill discuss later the possible effects of alternative fore-
eter ǫ appears clearly in the previous equations. It is useful ground removal methods.
to view some asymptotic limits of the solution. When the Even if in this work we chose to use the Verde et
parameterǫtendstozerowearebacktothefirstcaseofno al.(2003) likelihhood, our method can be generalised pro-
prior and the solution is going to be impossible to find for vided an analytical fit to the likelihood is given.
the reasons exposed above. If ǫ is taken very large and the ApplicationofBayestheoremandtherequirementthat
smoothing term dominates over the first term, the solution the estimated power spectrum should correspond to the
will be in the form that minimizes the first derivative, that maximum of the a posteriori probability reduces the mini-
is, a constant vector. mization problem to
4 Domenico Tocchini-Valentini, Yehuda Hoffman and Joseph Silk
∂ thewholeprocedureisrepeated untilthetwoequationsare
{S+ǫR}=0, (18)
∂x both satisfied.
where
S = 2(zth−zd)tQ−1(zth−zd)+ 2.3 Two types of errors
3
1(Wx−d)tN−1(Wx−d). (19) With theiterative procedure we can identify themaximum
3 oftheaposterioridistributionofthepowerspectrum.Given
undertheconstraint of the maximum and considering small deviations around the
maximum, then the distribution function can be approxi-
S[x]=Nd. (20) mated as Gaussian and can be fully defined by the error
HerethecovariancematrixNhasanexplicitdependenceon covariancematrix.Toestimatethestatisticalsignificanceof
x to account for the cosmic variance and N equal to 899, theestimatedpowerspectrumwedefinetwotypesoferrors.
d
According totheBayesian method,thecovariance ma-
thenumberof WMAP temperature data points.
trix that describes thescatter of thetruex around its esti-
Theminimization of(S+ǫR)cannot besolved analyt-
mator, x¯ is given by:
ically,duetothenon-Gaussianityofthelikelihoodfunction,
therefore anumerical method should beused; wefound the Σ = (x−x¯)(x−x¯)t = WtN−1W+ǫLtL −1. (25)
I
iterativeNewton-Raphsonmethodsuitableforourpurpose. D E (cid:2) (cid:3)
The m+1-th iteration of themethod is decribed by: Note that Σ provides an estimate of the deviation of the
I
truexfromitsestimator,anditisreferredtoasanerrorof
xm+1=xm− 1 H−1∂(S+ǫR) , (21) typeI.
i i 2Xj ij ∂xj (cid:12)(cid:12)(cid:12)xm Alternatively, one can ask for another measure of the
(cid:12) scatter, this time of frequentist nature. Let us assume for a
inwhichxm standsforthesolution(cid:12)i-thcomponentrelative
i momentthatthelikelihoodofthedataisGaussianandthat
to the m-th iteration. Here we have the following partial
theestimator can beexpressed as:
derivative:
x=Md, (26)
∂(S+ǫR) with
=
∂xi (cid:12)(cid:12)(cid:12)xm M= WtN−1W+ǫLtL −1WtN−1. (27)
32Xℓℓ′ Wℓi(cid:12)(cid:12)Nℓ−ℓ′1(Wxm+N)ℓ′(zth−zd)ℓ′+ Sreuaplipzo(cid:2)asteiotnhaotfatnheenpsermimb(cid:3)olerdoifaolbpsoewrveerrsspisemcterausmurixn,gathnedstahmaet
32 (zth−zd)ℓWℓiNℓ−ℓ′1(Wxm+N)ℓ′(zth−zd)ℓ′ + eisacchleaorbstehravtetrhmeeeanssuermesblaenadveersatigmedatveasluiteinoftthheeseasmtiemwataeyd.Ixt
Xℓℓ′ is given by:
31 WℓiNℓ−ℓ′1(Wxm−d)ℓ′ +ǫ(LtLxm)i− x = x¯ =MWx. (28)
Xℓℓ′ ensemble D E
1 (Wxm−d)2(2ℓ+1)f2W /(Wxm+N)3− The variance of the scatter of the estimated x from its en-
6Xℓ ℓ ℓ ℓi ℓ semble average is defined here as an error of typeII, and is
1 given by:
(zth−zd)2(2ℓ+1)f2W /(Wxm+N) . (22)
3Xℓ ℓ ℓ ℓi ℓ ΣII = (x¯−xensemble)(x¯−xensemble)t =MNMt. (29)
D E
Itisintendedthatinthelastequationzth=ln(Wxm+N) In other words, Σ gives the scatter in the estimator of x
II
ateachiteration.Furthermorethelasttwotermswerefound around its mean value for a given realization. This corre-
taking into account the derivative of the covariance matrix sponds to using the estimator of x as a surrogate for the
N−1,assumed,justfortheseterms,tobediagonalandwith
truemodel andtoemployingsyntheticdatabuiltaround it
diagonal elements given by:
to studythedistribution of theerrors.
Nℓ−ℓ1 = (2ℓ2+1)(Wxf+ℓ2N)2, (23) senteSdupbpyotsheetmhaattrtihxeMes,tidmoaestionnotminettrhoodducueseadnyhebriea,sreinprxe-,
ℓ then it follows that Σ provides a measure of how much x¯
II
in which f is the effective fraction of the sky computed by
ℓ deviates from thetrue value. It will be shown that the bias
theWMAP team (Verdeet al. (2003)).
is almost negligible for the very smooth concordance the-
FurthermorewehaveapproximatedtheHessianmatrix
oretical spectra and that the criterion in Eq.(20) tends to
with:
oversmooththepredictedsignal.ConsequentlyΣ presents
II
Hij = 21∂2∂(xSi+∂xǫjR)(cid:12)(cid:12)(cid:12)xm ≃(cid:0)WtN−1W+ǫLtL(cid:1)ij, (24) aorleotwiIcenarltllyhimeeixntpeoxencttsteehdcetsishocanap,tetMe.ronoftethCeaerlsotimsimatueldatxionfrsomgenitesrattheed-
(cid:12)
where we neglected(cid:12)difference terms that do not substan- around the concordance initial spectrum are used to show
tially alter the finalresult and may inducenumerical insta- that in the general non-Gaussian WMAP likelihood case,
bilities. At the(m+1)-th iteration stepthecovariancema- practicallynobiasispresentandthatthecovariancematrix
trix is evaluated at Wxm. Eq.(18) is solved for an assumed computed from the a posteriori distribution of the decon-
value of ǫ, upon convergence the value of ǫ is updated and volved samples is almost identical toΣ , as expected from
II
Non-parametricreconstruction ofthe primordialpowerspectrum athorizonscalesfrom WMAP data 5
errorpropagation.Furthermorethedistributionveryclosely fit values found combining WMAP temperature and SDSS
follows a Gaussian. data Tegmark et al. (2004): for the baryonic and matter
fractions, Ω h2 = 0.0231 and Ω h2 = 0.149; the Hubble
b m
constantinunitsof100Kms−1Mpc−1,h=0.685.Theopti-
2.4 Null Hypothesis Testing
caldepthwasfixedtoτ =0.166,assuggestedbytheWMAP
The smoothing prior necessarily induces a bias, that is temperature and cross correlated temperature-polarization
caused by the matrix (MW) being different with respect data,whenthepowerspectrumisdescribedbyapowerlaw
to unity. In other words, the deconvolved power spectrum, withspectralindexclose tounity(Spergeletal.2003). The
eveninabsenceofnoise,isinevitablygoingtobeasmoothed rangeoverwhichthetransferfunctionsareintegratedistyp-
version of the truespectrum. ically between 10−5 and 0.1Mpc−1, discretised in approxi-
However the actual concordance model power spectra mately500bins.Thischoicecorrespondsapproximativelyto
are predicted by the simplest inflation scenarios and are a marginalization if the precise question we are answering
given by very smooth shapes, namely a power law with an is whether there are deviations from a power law spectrum
exponent ns very close to 1, that result in a very flat com- with spectral index close to ns = 0.97, that was found in
bination kns−1. Therefore a smoothing prior seems reason- thebestfitofTegmark et al.(2004),whileassumingapower
able both on theoretical grounds (since it is in line with spectrumdefinedasapowerlawwiththespectralindexasa
inflationary predictions) and for practical purposes (since free parameter. In other words, themethod can be thought
it offers an effective way to extract informations given the of as a consistency test for the concordance model with a
limitations posed by thewidth of thetransfer function ker- powerlaw powerspectrum,sincebyfixingthecosmological
nel). Remarkably,for extremely flat configurations the bias parameters in this way, we should expect to find a power
caused by the smoothing prior adopted in our method can spectrum at least compatible with a power law of kns with
beconsiderednearlynegligible. Indeedthisiswhatwehave ns =0.97.Undertheseconditionsthepriorwilltendtopush
foundintheMonteCarlotestsexplainedinthenextsection, the reconstruction towards the above power law, if it rep-
whereweconsideredtwopowerlawspectrawithn =1and resentsthetrueunderlyingsignal. Whileifdeparturesfrom
s
0.97.Thefactthatthemethodhasnearlynegligiblebiasfor featureless initialconditionsarefoundwithenoughstatisti-
configurations close to theconcordance modelhas led usto cal significance, they may be due to a real signal or just to
consider an extremely effective Null Hypothesis Test that residual systematics still present in the data. We have also
consists of the following steps: implemented the analysis with a spectral index ns = 1 to
look for deviations from a scale free spectrum.
• the Null Hypothesis is chosen to be given, in turn, by As anticipated, an additional safety net is offered by
thetwo initial spectra with ns =1 and 0.97; the method: if large enough deviations truly exist, the esti-
• theCMBangularpowerspectraarecomputedfromthe mate obtained with the smoothing criterion in Eq.(20) will
initialspectraandfromthemrandomdeviatesaredrawnto in general tend to underestimate the signal by furnishing
get syntheticdata; anoversmoothed solution.This isaclear result oftheprior
• eachoftheMonteCarlorealizationsareusedasstarting bias that we have verified with a series of tests and is also
points for the reconstruction method and the deconvolved deduciblefromacomputationoftheeffectivenumberofde-
solutions areutilised tosampletheaposteriori distribution greesoffreedomcausedbysmoothing.Thesearedefined,as
of the initial spectrum around the NullHypothesis; in Tocchini-Valentini et al. (2005), by the trace of the ma-
• thereconstructionfromWMAPdataiscomparedwith trix(WM)in analogy with ordinary regression andamount
the a posteriori distribution and if strong deviations are to about 52 for the WMAP reconstruction. Eq.(20) should
found, then theNullTest is considered to havefailed. therefore be considered a rather conservativechoice.
To verify themethod we performed Monte Carlo tests.
Weremarkthatiftherealprimordialspectrumhasde-
Assuming a power law with spectral index n = 0.97 and
viations from a smooth shape, the bias introduced by the s
with n = 1 as an input, we calculated the corresponding
method will tend to conservatively underestimate them. s
CMBangularpowerspectrumandwecreated randomreal-
It is useful to mention that the errors of type II built
izations around it. Neglecting correlations, we sampled the
around the Null Hypothesis spectra are expected to be
probability distribution derived from the noise model pro-
equivalent to the errors that result from the scatter of a
posed in Knox (1995):
largenumberofMonteCarlorealizations,andindeedthisis
what wehaveobserved.Asspecified in thenextSection, to n V(n−2)/2e−V/2
quantify the probability of how likely the Null Hypothesis P(dℓ)= C +N 2n/2Γ(n/2) , (30)
ℓ ℓ
isfollowed inpre-definedwavenumberranges,wehaveinte-
grated over the distribution described by the Monte Carlo where Cℓ is the theoretical spectrum, n=fℓ2(2ℓ+1) and
samples. Clearly, all these considerations require that the
d +N
important systematics have been removed from the data; V =n ℓ ℓ. (31)
C +N
ℓ ℓ
a discussion on this delicate issue is presented in the next
Thisdistributionhasthemeanequaltothetheoreticalspec-
Section.
trum, d =C , and variance
ℓ ℓ
D E
2
3 RESULTS D(dℓ−Cℓ)2E= n(Cℓ+Nℓ)2. (32)
Since the numerical computations are rather intensive, we Inotherwordswehaveincorporated inthenoisemodelthe
fixed the late-time cosmological parameters to the best- fit of the diagonal of the Fisher matrix N−1 obtained by
6 Domenico Tocchini-Valentini, Yehuda Hoffman and Joseph Silk
11..44 11..44
11..22 11..22
kk 11 kk 11
(cid:144)(cid:144) (cid:144)(cid:144)
LL00..88 LL00..88
kk kk
00HH00..66 00HH00..66
PP PP
00..44 00..44
00..22 00..22
1100--44 1100--33 1100--22 1100--44 1100--33 1100--22
kk@@MMppcc--11DD kk @@MMppcc--11DD
Figure 1. Results from 1000 Monte Carlo realizations drawn Figure 2. Results from 1000 Monte Carlo realizations drawn
fromapowerlawpowerspectrum,givenbykns wherens=0.97, from a power law power spectrum, given by kns where ns = 1,
with the concordance WMAP-SDSS cosmological parameters of with the concordance WMAP-SDSS cosmological parameters of
Tegmarketal.(2004).Thethinsmoothlineinthecentreistheex- Tegmarketal.(2004).Thethinsmoothlineinthecentreistheex-
actinitialpowerlawandthedashedcurveistheaveragedrecon- actinitialpowerlawandthedashedcurveistheaveragedrecon-
struction.Alsoshownarethestandarddeviationbandscalculated struction.Alsoshownarethestandarddeviationbandscalculated
fromthescatter oftherealizationsaroundthemean.Practically fromthescatter oftherealizationsaroundthemean.Practically
nobiasappearsinthereconstruction,exceptatverylargescales, nobiasappearsinthereconstruction,exceptatverylargescales,
wherethe priorstartstodominate, andhowever thebiasiscon- wherethepriorstartstodominate, andhowever thebiasiscon-
siderably smaller than the error estimation. The plot is in units siderably smaller than the error estimation. The plot is in units
of2.95×10−9. of2.95×10−9.
the WMAP team (Verde et al. 2003) through monte carlo the estimated power spectrum from the WMAP temper-
realizationsoftheskythatincludednoise,beamandcutsky ature data (Bennett et al. 2003) at large scales together
effects. with the two type of errors. In Fig. 4 we show the WMAP
To speed up the computations, we considered just the reconstruction compared with the Monte Carlo results de-
diagonal elements of the inverse of the covariance matrix scribed previously for the power law spectrum case. The
N−1, and consequently of the inverse of the matrix Q−1 in figure can provide intuition of how unlikely in some wave
Eq.(17),and we created samples of thepower separately at numberranges thereconstruction could resemble astatisti-
each multipole. We verified that the reconstructions oper- calrandomdeviationfromthefiducialspectrum,anddepicts
atedwiththefullordiagonalmatricesN−1 andQ−1 donot agraphicalversionoftheNullHypothesisTestdescribedin
differ appreciably. Fig. 1 and Fig. 2 show the outcome of theprevious Section.
thetestanditisrelevanttonoticethatthebiasthatresults Assumingtheflatspectrumtodefinethefiducialinitial
from the reconstruction is negligible when compared with conditions,wefindevidenceforthethreefollowingpotential
typeIandeventypeIIerrors.Thisconclusionsupportsour deviations from a power spectrum without features consid-
consideration of the reliability of the less conservative type ering type I (type II) errors: a cutoff for the largest scales
II errors, when utilised to detect the departure from the 0.0001 < k < 0.001Mpc−1 at 1 (2) sigma c.l.; a dip at
flatandconcordancepowerlawmodels. Wealso notedthat 0.001 < k < 0.003Mpc−1 at 1.5 (3) sigma c.l.; a bump a
typeIIerrorsarealmostidenticaltothestandarddeviation 0.003 < k < 0.005Mpc−1 at 1 (2) sigma c.l.. While with
uncertaintiescomputedfrom theMonteCarlo samples;this thepowerlawasthefiducialspectrum,wefind,considering
providesa useful check for themethod. typeI(typeII)errors:acutoffat0.0001<k<0.001Mpc−1
For all the reconstructions from the Monte Carlo re- at 1.5 (2.5) sigma c.l.; a dip at 0.001<k<0.003Mpc−1 at
alizations, we fix the parameter ǫ to the value that results 1.5 (3) sigma c.l.; a bump a 0.003 < k < 0.005Mpc−1 at 1
from deconvolving the WMAP data according to Eq.(20). (2) sigma c.l..
WehavealsocheckedthatwhenchoosingǫfollowingEq.(20) Wenowprovideanotherwaytoestimatethedeviations,
for each individual Monte Carlo sample built from an ini- that is practically equivalent to integrating over the distri-
tialspectrumwithfeaturesadded,theresultingmeanrecon- butiondefinedbytheerrorsoftypeIIbuiltaroundtheNull
structionwasaconservativesmoothedversionofthestarting Hypothesis spectra in defined ranges in wavenumberspace.
one. Following Kogo et al. (2004a), we define a measure of the
The analysis of the Monte Carlo realizations validates distance between a generic spectrum (Pg) and the fiducial
thatthealgorithmusedhereindeedproducesanalmostun- one (Pf) in thewavenumberrange limited byk1 and k2:
biased estimator of power spectrum, when veryflat spectra k2 Pg(k) Pf(k) 2
are used as input. If the power spectrum has real features, D(k ,k ) = dk −
1 2 Z (cid:18) k k (cid:19)
then some bias will be inevitably present in the estimator, k1
but it will act in order to conservatively underestimate the i2 Pg(k ) Pf(k ) 2
featuWrees.verified that the Monte Carlo reconstructions on ≃ iX=i1∆ki(cid:18) ki i − ki i (cid:19) . (33)
allscales arewelldescribed byaGaussian distribution that We calculated the distance of all the Monte Carlo samples
follows typeII errors. and the WMAP reconstruction with respect to the fiducial
In Fig. 3 our most important result is plotted, namely spectrum and we counted how many realizations had adis-
Non-parametricreconstruction ofthe primordialpowerspectrum athorizonscalesfrom WMAP data 7
D
11..55 2K 1500
Μ
kk 11 @ 1000
L(cid:144)L(cid:144)kk HLCl
HH 00..55 500
00 Π
PP
2
00 L(cid:144) 0
1
+
l-500
H
1100--44 1100--33 1100--22 l
kk@@MMppcc--11DD 10 20 30 l 40 50 60 70
Figure 3. Reconstructed initial power spectrum in units of Figure 5. The thick dashed smooth line represents the re-
2.95×10−9 fromWMAPdatawiththeconcordance modelcos- constructed power spectrum from WMAP data convolved back
mological parameters. The irregular middle curve represent the to multipole space, while the power law with spectral index
mean and the surrounding curves that delimit the light shaded ns =0.97, best fit of the ΛCDM cosmology, is plotted as a con-
bandareobtainedbysummingandsubtractingthesquarerootof tinuous thick curve. The WMAP data are also shown as binned
thediagonalelementsofthecovariancematrixfortypeIIerrors. data points, with the error bars computed assuming the power
The borders of the larger dark shaded region are given by sum- lawspectrum.
mingandsubtractingthesquarerootofthediagonalelementsof
thecovariancematrixfortypeIerrors.Thesmoothlineisaref-
erencepowerlawpowerspectrumgivenbykns,wherens=0.97. methods such as ours offer the possibility to penetrate the
informational barrier built up by cosmic variance.
We plot in Fig. 5 the WMAP data together with the
11..44 convolved estimated power spectrum in multipole space to
illustrate the effect of smoothing. That interesting features
11..22
might have been present in the power spectrum could have
11
kk alreadybeenguessedfromaninspectionoftheWMAPdata
(cid:144)(cid:144)00..88
LLkk confronted with a power law power spectrum ΛCDM pre-
HH00..66 diction.Howeverhowtheinformation (includingtheexper-
00
PP00..44 imental uncertainties) in multipole space was going to be
00..22 transferred in detail to wave number space could not have
00 been predicted reliably without an analysis that had taken
1100--44 1100--33 1100--22 properly into account the complicated transfer functions.
kk@@MMppcc--11DD Furthermore, the processed information is now in a format
that permits a direct comparison with the theoretical pre-
dictions from inflation.
Figure 4. Results from 1000 Monte Carlo realizations drawn
Even though we compute the full covariance matrices,
fromapowerlawpowerspectrum,givenbykns wherens=0.97, in the plots we show the errors given by the square root of
with the concordance WMAP-SDSS cosmological parameters of
theinversediagonal elementsoftherelativecovariancema-
Tegmarketal.(2004).Thethinsmoothlineinthecentreistheex-
trix for illustrational purposes. However it is important to
actinitialpowerlawandthedashedcurveistheaveragedrecon-
struction.Alsoshownarethestandarddeviationbandscalculated assess if the deviations detected and their magnitude take
from the scatter of the realizations around the mean. The thick their origin from error correlation. We implemented some
continuousline,thereconstructionfromWMAPdata,isalsoplot- tests to check this relevant issue. In turn, we replaced the
tedtoshowhowlikelyitisthatthedeconvolutioncouldrepresent WMAPmultipolesthatcorrespondtothescalesofthethree
arandomdeviateofthepowerlawfiducialmodel,assumedtobe detected features with the concordance model predictions
ourNullHypothesis.Theplotisinunitsof2.95×10−9. without any noise. From the reconstructions in Fig. 6, we
deducethattheevidenceforthefeaturesisnottheproduct
of correlation, since their significance is only slightly wors-
tancelargerthantheonecomputedfortheWMAPdatade- ened. It is interesting to note that, as expected, the low
convolution.Inthisway wecouldquantifytheprobabilities values of the dipole and the quadrupole are the principal
ofexceedingtheobserveddeviationsinWMAPdatarespect cause for the observed cutoff. Furthermore the wiggles in
to the fiducial models. We detected the following potential the WMAP data around the multipoles of 20 and 40 show
departures from a scale free (spectral index n =0.97) ini- up clearly as the bumpand thedip in ourreconstruction.
s
tial spectrum: a cut-off at 0.0001 < k < 0.001Mpc−1 at We show Fig. 7 how the reconstruction changes if we
79.5% (92%), a dip at 0.001 < k < 0.003Mpc−1 at 87.2% adopt the analysis of Tegmark et al. (2003), based on an
(98%) and a bump at 0.003 < k < 0.004Mpc−1 at 90.3% all-sky foreground cleaned map derived from WMAP data.
(55.5%). Forrepresentativepurposes,weutilisetheWMAPdatalike-
Weaddthattheprior,similarlytotheWeinerfilter(see lihoodwiththemeanvaluesofthequadrupoleandoctopole
e.g.Zaroubi et al.(1995)),helpsinretrievingusefulinforma- liftedtorespectively202and870µK2,ratherthanusingthe
tionatthecrossoverbetweenthesignalandnoisedominated original WMAP values of 123 and 612 µK2. It can be seen
regimes,ifcombinedwithMonteCarlotesting.Furthermore that, with respect to the concordance power law spectrum
8 Domenico Tocchini-Valentini, Yehuda Hoffman and Joseph Silk
1.5
11..55
k
(cid:144) 1.
Lk
0HP0.5 kk 11
(cid:144)(cid:144)
0. LL
kk
HH 00..55
1.5 00
PP
k
(cid:144) 1. 00
Lk
0HP0.5
0. 1100--44 1100--33 1100--22
10-4 k10@-3Mpc-110D-2 10-4 k10@-3Mpc-110D-2 kk @@ MMppcc--11DD
Figure 6. Thefigureshows theeffects ofcorrelationandsheds Figure 7. The result of considering quadrupole and octopole
light on the multipole space origin of the features detected. To meanvaluesraisedtotheTegmarketal.(2003)all-skyforeground
servesuchapurpose,wereplacedsomeWMAPdatapointswith subtracted map determination. The smooth curve represents a
thevaluesofpredictionoftheΛCDMconcordance modelwitha power-lawpowerspectrum givenbykns,wherens=0.97,while
power law power spectrum given by kns, where ns = 0.97. The the dotted line is the deconvolution from the original WMAP
dotted curve is the WMAP reconstruction and the error bands data. The continuous curve is the reconstruction with the new
are derived from type II errors (see text). Going from top left data, the errorbands come fromthetype I(darker) andtype II
andclockwisethepowerwasmodifiedatthefollowingmultipoles: (lighter) errors. The smoothing parameter ǫ was fixed according
ℓ=2;ℓ=2,3;156ℓ626;266ℓ656. toEq.(20)usingthenewdataset.
with spectral index n = 0.97, a cut-off detection is still
s
presentatabout2σ infrequentistterms,whiletheBayesian 11..55
estimatefindsthexoftheNullHypothesis1σawayfromthe
estimate x¯. We report that some other of the post-WMAP kk 11
(cid:144)(cid:144)
papers on the estimation of the quadrupole point towards LL
kk
asomewhat largervaluerespect totheoriginal WMAPone HH 00..55
00
(e.g. Efstathiou 2004 and Slosar et al. 2004). For example PP
Slosaretal.(2004)findabroadquadrupolelikelihoodfunc- 00
tion with a tail that favours greater values, as a result of
havingmarginalizedovertheforegroundtemplates.Thisdif-
fers from the WMAP analysis that consisted in fitting and 1100--44 1100--33 1100--22
subtracting out the templates from the data. Clearly both kk @@ MMppcc--11DD
the estimated mean and errors of the reconstructed initial
powerspectrumwouldbeaffectedbythemodificationofthe Figure8. Weshowthereconstructionforthecosmologicallate-
CMB dataset. time parameters given in Spergel et al. (2003), where an optical
WealsopresentinFig.8areconstructionwiththebest depthτ =0.1isconsidered.Thesmoothcurverepresentsapower
fitparametersfoundinSpergeletal.(2003),whenconsider- law power spectrum given by kns, where ns = 0.97, while the
ing a power law spectrum: Ω h2 =0.023 and Ω h2 =0.13; dotted line is the deconvolution from the original WMAP data.
b m
theHubbleconstantinunitsof100Kms−1Mpc−1,h=0.68; Thecontinuouscurveisthereconstructionwiththenewparame-
ters,theerrorbandscomingfromthetypeI(darker)andtypeII
the optical depth, τ =0.1. The main effect with respect to
(lighter) errors. The smoothing parameter ǫ was fixed according
themodelconsideredpreviouslyisareductionofthecut-off
to Eq.(20), taking into account the new transfer function. The
significance due to a smaller optical depth, as the main ef-
smalleropticaldepthcausesareductioninthestatistical signifi-
fectwouldbetoreducethepoweratintermediateandsmall
canceofthecutoff.
scales while leaving the power unaltered at large scales. A
conspiracybetweenalowopticaldepthandenhancedpower
in the quadrupoleand octopole could in principle consider- tionsfromtheconcordancefeaturlesspowerlawpowerspec-
ably decrease thedetection of acut-off;such measurements trum, with spectral index n approximately equal to 0.97.
s
should therefore be followed closely in futureexperiments. We also implemented the analysis with a scale free spec-
trum. The horizon scale for the model we are considering
correspondstothewavenumberk =4.52×10−4Mpc−1.By
h
creating and reconstructing from Monte Carlo realizations
4 DISCUSSION
of CMB data, we have verified that our proposed estima-
GiventhetemperaturedatafromtheWMAPexperimentwe tor is effectively unbiased if a scale free or power law with
have reconstructed at high resolution the primordial power spectral index n = 0.97 power spectra are assumed. This
s
spectrumatlargescalesbyadoptinganiterativesmoothing hasled usto thedetection of thefollowing threedeviations
algorithm,builttodealwiththenon-gaussianityofthedata fromascalefree(spectralindexn =0.97)initialspectrum:
s
errors caused mainly bycosmic variance. Wehavefixedthe acut-offat0.0001<k<0.001Mpc−1at79.5%(92%),adip
parameterstothebestfitlate-timecosmologicalparameters at0.001<k<0.003Mpc−1 at87.2% (98%)andabumpat
foundbyTegmark et al.(2004)inordertosearchfordevia- 0.003<k<0.004Mpc−1 at 90.3% (55.5%).
Non-parametricreconstruction ofthe primordialpowerspectrum athorizonscalesfrom WMAP data 9
Wehavealsoderivedusingtwoalternativewaysamea- Matsumiya M., Sasaki M., Yokoyama J., 2003, J. Cosm.
sureofthelocal uncertaintiesin theestimates ofdeviations Astropart. Phys.,0302, 003
and effectivelocation of theinitial powerspectrum, labeled Mukherjee P., Wang Y.,2003, ApJ, 599, 1
byerrorsoftypeIandII.Thefrequentistanalysisfindsthe Mukherjee P., Wang Y.,2005, astro-ph/0502136
lowkcutoffoftheestimatedpowerspectrumtobeatabout Peiris H. V.et al., 2003, ApJS,148, 213
2.5σ awayfromthen =0.97model,whileintheBayesian Sealfon C., VerdeL., Jimenez R., 2005, astro-ph/0506707
II s
analysis the model is about 1.5σ away from the estimated Seljak U., Zaldarriaga M., 1996, ApJ, 469, 437
I
spectrum. ShafielooA.,SouradeepT.,2004,Phys.Rev.D,70,043523
The convergence in the observed low power at large Slosar A., Seljak U, Makarov A., 2004, Phys. Rev. D, 69,
scales by the WMAP and COBE DMR experiments is cer- 123003
tainly interesting and merits attention. Since further high Spergel D. N.et al., 2003, ApJS, 148, 175
qualitylarge-scale temperatureand polarization CMB data Tegmark M., Zaldarriaga M., 2002, Phys. Rev. D, 66,
are expected in the near future, for example from the next 103508
datareleasesoftheWMAPteamandthefuturePlanckmis- Tegmark M., de Oliveira-Costa A., Hamilton A., 2003,
sion,anymethod thatcould shedlight on thenatureof the Phys.Rev.D, 68, 123523
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for improving our understanding of the mechanism that is Tocchini-ValentiniD., DouspisM., Silk J., 2005, MNRAS,
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ACKNOWLEDGMENTS
DTV would like to thank the Hebrew University, where ThispaperhasbeentypesetfromaTEX/LATEXfileprepared
part of this work has been carried out, for its hospitality. bythe author.
DTV also acknowledges a Scatcherd Scholarship from the
Universityof Oxford.DTVthanksChuckBennett,Andrew
Jaffe and Glenn Starkman for useful discussions. This re-
search has been partially supported by the ISF-143/02 and
theSheinhornFoundation(YH).YHacknowledgesaLever-
hulmevisitingProfessorship (UniversityofOxford)andthe
Mercator Gastprofessur (Astrophysikalisches Institut Pots-
dam).
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