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:dtv@astro.ox.ac.uk, dtv@pha.jhu.edu from CMB temperature data in the wave number range † E-mail:hoffman@huji.ac.il 0.01 < k < 0.1hMpc−1, with h standing for the Hubble ‡ E-mail:silk@astro.ox.ac.uk 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 reconstruction 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

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

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