arXiv:astro-ph/0309662 v1 24 Sep 2003 - iucaa

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arXiv:astro-ph/0309662 v1 24 Sep 2003 - iucaa

Reconstructing the primordial spectrum from WMAP databy the cosmic inversion methodNoriyuki Kogo 1 , Makoto Matsumiya 2 , Misao Sasaki 3 , and Jun’ichi Yokoyama 4arXiv:astro-ph/0309662 v1 24 Sep 20031,3,4 Department of Earth and Space Science, Graduate School of Science, Osaka University,Toyonaka 560-0043, Japan2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan1 kogo@vega.ess.sci.osaka-u.ac.jp, 2 matumiya@vega.ess.sci.osaka-u.ac.jp,3 misao@yukawa.kyoto-u.ac.jp, 4 yokoyama@vega.ess.sci.osaka-u.ac.jpABSTRACTWe reconstruct the primordial spectrum of the curvature perturbation, P(k)from the observational data of the Wilkinson Microwave Anisotropy Probe(WMAP) by the cosmic inversion method developed recently. In contrast toconventional parameter-fitting methods, our method can reproduce small featuresin P(k) with good accuracy. To take the observational errors into account,we perform Monte Carlo simulations with fixed values of the cosmological parameters.As a result, we obtain an oscillatory P(k). We confirm that thusreconstructed P(k) recovers the WMAP angular power spectrum with the resolutionof ∆l ≃ 5. However, a very similar oscillatory behavior is found in testcalculations using artificial CMB data generated from a scale-invariant primordialspectrum by adding random numbers which mimic the observational errors.Thus, the oscillation found in P(k) for the WMAP data may be simply causedby the dispersion of data.Subject headings: cosmic microwave background—cosmology: theory1. IntroductionThe Wilkinson Microwave Anisotropy Probe (WMAP) satellite have brought us interestinginformation of our Universe (Bennett et al. 2003). From their remarkably preciseobservation of the angular power spectrum, one can obtain not only accurate values of theglobal cosmological parameters, but also invaluable information of the properties of the primordialfluctuations (Bennett et al. 2003; Spergel et al. 2003; Peiris et al. 2003). Although


– 2 –their results as a whole support the standard ΛCDM model with scale-invariant adiabaticfluctuations, some features that cannot be explained by the standard model have also beenpointed out, such as (i) the lack of power on large scales, (ii) running of the spectral indexn s from n s > 1 on larger scales to n s < 1 on smaller scales, and (iii) oscillatory behaviors ofthe power spectrum on intermediate scales. In fact, (i) was known already with COBE data(Bennett et al. 1996) and possibe lexplanation was proposed (Yokoyama 1999). There aremany new proposals these days (de Deo et al. 2003; Efstathiou 2003; Uzan et al. 2003; Contaldiet al. 2003; Cline et al. 2003; Feng and Zhang 2003; Kawasaki and Takahashi 2003).A number of inflation models that can account for (ii) have also been proposed recently(Feng et al. 2003; Kyae and Shafi 2003; Kawasaki et al. 2003; Huang and Li 2003; Chunget al. 2003; Yamaguchi and Yokoyama 2003). On the other hand, (iii), namely, possibleoscillatory behaviors around a simple power-law spectrum, are more difficult to quantify(Peiris et al. 2003). Although several attempts to reconstruct the primordial spectrum weremade combining with other independent observational data such as 2dFGRS and Lyman-αforest data, and reconstructed a wide range of the primordial spectrum (Wang et al. 1998,1999; Hannestad 2001; Wang and Mathews 2002; Bridle et al. 2003; Mukherjee and Wang2003), they all employed the binning or wavelet band powers method to the data. Thesemethods cannot detect possible oscillatory behaviors if their scale is smaller than the binningscale. It is preferable to use a method which can restore the primordial spectrum as acontinuous function without any ad hoc filtering scale. Fortunately, such a method has beenproposed by Matsumiya et al. (2002, 2003) and test calculations using artificial data haveshown that this method can reproduce possible small dips and peaks off a scale-invariantspectrum quite well, in the spatially flat universe with the adiabatic initial condition. Inthis paper, we attempt to reconstruct P(k) from the WMAP data by this method. To applyit to a real situation, we consider the observational errors by Monte Carlo simulations, anduse the numerical code based on a modified CMBFAST 1 which can compute angular powerspectra more accurately than the original code. In the present work, as a first step, we fixthe cosmological parameters to the best-fit values of WMAP for the scale-invariant P(k)although their uncertainty must be taken into account eventually.2. Inversion methodFirst, we briefly review the cosmic inversion method proposed by Matsumiya et al.(2002, 2003). We assume the spatially flat universe and the adiabatic initial condition, bothof which have been supported by the WMAP data. In the end, we obtain a first-order1 http://www.cmbfast.org/


– 3 –differential equation for the primordial spectrum P(k).The CMB anisotropy is quantified by the angular correlation function defined asC(θ) ≡ 〈Θ(ˆn 1 )Θ(ˆn 2 )〉 =∞∑l=02l + 14π C lP l (cos θ), cosθ = ˆn 1 · ˆn 2 , (1)where Θ(ˆn) is the temperature fluctuation in the direction ˆn. We decompose the Fouriercomponents of the temperature fluctuations Θ(η, k) into multipole moments,Θ(η, k, µ) =∞∑(−i) l Θ l (η, k)P l (µ). (2)l=0where µ ≡ ˆk · ˆn, k is the comoving wavenumber, and η is the conformal time with its presentvalue being η 0 . Using Θ l (η, k), the angular power spectrum is expressed as2l + 14π C l = 12π 2 ∫ ∞0dkkk 3 〈|Θ l (η 0 , k)| 2 〉. (3)2l + 1The Boltzmann equation for Θ(η, k) can be transformed into the following integral form (Huand Sugiyama 1995).(Θ + Ψ)(η 0 , k, µ) =∫ η00dη{[Θ 0 + Ψ − iµV b ]V(η) + ( ˙Ψ − ˙Φ)e}−τ(η) e ikµ(η−η0) , (4)where the overdot denotes the derivative with respect to the conformal time. Here, Ψ andΦ are the Newton potential and the spatial curvature perturbation in the Newton gauge,respectively (Kodama and Sasaki 1984), andV(η) ≡ ˙τe −τ(η) ,τ(η) ≡∫ η0η˙τdη, (5)are the visibility function and the optical depth for Thomson scattering, respectively. In thelimit that the thickness of the last scattering surface (LSS) is negligible, we have V(η) ≈δ(η −η ∗ ), e −τ(η) ≈ θ(η −η ∗ ), where η ∗ is the recombination time when the visibility functionis maximum (Hu and Sugiyama 1995). Taking the thickness of the LSS into account, wehave a better approximation for Eq. (4) as(Θ + Ψ)(η 0 , k, µ) ≈∫ η∗endη ∗startdη{[Θ 0 + Ψ − iµΘ 1 ]V(η) + ( ˙Ψ − ˙Φ)e −τ(η) }e −ikµd ≡ Θ app + Ψ, (6)where d ≡ η 0 − η ∗ is the conformal distance from the present to the LSS, η ∗start and η ∗endare the time when the recombination starts and ends, respectively. Here, we introduce the


– 4 –transfer functions, f(k) and g(k), defined by∫ η∗enddη[(Θ 0 + Ψ)(η, k)V(η) + ( ˙Ψ − ˙Φ)(η,]k)e −τ(η)η ∗start∫ η∗end≡ f(k)Φ(0,k), (7)η ∗startdη Θ 1 (η, k)V(η) ≡ g(k)Φ(0,k) . (8)We can calculate f(k) and g(k) numerically, which depend only on the cosmological parameters,for we are assuming that only adiabatic fluctuations are present. Then, we find theapproximated multipole moments asΘ appl(η 0 , k) = (2l + 1) [f(k)j l (kd) + g(k)j l ′ (kd)]Φ(0,k), (9)and the approximated angular correlation function asC app (r) =l∑maxl=l min2l + 14π Capp( )lP l 1 − r2, (10)2d 2where C applis obtained by putting Eq. (9) into Eq. (3), r is defined as r = 2d sin(θ/2) onthe LSS, and l min and l max are lower and upper bounds on l due to the limitation of theobservation. In the small scale limit r ≪ d, using the Fourier sine formula, we obtain afirst-order differential equation for the primordial spectrum of the curvature perturbation,P(k) ≡ 〈|Φ(0,k)| 2 〉,−k 2 f 2 (k)P ′ (k) + [ −2k 2 f(k)f ′ (k) + kg 2 (k) ] P(k) = 4π∫ ∞0dr 1 r∂∂r {r3 C app (r)} sinkr ≡ S(k).(11)Since f(k) and g(k) are oscillatory functions around zero, we can find values of P(k) at thezero-points of f(k) asP(k s ) = S(k s)k s g 2 (k s )for f(k s ) = 0, (12)assuming that P ′ (k) is finite at the singularities, k = k s . If the cosmological parameters andthe angular power spectrum are given, we can solve Eq. (11) as a boundary value problembetween singularities.However, because Eq. (11) is derived by adopting the approximation (6), C applis differentfor the same initial spectrum. The errors caused by thefrom the exact angular spectrum C exlapproximation, or the relative differences between C appland Cl ex are as large as about 30%.directly in Eq. (10). Instead,that would be obtained for the real P(k). Although this is impossibleThus, we should not use the observed power spectrum C obslwe must find C appl


– 5 –in the rigorous sense, since it is the real P(k) that we are to reconstruct, it turns out to bepossible with accuracy high enough for our present purpose. The crucial observation is thatthe ratio,b l ≡ Cex lC appl, (13)is found to be almost independent of P(k) (Matsumiya et al. 2003). Using this fact, we firstcalculate the ratio, b (0)l= C ex(0)l, for a known fiducial initial spectrum P (0) (k) such/C app(0)las the scale-invariant one. Then, inserting Cl obs /b (0)l, which is much closer to the actual C appl,into the source term of Eq. (11), we may solve for P(k) with good accuracy.In practice, we cannot take the upper bound of the integration in the right-hand side ofEq. (11) to be infinite. The integrand in Eq. (11) is oscillating with its amplitude increasingwith r. We therefore introduce a cutoff scale r cut . But, this inevitably introduces a smoothingscale to our method. As the cutoff scale is made larger, the rapid oscillations of the integrandwith increasing amplitude become numerically uncontrollable. On the other hand, if thecutoff scale is made smaller, the resolution in the k-space becomes worse as ∆k ≃ π/r cut . Inthe actual calculations, to maintain the numerical accuracy as good as possible and to obtainsimultaneously the resolution in the k-space as fine as possible, we convolve an exponentiallydecreasing function into the integration of the Fourier sine transform with the cutoff scaleof r cut ≃ 0.5d, corresponding to θ ≃ 30 ◦ or ∆l ∼ 6. Thus, the resolution of the Fourier sinetransform is limited to ∆kd ≃ 6. Note that there is an absolute theoretical limit r cut 2dor ∆kd 1.5 due to the finiteness of the LSS sphere.In this paper, we also take account of the effect of observational errors on the reconstructedP(k). First, we generate random Gaussian numbers with vanishing mean and withvariance given by the 1σ observational error for each C l and add it to the mean value of eachobserved C l , and then reconstruct P(k) for each realization. After repeating this procedure1000 times, we estimate the mean value and the variance of the reconstructed P(k) at eachk. We find that the errors caused by our inversion method, whose magnitude is estimated byinverting C l ’s calculated from known P(k)’s, are much smaller than the observational errorsof WMAP, except around the singularities where the numerical errors are amplified.To calculate C exl , we use a modified CMBFAST with much finer resolutions than theoriginal code for both k and l. We adopt the scale-invariant spectrum as the fiducial initialspectrum, P (0) (k). Although changes in the cosmological parameters affect the shape of thereconstructed P(k), we fix the cosmological parameters to the best-fit values of WMAP forthe scale-invariant spectrum, k 3 P(k) = (const.). Namely, we assume τ = 0.17, h = 0.72,Ω b = 0.047 and Ω Λ = 0.71. We limit C l in the range 20 ≤ l ≤ 700 in order not to use thedata which have large observational errors due to the cosmic variance at small l and the


– 6 –detector noise at large l. The positions of the singularities (12) are at kd ≃ 70, 430, · · ·,where d ≃ 13400Mpc. Because the reconstructed P(k) around the singularities has largenumerical errors, we can obtain P(k) with good accuracy in the limited range 120 kd 380or 9.0 × 10 −3 Mpc −1 k 2.8 × 10 −2 Mpc −1 . We note that the large numerical errors at thesingularities may be due to inappropriate choice of the cosmological parameters as mentionedin Matsumiya et al. (2003). This search for a better set of the cosmological parameters isleft for future work.3. ResultsWe show P(k) reconstructed from the WMAP data in Fig. 1. We can see the oscillationswhose amplitude is about 20−30% of the mean value with frequencies (∆kd) −1 ≃ 1/15−1/10.Note that P(k) becomes negative in the vicinity of the singularities, which is perhaps dueto numerical errors. To check whether our method works correctly, we recalculate C l fromthe obtained P(k) in the range 120 kd 380, assuming the scale-invariance outside ofthis range, and compare it with the observational data of WMAP. Then, we find that therecalculated C l agrees with the binned original data whose bin size is ∆l = 5 as shownin Fig. 2. As mentioned in Sec. 2, our method reconstructs P(k) with a finite resolution∆l ∼ ∆kd ≃ 6 or ∆k ≃ 4.5 × 10 −4 Mpc −1 , which is caused by the resolution of the Fouriersine transform. The agreement in Fig. 2 is quite impressive.To examine if the oscillatory behavior is real, we performed test calculations as follows.First, we assume a scale-invariant P(k) and calculate the theoretical C l for a fixed set of thecosmological parameters. Then, we make artificial observational data with the same errorsas those of the WMAP data, by adding a random Gaussian number to each C l . We usethese artificial data to reconstruct P(k), assuming the cosmological parameters are known.The resulting P(k) for several different realizations are shown in Fig. 3. We see that theyalso have an oscillatory feature whose amplitude and frequency are almost the same as P(k)from the WMAP data. This suggests that with the present accuracy of the WMAP data, wecannot tell if the oscillatory behavior in P(k) is a real one or simply caused by the dispersionof the data. Thus, we need more quantitative arguments in order to conclude whether ornot there are sigfinicant deviations from the scale-invariance. This issue is currently understudy.


– 7 –4. ConclusionWe reconstructed the shape of the primordial spectrum, P(k), from the WMAP angularpower spectrum data, C l , by the inversion method proposed by Matsumiya et al. (2002,2003). We assumed that the Universe is spatially flat and the primordial fluctuations arepurely adiabatic. We fixed the cosmological parameters as τ = 0.17, h = 0.72, Ω b = 0.047,Ω Λ = 0.71, which are the best-fit values for the WMAP data for the scale-invariant P(k). Toestimate the effect of observational errors on P(k), we performed Monte Carlo simulations.As a result, we obtained the oscillatory P(k) whose amplitude is about 20 − 30% of themean value and whose frequency is about (∆kd) −1 ≃ 1/15 − 1/10. We confirmed that P(k)is reconstructed with good accuracy in the range 120 kd 380 or 9.0 × 10 −3 Mpc −1 k 2.8 × 10 −2 Mpc −1 , with the resolution of ∆kd ≃ 5 or ∆k ≃ 3.7 × 10 −4 Mpc −1 . Thus, ourinversion method can reconstruct P(k) with a much finer resolution than any other methodsuch as the binning or wavelet band powers method (Wang et al. 1998, 1999; Hannestad2001; Wang and Mathews 2002; Bridle et al. 2003; Mukherjee and Wang 2003). However,test calculations for scale-invariant spectrum with the observational errors which are givenartificially, yielded P(k) with almost the same oscillatory feature as P(k) reconstructed fromWMAP. Thus, no significant deviation from the scale-invariance was detected.In the future work, we should study some issues as follows. First, we need to quantifydeviations from the scale-invariance of P(k). It seems to be difficult because the reconstructedP(k) is the continuous function and we are interested in its shape. Second, weshould take the uncertainty of the cosmological parameters into account. As pointed out inthe previous work, if the cosmological parameters are different from real values in the inversion,the reconstructed P(k) is distorted from real one, especially around the singularities(Matsumiya et al. 2003). Note that the reconstructed P(k) from WMAP data becomes negativeat 350 kd 380 while those from the artificial data remain positive, from which wecan speculate that the cosmological parameters we assumed differ from real values. Hence,in order to deal with the uncertainty of the cosmological parameters, we should first make anumber of different sets of these parameters and then perform the same statistical analysis asabove for each set. Third, we should extend the formalism to include the CMB polarizationso that it may be applied not only to the WMAP data but also to future CMB observationssuch as PLANCK 2 . Inclusion of the CMB polarization will constrain P(k) more severely, andif the B-mode polarization is detected, we can investigate the tensor mode of the primordialperturbations (Starobinsky 1979; Rubakov et al. 1982; Abbott and Wise 1984; Crittendenet al. 1993; Polnarev 1985; Crittenden et al. 1993).2 http://astro.estec.esa.nl/SA-general/Projects/Planck/


– 8 –We would like to thank Eiichiro Komatsu for fruitful discussions. This work wassupported in part by JSPS Grant-in-Aid for Scientific Research Nos. 12640269(MS) and13640285(JY), and by Monbu Kagakusho Grant-in-Aid for Scientific Research (S) No. 14102004(MS).NK is supported by Research Fellowships of JSPS for Young Scientists (No. 04249).REFERENCESAbbott, L. F., and Wise, M. 1984, Nucl. Phys., B244, 541Bennett, C. L., Banday, A. J., Gorski, K. M., Hinshaw, G., Jackson, P., Keegstra, P., Kogut,A., Smoot, G. F., Wilkinson, D. T., Wright, E. L. 1996, ApJ, 464, L1Bennett, C. L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S.S., Page, L., Spergel, D. N., Tucker, G. S., Wollack, E., Wright, E. L., Barnes, C.,Greason, M. R., Hill, R. S., Komatsu, E., Nolta, M. R., Odegard, N., Peirs, H. V.,Verde, L., Weiland J. L. 2003, ApJS, 148, 1Bridle, S. L., Lewis, A. M., Weller, J., and Efstathiou, G. 2003, Mon. Not. Roy. Astron. Soc.,342, L72Chung, D. J. H., Shiu, G., and Trodden, M., astro-ph/0305193Cline, J. M., Crotty, P., and Lesgourgues, J., astro-ph/0304558Contaldi, C. R., Peloso, M., Kofman, L., and Linde, A. 2003, JCAP, 0307, 002Crittenden, R., Bond, J. R., Davis, R. L., Efstathiou, G., and Steinhardt, P. J. 1993,Phys. Rev. Lett., 71, 324Crittenden, R., Davis, R. L., and Steinhardt, P. J. 1993, ApJ, 417, L13de Deo, S., Caldwell, R. R., and Steinhardt. P. J. 2003, Phys. Rev. D, 67, 103509Efstathiou, G. 2003, Mon. Not. Roy. Astron. Soc., 343, L95Feng, B., Li, M., Zhang, R. J., and Zhang, X., astro-ph/0302479Feng, B., and Zhang, X., astro-ph/0305020Hannestad, S. 2001, Phys. Rev. D, 63, 043009Hu, W., and Sugiyama, N. 1995, ApJ, 444, 489


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– 10 –21.5k 3 P(k)10.50100 150 200 250 300 350 400kdFig. 1.— The primordial spectrum P(k) reconstructed from the WMAP data. The solidcurve and the dashed curve represent the mean and the variance, respectively, of the reconstructedP(k). kd corresponds roughly to l.l(l+1)C l / 2π700060005000400030002000binned datareconstructedrelative error (%)20151050-5-101000-150100 150 200 250 300 350 400l-20100 150 200 250 300 350 400lFig. 2.— An accuracy check of our reconstruction method. The left panel shows comparisonof the WMAP data binned by ∆l = 5, Clbin (crosses with error bars), with the angular powerspectrum, Clre (solid curve), recovered from the reconstructed P(k) shown in Fig. 1. Theright panel shows the relative errors (Clbin −Cl re)/Crel . The relative errors are small for mostof the bins except for those near the singularities.


– 11 –221.51.5k 3 P(k)1k 3 P(k)10.50.50100 150 200 250 300 350 400kd0100 150 200 250 300 350 400kd221.51.5k 3 P(k)1k 3 P(k)10.50.50100 150 200 250 300 350 400kd0100 150 200 250 300 350 400kdFig. 3.— The primordial spectra P(k) reconstructed from artificial CMB data for four differentrealizations. The original spectrum is taken to be scale-invariant, and each realizationis generated by adding a random number to each C l with the same error as the WMAPdata. The same oscillatory feature as shown in Fig. 1 is seen.

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