# Reconstruction of the primordial fluctuation spectrum
from the five-year WMAP data

by the cosmic inversion method with band-power decorrelation analysis

###### Abstract

The primordial curvature fluctuation spectrum is reconstructed by the cosmic inversion method using the five-year Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background temperature anisotropy. We apply the covariance matrix analysis and decompose the reconstructed spectrum into statistically independent band-powers. The statistically significant deviation from a simple power-law spectrum suggested by the analysis of the first-year data is not found in the five-year data except possibly at one point near the border of the wavenumber domain where accurate reconstruction is possible.

###### pacs:

98.70.Vc, 95.30.-k, 98.80.Es^{†}

^{†}preprint: RESCEU-60/08

## I Introduction

Inflationary cosmology inf1 ; inf2 ; inf3 explains the origin of cosmic structure on various scales in an unified way. The expansion history during the inflationary stage of the early universe is recorded in the spectrum of seed structure which can be revealed by modern cosmological observations such as the Wilkinson Microwave Anisotropy Probe (WMAP) observation of the cosmic microwave background (CMB) WMAPBASIC ; WMAPTEMP ; WMAPCOSMO ; WMAPINF ; WMAPTTTE ; loglike ; WMAP3TEMP ; WMAP5 ; WMAP5ONLY ; WMAP5COSMO . The WMAP mission evaluated every multipole moment of the CMB anisotropy spectrum in a wide range of scales which is potentially a record of the inflationary expansion history with high time resolution. It is a feasible challenge to probe the highly time-resolved behavior of the inflaton field (s) which drives inflation.

Since the initial release of the WMAP results WMAPBASIC ; WMAPTEMP ; WMAPCOSMO ; WMAPINF ; WMAPTTTE , it has been argued that the CMB temperature anisotropy spectrum has nontrivial features such as running of the spectral index, oscillatory behaviors on intermediate scales, and lack of power on large scales INFRA ; JF94 ; COBE ; JY99 , which cannot be explained by a power-law primordial spectrum that is a generic prediction of simplest inflation models. These features may provide clues to unnoticed physics of inflation. Some of these anomalous structures disappeared on the three-year spectrum, however several anomalies are still existing WMAP3TEMP ; COSPA07 . To explain these features, a number of inflation models have been proposed CPKL03 ; CCL03 ; FZ03 ; KT03 ; HM04 ; LMMR04 ; KYY03 ; YY03 ; KS03 ; YY04 ; MR04A ; MR04C ; KTT04 ; HS04 ; HS07 ; CHMSS06 .

As an observational approach to these possible nontrivial features, which can be an alternative to model fitting, there have been several attempts to reconstruct the primordial spectrum using CMB anisotropy data without any prior assumptions about the shape of the primordial spectrum WSS99 ; BLWE03 ; MW03A ; MW03B ; MW03C ; MW03D ; SH01 ; SH04 ; BLH07 ; SVJ05 ; WMAP3COSMO ; LSV08 ; VP08 . One such attempt to reconstruct the primordial spectrum is a filtering method where the primordial spectrum is characterized by amplitudes on a few number of representative scales. While such methods have an advantage for reconstructing the global structure of the primordial spectrum, they may miss possible fine structures if their scale width is narrower than the filtering scale, which has been chosen rather arbitrarily so far.

On the other hand, there exist other methods which can reconstruct the primordial spectrum as a continuous function without any ad hoc filtering scale to investigate detailed features such as the cosmic inversion method MSY02 ; MSY03 ; KMSY04 ; KSY04 ; KSY05 , the Richardson-Lucy method SS03 ; SS06 ; SS07 , or a nonparametric method TDS04 ; THS05 . The cosmic inversion method has proved its ability of reconstructing the modulations off a power-law spectrum quite well by the analysis of mock anisotropy data. In the analysis of the first-year WMAP data, we pointed out the possibility of nontrivial structures in the primordial spectrum around the scales of and .

Theoretically, according to the standard inflation paradigm, each wavenumber (-) mode of the power spectrum is mutually independent. On the other hand, each -mode of the power spectrum reconstructed from the observed CMB anisotropy has a strong correlation with neighboring modes because each multipole of CMB anisotropy, , depends on the -modes in the wide range around where is the distance to the last scattering surface. In order to extract real features, therefore, it is important to decompose the reconstructed spectrum to mutually independent band-powers keeping resolution as high as possible.

The purpose of this work is to apply the cosmic inversion method to the five-year WMAP temperature anisotropy spectrum WMAP5 , update the reconstructed primordial curvature spectrum, and perform the above-mentioned band-power analysis by diagonalizing the covariance matrix. Then, we revisit the possibility of fine structures in the primordial spectrum. Because of the arbitrariness of the primordial spectrum, we inevitably incorporate infinite degree of freedom to our analysis, which results in degeneracy among spectral shape and cosmological parameters KNN01 ; SBKET . In this paper, we consider the concordance adiabatic CDM model, where the cosmological parameters (except for the ones characterizing the primordial spectrum) are those of the WMAP team’s best-fit power-law model WMAP5ONLY ; WMAP5COSMO , and instead focus on the detailed shape of the primordial spectrum. Note that, as shown in KMSY04 , different choices of cosmological parameters affect only the overall shape. The fine structures of the reconstructed power spectrum remain intact in the relatively small wavenumber range we probe.

This paper is organized as follows: In Sec. II, the overview of our analysis is described. In Sec. III, we show the reconstructed primordial power spectrum from the five-year WMAP data and discuss its implication. Finally, Sec. IV is devoted to the conclusion.

## Ii Inversion method

### ii.1 Basic formulas

Before presenting the inversion method we first list basic formulas to be inverted. Although we only express formulas related to temperature anisotropy here, the same procedure can be repeated to polarization anisotropy as well KSY04 , which will give us additional information in the future.

The temperature anisotropy of photons coming from direction observed at is decomposed to Fourier modes and multipole moments as

(1) | |||||

where and is the conformal time. In terms of multipole moment in the Fourier space, , which is defined by

(2) |

is expressed as

(3) |

Thanks to the assumption that there exist only adiabatic fluctuations, we can define the transfer function, , from the Fourier mode of primordial comoving curvature perturbation, , to by , where is the present conformal time. Then the angular power spectrum, , of temperature anisotropy and the power spectrum of curvature perturbation, , are related by

(4) | |||||

This is the master equation we wish to invert. Note that depends on the cosmological parameters, too.

### ii.2 Cosmic inversion method

Let us introduce the cosmic inversion formula which relates the observational CMB anisotropy spectrum to the primordial curvature fluctuation spectrum by a first-order differential equation. Working in the longitudinal gauge,

(5) |

the Boltzmann equation for can be transformed into the following integral form (HS95, ).

(6) |

where the overdot denotes the derivative with respect to the conformal time. Here, and are the gauge-invariant quantities representing the Fourier transform of the Newton potential and the spatial curvature perturbation in this gauge, respectively (B80, ; KS84, ), and

(7) |

are the visibility function and the optical depth for Thomson scattering, respectively. In the limit that the thickness of the last scattering surface (LSS) is negligible, we find and where is the recombination time when the visibility function is maximal HS95 . Taking the thickness of the LSS into account, we have a better approximation for Eq. (6) as

(8) |

where is the conformal distance from the present to the LSS and and are the time when the recombination starts and ends, respectively. Here, we introduce the transfer functions, and , defined by

(9) | |||

(10) |

We can calculate and numerically, which depend only on the cosmological parameters, for we are assuming that only adiabatic fluctuations are present.

Then, we find the approximated multipole moments as

(11) |

and the approximated angular correlation function as

(12) |

where is obtained by putting Eq. (11) into Eq. (4), is defined as on the LSS, and and are lower and upper bounds on due to the limitation of the observation. In the small scale limit , using the Fourier sine formula, we obtain a first-order differential equation for the primordial spectrum of the curvature perturbation,

Since and are oscillatory functions around zero, we can find values of at the zero-points of as

(14) |

assuming that is finite at the singularities, . If the cosmological parameters and the angular power spectrum are given, we can solve Eq. (LABEL:DIFF) as a boundary value problem between singularities.

However, because Eq. (LABEL:DIFF) is derived by adopting the approximation (8), is different from the exact angular spectrum for the same initial spectrum. The errors caused by the approximation, or the relative differences between and are as large as about 30%. Thus, we should not use the observed power spectrum directly in Eq. (12). Instead, we must find that would be obtained for the real . Although this is impossible in the rigorous sense, we have found an empirical remedy to find corresponding to in the following way. The crucial observation is that the ratio,

(15) |

is found to be almost independent of (MSY03, ). Using this fact, we first calculate the ratio, , for a known fiducial initial spectrum such as the WMAP team’s best-fit power-law spectrum. Then, inserting , which is much closer to the actual , into the source term of Eq. (LABEL:DIFF), we may solve for with good accuracy. We may continue this procedure iteratively.

### ii.3 Numerical calculation

Given an initial condition and cosmological parameters, we can calculate the transfer functions and numerically. Then, with the angular correlation function (or equivalently anisotropy spectrum), Eq.(LABEL:DIFF) is solved as a boundary value problem between the neighboring singularities, and hence is reconstructed. Hereafter, we treat instead of itself for the comprehensive display purpose and consistency with the common normalization of fluctuation amplitude.

We adopt the adiabatic initial condition and fiducial cosmological parameter set which is the WMAP team’s best-fit power-law model WMAP5ONLY ; WMAP5COSMO to calculate the transfer functions. For the reconstruction from the five-year WMAP data, the cosmological parameters are , , , , and . In this case, the positions of the singularities given by Eq. (14) are , where . Around the singularities, the reconstructed spectrum has large numerical errors that are amplified by the observational errors. Using various model power spectra to investigate the accuracy of our inversion formula, we found that we can achieve the reconstruction with good accuracy in the limited range or . In this region, the errors due to our inversion method turn out to be much smaller than those due to observational errors including the cosmic variance.

In practice, we cannot take the upper bound of the integration in the right-hand side of Eq. (LABEL:DIFF) to be infinite. The integrand in Eq. (LABEL:DIFF) is oscillating with its amplitude increasing with . To evaluate the right-hand side of Eq.(LABEL:DIFF) as finite, we convolve an exponentially decreasing function with a cutoff scale into the integration of the Fourier sine transform. As the cutoff scale is made larger, the rapid oscillations of the integrand with increasing amplitude become numerically uncontrollable. On the other hand, if the cutoff scale is made smaller, the resolution in the -space becomes worse as . For both numerical stability and resolution in -space, we adopt the optimized cutoff scale of .

For implementing the inversion scheme, we employ the routines of CMBFAST code SZ96 to calculate the transfer functions, but we have modified it to adopt much finer resolution than the original one in both and so that we can compute the fine structure of angular power spectra accurately.

### ii.4 Monte-Carlo simulation

In order to incorporate observational errors and the cosmic variance and to obtain mutually independent band-powers, we employ Monte-Carlo method to calculate the covariance matrix of the reconstructed power spectrum. Producing 50000 realizations of a temperature anisotropy spectrum based on the WMAP team’s best-fit power-law model whose statistics obey the likelihood function provided by WMAP team with good precision, we obtain 50000 realizations of a reconstructed primordial spectrum. (The prescription of simulating anisotropy spectra is described in Appendix.) About 1000 samples are sufficient for convergence of the covariance matrix introduced below, which assures the robustness of our conclusion.

The covariance matrix of the reconstructed power spectrum is defined by

(16) | |||||

where represents the value of the reconstructed power spectrum at in the -th realization, and as mentioned above. The resultant is a square matrix with its dimension equal to the number of sampling points , . When we solve the cosmic inversion equation, (LABEL:DIFF), we discretize the relevant range of to more than 2400 points. We calculate at each point to estimate the error of there. In practice, however, the neighboring -modes are strongly correlated with each other, as mentioned above, and so the number of independent modes are much smaller. Hence we do not need to, and in fact, should not take so many points to calculate .

Since is a real symmetric matrix, it can be diagonalized by a real unitary matrix , to yield

(17) |

where ’s are the eigenvalues of . We find they are positive definite as they should be, provided that we take small enough so that neighboring modes are not degenerate with each other. In the present case, we find that if we take , the covariance matrix is well behaved in the sense that the following procedure is possible with positive definite and the well-behaved window matrix defined below. In terms of

(18) |

we define inverse square root of as .

We also define a window matrix by

(19) |

which satisfies the normalization condition . Convolving with this window function, we define a new statistical variable as

(20) |

whose correlation matrix is diagonal and reads

(21) |

where denotes a transposed matrix.

Note that in the previous band-power analysis of the power spectrum in the literature, decomposition into band-powers or wavelets is done by hand without calculating the covariance matrix; hence they result in either undersampling to lack traceability of fine structures or oversampling which generates unwanted correlations between different modes.

## Iii Results of reconstruction

### iii.1 Reconstructed primordial spectrum

Figure 1 shows the primordial spectrum reconstructed from the five-year WMAP data by the cosmic inversion method. In this figure, the solid wavy curve depicts the result of reconstruction and the solid straight line is the best-fit power-law CDM model obtained by the five-year WMAP observations, namely, the power-law spectrum with and , where is the amplitude of curvature perturbation at . The dotted curves around the power-law are associated standard errors which correspond to the diagonal elements of the error covariance matrix (Fig. 3) calculated by Monte Carlo simulation.

The modulations of the reconstructed spectrum roughly fit inside the borders; therefore this oscillatory deviation from the power-law spectrum is attributed to the fact that we are analyzing only one random realization of an ensemble of a simple power-law spectrum. It is not necessarily required that the inflation model responsible for creation of our Universe should predict such a highly nontrivial spectrum, for our Universe is merely one of the realizations of quantum ensemble that is accompanied by significant fluctuations.

Once the possibility of prominent dips around and 350 was pointed out in the analysis of first-year WMAP data KMSY04 . As seen in Fig. 1, the modulations on such scales are fairly degraded in the reconstructed spectrum from the five-year data. Since many of the glitches and bites seen in the first-year WMAP anisotropy spectrum have disappeared in the five-year WMAP spectrum, the updated primordial spectrum is much more smoothed.

### iii.2 Restored anisotropy spectrum

Figure 2 illustrates the CMB temperature anisotropy spectrum which we restored from the updated primordial spectrum by adopting the cosmological parameters of the best-fit power-law model and grafting the best-fit power-law spectrum outside of the investigated range. The effective value in the range of for this restored anisotropy spectrum, which we calculate using the likelihood tool provided by the WMAP team LAMBDA , is 246, while that for the best-fit power-law model is 273. Although the degree of fit is improved significantly, it does not have the original statistical meaning because we have incorporated a functional degree of freedom.

### iii.3 Band-power analysis

Despite the fact that our reconstructed power spectrum restores the fine structures of the observed angular power spectrum of CMB anisotropy well, the oscillations observed in the raw reconstructed spectrum may not have true statistical meaning. As mentioned in the previous section and we can see in the covariance matrix Fig. 3 explicitly, neighboring -modes of the reconstructed power spectrum are correlated with each other up to the range . The origin of the correlation is the convolution with the transfer function between and . In particular, mapping the primordial spectrum to erases the information that is responsible for the fine structure whose characteristic scale is much below the correlation width of .

It is important to decompose the power spectrum into mutually independent modes for evaluating the statistical significance of the oscillations in the reconstructed spectrum. Here we construct band-powers using the window matrix defined in Sec. II.4, which diagonalizes the covariance matrix and gives mutually independent errors. We take the dimension of the window matrix to and decompose in the region to 40 band-powers. Figure 4 shows the window functions for each mode.

Figure 5 is the result of band-power analysis of the five-year WMAP data. In this graph, -th data point indicates the value of

(22) |

and the vertical error bar represents the variance

(23) |

Here is the location of the peak of the -th line of the window matrix . The horizontal bar, on the other hand, indicates the width of the window matrix, where the dispersion of the fitted Gaussian is shown (see Fig. 4). We find their typical full width is and the neighboring horizontal bars barely overlap with each other. As is seen clearly in Fig. 5, our band-power reconstruction of the five-year power spectrum basically agrees with the best-fit power-law as a whole.

The -th band-power depends on the multipole moment on the relevant angular scale of and also on the surrounding multipoles of . Indeed, we found that the statistics of each band-power subject to Gaussian distribution due to the superposition of several multipole moments even though the distribution of is non-Gaussian (see Appendix). By virtue of our band-power analysis, we can also estimate the statistical significance of the reconstructed spectrum itself by evaluating the deviation from the best-fit power-law spectrum at every band simultaneously. We estimated the whole statistical significance of the central 34 bands which correspond to the scales of and found that, in terms of reduced value which is , the reconstructed spectrum from the five-year WMAP data is realization. It is quite consistent with the best-fit power-law spectrum.

Here the absence of the large modulations around and is confirmed again. In the band-power analysis, we find dips around and with the statistical significance about 1. The nontrivial features reported previously have practically disappeared in the reconstructed spectrum from the five-year data.

Note, however, that in Fig. 5, we find a prominent deviation around which would be a true signal of deviation from a simple power-law spectrum if our reconstruction method could be trusted there. While it is possible that this peak is indeed a real signal associated with the feature observed around in the angular power spectrum of the temperature anisotropy, it may simply be an artifact because is so close to the singularity that we can count on our method only marginally there.

## Iv Conclusion

Using the cosmic inversion method, we have reconstructed the primordial power spectrum of curvature perturbations assuming the absence of isocurvature modes and the best-fit values of cosmological parameters for the power-law CDM model. While the range of accurate reconstruction is rather narrow, , we can reproduce the fine structures of modulations off a simple power-law with which we can recover the highly oscillatory features observed in .

The statistical significance of the oscillatory structures in the reconstructed power spectrum is difficult to quantify due to the strong correlation among the neighboring -modes. We have therefore performed the covariance matrix analysis to calculate the window matrix, , which diagonalizes the covariance matrix into statistically independent modes. We have chosen the large enough number of band-powers, , to probe the feature of the primordial spectrum as precisely as possible while keeping the overlap of the window functions small enough.

As a result we have found all the independent modes are consistent with the best-fit power-law CDM model for the five-year WMAP data except possibly for a point around . Whereas, the possible deviation from a simple power-law around and reported in the analysis of the first-year WMAP data KMSY04 has disappeared. This difference is due to the fact that the observed in the five-year WMAP data has become much smoother than the first-year counterpart around the first peak.

On the other hand, there still remain some nontrivial deviations from the expected of a simple power-law spectrum around even in the five-year data. Unfortunately, the cosmic inversion method cannot probe the primordial spectrum in this region. So we need to develop a different method which can probe the power spectrum for smaller wavenumber region precisely.

From the covariance matrix analysis, we have shown that statistically independent bands of the primordial spectrum have an effective width . This means that it is difficult to probe the possible fine structures on predicted by, say, trans-Planckian processes Martin:2000xs , if the scale width of their characteristic modulation is narrower than . In other words, even if our result is consistent with a simple power-law spectrum, this does not rule out such possibility of narrow modulation of .

###### Acknowledgements.

We would like to thank Noriyuki Kogo for his help in numerical computation. We are also grateful to Fran\accois Bouchet, David Spergel, and Masahiro Takada for useful comments. This work was partially supported by JSPS Grant-in-Aid for Scientific Research Nos. 16340076(JY) and 19340054(JY), JSPS-CNRS Bilateral Joint Project “The Early Universe: A Precision Laboratory for High Energy Physics,” and JSPS Core-to-Core Program “International Research Network for Dark Energy.” *## Appendix A Prescription of generating mock anisotropy spectra

In order to perform a proper error analysis of the reconstructed power spectrum, we must prepare a number of realizations of ’s which obey the correct statistical distribution function. In the ideal situation with full-sky, uncontaminated observation, each multipole coefficient is Gaussian distributed and mutually independent if primordial perturbation is random Gaussian. Consequently each angular multipole is distributed with degrees of freedom, and different multipoles are uncorrelated. In practice, however, reliable observation can be made in only a finite fraction, , of the sky and different modes are somewhat correlated. In this situation, the following form of likelihood function has been proposed loglike and used in the statistical analysis of WMAP data WMAP5 .

(24) |

where and with . Here is the Fisher matrix of and is related with as

(25) |

Superscripts and denote data and the theoretical model, respectively.

The value of each component of Fisher matrix is given in LAMBDA where we find diagonal components are larger than neighboring off-diagonal elements typically by a factor of . When we depict ’s, these diagonal components of the Fisher matrix are used to indicate error bars associated with the respective multipole. However, since different multipoles are correlated we would obtain an erroneous result if we created random samples based on these errors only. We should first find a basis consisting of linear combinations of ’s which diagonalize the Fisher matrix to obtain statistically independent quantities.

Here we describe the prescription to find an appropriate basis to diagonalize the Fisher matrix. First we note that, since the Fisher matrix is a real symmetric matrix, it can be diagonalized by a real unitary matrix which we denote by , namely,

(26) |

where ’s are eigenvalues of and they are all positive. In terms of

(27) |

we define matrices

(28) |

and

(29) |

that is, is a “square root” of the Fisher matrix with , and is a diagonal matrix with their diagonal components identical to those of . We find

(30) |

that is, can also be diagonalized if it is sandwiched by and . Note, however, that ’s are not eigenvalues of because is not a unitary matrix. Nevertheless, defining

(31) |

and

(32) |

we find the likelihood function is diagonalized as follows.

(33) |

Now and are independent of each other. On the other hand, ’s are dependent on ’s. Reflecting the properties of the Fisher matrix and thanks to the normalization by , we find that each diagonal component of is equal to unity. Among the off-diagonal components in each line, have the largest magnitudes and their magnitude is no larger than . The other off-diagonal components are even smaller. Therefore, with a good approximation, we may put and . In fact, we do not adopt these approximations but assume a relation

(34) |

which is inferred from the above approximate relations, so that each multipole is completely isolated in the likelihood function (33). With this approximation we can generate random numbers for each separately obeying a probability distribution function (PDF)

(35) |

Without the approximation (34), it would be computationally formidable to generate many random samples of an angular power spectrum with the appropriate statistical distribution.

We generate random numbers for each satisfying the above PDF (35) from which we constitute realizations of according to

(36) |

which have the desired correlation properties and PDFs.

We make 50000 random realizations based on the WMAP team’s best-fit power-law angular spectrum according to the above prescription and perform inversion using the cosmic inversion method to calculate the error covariance matrix of the reconstructed primordial spectrum.

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