# A new search for features in the primordial power spectrum

###### Abstract

We develop a new approach toward a high resolution non-parametric reconstruction of the primordial power spectrum using WMAP cosmic microwave background temperature anisotropies that we confront with SDSS large-scale structure data in the range . We utilise the standard CDM cosmological model but we allow the baryon fraction to vary. In particular, for the concordance baryon fraction, we compare indications of a possible feature at in WMAP data with suggestions of similar features in large scale structure surveys.

###### keywords:

cosmology: early universe – cosmic microwave background – large-scale structure of Universe – cosmological parameters – methods: data analysis.^{†}

^{†}pagerange: A new search for features in the primordial power spectrum–A new search for features in the primordial power spectrum

^{†}

^{†}pubyear: 2004

## 1 Introduction

The Wilkinson Microwave Anisotropy Probe (WMAP) experiment has recently provided a high resolution full sky cosmic microwave background (CMB) map that provides an unprecedented opportunity to probe the structure and the contents of the universe (Spergel et al. 2003). The corresponding angular power spectrum has been used to set constraints on cosmological parameters. In the framework of CDM cosmologies, such constraints are strong if the initial conditions are parametrized by primordial power spectra in the form of power laws with or without running spectral index. However parameter estimation can be weakened and biased if more complex forms of initial conditions are adopted, cf. Bucher et al. (2004) and Blanchard et al. (2003). We explore another such possibility here.

Recent papers from the Sloan Digital Sky Survey (SDSS) team report measurements of the matter power spectrum and the cosmological parameters that give a best fit to its shape (Tegmark et al. 2003a,b; Pope et al. 2004). In particular Pope et al. (2004) finds that for the for a value of the matter density, and baryon fraction, at 1 . These determinations, in particular that of the baryon density, are in marginal conflict with the data from both WMAP (Spergel et al. 2003) and primordial nucleosynthesis (BBN) (Cyburt et al. 2003; Cuoco et al. 2003), which, together with the Hubble constant (Freedman et al. 2001) determination, prefer a lower baryon fraction, when a power law primordial power spectrum in a CDM cosmology is adopted.

Such a discrepancy might be due to the presence of intrinsic features in the power spectrum that affect the SDSS (and other survey) data, such as for example a dip and a bump at and (see also Atrio-Barandela et al. 2001; Barriga et al. 2001). It is useful to explore the implications of such a possibility, even though the data at present is not compelling, because very similar features in the matter power spectrum have been detected in independent large-scale structure surveys, notably the 2dF, Abell/ACO cluster and PSCz surveys (Einasto et al. 1997; Percival et al. 2001; Miller & Batusky 2001; Hamilton et al. 2000). In this work, we show that, if the power law hypothesis for the shape of the initial power spectrum is relaxed, a still good or even better agreement between the different data sets can be obtained. Our strategy is principally comprised of two steps: a high definition reconstruction of the primordial power spectrum from WMAP data (for given sets of cosmological parameters) and a convolution of the initial power spectrum with the SDSS window functions to determine the statistical agreement between the two data sets. We choose to study only these two data sets in a limited range of space in order to make a preliminary exploration of the possible influence of primordial features. We find that for reasonable choices of the cosmological parameters, the WMAP-reconstructed matter power spectrum shows oscillations that are very similar to those of SDSS once the convolution is performed. This results in an improved concordance for the inferred baryon fraction and at the same time is suggestive of a deviation from a power-law primordial power spectrum.

## 2 Method

The temperature CMB angular power spectrum is related to the primordial power spectrum, , by a convolution with a window function that depends on the cosmological parameters:

(1) |

where the last term is in matrix notation and represents a numerical approximation to the integral, calculated using a modified version of CMBFAST (Seljak & Zaldarriaga 1996). The spectrum then contains information about both the cosmological parameters and the initial power spectrum.

Various papers have presented reconstructions of the power spectrum from WMAP data (e.g. Bridle et al. 2003; Mukherjee & Wang 2003; Matsumiya et al. 2003; Shafieloo & Souradeep 2004; Kogo et al. 2004; Hannestad 2004). Most of these have confronted an a priori parametrised power spectrum with the data to obtain information about its shape and amplitude. Following the spirit of Gawiser & Silk (1998) and Tegmark & Zaldarriaga (2002), our power spectrum is not described by a small set of parameters that incorporate features, but is finely discretized in -space (see also Shafieloo & Souradeep 2004; Kogo et al. 2004). Our work represents the first attempt to reconstruct the power spectrum at high resolution in the full range together with an estimation of the error covariance matrix. Our findings are consistent with the results obtained with a different method by Kogo et al. (2004) that are limited to , corresponding to .

Generally speaking, it is not possible to invert the matrix to solve for the power spectrum, because each embraces information about a limited range in -space. However the inversion is feasible under some assumptions. Basically the lack of information can be remedied by the introduction of priors, such as for example the requirement of a certain degree of smoothness in the solution. To be specific, we consider a solution of the form

(2) |

with an error covariance matrix given by

(3) |

Here: ; are the mean values of WMAP data; is the WMAP covariance matrix (Verde et al. 2003) and is a discrete approximation to the first derivative operator. Our approach follows the spirit of a similar solution proposed in Tegmark & Zaldarriaga (2002); however we replace the identity operator by . The parameter regulates the degree of solution smoothness, since the derivative operator acts in such a way to minimize the norm of the solution derivative. This solution can be effectively thought of as a linear least squares solution modified to accommodate smoothing.

In order to select a value for the parameter , we calculate the angular power spectrum resulting from . Then we form the expression

(4) |

It is reasonable to fix the parameter to render the in the previous equation equal to the number of WMAP temperature data points, namely 899. In other words, a solution is chosen that gives an acceptably good fit to the data. Moreover, as will be explained below, this represents a conservative choice since the smoothing decreases the effective number of degrees of freedom.

To test the method, we reconstruct the power spectrum by using angular power spectra calculated from various power spectrum shapes with and without added errors. We obtain very good results, except when there are features of extension comparable to our -space gridding , which we took to be approximately in the range . The integration range is typically between and and comprises about 1500 points. We also verify that the signals detected in the actual WMAP data (see below) do not depend on discretization issues by testing different resolutions. Our method yields a -space resolution that is limited only by computing resources, at the price however of introducing correlations between neighboring points that we take fully into account into our statistical analysis.

In analogy with ordinary regression it is possible to define the effective number of degrees of freedom caused by smoothing by the expression . It is easy to recognize that without smoothing this quantity would have been equal to the number of bins in -space. We have found that our choice of the parameter implies the effective usage of approximately 45 constraining informations out of the more than 1000 amplitudes that specify the power spectrum.

The procedure just described must be repeated for each given set of cosmological parameters. We proceed by first evolving the primordial power spectrum to redshift zero by multiplying it by the appropriate transfer function calculated by CMBFAST. Then we convolve the derived matter power spectrum with the SDSS window functions given by Tegmark et al. (2003a). Finally, we evaluate :

(5) |

where and represent the SDSS mean data and (diagonal) covariance matrix (Tegmark et al. 2003a), and are the SDSS-convolved WMAP-reconstructed matter power spectrum and covariance matrix (calculated by error propagation) and is the bias parameter. We follow the common practice of considering a constant bias which we set at the optimal value.

## 3 Results

In performing the analysis indicated by Eq. (5), we restrict the computation to the SDSS experimental data corresponding to the 11 points with lower -values, such that , in order to avoid effects caused by the limitation in -range that arises since we are extracting information from the CMB power spectrum and the quality of the WMAP data degrades significantly above .

We select a few choices of the cosmological parameters, for which we perform the reconstruction. The presence of features in the power spectrum is associated with the deviations from a power law angular power spectrum noticed by Spergel et al. (2003). These authors mention that the first release of data did not include effects contributing roughly 0.5-1% to the power spectrum covariance, mainly due to gravitational lensing, beam uncertainties and non-Gaussianity in the noise maps. Therefore, to verify if they are real, a cross-check with an independent dataset is needed. The SDSS data serve such a purpose.

In Table 1, we list the models used, defined in terms of the baryonic and total matter content, Hubble constant (in units of ) and bias . The corresponding for a selection the first 11 SDSS data points with lowest wavenumber are also given and will be used for the discussion below. The optical depth is set equal to the best WMAP case, 0.166. We have chosen to concentrate on a few cases that differ mainly with respect to the baryonic fraction, since this is the most interesting quantity to probe in light of the SDSS analysis that uses a power-law power spectrum.

Model | ||||||
---|---|---|---|---|---|---|

1 | 0.171 | 0.194 | 0.024 | 0.72 | 1.04 | 4.7/10 |

2 | 0.286 | 0.264 | 0.054 | 0.72 | 0.97 | 26.4/10 |

3 | 0.155 | 0.215 | 0.023 | 0.695 | 1.10 | 3.3/10 |

4 | 0.200 | 0.209 | 0.030 | 0.72 | 1.11 | 3.6/10 |

5 | 0.143 | 0.194 | 0.020 | 0.72 | 1.00 | 6.4/10 |

We label the models as: model 1, the WMAP mean best case (Spergel et al. 2003); model 2, the SDSS best case (Pope et al. 2004) (in marginal conflict with the previous one); model 3, close to the joint WMAP-SDSS best case (Tegmark et al. 2003b); model 4 and model 5, characterized by a relatively high and low baryonic content. The “best cases” we refer to were all obtained for a power-law power spectrum. The latest constraints from primordial nucleosynthesis give, at 1 : (Cyburt et al. 2003; Cuoco et al. 2003), which excludes model 2 with .

Our first result is a reconstruction of the initial power spectrum shape from the CMB angular power spectrum with the associated error covariance matrix. Fig. 1 shows the corresponding initial spectrum reconstructed from the WMAP data and assuming the cosmological parameters of model 3, in the range . The effective range probed in is approximately 140-800. By reconvolving the power spectrum back to multipole space, it is possible to note in Fig. 2 the action of smoothing and appreciate the accuracy of the inversion. Even if not far from a power law, small features are present in the initial power spectrum at all scales; we shall devote our attention mostly to the succession of bumps and dips around .

It is useful to implement Monte Carlo simulations to look for deviations and systematic biases in the reconstruction. We generated 5000 random realizations around the angular power spectrum calculated from a power law primordial power spectrum assuming a gaussian distribution, which is reasonable in the probed multipole range. The case for model 3 is presented in Fig. 3. The averaged reconstruction doesn’t show strong bias and deviations bigger that the one sigma band, obtained from the standard deviation respect to the averaged reconstruction, are also evident. Again, there are indications of possible features at and other wave numbers.

Models 1 and 2 reconstructed matter power spectra are plotted in Fig. 4. Model 2, constructed to fit the SDSS data, is marginally excluded by WMAP data under the initial power law assumption: it requires too many baryons in order to fit the features (oscillations) of the SDSS reconstructed matter power spectrum. If we relax this assumption by using our method and keeping the same baryon fraction, we see that the WMAP features translated into k-space do not match the SDSS features (). Baryonic oscillations and CMB features do not converge to reproduce the SDSS behaviour. As noted in Pope et al. (2004), oscillations induced by the prefered baryon fraction value of CMB are out of phase with SDSS “oscillations”.

Conversely, assuming a power law initial power spectrum, model 1 gives a poor fit to CMB data (Spergel et al. 2003) and a good fit to the selected SDSS data (). When we allow for a free shape for the initial power spectrum, using our method, model 1 becomes by definition a good fit to WMAP ’s and decently fits the 11 points selected from SDSS (). By taking into account the features in the CMB we improve not only the fit to the CMB but also to the SDSS data.

However we found other models that can achieve a better fit. If we consider the cosmological parameter of model 3 (close to the best WMAP+SDSS fit), our reconstructed matter power spectrum gives a better fit (, see Fig. 5). In doing this, we show that the same model could fit both the WMAP ’s and SDSS matter power spectrum shape. This last point is our principal result, and suggests that an acceptable compromise that simultaneously fits both WMAP and SDSS may require a change respect to the concordance model, such as deviations from a power-law initial power spectrum.

We then investigated the effect of varying the baryonic fraction in model 3 and find that features from the CMB angular spectrum must be combined with the appropriate amount of baryons in order to fit the SDSS “oscillations” (see Fig. 5). Model 5, with a low baryonic content, is a relatively bad fit because the WMAP features seem almost out of phase with the SDSS features. While model 4 leads to a better agreement since the shapes of the SDSS and features are mimicked reasonably well. We furthermore note that model 3 falls in the observational range of both (, Freedman et al. 2003) and (, Cyburt et al. 2003; Cuoco et al. 2003).

Finally, looking at Fig. 6 suggests that the feature seen at in the SDSS is well reproduced by the reconstructed WMAP data with model 3 cosmological parameters.

More precisely, taking into account the effective number of degrees of freedom defined above, it is feasible to compare the probability of model 3 with the power law case, considering both the WMAP and SDSS data. For a power law the probability is 3.9%, considering as free parameters , , , , , and the amplitude of the fluctuations. While for model 3, with free parameters , , , , and the effective contribution from the smoothed amplitudes that amounts to 45.2, the probability is 11.9%.

In addition to a global goodness of fit, a statistical test to search for local features can be implemented. We combined the SDSS data and the SDSS-convolved WMAP-reconstructed power spectrum for model 3 to show more clearly where it is more likely to have deviations from a power law. The resulting power spectrum is depicted in Fig. 7 and its error covariance matrix and mean are given by

(6) |

(7) |

The feature at stands out as a deviation from a power law at a significance of one sigma. Also note that the interesting eighth and ninth points are positively correlated at a level of approximately 20%. Similar detections in other surveys suggest that this feature may be real and certainly encourage further searches in future experiments.

The resolution of the released SDSS data does not allow us to test the reality of various fine features that seem to be present in the reconstructed power spectrum. Along with progress in large scale structure measurements, a detailed understanding in forthcoming CMB experiments of possible systematics in beam uncertainties and eventual non-gaussianities coming from foregrounds and point-sources residues would definitely increase the chances of detection of features in the primordial power spectrum.

## 4 Discussion

In summary we have shown that a slight discrepancy in the baryonic fraction that arises from SDSS and WMAP data could be resolved if the primordial power spectrum is not fixed a priori in the form of a power law, but is constrained to satisfy CMB data when confronted with large-scale structure information and letting key cosmological parameters vary. Our method is feasible due to a high resolution reconstruction of the power spectrum from the CMB angular power spectrum, that is carried out for the first time in the full range considered, and provides evidence in favour of intrinsic features in the primordial power spectrum. In particular a feature at seems to occur in both the WMAP and SDSS data sets. This kind of technique offers an effective way to break the degeneracy between the baryon abundance and the shape of the power spectrum (in the conventional approach parametrized by the spectral index, ) and relevantly brings also to a measure of the galaxy bias.

A power spectrum with the observed features brings to a good fit of SDSS data without the necessity of a large baryon fraction. More generally, if the initial power law hypothesis is relaxed, deviations seem to induce a somewhat better concordance picture from the combination of CMB, large-scale structure, BBN and Hubble constant measurements. This could have important consequences for fundamental physics.

## Acknowledgments

DTV would like to thank Y. Hoffman, P. Ferreira, C. Skordis and K. Moodley for useful discussions. DTV acknowledges a Scatcherd Scholarship. MD acknowledges financial support provided through the E. U. Human Potential Program under contract HPRN-CT-2002-00124, CMBNET.

## References

- Atrio-Barandela et al. (2001) Atrio-Barandela F., et al., 2001, ApJ, 559, 1
- Barriga et al. (2001) Barriga J., Gatzanaga E., Santos M. G., Sarkar S., 2001, MNRAS, 324, 977
- Blanchard et al. (2003) Blanchard A., Douspis M., Rowan-Robinson M., Sarkar S., 2003, A&A, 412, 35
- Bridle et al. (2003) Bridle S. L., et al., 2003, MNRAS, 342, L72
- Bucher et al. (2004) Bucher M., Dunkley J., Ferreira P. G., Moodley K., Skordis C., 2004, astro-ph/0401417
- Cuoco et al. (2003) Cuoco A., et al., 2003, astro-ph/0307213
- Cyburt et al. (2003) Cyburt R. H., et al., 2003, Phys. Lett. B, 567, 227
- Einasto et al. (1997) Einasto J., et al., 1997, Nature, 385, 139
- Freedman et al. (2001) Freedman W. L., et al., 2001, ApJ, 553, 47
- Gawiser & Silk (1998) Gawiser E., Silk J., 1998, Science, 280, 1405
- Hamilton et al. (2000) Hamilton A. J. S., et al., 2000, MNRAS, 317, L23
- Hannestad (2004) Hannestad S., 2004, JCAP, 0404, 002
- Kogo et al. (2004) Kogo N., Matsumiya M., Sasaki M., Yokoyama J., 2004, ApJ, 607, 32
- Martin & Ringeval (2004) Martin J., Ringeval C., 2004, astro-ph/0402609
- Matsumiya et al. (2003) Matsumiya M., Sasaki M., Yokoyama J., 2003, JCAP, 0302, 003
- Miller & Batuski (2001) Miller C. J., Batuski D. J., 2001, ApJ, 551, 635
- Mukherjee & Wang (2003) Mukherjee P., Wang Y., 2003, ApJ, 599, 1
- Percival et al. (2001) Percival W. J., et al., 2001, MNRAS, 327, 1297
- Pope et al. (2004) Pope A. C., et al., 2004, ApJ, 607, 655
- Seljak & Zaldarriaga (1996) Seljak U., Zaldarriaga M., 1996, ApJ, 469, 437
- Shafieloo & Souradeep (2004) Shafieloo A., Souradeep T., 2004, Phys. Rev. D, 70, 043523
- Spergel et al. (2003) Spergel D. N., et al., 2003, ApJS, 148, 175
- Tegmark et al. (2003a) Tegmark M., et al., 2003a, ApJ, 606, 702
- Tegmark et al. (2003b) Tegmark M., et al., 2003b, Phys. Rev. D, 69, 103501
- Tegmark & Zaldarriaga (2002) Tegmark M., Zaldarriaga M., 2002, Phys. Rev. D, 66, 103508
- Verde et al. (2003) Verde L., et al., 2003, ApJS, 148, 195