The correlation of peaks in the microwave background

We present accurate small-angle predictions of the correlation function of hotspots in the microwave background radiation for Gaussian theories such as those predicted in most inflation models. The correlation function of peaks above a certain threshold depends only on the threshold and the power spectrum of temperature fluctuations. Since these are both potentially observable quantities in a microwave background map, there are no adjustable parameters in the predictions. These correlations should therefore provide a powerful test of the Gaussian hypothesis, and provide a useful discriminant between inflation and topological defect models such as the cosmic string model. The correlations have a number of oscillatory features, which should be detectable at high signal-to-noise ratio with future satellite experiments such as MAP and Planck .     

Full-sky correlations of peaks in the microwave background

We compute precise predictions for the two-point correlation function of local maxima (or minima) in the temperature of the microwave background, under the assumption that it is a random Gaussian field. For a given power spectrum and peak threshold there are no adjustable parameters, and since this analysis does not make the small-angle approximation of Heavens & Sheth, it is essentially complete. We find oscillatory features which are absent in the temperature autocorrelation function, and we also find that the small-angle approximation to the peak–peak correlation function is accurate to better than 0.01 on all scales. These high-precision predictions can form the basis of a sensitive test of the Gaussian hypothesis with upcoming all-sky microwave background experiments MAP and Planck , affording a thorough test of the inflationary theory of the early Universe. To illustrate the effectiveness of the technique, we apply it to simulated maps of the microwave sky arising from the cosmic string model of structure formation, and compare the two-point correlation function of peaks with the bispectrum as a non-Gaussian discriminant. We also show how peak statistics can be a valuable tool in assessing and statistically removing contamination of the map by foreground point sources.     

Large-scale structure, the cosmic microwave background and primordial non-Gaussianity

Cosmic microwave background and large-scale structure data will shortly improve dramatically with the Microwave Anisotropy Probe and Planck Surveyor , and the Anglo-Australian 2-Degree Field and Sloan Digital Sky Survey. It is therefore timely to ask which of the microwave background and large-scale structure will provide a better probe of primordial non-Gaussianity. In this paper we consider this question, using the bispectrum as a discriminating statistic. We consider several non-Gaussian models and find that in each case the microwave background will provide a better probe of primordial non-Gaussianity. Our results suggest that if microwave background maps appear Gaussian, then apparent deviations from Gaussian initial conditions in galaxy surveys can be attributed with confidence to the effects of biasing. We demonstrate this precisely for the spatial bispectrum induced by local non-linear biasing.     

Cosmic microwave background bispectrum and slow-roll inflation

Recent tentative findings of non-Gaussian structure in the COBE -DMR data set have triggered renewed attention on candidate models from which such intrinsic signature could arise. In the framework of slow-roll inflation with built-in non-linearities in the inflaton field evolution, we present expressions for both the cosmic microwave background (CMB) skewness and the full angular bispectrum _ℓ1ℓ₂ℓ₃ in terms of the slow-roll parameters. We use an estimator for the angular bispectrum recently proposed in the literature and calculate its variance for an arbitrary ℓ _i multipole combination. We stress that a real detection of non-Gaussianity in the CMB would imply that an important component of the anisotropies arises from processes other than primordial quantum fluctuations. We further investigate the behaviour of the signal-to-(theoretical) noise ratio and demonstrate for generic inflationary models that it decreases in the limited range of small ℓs considered for increasing multipole ℓ, while the opposite applies for the standard _ℓs.     

Cosmic microwave background anisotropy: deviations from Gaussianity caused by non-linear gravity

Non-linear evolution of cosmological energy density fluctuations triggers deviations from Gaussianity in the temperature distribution of the cosmic microwave background. A method to estimate these deviations is proposed. N -body simulations – in a Λ cold dark matter cosmology – are used to simulate the strongly non-linear evolution of cosmological structures. It is proved that these simulations can be combined with the potential approximation to calculate the statistical moments of the cosmic microwave background anisotropies produced by non-linear gravity. Some of these moments are computed and the resulting values are different from those corresponding to Gaussianity.     

Fast parameter estimation from the cosmic microwave background power spectrum

The statistical properties of a map of the primary fluctuations in the cosmic microwave background (CMB) may be specified to high accuracy by a few thousand power spectra measurements, provided the fluctuations are Gaussian, yet the number of parameters relevant for the CMB is probably no more than ∼10–20. Consequently, there is a large degree of redundancy in the power spectrum data. In this paper, we show that the moped data compression technique can reduce the CMB power spectrum measurements to ∼10–20 numbers (one for each parameter), from which the cosmological parameters can be estimated virtually as accurately as from the complete power spectrum. Combined with recent advances in the speed of generation of theoretical power spectra, this offers opportunities for very fast parameter estimation from real and simulated CMB skies. The evaluation of the likelihood itself, at Planck resolution, is speeded up by factors up to ∼10⁸, ensuring that this step will not be the dominant part of the data analysis pipeline.     

Features in the primordial power spectrum: constraints from the cosmic microwave background and the limitation of the 2dF and SDSS redshift surveys to detect them

We allow a more general (step-function) form of the primordial power spectrum than the usual featureless power-law Harrison–Zeldovich (with spectral index   n =1)  power spectrum, and fit it to the latest cosmic microwave background data sets. Although the best-fitting initial power spectrum can differ significantly from the power-law shape, and contains a dip at scales   k ∼0.003  h  Mpc^-1  , we find that  Ω_m≈0.24  , consistent with previous analyses that assume power-law initial fluctuations. We also explore the feasibility of the early releases of the 2dF and Sloan Digital Sky Survey (SDSS) galaxy redshifts surveys to see these features, and we find that even if features exist in the primordial power spectrum, they are washed out by the window functions of the redshift surveys on scales   k <0.03  h  Mpc^-1  .     

Lines in the cosmic microwave background spectrum from the epoch of cosmological hydrogen recombination

We compute the spectral distortions of the cosmic microwave background (CMB) arising during the epoch of cosmological hydrogen recombination within the standard cosmological (concordance) model for frequencies in the range 1–3500 GHz. We follow the evolution of the populations of the hydrogen levels including states up to principle quantum number   n = 30  in the redshift range  500 ≤ z ≤ 3500  . All angular momentum substates are treated individually, resulting in a total number of 465 hydrogen levels. The evolution of the matter temperature and the fraction of electrons coming from He  ii are also included. We present a detailed discussion of the distortions arising from the main dipolar transitions, for example Lyman and Balmer series, as well as the emission due to the two-photon decay of the hydrogen 2s level. Furthermore, we investigate the robusteness of the results against changes in the number of shells considered. The resulting spectral distortions have a characteristic oscillatory behaviour, which might allow experimentalists to separate them from other backgrounds. The relative distortion of the spectrum exceeds a value of 10⁻⁷ at wavelengths longer than 21 cm. Our results also show the importance of detailed follow-up of the angular momentum substates, and their effect on the amplitude of the lines. The effect on the residual electron fraction is only moderate, and mainly occurs at low redshifts. The CMB angular power spectrum is changed by less than 1 per cent. Finally, our computations show that if the primordial radiation field is described by a pure blackbody, then there is no significant emission from any hydrogen transition at redshifts greater than   z ∼ 2000  . This is in contrast to some earlier works, where the existence of a 'pre-recombination' peak was claimed.     

Tests for primordial non-Gaussianity

We investigate the relative sensitivities of several tests for deviations from Gaussianity in the primordial distribution of density perturbations. We consider models for non-Gaussianity that mimic that which comes from inflation as well as that which comes from topological defects. The tests we consider involve the cosmic microwave background (CMB), large-scale structure, high-redshift galaxies, and the abundances and properties of clusters. We find that the CMB is superior at finding non-Gaussianity in the primordial gravitational potential (as inflation would produce), while observations of high-redshift galaxies are much better suited to find non-Gaussianity that resembles that expected from topological defects. We derive a simple expression that relates the abundance of high-redshift objects in non-Gaussian models to the primordial skewness.     

Global fits for the spectral index of the cosmological curvature perturbation

Best-fitting values of the spectral index of the curvature perturbation are presented, assuming the ΛCDM cosmology. Apart from the spectral index, the parameters are the Hubble parameter, the total matter density and the baryon density. The data points are intended to represent all measurements that are likely to affect the result significantly. The cosmic microwave anisotropy is represented by the COBE normalization, and heights of the first and second peaks are given by the latest Boomerang and Maxima data. The slope of the galaxy correlation function and the matter density contrast on the 8  h ⁻¹ Mpc scale are each represented by a data point, as are the expected values of the Hubble parameter and matter density. The 'low-deuterium' nucleosynthesis value of the baryon density provides a final data point, the fit giving a value higher by about one standard deviation. The reionization epoch is calculated from the model by assuming that it corresponds to the collapse of a fraction f ≳10⁻⁴ of matter. We consider the case of a scale-independent spectral index, and also the scale-dependent spectral index predicted by running mass models of inflation. In the former case, the result is compared with the prediction of models of inflation based on effective field theory, in which the field value is small on the Planck scale. A detailed comparison is made with other fits, and other approaches to the comparison with theory.     

A measurement at the first acoustic peak of the cosmic microwave background with the 33-GHz interferometer

This paper presents the results from the Jodrell BankInstituto de Astrofisicia de Canarias (IAC) two-element 33-GHz interferometer operated with an element separation of 32.9 wavelengths and hence sensitive to 1°-scale structure on the sky. The level of cosmic microwave background (CMB) fluctuations, assuming a flat CMB spatial power spectrum over the range of multipoles =208±18, was found using a likelihood analysis to be at the 68 per cent confidence level, after the subtraction of the contribution of monitored point sources. Other possible foreground contributions have been assessed and are expected to have negligible impact on this result.     

Cosmological parameters from velocities, cosmic microwave background and supernovae

We compare and combine likelihood functions of the cosmological parameters Ω_m, h and σ ₈, from peculiar velocities, cosmic microwave background (CMB) and type Ia supernovae. These three data sets directly probe the mass in the Universe, without the need to relate the galaxy distribution to the underlying mass via a 'biasing' relation. We include the recent results from the CMB experiments BOOMERANG and MAXIMA-1. Our analysis assumes a flat Λ cold dark matter (ΛCDM) cosmology with a scale-invariant adiabatic initial power spectrum and baryonic fraction as inferred from big-bang nucleosynthesis. We find that all three data sets agree well, overlapping significantly at the 2 σ level. This therefore justifies a joint analysis, in which we find a joint best-fitting point and 95 per cent confidence limits of     (0.17,0.39),     (0.64,0.86) and     (0.98,1.37). In terms of the natural parameter combinations for these data     (0.40,0.73),     (0.16,0.27). Also for the best-fitting point,     and the age of the Universe is 13.2 Gyr.