Temperature-polarization Cross-power Spectrum Research Articles

Effect of the length of inflation on angular TT and TE power spectra in power-law inflation

The effect of the length of inflation on the power spectra of scalar and tensor perturbations is estimated using the power-law inflation model with a scale factor of a(η) = (−η)p = tq. Considering various pre-inflation models with radiation-dominated or scalar matter-dominated periods before inflation in combination with two matching conditions, the temperature angular power spectrum (TT) and temperature–polarization cross-power spectrum (TE) are calculated and a likelihood analysis is performed. It is shown that the discrepancies between the Wilkinson microwave anisotropy probe (WMAP) data and the ΛCDM model, such as suppression of the spectrum at l = 2, 3 and oscillatory behaviour, may be explained by the finite length of inflation model if the length of inflation is near 60 e-folds and q ⩾ 300. The proposed models retain similar values of χ2 to that achieved by the ΛCDM model with respect to fit to the WMAP data, but display different characteristics of the angular TE power spectra at l ⩽ 20.

First Year WMAP Observations

The results of the first year WMAP sky survey are full sky microwave maps in five frequency bands from 23 to 94 GHz. Calibration errors are < 0.5% and the low systematic error level is well specified. The cosmic microwave background (CMB) is separated from the foregrounds using the multifrequency data. The 2 ≤ l ≤ 900 anisotropy power spectrum is cosmic variance limited for l < 354 with a signal-to-noise ratio >1 per mode to l = 658. The temperature-polarization cross-power spectrum reveals both acoustic features and a large angle correlation from reionization. The optical depth of reionization is τ = 0.17 ± 0.04. A best-fit cosmological model to the CMB and other measures of large scale structure works remarkably well with only a few parameters. The age of the best-fit universe is t0 = 13.7 ± 0.2 Gyr old. The matter density is Ωmh2 = 0.135+0.008--0.009, the baryon density is Ωbh2 = 0.0224 ± 0.0009, and the total mass-energy of the universe is Ωtot = 1.02±0.02. For WMAP data alone, ns = 0.99 ± 0.04. The lack of CMB fluctuation power on the largest angular scales reported by COBE and confirmed by WMAP is intriguing. WMAP continues to operate, so results will improve.

Scaling defect decay and the reionization history of the Universe

We consider a model for the reionization history of the Universe in which a significant fraction of the observed optical depth is a result of direct reionization by the decay products of a scaling cosmic defect network. We show that such network can make a significant contribution to the reionization history of the Universe even if its energy density is very small (the defect energy density has to be greater than about ${10}^{\ensuremath{-}11}$ of the background density). We compute the Cosmic Microwave Background temperature, polarization and temperature-polarization cross power spectrum and show that a contribution to the observed optical depth due to the decay products of a scaling defect network may help to reconcile a high optical depth with a low redshift of complete reionization suggested by quasar data. However, if the energy density of defects is approximately a constant fraction of the background density then these models do not explain the large scale bump in the temperature-polarization cross power spectrum observed by Wilkinson Microwave Anisotropy Probe.

Cross-correlation studies with CMB polarization maps

The free-electron population during the reionized epoch rescatters the cosmic microwave background (CMB) temperature quadrupole and generates a now well-known polarization signal at large angular scales. While this contribution has been detected in the temperature-polarization cross power spectrum measured with Wilkinson Microwave Anisotropy Probe data, due to the large cosmic variance associated with anisotropy measurements at angular scales of tens of degrees only limited information related to reionization, such as the optical depth to electron scattering, can be extracted. The inhomogeneities in the free-electron population lead to an additional secondary polarization anisotropy contribution at arcminute scales. While the fluctuation amplitude, relative to dominant primordial fluctuations, is small, we suggest that a cross correlation between the arcminute scale CMB polarization data and a tracer field of the high redshift universe, such as through fluctuations captured by the 21 cm neutral hydrogen background or those in the infrared background related to the first protogalaxies, may allow one to study additional details related to reionization. For this purpose, we discuss an optimized higher order correlation measurement, in the form of a three-point function, including information from large angular scale CMB temperature anisotropies in addition to the arcminute scale polarization signal related to inhomogeneous reionization. We suggest that the proposed bispectrum can be measured with a substantial signal-to-noise ratio and does not require all-sky maps of CMB polarization or that of the tracer field. A measurement such as the one proposed may allow one to establish the epoch when CMB polarization related to reionization is generated and to address the question whether the universe was reionized once or twice.

First‐YearWilkinson Microwave Anisotropy Probe(WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks

The CMB has distinct peaks in both its temperature angular power spectrum (TT) and temperature-polarization cross-power spectrum (TE). From the WMAP data we find the first peak in the temperature spectrum at l = 220.1 +- 0.8 with an amplitude of 74.7 +- 0.5 microK; the first trough at l = 411.7 +- 3.5 with an amplitude of 41.0 +- 0.5 microK; and the second peak at l = 546 +- 10 with an amplitude of 48.8 +- 0.9 microK. The TE spectrum has an antipeak at l = 137 +- 9 with a cross-power of -35 +- 9 microK^2, and a peak at l = 329 +- 19 with cross-power 105 +- 18 microK^2. All uncertainties are 1 sigma and include calibration and beam errors. An intuition for how the data determine the cosmological parameters may be gained by limiting one's attention to a subset of parameters and their effects on the peak characteristics. We interpret the peaks in the context of a flat adiabatic LambdaCDM model with the goal of showing how the cosmic baryon density, Omega_b h^2, matter density, Omega_m h^2, scalar index, n_s, and age of the universe are encoded in their positions and amplitudes. To this end, we introduce a new scaling relation for the TE antipeak-to-peak amplitude ratio and recompute known related scaling relations for the TT spectrum in light of the WMAP data. From the scaling relations, we show that WMAP's tight bound on Omega_b h^2 is intimately linked to its robust detection of the first and second peaks of the TT spectrum.

First‐Year Wilkinson Microwave Anisotropy Probe ( WMAP ) Observations: Implications For Inflation

We confront predictions of inflationary scenarios with the Wilkinson Microwave Anisotropy Probe (WMAP) data, in combination with complementary small-scale cosmic microwave background (CMB) measurements and large-scale structure data. The WMAP detection of a large-angle anticorrelation in the temperature-polarization cross-power spectrum is the signature of adiabatic superhorizon fluctuations at the time of decoupling. The WMAP data are described by pure adiabatic fluctuations: we place an upper limit on a correlated cold dark matter (CDM) isocurvature component. Using WMAP constraints on the shape of the scalar power spectrum and the amplitude of gravity waves, we explore the parameter space of inflationary models that is consistent with the data. We place limits on inflationary models; for example, a minimally coupled λϕ4 is disfavored at more than 3 σ using WMAP data in combination with smaller scale CMB and large-scale structure survey data. The limits on the primordial parameters using WMAP data alone are ns(k0 = 0.002 Mpc-1) = 1.20, dns/d ln k = -0.077, A(k0 = 0.002 Mpc-1) = 0.71 (68% CL), and r(k0 = 0.002 Mpc-1) < 1.28 (95% CL).

Cosmological perturbations and the reionization epoch

We investigate the dependence of the epoch of reionization on the properties of cosmological perturbations, in the context of cosmologies permitted by WMAP. We compute the redshift of reionization using a simple model based on the Press–Schechter approximation. For a power-law initial spectrum we estimate that reionization is likely to occur at a redshift zreion= 17+10−7, consistent with the WMAP determination based on the temperature-polarization cross power spectrum. We estimate the delay in reionization if there is a negative running of the spectral index, as weakly indicated by WMAP. We then investigate the dependence of the reionization redshift on the nature of the initial perturbations. We consider chi-squared probability distribution functions with various degrees of freedom, motivated both by non-standard inflationary scenarios and by defect models. We find that in these models reionization is likely to occur much earlier, and to be a slower process, than in the case of initial Gaussian fluctuations. We also consider a hybrid model in which cosmic strings make an important contribution to the seed fluctuations on scales relevant for reionization. We find that in order for that model to agree with the latest WMAP results, the string contribution to the matter power spectrum on the standard 8 h−1 Mpc scale is likely to be at most at the level of 1 per cent, which imposes tight constraints on the value of the string mass per unit length.

First‐Year Wilkinson Microwave Anisotropy Probe ( WMAP ) Observations: Preliminary Maps and Basic Results

We present full sky microwave maps in five frequency bands (23 to 94 GHz) from the WMAP first year sky survey. Calibration errors are less than 0.5% and the low systematic error level is well specified. The cosmic microwave background (CMB) is separated from the foregrounds using multifrequency data. The sky maps are consistent with the 7 in. full-width at half-maximum (FWHM) Cosmic Background Explorer (COBE) maps. We report more precise, but consistent, dipole and quadrupole values. The CMB anisotropy obeys Gaussian statistics with -58 less than f(sub NL) less than 134 (95% CL). The 2 less than or = l less than or = 900 anisotropy power spectrum is cosmic variance limited for l less than 354 with a signal-to-noise ratio greater than 1 per mode to l = 658. The temperature-polarization cross-power spectrum reveals both acoustic features and a large angle correlation from reionization. The optical depth of reionization is tau = 0.17 +/- 0.04, which implies a reionization epoch of t(sub r) = 180(sup +220, sub -80) Myr (95% CL) after the Big Bang at a redshift of z(sub r) = 20(sup +10, sub -9) (95% CL) for a range of ionization scenarios. This early reionization is incompatible with the presence of a significant warm dark matter density. A best-fit cosmological model to the CMB and other measures of large scale structure works remarkably well with only a few parameters. The age of the best-fit universe is t(sub 0) = 13.7 +/- 0.2 Gyr old. Decoupling was t(sub dec) = 379(sup +8, sub -7)kyr after the Big Bang at a redshift of z(sub dec) = 1089 +/- 1. The thickness of the decoupling surface was Delta(sub z(sub dec)) = 195 +/- 2. The matter density of the universe is Omega(sub m)h(sup 2) = 0.135(sup +0.008, sub -0.009) the baryon density is Omega(sub b)h(sup 2) = 0.0224 +/- 0.0009, and the total mass-energy of the universe is Omega(sub tot) = 1.02 +/- 0.02. There is progressively less fluctuation power on smaller scales, from WMAP to fine scale CMB measurements to galaxies and finally to the Ly-alpha forest. This is accounted for with a running spectral index, significant at the approx. 2(sigma) level. The spectral index of scalar fluctuations is fit as n(sub s) = 0.93 +/-0.03 at wavenumber k(sub o) = 0.05/Mpc ((sub eff) approx. = 700), with a slope of dn(sub s)/d I(sub nk) = -0.031(sup + 0.016, sub -0.018) in the best-fit model.

Gravitational time delay effects on cosmic microwave background anisotropies

We study the effect of gravitational time delay on the power spectra and bispectra of the cosmic microwave background (CMB) temperature and polarization anisotropies. The time delay effect modulates the spatial surface at recombination on which temperature anisotropies are observed, typically by $\ensuremath{\sim}1$ Mpc. While this is a relatively large shift, its observable effects in the temperature and polarization fields are suppressed by geometric considerations. The leading order effect is from its correlation with the closely related gravitational lensing effect. The change to the temperature-polarization cross power spectrum is of order $0.1%$ and is hence comparable to the cosmic variance for the power in the multipoles around $l\ensuremath{\sim}1000.$ While unlikely to be extracted from the data in its own right, its omission in modeling would produce a systematic error comparable to this limiting statistical error and, in principle, is relevant for future high precision experiments. Contributions to the bispectra result mainly from correlations with the Sachs-Wolfe effect and may safely be neglected in a low density universe.