Cosmological parameters from WMAP 5-year temperature maps

I calculate a hybrid cross-power spectrum estimator from the WMAP 5-year CMB temperature maps, discuss the goodness of fit, and then constrain cosmological parameters. The spectrum and results are generally consistent with previous results, though the power spectrum error bars are slightly smaller and there are small shifts at high $l$. The small improvement in error bars is obtained at very low numerical cost but does not significantly improve parameter constraints. I discuss the accuracy of the likelihood model and how constraints on the optical depth translate into constraints on the reionization history allowing for helium reionization. In the appendices I propose a simple reionization parametrization that determines the history in terms of a midpoint reionization redshift, and suggest a new likelihood approximation for chi-squared-like distributions with varying skewness.

Figure 1

The hybrid power spectrum (blue) compared with the WMAP estimators (red). The top panel is binned with ${\Delta }_l=31$, and error bars show $\pm 1\sigma$ error bars slightly offset for clarity; the dash-dotted line is $N_l^{\mathrm{eff}}$. The lower panel shows the estimators at each $l$, with the lines showing $2\sigma$ symmetric posterior diagonal errors from the assumed fiducial model; the slightly broader dashed lines are the corresponding diagonal errors from the WMAP likelihood code.

Figure 2

Diagonal contributions to the likelihood from each $l$ of the observed cross-spectrum ${\widehat{C}}_l$. The lines show the smoothed rms, 0.01 and 0.001 confidence limits estimated from simulations of the combined map spectra (bottom to top). Points move around a bit depending on which model is used to calculate the contribution to the likelihood.

Figure 3

Marginalized parameter constraints of this paper (solid lines) compared to using the WMAP likelihood code with the spectrum and covariance from this paper (dashed line) and the result from the original WMAP likelihood code and spectrum (dot-dashed line). The bottom five parameters (except $A_{\mathrm{SZ}}$) are derived from the other parameters.

Figure 4

The fractional difference in diagonal error as a function of $l$ (i.e. the square root of the diagonal of the ${\widehat{C}}_l$ covariance) compared to that from 1500 simulations of the hybrid cross-spectrum estimator. The solid line shows the analytic approximation for the combined-map hybrid spectrum (having smoothed the input power spectrum with ${\Delta }_l\sim 25$ for $l>200$). The dashed line is from 5000 simulations of the combined-map hybrid estimator. The dash-dotted line shows the result used by the WMAP likelihood code supplied on LAMBDA (which is discontinuous at $l=500$ where they switch from uniform to inverse-noise weighting). Simulation results are smoothed over ${\Delta }_l=20$ to reduce sampling noise. The analytic result is accurate on small scales because the noise dominates.

Figure 5

The skewness (${\kappa }^{(3)}/[{\kappa }^{(2)}{]}^{3/2}$) as a fraction of that expected from a reduced chi-squared distribution on the full sky. The solid lines is the prediction for the hybrid estimator based on assuming a reduced chi-squared distribution determined by the variance and degrees of freedom. Points are binned estimates from 10 000 simulations with error bars estimated from the values in each bin. The dashed line is a possible improved model with $\alpha =1-f_{\mathrm{sky}}^{\mathrm{eff}}$.

Figure 6

Three recombination histories all with $\tau =0.09$. The dashed line is the model typically used by CMBFAST and CAMB prior to March 2008 with $f=1$. The black line is the new model with ${\Delta }_z=0.5$, the red line with ${\Delta }_z=1.5$.