Cosmological parameter estimation and the inflationary cosmology

TL;DR

The paper analyzes strategies for cosmological parameter estimation within inflationary cosmology, emphasizing the necessity of accurate initial spectra ${\cal P}_{\cal R}(k)$ and ${\cal P}_h(k)$. It contrasts inflation-agnostic power-law fits with slow-roll–informed spectra, quantifying when second-order slow-roll corrections are essential for Planck-level precision. By deriving analytic slow-roll spectra up to second order and introducing robust error diagnostics, it shows that spectral-shape matters for certain inflation models and that a pipeline incorporating slow-roll predictions yields more reliable inflationary parameter inferences. The proposed testing framework enables consistent checks of slow-roll inflation, retrieval of the inflationary energy scale $H$, and a principled path to potential reconstruction, while also outlining how to interpret non-detections of tensor modes.

Abstract

We consider approaches to cosmological parameter estimation in the inflationary cosmology, focussing on the required accuracy of the initial power spectra. Parametrizing the spectra, for example by power-laws, is well suited to testing the inflationary paradigm but will only correctly estimate cosmological parameters if the parametrization is sufficiently accurate, and we investigate conditions under which this is achieved both for present data and for upcoming satellite data. If inflation is favoured, reliable estimation of its physical parameters requires an alternative approach adopting its detailed predictions. For slow-roll inflation, we investigate the accuracy of the predicted spectra at first and second order in the slow-roll expansion (presenting the complete second-order corrections for the tensors for the first time). We find that within the presently-allowed parameter space, there are regions where it will be necessary to include second-order corrections to reach the accuracy requirements of MAP and Planck satellite data. We end by proposing a data analysis pipeline appropriate for testing inflation and for cosmological parameter estimation from high-precision data.

Figures (12)

• Figure 1: The top panel shows the power spectra of scalar (upper lines) and tensor (lower lines) perturbations for our three models. The scalar spectra are normalized to ${\cal P}_{{\cal R}} = 2\times10^{-9}$ at the scale $k_* = 0.01h\hbox{Mpc}^{-1}$, which approximately matches the COBE normalization. The bottom panel shows the corresponding $C_{\ell}$ curves for a flat cosmological model with $\omega_{{\rm b}}=0.0200$, $\omega_{{\rm m}}=0.1268$ and $\omega_{\Lambda}=0.2958$ (implying $h=0.65$), and reionization optical depth $\tau = 0.05$, with the upper lines again the scalar contribution and the tensors considerably subdominant. Only the sum of the two can be detected, though they contribute differently to polarization anisotropies.
• Figure 2: Error curves for various fits to the scalar power spectrum for the false vacuum model. While the power-law fit is acceptable for fitting to present data, neglecting running affects the estimate of the power spectrum amplitude at the pivot point at the percent level.
• Figure 3: As Fig. \ref{['fig:errorS_fit_fv']} but for the arctan model. For fitting to data of the present quality, the inclusion of the running is required.
• Figure 4: As Fig. \ref{['fig:errorS_fit_fv']} but for the chaotic inflation model tensor spectrum. Although the percentage error is large for the scale-invariant fit, the absolute error is small compared to the scalar spectrum, and so the scale-invariant fit is still acceptable.
• Figure 5: Fitting the slow-roll shape to the arctan model. The errors should be compared with the errors in Fig. \ref{['fig:errorS_fit_wang']}. For this model, the second-order slow-roll shape provides a better fit that the power-law plus running shape.