Four-year COBE normalization of inflationary cosmologies

TL;DR

This work addresses normalizing inflationary perturbations to the four-year COBE data, explicitly including the gravitational-wave contribution and the scalar–tensor link via slow-roll parameters. It develops fitting functions for the COBE normalization $\delta_H(n,r)$ and the inflationary energy scale $V_*^{1/4}$, using a pivot scale $k_*=7 a_0 H_0$ and next-order corrections, while assessing the validity of a power-law spectrum over COBE scales. The paper provides a worked example with the quadratic potential, demonstrates robustness to reasonable parameter changes, and extends the analysis to nonzero $\Lambda$ in an Appendix. The results offer a practical, accurate toolkit for normalizing the present-day matter power spectrum for large-scale structure studies, accommodating tilted spectra and tensor modes from inflation. Overall, it delivers compact, widely usable formulae that link inflationary microphysics to observables on large scales, with extensions to cosmological-constant scenarios.

Abstract

We supply fitting formulae enabling the normalization of slow-roll inflation models to the four-year COBE data. We fully include the effect of the gravitational wave modes, including the predicted relation of the amplitude of these modes to that of the density perturbations. We provide the normalization of the matter power spectrum, which can be directly used for large-scale structure studies. The normalization for tilted spectra is a special case. We also provide fitting functions for the inflationary energy scale of COBE-normalized models and discuss the validity of approximating the spectra by power-laws. In an Appendix, we extend our analysis to include models with a cosmological constant, both with and without gravitational waves.