Determination of inflationary observables by cosmic microwave background anisotropy experiments

Abstract

Inflation produces nearly scale-invariant scalar and tensor perturbation spectra which lead to anisotropy in the cosmic microwave background (CMB). The amplitudes and shapes of these spectra can be parametrized by ${\mathit{Q}}_{\mathit{S}}^{2}$, r\ensuremath{\equiv}${\mathit{Q}}_{\mathit{T}}^{2}$/${\mathit{Q}}_{\mathit{S}}^{2}$, ${\mathit{n}}_{\mathit{S}}$, and ${\mathit{n}}_{\mathit{T}}$ where ${\mathit{Q}}_{\mathit{S}}^{2}$ and ${\mathit{Q}}_{\mathit{T}}^{2}$ are the scalar and tensor contributions to the square of the CMB quadrupole and ${\mathit{n}}_{\mathit{S}}$ and ${\mathit{n}}_{\mathit{T}}$ are the power-law spectral indices. Even if we restrict ourselves to information from angles greater than one-third of a degree, three of these observables can be measured with some precision. The combination ${105}_{\mathit{S}}^{1\mathrm{\ensuremath{-}}\mathit{n}}$${\mathit{Q}}_{\mathit{S}}^{2}$ can be known to better than \ifmmode\pm\else\textpm\fi{}0.3%. The scalar index ${\mathit{n}}_{\mathit{S}}$ can be determined to better than \ifmmode\pm\else\textpm\fi{}0.02. The ratio r can be known to about \ifmmode\pm\else\textpm\fi{}0.1 for ${\mathit{n}}_{\mathit{S}}$\ensuremath{\simeq}1 and slightly better for smaller ${\mathit{n}}_{\mathit{S}}$. The precision with which ${\mathit{n}}_{\mathit{T}}$ can be measured depends weakly on ${\mathit{n}}_{\mathit{S}}$ and strongly on r. For ${\mathit{n}}_{\mathit{S}}$\ensuremath{\simeq}1, ${\mathit{n}}_{\mathit{T}}$ can be determined with a precision of about \ifmmode\pm\else\textpm\fi{}0.056(1.5+r)/r. A full-sky experiment with a 20 arc min beam using technology available today, similar to those being planned by several groups, can achieve the above precision. Good angular resolution is more important than high signal-to-noise ratio; for a given detector sensitivity and observing time a smaller beam provides more information than a larger beam. The uncertainties in ${\mathit{n}}_{\mathit{S}}$ and r are roughly proportional to the beam size. We briefly discuss the effects of uncertainty in the Hubble constant, baryon density, cosmological constant, and ionization history.