Natural inflation: Particle physics models, power-law spectra for large-scale structure, and constraints from the Cosmic Background Explorer

Abstract

We discuss the particle physics basis for models of natural inflation with pseudo Nambu-Goldstone bosons and study the consequences for large-scale structure of the nonscale-invariant density fluctuation spectra that arise in natural inflation and other models. A pseudo Nambu-Goldstone boson, with a potential of the form $V(\ensuremath{\varphi})={\ensuremath{\Lambda}}^{4}[1\ifmmode\pm\else\textpm\fi{}cos(\frac{\ensuremath{\varphi}}{f})]$, can naturally give rise to an epoch of inflation in the early Universe, if $f\ensuremath{\sim}{M}_{\mathrm{Pl}}$ and $\ensuremath{\Lambda}\ensuremath{\sim}{M}_{\mathrm{GUT}}$. Such mass scales arise in particle physics models with a gauge group that becomes strongly interacting at the grand unified theory scale. We work out a specific particle physics example based on the multiple gaugino condensation scenario in superstring theory. We then study the cosmological evolution of and constraints upon these inflation models numerically and analytically. To obtain sufficient inflation with a probability of order 1 and a high enough post-inflation reheat temperature for baryogenesis, we require $f\ensuremath{\gtrsim}0.3{M}_{\mathrm{Pl}}$. The primordial density fluctuation spectrum generated by quantum fluctuations in $\ensuremath{\varphi}$ is a non-scale-invariant power law $P(k)\ensuremath{\propto}{k}^{{n}_{s}}$, with ${n}_{s}\ensuremath{\simeq}1\ensuremath{-}(\frac{{M}_{\mathrm{Pl}}^{2}}{8\ensuremath{\pi}{f}^{2}})$ leading to more power on large length scales than the ${n}_{s}=1$ Harrison-Zeldovich spectrum. (For the reader primarily interested in large-scale structure, the discussion of this topic is presented in Sec. IV and is intended to be nearly self-contained.) We pay special attention to the prospects of using the enhanced power to explain the otherwise puzzling large-scale clustering of galaxies and clusters and their flows. We find that the standard cold dark matter (CDM) model with $0\ensuremath{\lesssim}{n}_{s}\ensuremath{\lesssim}0.6$ could in principle explain these data. However, the microwave background anisotropies recently detected by the Cosmic Background Explorer (COBE) imply such low primordial amplitudes for these CDM models (that is, bias factors ${b}_{8}\ensuremath{\gtrsim}2$ for ${n}_{s}\ensuremath{\lesssim}0.6$) that galaxy formation would occur too late to be viable and the large-scale galaxy velocities would be too small. In fact, combining the COBE results with the requirement of sufficiently early galaxy formation (${z}_{\mathrm{GF}}>2$) leads to the constraint ${n}_{s}\ensuremath{\gtrsim}0.63$, which corresponds to $f\ensuremath{\gtrsim}0.3{M}_{\mathrm{Pl}}$ for natural inflation (virtually the same as the sufficient reheating constraint). A comparable bound ${n}_{s}\ensuremath{\gtrsim}0.72$ arises by combining COBE with the inferred large-scale flows. For other inflation models, such as extended inflation and inflation with exponential potentials, which give rise to initial fluctuation spectra that are power laws through the 3 decades in wavelength probed by large-scale observations, gravity waves can produce a significant fraction of the COBE signal (while they are negligible for natural inflation); for these models, our corresponding COBE constraints on ${n}_{s}$ are therefore tighter, ${n}_{s}>0.76$ (from ${z}_{\mathrm{GF}}>2$) and ${n}_{s}>0.89$ (from large-scale flows). Combined with other constraints on the Brans-Dicke parameter (which effectively imply ${n}_{s}<0.77\ensuremath{-}0.84$), this leaves little or no room for most extended inflation models. Chaotic inflation models with power-law potentials have ${n}_{s}\ensuremath{\gtrsim}0.95$ over observable wavelengths and so are not affected. Although no single value of the spectral index ${n}_{s}$ in the standard cold dark matter model universally fits the data, a value ${n}_{s}\ensuremath{\le}1$ may be combined with other variations of the standard CDM framework to explain the large-scale structure. For example, if the baryon density is as high as ${\ensuremath{\Omega}}_{B}=0.1$ or the Hubble parameter as low as ${H}_{0}=40$ km/(sec/Mpc), then ${n}_{s}\ensuremath{\sim}0.7$ with CDM would be at least marginally consistent with the large-scale structure data [e.g., the automatic plate measuring (APM) survey angular correlation function], COBE, large-scale velocities, and the requirement of sufficiently early structure formation.