Cosmological Perturbations in an Inflationary Universe

TL;DR

The work develops a comprehensive, gauge-invariant framework for cosmological perturbations in FRW spacetimes with multiple fluids and energy transfer, deriving exact, non slow-roll results for both adiabatic and entropic modes. By rotating field space in assisted inflation, it identifies a weighted mean field that drives a tractable, analytic attractor solution and yields explicit perturbation spectra around the attractor, including the curvature perturbation on large scales. The analysis demonstrates that, for the simplest preheating scenario, large-scale curvature perturbations remain effectively conserved, with any isocurvature or non-adiabatic contributions being suppressed or confined to small scales. Together, these results provide robust predictions for the evolution of curvature perturbations from inflation through preheating, and they outline clear extensions to multi-fluid dynamics and higher-dimensional cosmologies with potential observational consequences.

Abstract

After introducing gauge-invariant cosmological perturbation theory we give an improved set of governing equations for multiple fluids including energy transfer. Having defined adiabatic and entropic perturbations we derive the ``conservation law'' for the curvature perturbation on large scales using only the energy conservation equation. We then investigate the dynamics of assisted inflation. By choosing an appropriate rotation in field space we can write down explicitly the potential for the weighted mean field along the scaling solution and for fields orthogonal to it. This allows us to present analytic solutions describing homogeneous and inhomogeneous perturbations about the attractor solution without resorting to slow-roll approximations. Finally we analyze the simplest model of preheating analytically, and show that in linear perturbation theory the effect of preheating on the amplitude of the curvature perturbation on large scales is negligible. We end with some concluding remarks, possible extensions and an outlook to future work.

Figures (5)

• Figure 1: A schematic illustration of the separate universes picture, with the symbols as identified in the text.
• Figure 2: Evolution of four fields $\varphi_1$, $\varphi_2$, $\varphi_3$ and $\varphi_4$ (from bottom to top) during assisted inflation with $p_1=0.3$, $p_2=1$, $p_3=2$ and $p_4=7$.
• Figure 3: The field rotation in the two field case. The $\bar{\varphi}_2$ direction is along the attractor, $\bar{\sigma}_2$ orthogonal to it.
• Figure 4: Evolution of the fields $\bar{\sigma}_1$, $\bar{\sigma}_2$, and $\bar{\sigma}_3$, orthogonal to the scaling solution, in the assisted inflation model shown in Fig. \ref{['Fone']}.
• Figure 5: The power spectrum of the non-adiabatic curvature perturbation ${\cal P}_{\zeta_{{\rm nad}}}$, shown at four different times: from bottom to top $m\Delta t = 0$, $50$, $100$ and $150$. The parameters used were $g=10^{-3}$, $m=10^{-6} m_{{\rm Pl}}$ and $k_{\rm cut}=k_{\rm max}$.