A universal bound on the duration of a kination era

TL;DR

The paper analyzes how primordial curvature fluctuations source a kination era with $\omega=1$ and shows that scalar-fluctuation growth generates a radiation-like component that ends kination after about $N_{\rm KD}\approx 11$ e-folds. By connecting the induced energy-density fluctuations to $\Delta N_{\rm eff}$, the authors derive novel, $r$-independent bounds on the duration of kination from scalar modes and fluctuation domination, with the Planck constraint $\Delta N_{\rm eff}\lesssim 0.34$ implying $N_{\rm KD}\lesssim 10$. They also contrast these with the traditional tensor-based bound and show the scalar-bound is typically stronger, especially when $\epsilon(k_{\rm KD})\ll 1$. The work highlights how pre-BBN stiff phases are tightly constrained by CMB measurements and sketches how future observations could further tighten these limits.

Abstract

We show that primordial adiabatic curvature fluctuations generate an instability of the scalar field sourcing a kination era. We demonstrate that the generated higher Fourier modes constitute a radiation-like component dominating over the kination background after about $11$ e-folds of cosmic expansion. Current constraints on the extra number of neutrino flavors $ΔN_{\rm eff}$ thus imply the observational bound of approximately 10 e-folds, representing the most stringent bound to date on the stiffness of the equation of state of the pre-Big-Bang-Nucleosynthesis universe.

Figures (2)

• Figure 1: During kination, inflationary curvature fluctuations (green) source kination fluctuations (red), which either terminate the kination era if $a_{\rm RD}>a_{\rm NL}$ or contribute to $N_{\rm eff}$ if $a_{\rm RD}<a_{\rm NL}$. The former is excluded by Eq. \ref{['eq:N_KD_bound_fluctuations']}. We consider kination (blue) either immediately after inflation (purple) if $a_{\rm KD}=a_{\rm end}$, as in quintessential inflation scenario Peebles:1998qnDimopoulos:2001ix, or after unspecified post-inflationary dynamics (orange), which is modeled to be radiation-like for simplicity and a conservative result. $\delta \rho_1$ and $\delta \rho_2$ are the linear and quadratic corrections to the kination background defined in Eqs. \ref{['eq:delta_phi_1']} and \ref{['eq:delta_phi_2']}. The latter implies that kination fluctuations have a non-vanishing volume average and contributes to $\Delta N_{\rm eff}$. Super-horizon energy fluctuations (green) evolve as $\delta \rho_{\mathbf{k}} \propto (1+\omega)\,\mathcal{R}_{\mathbf{k}}/a^2$, see Eqs. \ref{['eq:R_vs_Phi']} and \ref{['eq:Poisson']}. A sudden rise occurs at the end of inflation when $1+\omega = 2\epsilon/3$ initially $\ll 1$ grows to $\mathcal{O}(1)$, with $\epsilon$ being the slow-roll parameter, cf. Eq. \ref{['eq:epsilon_def']}. We picture the super-horizon fluctuations in dashed since they depend on the gauge, here chosen to be the Comoving gauge. We show the mode $k=\mathcal{H}_{\rm KD}$ that re-enters the horizon when kination starts since it is the mode that dominates the contribution to the volume average $\left<\delta \rho_2\right>$, cf. Eqs. \ref{['eq:rho_fluct_main']} and \ref{['eq:Omega_deltaphi']}, and therefore to $\Delta N_{\rm eff}$. The duration of kination is $N_{\rm KD}\equiv \ln(a_{\rm RD}/a_{\rm KD})$. The quantities $a_{\rm end}$, $a_{\rm KD}$, $a_{\rm RD}$, $a_{\rm NL}$ are the scales factors at the end of inflation, start of kination, end of the kination, and start of non-linear regime.
• Figure 2: Observational bounds on the duration of a kination era due to the linear growth of tensor (Eq. \ref{['eq:N_KD_bound_tensor']} and blue region) and scalar modes (Eq. \ref{['eq:N_KD_bound_scalar']} and brown region) violating the observational contraints $\Delta N_{\rm eff}\lesssim 0.34$ ($2\sigma$) Planck:2018vyg. The purple region is a physical upper limit on the kination duration due to domination by the radiation-like fluctuations, cf. Eq. \ref{['eq:N_KD_bound_fluctuations']}. The power spectra $\mathcal{P}_t(k)$ and $\mathcal{P}_{\mathcal{R}}(k)$ are either given by the power-laws in Eqs. \ref{['eq:Delta_R_k']} and \ref{['eq:Delta_tensor_k']} for the solid lines or by the predictions from Starobinsky inflation model in Eq. \ref{['eq:power_spectra_scalar_tensor_slow_roll']} for the dashed lines. The brown arrow indicates the uncertainty in the spectral-index running $\alpha_s$ in Eq. \ref{['eq:alpha_s_posterior']}, for extrapolating $\mathcal{P}_{\mathcal{R}}(k)$ above the CMB pivot scale $k_{\star}=0.05~\rm Mpc^{-1}$. The solid brown and purple lines assumes $\alpha_s$ to be given by the mean of Eq. \ref{['eq:alpha_s_posterior']}. In the green region, kination would end after the onset of BBN around $T_{\rm BBN}\simeq 5~\rm MeV$. In the red region, kination would start at a Hubble scale larger than the value $H_{\rm end}$ predicted in Starobinsky inflation (dashed) or its maximal value allowed by the BICEP/Keck bound $r\lesssim 0.036$BICEP:2021xfz (solid).