A critical value of the inflationary tensor-to-scalar ratio from inhomogeneous inflation

TL;DR

The paper addresses whether inflation can robustly start from generic inhomogeneous initial conditions in the context of α-attractor T-models. It employs full GR numerical relativity with equipartition initial data, mapping the characteristic scale μ to the tensor-to-scalar ratio via $r=\frac{16\pi\,μ^2}{m_P^2\,N_*^2}$ with $N_*\\approx60$, and identifies a critical μ, $μ_{crit}\approx0.020$–$0.025\,m_P$, corresponding to a lower bound on $r$ of order $\sim$ few × $10^{-6}$. The results show that inhomogeneities push inflation's viability to higher μ (and hence higher $r$), with the kinetic-energy component increasing robustness relative to gradient-only cases; a slingshot mechanism can, in some μ ranges, enhance $N_{ m max}$ beyond the homogeneous expectation. These findings suggest a measurable link between the early-universe initial conditions and inflationary observables, albeit with caveats about mode selection and the mean-field initialization. Overall, the work provides a principle for when a given inflationary model remains viable under generic inhomogeneities and how that viability translates into a lower bound on $r$, offering targets for future observations and further exploration of initial-condition measures in inflationary theory.

Abstract

We show that, for a given fixed value of the number of e-folds of the homogeneous solution, inflation succeeds with order unity inhomogeneities in the initial conditions above a characteristic value of the tensor-to-scalar ratio $r$. In practice, we work with an $α$-attractor $T$-model and vary its characteristic scale $μ$, keeping the initial inhomogeneities in both gradient and kinetic fields of order unity of the inflationary energy scale. Under these conditions, and assuming 100 e-folds for the homogeneous solution, the requirement for 60 e-folds of inflation occurs at a critical characteristic scale $μ_{crit} \approx 0.02m_{P}$, corresponding to an $r_{crit} \approx 10^{-6}$. Since increasing the amplitude of the inhomogeneities will make inflation less robust and hence require a higher characteristic scale in order for inflation to succeed, for a given number of e-folds achieved by the homogeneous solution $r_{crit}$ is a lower bound.

Figures (5)

• Figure 1: The dependence of the shape of the potential $V(\phi)$ for $T$-model $\alpha$-attractor with respect to $\mu$ values.
• Figure 2: Dependence of the maximum number of $e$-folds, $N_{\text{max}}$, on the characteristic scale $\mu$, with initial conditions $\langle\rho_{\nabla,0}\rangle = \langle\rho_{K,0}\rangle = V(\phi_0)$. The red-shaded area indicate a maximum number of e-folds $N_{\text{max}}<60$ which we define as failure of inflation, whereas the green-shaded area shows $N_{\text{max}}>60$ and represents successful inflation. Here the red branch corresponds to out-of-phase initial perturbations ($\theta = \pi$), and the green branch to in-phase ones ($\theta = 0$). The left panel shows the results for $0.014m_\mathrm{P}\leq\mu < 0.026m_\mathrm{P}$ and the right panel for $0.014m_\mathrm{P}\leq\mu \leq \sqrt{3/16\pi}m_\mathrm{P}$. We observe that the critical values for which there is a successful amount of inflation are $\mu = 0.02m_\mathrm{P}$ and $\mu = 0.025m_\mathrm{P}$ for the out-of-phase and in-phase perturbations respectively. These correspond to the critical tensor-to-scalar ratios $r = 5.6\times10^{-6}$ and $r = 8.7\times10^{-6}$.
• Figure 3: Evolution of the field profiles across the largest diagonal of the simulation $(0,0,0)\xrightarrow{}(L,L,L)$ for both the in-phase (left) and out-of-phase (right) initial field configurations with $\mu = 0.03 m_\mathrm{P}$. The shaded red region indicates the field values for which the potential cannot support slow-roll i.e. $\epsilon \geq 1$. For both cases we observe inflation succeeding in the centre but failing at the corners. However, for the in-phase case the central field initially explores higher up the positive-$\phi$ side of the potential, before being slingshot up the plateau by steep restorative field gradients to subsequently yield $> 600$e-folds of inflation.
• Figure 4: Results for the dependence of the maximum number of e-folds $N_{\text{max}}$ on the characteristic scale $\mu$ for the case without kinetic inhomogeneities. The red-shaded area shows a maximum number of e-folds $N_{\text{max}}<60$ which we define as failure, whereas the green-shaded area shows $N_{\text{max}}>60$ and represents successful inflation. Each dot shows the $N_{\text{max}}$ of that simulation, and we present with different colours three branches of fixed ratio $\langle \rho_{\nabla\,0}\rangle/V(\phi_0)$. For $\mu \gtrsim 0.017m_\mathrm{P}$ simulations with initial conditions such that $\langle\rho_{\nabla}\rangle= V(\phi_0)$ (black line) give $N_{\text{max}}>60$. This suggests a smaller lower bound on $\mu$ and thus to the $r_{\text{crit}} = 4.0\times10^{-6}$ compared to the case where also kinetic inhomogeneities are included. The blue branch represents initial conditions such that $\langle\rho_{\nabla\,0}\rangle =10V(\phi_0)$ and we obtain successul inflation for $\mu\geq0.03m_\mathrm{P}$ which gives $r_{\text{crit}} = 1.3\times10^{-5}$. We note that this seemingly monotonic behaviour between energy ratios and $\mu_{\text{crit}}$ holds only over large energy ranges -- small differences in energy ratios around unity do not necessarily lead to monotonic behaviour due to the presence of the minimum in the potential.
• Figure 5: The covergence of the value of $\phi$ at the centre of the simulation domain for $\mu = 0.03m_\mathrm{P}$ and $\langle\rho_{\nabla\, 0}\rangle = \langle\rho_{K\,0}\rangle = V(\phi_0)$ and $\theta = 0$.