Causality Bounds on the Primordial Power Spectrum

TL;DR

This work analyzes how causality constrains the EFT of inflation by requiring causal propagation for the comoving curvature perturbation, encoded in phase shifts of high-energy modes and their associated spatial shifts. By deriving bounds on the dispersion relation with $c_s^2$ and a higher-derivative term $\alpha$, the authors show that large growth of the primordial power spectrum at small scales is generically restricted by EFT validity and weak coupling, making substantial PBH or scalar-induced GW production difficult for natural Wilson coefficients. They study several transition parametrizations (Gaussian, tanh, split Gaussian) and demonstrate universal features: a luminal $c_s$ implies a free theory, and slow transitions can induce acausality unless tightly constrained. The analysis is extended to UV completions, where a causal two-field model preserves causality in the resulting single-field EFT, reinforcing the notion that UV physics can protect against causality violations. Overall, causality bounds significantly constrain the parameter space of inflationary EFTs aiming to generate features in the small-scale power spectrum.

Abstract

Effective field theories (EFTs) parametrize our ignorance of the underlying UV theory through their Wilson coefficients. However, not all values of these coefficients are consistent with fundamental physical principles. In this paper, we explore the consequences of imposing causal propagation on the comoving curvature perturbation in the EFT of inflation, particularly its impact on the primordial power spectrum and the effective sound speed $c_s^\text{eff}$. We investigate scenarios where $c_s^\text{eff}$ undergoes a transition, remaining consistent with CMB constraints at early times but later experiencing a drastic change, becoming highly subluminal. Such scenarios allow the primordial power spectrum to grow at small scales, potentially leading to the formation of primordial black holes or the generation of scalar-induced gravitational waves. We find the generic feature that in a causal theory, luminal sound speeds imply a free theory, effectively constraining the dynamics. Additionally, we obtain that when considering natural values for the Wilson coefficients, maintaining the validity of the EFT and the weakly coupled regime, and enforcing causal propagation of the EFT modes, the power spectrum cannot increase drastically. This imposes significant constraints on the parameter space of models aiming to produce such features.

Figures (14)

• Figure 1: The yellow region shows the allowed parameter space by the causality bounds from \ref{['eq:cbound']}. The blue region corresponds to the parameter space where the EFT is valid, $|\alpha k^2 \tau_{\mathrm{ini}}^2|<1$. We have set $\tau_{\mathrm{final}}=-1.5$ and $\tau_{\mathrm{ini}}=-5$.
• Figure 2: In these plots we consider the values $\epsilon=10^{-4}$, $\gamma=\eta=0$, $A_\alpha=10^{-7}$, $A_{c_s}=0.7$, $\sigma=1$, $N^*=5$, and $k/H=10^{4}$. LHS: Plot of the time evolution of $\alpha$ and $c_s^2$. The vertical black lines are placed at $N^*\pm 3 \sigma$. RHS: Comparison of WKB and numerical solutions to the equation of motion in Eq. \ref{['eq:eomNnofric']}. The WKB approximation shows the result using the leading order phase shift. The black line marks the breakdown of the WKB approximation at $N_\text{break}=0.9 N_{W_k=0}$ where $N_{W_k=0}=8.8$ is the e-fold where $W_k=0$.
• Figure 3: On the LHS we see the plot for the effective sound speed with Eq. \ref{['eq:gaussians']} for varying values of $c_s^2(N^*)=1-A_{c_s}$ and fixed $A_\alpha=10^{-5}$, $N^*=5$, $\sigma=2/3$, and $\epsilon=10^{-4}, \eta=\gamma=0$. The RHS shows the plot for the integrand in the causality bound as written in Eq. \ref{['eq:cboundIntegrand']} for the same set of parameters.
• Figure 4: Bounds on the $\alpha-c_s^2$ plane for different widths of the transition with $\epsilon=10^{-4}, \eta=\gamma=0$. We see that a luminal speed of sound leads to acausal propagation. This can be avoided if the width is small enough, but in that case, the EFT is no longer valid. The green region corresponds to the values of $\alpha$ and $c_s^2$ for which Eq. \ref{['alphacond']} is violated in the integration region, that is, the modes that are within the horizon before the transition is over cannot be described by the EFT.
• Figure 5: LHS: Bounds on the $\alpha-$width plane for $c_s^2(N^*)=0.001$. The contours show the maximum value of the power spectrum, which has been normalized to $2 \times 10^{-9}$ at the initial integration e-fold. We observe that as we increase the width, the maximum of the power spectrum grows, but around 9 e-folds, the theory becomes acausal and $\max{(\Delta_\zeta^2)}\lesssim 4 \times 10^{-7}$ for a causal theory. RHS: Plot of the power spectrum for different values of momentum normalized to the scale where the power spectrum peaks. The free parameters have the same values as in the LHS plot and additionally, $\alpha=10^{-10}$ and $\text{width}=7$.