Higher-order gravity models: corrections up to cubic curvature invariants and inflation

TL;DR

This work addresses inflation within gravity theories extended by higher-order curvature invariants up to mass-dimension six. Starting from the Jordan frame, the authors derive the full set of field equations and, on a flat FLRW background, recast the dynamics as a four-dimensional autonomous system; they then specialize to the inflationary model $R+R^{2}+RR_{\mu\nu}R^{\mu\nu}$ by setting $\alpha_{0}=\beta_{0}=0$ and linearize in the small parameter $|\gamma_{0}|$. In the Einstein frame, the model exhibits a slow-roll attractor and a stable end-point analogous to Starobinsky inflation, with an effective potential $V(\varphi)\approx \frac{1}{6}\left(1-e^{-\varphi}-\frac{13}{144}\gamma_{0}e^{\varphi}\right)^{2}$ and slow-roll parameters that yield $n_{s}$ and $r$ close to observational bounds. The analytical expressions for $n_{s}$ and $r$ show consistency with Planck, BICEP/Keck, and BAO data for $|\gamma_{0}|\lesssim 10^{-3}$, while negative $\gamma_{0}$ can improve alignment with ACT, Planck, and DESI results, suggesting higher-order corrections may refine inflationary cosmology. The work points toward further studies of non-linear $\gamma_{0}$ effects and perturbative analyses to better constrain these higher-order corrections against upcoming observations.

Abstract

We construct a higher-order gravity model including all corrections up to mass dimension six. Starting from the Jordan frame, we derive the field equations and specialize to the FLRW background, where the dynamics take the form of a four-dimensional autonomous system. Focusing on the $R+R^{2}+RR_{μν}R^{μν}$ case, we obtain linearized equations in the parameter $γ_{0}$ and analyze the resulting phase space. The model exhibits the main desirable features of an inflationary regime, with a slow-roll attractor and a stable critical point corresponding to the end of inflation. Analytical expressions for the scalar spectral index $n_{s}$ and the tensor-to-scalar ratio $r$ show that the model is consistent with Planck, BICEP/Keck, and BAO data if $|γ_{0}|\lesssim 10^{-3}$. Moreover, negative values of $γ_{0}$ restore compatibility with recent ACT, Planck, and DESI results, suggesting that higher-order corrections may be relevant in refining inflationary cosmology.

Figures (2)

• Figure 1: Phase spaces of the $R+R^{2}+RR_{\mu\nu}R^{\mu\nu}$ model for $\gamma_{0}=10^{-3}$ (Left) and $\gamma_{0}=-10^{-3}$ (Right). The blue point in the second plot represents the critical point $P_{c}=\left( 3.83,0\right)$.
• Figure 2: The blue contours correspond to $68\%$ and $95\%$ C.L. constraints on $n_{s}\times r_{0.002}$ given by Planck plus BICEP3/Keck plus BAO data BICEP3. The black circles represent the Starobinsky model ($\gamma_{0}=0$) for $N=50$ (smaller one) and $N=60$ (bigger one). As $\gamma_{0}$ increases (decreases), the curves move to the left (right) light green (light purple) region. The grey circles represent the upper limits for $\gamma_{0}$ associated with $95\%$ C.L.. Table \ref{['tab:grey_circles']} provides the $\gamma_{0}$ numerical values associated with the grey circles.