Inflation in theories with broken diffeomorphisms

TL;DR

This work investigates inflation in theories where diffeomorphism invariance is broken to transverse diffeomorphisms (TDiff) in the inflaton sector. By adopting a power-law volume function $f(g)=g^{\alpha}$, the authors derive a modified slow-roll framework with $H_K(Y)=Y^{1-2\alpha}$, yielding new pre-factors in the slow-roll parameters and a distinct evolution for the auxiliary variable $Y$. They compute the primordial power spectrum through a Mukhanov-Sasaki analysis, obtaining a modified spectral index $n_S$ and amplitude $A_S$ that depend on $\alpha$, $\varepsilon$, and $\delta$, and they compare these predictions with Planck and ACT data for power-law potentials $V(\phi)=\lambda\phi^p$, finding that $\alpha>1/2$ can improve agreement for some $p$ while quadratic models remain challenged. Beyond inflation, the TDdiff framework imposes a strong constraint on post-inflationary dynamics, eliminating inflaton oscillations and leading to a strong TDiff (STR) regime with phase-space structures such as brick-wall and bifurcation points; numerical and asymptotic analyses for a quadratic potential reveal a universal late-time behavior where the condensate acts as matter with $w\to 0$ and $H(t)\sim t^{-1}$. Overall, the paper demonstrates that TDdiff inflation yields distinctive observables and rich post-inflationary dynamics, motivating further exploration of alternative TDdiff functions and reheating mechanisms.

Abstract

We analyze the impact of breaking diffeomorphism invariance in the inflaton sector. In particular, we consider inflaton models which are invariant under the subgroup of transverse diffeomorphisms and address the possibility of implementing a slow-roll phase. We obtain the corresponding expressions for relevant quantities such as the slow-roll parameters and the number of $e$-folds, and derive the primordial power-spectrum of curvature perturbations. The scalar spectral index features modifications which are confronted with CMB data from Planck and ACT. We study in detail the quadratic potential model, combining asymptotic and numerical analysis. We show that the post-inflationary behavior can be drastically different from the diffeomorphism-invariant case, exhibiting novel dynamical regimes.

Figures (6)

• Figure 1: Excluded and allowed values of $\alpha$ as a function of the power $p$ for a given number of $e$-folds $N$. The striped area represents the excluded $\alpha$ values, which corresponds to $N>N_{\rm max}$. The colored region represents the allowed $\alpha$ values. The blue colored line is the border of the region given by \ref{['eq:condition alpha']}, presenting a vertical asymptote at $p=(2N-1)/(N-1)$ and a horizontal asymptote at $\alpha=N/[2(N-1)]$.
• Figure 2: Constraints on the scalar and tensor primordial spectra at $k_\ast=0.05 \,\,\text{Mpc}^{-1}$ for 95% CL region (solid line) and 68% CL (dashed line), represented in the $n_S$--$r$ plane. The panel combines datasets from ACT (blue), re-run data of Planck 2018 with Sroll2 dataset (orange) and P-ACT (purple). All datasets make use of DESI Data Release 1 (DR1) and the contours for DESI DR2 have not been included, as the changes for P-ACT are hardly noticeable. In all cases, the dataset includes measurements of CMB lensing, BAO (LB) and CMB B-modes of polarization (BK18). The quadratic potential has been represented along variations of the TDiff parameter $\alpha$ for the number of $e$-folds of inflation $N\in[50,60]$. The Diff case is represented in black and overlaps with values of $\alpha\leq1/2$. The teal colored line represents values of $\alpha>\frac{1}{2}$.
• Figure 3: Constraints on the scalar and tensor primordial spectra at $k_\ast=0.05 \,\,\text{Mpc}^{-1}$ for 95% CL region (solid line) and 68% CL (dashed line), represented in the $n_S$--$r$ plane. The various colored bands show examples of power-law potentials along variations of the TDiff parameter $\alpha$ for the number of $e$-folds of inflation $N\in[50,60]$. The black colored lines correspond in every panel to the Diff case. The curves with $\alpha_p$ correspond to the transition value which indicates the change of the dominance for the slow-roll parameters. The $V\propto\phi$ plot lacks this line, as $\varepsilon$ always dominates.
• Figure 4: Constraints on the scalar and tensor primordial spectra at $k_\ast=0.002 \,\,\text{Mpc}^{-1}$ for 95% CL region (solid line) and 68% CL (dashed line), represented in the $n_S$--$r$ plane. The panels combine datasets from Planck 2018: temperature--temperature power spectra (TT), temperature-E mode polarization power spectra (TE) and polarization-polarization power spectra (EE); low multipole E-mode polarization (lowE), CMB lensing (lensing), B-mode polarization (BK15), from BICEP-Keck 2015; and BAO. The color bands represent power-law potentials along variations of the TDiff parameter $\alpha$ for the number of $e$-folds of inflation $N\in[50,60]$. The Diff case is represented in black. The quadratic potential behavior for $\alpha\leq1/2$ overlaps with the region $\alpha\to0$ (brown colored dotted line) of the rest of represented potentials; for $\alpha>1/2$ it decreases following the teal colored line.
• Figure 5: Phase portraits in the $\phi$--$\dot\phi$ plane, featuring $\{c_2,\alpha\}$ for each model. Left column: potential domination $c_2=-1$. Center column: kinetic domination $c_2=+1$. Right column: on top right, the limiting case $c_2=0$; on bottom right, the Diff oscillator case. The direction of the arrow indicates the evolution of the field regarding the sign of the velocity. The points marked as $P_i$ and $Q_i$ ($i=1,2$) represent bifurcation and "brick-wall" points, respectively. The different colors identify branches which cannot be connected.