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Quintessential Inflation in Palatini $F(R,X)$ gravity

Konstantinos Dimopoulos, Christian Dioguardi, Gert Hütsi, Antonio Racioppi

TL;DR

This paper investigates quintessential inflation within Palatini $F(R,X)$ gravity, showing that a quadratic $F(R,X)$ can render the Peebles–Vilenkin potential observationally viable by generating an inflationary plateau and a subsequent kination phase. It derives inflationary observables, analyzes reheating and the GW-overproduction constraints during kination, and demonstrates that a PV-like potential with $k=q=4$ works in the quadratic case only if reheating occurs so as to avoid GW overproduction, potentially requiring heavy-particle production. The authors also explore higher-order $F(R,X)_{>2}$ models, finding that an exponential PV tail can yield viable inflation and a quintessence tail, but similar GW/reheating constraints persist, often demanding additional mechanisms. Overall, Palatini $F(R,X)$ gravity offers a promising route to unify early inflation with late-time acceleration, though achieving all cosmological constraints simultaneously often requires nonstandard reheating or new particle physics ingredients. Simple exponential tails, by contrast, struggle to supply a consistent quintessence tail within these frameworks without extra features.

Abstract

Palatini $F(R,X)$ gravity, with $X$ the inflaton kinetic term, proved to be a powerful framework for generating asymptotically flat inflaton potentials. Here we show that a quadratic Palatini $F(R,X)$ restores compatibility with the observational data of the Peebles-Vilenkin quintessential inflation model. Moreover, the same can be achieved with an exponential version of the Peebles-Vilenkin potential if embedded in a Palatini $F(R,X)$ of order higher than two.

Quintessential Inflation in Palatini $F(R,X)$ gravity

TL;DR

This paper investigates quintessential inflation within Palatini gravity, showing that a quadratic can render the Peebles–Vilenkin potential observationally viable by generating an inflationary plateau and a subsequent kination phase. It derives inflationary observables, analyzes reheating and the GW-overproduction constraints during kination, and demonstrates that a PV-like potential with works in the quadratic case only if reheating occurs so as to avoid GW overproduction, potentially requiring heavy-particle production. The authors also explore higher-order models, finding that an exponential PV tail can yield viable inflation and a quintessence tail, but similar GW/reheating constraints persist, often demanding additional mechanisms. Overall, Palatini gravity offers a promising route to unify early inflation with late-time acceleration, though achieving all cosmological constraints simultaneously often requires nonstandard reheating or new particle physics ingredients. Simple exponential tails, by contrast, struggle to supply a consistent quintessence tail within these frameworks without extra features.

Abstract

Palatini gravity, with the inflaton kinetic term, proved to be a powerful framework for generating asymptotically flat inflaton potentials. Here we show that a quadratic Palatini restores compatibility with the observational data of the Peebles-Vilenkin quintessential inflation model. Moreover, the same can be achieved with an exponential version of the Peebles-Vilenkin potential if embedded in a Palatini of order higher than two.

Paper Structure

This paper contains 13 sections, 63 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Palatini $F(R,X)$ framework
  3. Quintessential inflation for $F(R_X)=R_X + \alpha R_X^2$
  4. Inflation
  5. Kination
  6. Reheating
  7. Quintessence
  8. The case of the exponential tail
  9. Quintessential inflation for $F(R_X)_{>2}$
  10. Exponential PV potential
  11. A note on the exponential tail
  12. Conclusions
  13. Constraints from the overproduction of GWs

Figures (5)

  • Figure 1: The PV potential (black), and the modified PV potential (blue) with $\alpha =2\cdot 10^8$, $q=k=4$, predicting viable CMB observables $r = 0.017$, $n_s=0.966$ (at $N_e = 60$) and a quintessential tail the solves the coincidence problem at the mass scale $M=1.38\cdot 10^{-13}$. We also show $\phi_N$ (star) and $\phi_{end}$ (dot) in the same color code. $\phi_N$ is not visible for the original PV potential as it lays at $V(\phi_N)\sim 10^{-8}$. We notice that while the potential is modified at the inflation scale, it remains unchanged in the tail.
  • Figure 2: $r$ vs. $n_s$ (a), $r$ vs. alpha (b), $\alpha$ vs. $n_s$ (c), $\lambda$ vs. $\alpha$ (d) for the PV potential with $k=2$ (blue), $k=4$ (red), $k=8$ (green) for $N_e =60$ (thick) and $N_e = 70$ (dashed). The dots represents the predictions of the of the original PV potential in the same color code. For each value of $k$ the plot is obtained fixing $\lambda$ by imposing the condition on the observed amplitude of the scalar perturbations $A_s = 2.1 \cdot 10^{-9}$ and varying $\alpha$ in the range $0<\alpha<10^{13}$ (i.e. from small to large couplings of the higher order term). The gray regions indicate the 95% (dark-gray) and 68% (light-gray) confidence levels (CL), respectively, based on the latest combination of Planck, BICEP/Keck, and BAO data BICEP:2021xfz.
  • Figure 3: a) Evolution of the barotropic parameter of the universe in terms of the elapsing $e$-folds number $N_e$ for the benchmark potential in Fig.\ref{['fig:U_peebles_potential']} with $\alpha = 2\cdot 10^8$, $q=k=4$ and $M = 1.38\cdot 10^{-13}$. The plot shows the natural appearance of a kination phase $w=1$ right after the end of inflation (Kin). As radiation domination (RD) begins $w$ drops to $1/3$. After that we have matter domination (MD) with $w=0$. Finally in the recent time the scalar field energy density becomes again the dominant source of energy-density in the universe (DE). Today we have $w =-\Omega_\Lambda \simeq -0.7$. The vertical lines denote the corresponding $e$-folds number $N_e$ for transitions between different epochs. b) Evolution of the energy densities for the scalar field $\rho_\phi$, the radiation fluid $\rho_r$ and the matter fluid $\rho_M$ in units of $m_P^4=1$. The vertical lines denote the corresponding $e$-folds number $N_e$ for the transitions between different epochs. It is assumed that radiation is originally generated at the end of inflation (e.g. by Ricci reheating Dimopoulos:2018wfgOpferkuch:2019zbdBettoni:2021zhq).
  • Figure 4: $r$ vs. $n_s$ (a), $r$ vs. $\alpha$ (b), $\alpha$ vs. $n_s$ (c), $\lambda$ vs. $\alpha$ (d) for the quadratic model with $V(\phi) = V_0 e^{-\lambda \phi}$ for $N_e =60$. The relation between $\alpha$ and $\lambda$ is imposed by fixing $A_s \sim 2.1\cdot 10^{-9}$. The gray regions indicate the 95% (dark-gray) and 68% (light-gray) confidence levels (CL), respectively, based on the latest combination of Planck, BICEP/Keck, and BAO data BICEP:2021xfz.
  • Figure 5: Exponential potential for $M=10^{-3}$ (blue), $M=10^{-5}$ (red), $M=10^{-8}$ (green) and $\alpha = 10^{10}$ and corresponding $\lambda =2.05$ fixed by setting $A_s \sim 2.1 \cdot 10^{-9}$. We also show $\phi_N$ (star) and $\phi_{end}$ (dot) in the same color code. Changing $M$ amounts to a shift in the potential along $\phi$-axis.