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Temperature definitions and phase transitions within non-minimal large and small inflationary potentials

Jesus Anaya-Galeana, Orlando Luongo, Hernando Quevedo

Abstract

We explore and compare two distinct temperature definitions for scalar field inflation in the context of small- and large-field potentials. The first is based on a real gas, fluid-like temperature, $T_{RG}$, while the second corresponds to a relativistic species-like temperature, $T_{RS}$. We derive the fundamental thermodynamic relations for both and analyze their implications for the most viable inflationary potentials, consistent with Planck constraints. We also investigate non-minimally coupled scenarios, finding that $T_{RS}$ is the most self-consistent choice, as it decreases during inflation, satisfies standard thermodynamic laws, and exhibits frame-independent behavior in both the Jordan and Einstein frames. Remarkably, the $T_{RS}$ approach shows that the inflaton's dynamics is well-described by Van der Waals-like isotherms, linking inflationary evolution to thermodynamic phase transitions. We find that the onset of inflation is associated with a phase transition acting as the ``trigger'' of the inflationary epoch. Our analysis highlights inconsistencies in the hilltop potential and, more generally, in small-field potentials unless a non-minimal coupling is introduced. Conversely, the Starobinsky and $α$-attractor models emerge as the most suitable paradigms. We further show that \emph{frame independence} is achieved only for coupling values $ζ\leq 1/6$, supporting very small values. Finally, our study of natural inflation with non-minimal coupling reveals a strong dependence on the coupling parameter, where bounds associated with thermodynamic phase transitions coincide with observationally viable ranges, suggesting that thermodynamic considerations may provide an additional criterion to discriminate among inflationary scenarios.

Temperature definitions and phase transitions within non-minimal large and small inflationary potentials

Abstract

We explore and compare two distinct temperature definitions for scalar field inflation in the context of small- and large-field potentials. The first is based on a real gas, fluid-like temperature, , while the second corresponds to a relativistic species-like temperature, . We derive the fundamental thermodynamic relations for both and analyze their implications for the most viable inflationary potentials, consistent with Planck constraints. We also investigate non-minimally coupled scenarios, finding that is the most self-consistent choice, as it decreases during inflation, satisfies standard thermodynamic laws, and exhibits frame-independent behavior in both the Jordan and Einstein frames. Remarkably, the approach shows that the inflaton's dynamics is well-described by Van der Waals-like isotherms, linking inflationary evolution to thermodynamic phase transitions. We find that the onset of inflation is associated with a phase transition acting as the ``trigger'' of the inflationary epoch. Our analysis highlights inconsistencies in the hilltop potential and, more generally, in small-field potentials unless a non-minimal coupling is introduced. Conversely, the Starobinsky and -attractor models emerge as the most suitable paradigms. We further show that \emph{frame independence} is achieved only for coupling values , supporting very small values. Finally, our study of natural inflation with non-minimal coupling reveals a strong dependence on the coupling parameter, where bounds associated with thermodynamic phase transitions coincide with observationally viable ranges, suggesting that thermodynamic considerations may provide an additional criterion to discriminate among inflationary scenarios.

Paper Structure

This paper contains 30 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inflationary set up
  3. Minimally coupled scalar field
  4. Non-minimally coupled scalar field
  5. Temperature definitions for the inflaton field
  6. Real gas fluid-like temperature
  7. Relativistic species-like temperature
  8. Fundamental Equation for $T_{RG}$
  9. Isotherms
  10. Stability Conditions
  11. Critical Points
  12. Fundamental Equation for $T_{RS}$
  13. Isotherms
  14. Stability Conditions
  15. Critical Points
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Real gas fluid-like temperature (black line) and relativistic species-like temperature (dashed line) as a function of $x=\phi / M_{Pl}$. The temperature is fixed so that at the end of inflation we have $T_{RG}(x_f) = T_{RS}(x_f) = T_f$ . The blue dashed-dotted vertical represents the value of $x$ when inflation started (we suppose $60$ e-foldings), whereas the black dotted vertical line represents the value of $x$ when inflation ends. On the top left Starobinsky, top right hilltop ($n=4$), lower left non-minimally coupling in the Einstein frame, lower right non-minimally coupling in the Jordan frame
  • Figure 2: Isotherms of the relativistic species-like temperature for different temperature values, presented on a log-log scale. The blue dashed-dotted vertical line indicates the volume $V$ at the beginning of inflation (assuming $60$ e-folds), while the black dotted vertical line indicates the volume $V$ at the end of inflation. The top panel shows the non-minimally coupled potential in the Einstein frame, and the bottom panel shows it in the Jordan frame. The isotherms in both frames exhibit a shape characteristic of Van der Waals fluids, suggesting that thermodynamic phase transitions may occur.