[Paper Review] CMB Constraints on Reheating Models with Varying Equation of State

R. C. Freitas, S. V. B. Gonçalves|arXiv (Cornell University)|Sep 28, 2015

Gas Dynamics and Kinetic Theory24 references3 citations

TL;DR

This paper investigates cosmological constraints on reheating models with time-varying equation of state (EoS) parameters during the post-inflationary era, using CMB data to link reheating parameters (e.g., reheating temperature $T_{\text{re}}$, e-folding number $N_{\text{re}}$) to inflationary observables like the scalar spectral index $n_S$. It shows that varying EoS models—particularly continuous and discontinuous transitions from $\omega_{\text{re}}=0$ to $\omega_{\text{re}}\approx1/3$—yield tighter, narrower constraints on $n_S$ compared to constant-EoS models, with the continuous model being robust to the sharpness parameter $\kappa$, and confirms that constant-EoS approximations are reasonable but miss subtle features.

ABSTRACT

The temperature at the end of reheating and the length of this cosmological phase can be bound to the inflationary observables if one considers the cosmological evolution from the time of Hubble crossing until today. There are many examples in the literature where it is made for single-field inflationary models and a constant equation of state during reheating. We adopt two simple varying equation of state parameters during reheating, combine the allowed range of the reheating parameters with the observational limits of the scalar perturbations spectral index and compare the constraints of some inflationary models with the case of a constant equation of state parameter during reheating.

Motivation & Objective

• To improve constraints on reheating parameters by modeling the equation of state (EoS) during reheating as time-varying rather than constant.
• To investigate how different functional forms of the EoS—discontinuous and continuous transitions—impact the link between reheating and inflationary observables.
• To test whether the standard assumption of constant EoS during reheating is sufficient or if time-varying EoS introduces measurable, observable features in CMB data.

Proposed method

• Modeling reheating with a step-function EoS parameter $\omega_{\text{re}}$ that jumps from 0 to 0.2–0.3 at a critical time.
• Introducing a continuous EoS transition via a parameter $\kappa$ controlling the sharpness of the jump from $\omega_{\text{pre}}=0$ to $\omega_{\text{re}}=1/3$, with the limit $\kappa \to \infty$ recovering the discontinuous case.
• Deriving the evolution of the scale factor and Hubble parameter using the Friedmann equations with time-dependent $\omega_{\text{re}}$, linking reheating to inflationary observables.
• Using the CMB power spectrum and observational constraints on the scalar spectral index $n_S$ to derive bounds on $T_{\text{re}}$ and $N_{\text{re}}$ for different EoS models.
• Comparing results from three models: constant $\omega_{\text{re}}$, discontinuous jump, and continuous transition, with the latter two introducing an additional parameter $\kappa$.
• Applying conformal transformations to map non-minimally coupled Higgs inflation to the Starobinsky model, enabling direct comparison of predictions across models.

Experimental results

Research questions

• RQ1 How do varying equation of state parameters during reheating affect the constraints on the scalar spectral index $n_S$ derived from CMB data?
• RQ2 Can a continuous transition in the reheating EoS parameter $\omega_{\text{re}}$ from 0 to $1/3$ yield results robust to the transition sharpness parameter $\kappa$?
• RQ3 How do the allowed regions in the $N_{\text{re}} \times n_S$ and $T_{\text{re}} \times n_S$ planes differ between constant-EoS and time-varying EoS models?
• RQ4 Does the discontinuous EoS model recover the expected behavior in the limit of instantaneous reheating ($N_{\text{re}} \to 0$), and if not, how does the continuous model resolve this?
• RQ5 To what extent do varying EoS models refine or alter the standard constraints derived under the assumption of constant $\omega_{\text{re}}$?

Key findings

• The regions in the $N_{\text{re}} \times n_S$ and $T_{\text{re}} \times n_S$ planes are significantly narrower when using time-varying EoS models compared to the standard constant-EoS assumption.
• The continuous EoS model is robust to the transition sharpness parameter $\kappa$, as the final results do not depend on $\kappa$ for large $\kappa$, confirming the physical consistency of the model.
• The discontinuous EoS model does not converge to the same point in the $N_{\text{re}} \to 0$ limit, but this issue is resolved in the continuous model, which smoothly approaches the expected behavior.
• A bend in the $n_S$-trajectory occurs at the fixed value of $N_{\text{pre}}$, corresponding to the transition point from $\omega_{\text{pre}}=0$ to $\omega_{\text{re}}=0.2$--$0.3$, indicating a physical signature of the EoS change.
• The continuous EoS model slightly dislocates the $n_S$ value in favor of a larger scalar spectral index compared to the discontinuous case, showing a measurable correction.
• The results confirm that while constant-EoS models are a good approximation, time-varying EoS models reveal subtle but non-negligible features in the reheating-inflation connection, especially in the shape and width of allowed parameter regions.