Weakly model-independent determination of total expansion during inflation

1. Problem Statement

The paper addresses a fundamental gap in connecting the inflationary epoch to the standard Hot Big Bang cosmology: the reheating epoch.

• The Challenge: Inflation ends with the inflaton field dominating the energy density. To transition to the radiation-dominated era (necessary for Big Bang Nucleosynthesis), the inflaton must decay into standard model particles. This process (reheating) is highly model-dependent and poorly understood. It involves complex non-perturbative dynamics (e.g., preheating, rescattering) or unknown decay widths.
• The Consequence: The total number of e-folds ($N_k$) a perturbation mode experiences from horizon exit during inflation to the end of inflation depends critically on the duration and dynamics of reheating. Without a precise model of reheating, determining $N_k$ (and thus constraining inflation models) is ambiguous.
• The Goal: To derive a systematic, weakly model-independent expression for the total expansion ($N_k$) that isolates the unknown reheating dynamics from observable quantities and model-independent parameters.

2. Methodology

The authors develop a formalism that separates the total expansion into model-independent observational inputs and model-dependent reheating dynamics.

A. Decomposition of $N_k$

Starting from the ratio of the comoving Hubble scale at horizon crossing ($k$) to the present day, the authors derive the standard relation for the number of e-folds ($N_k$):
$$N_k = -\ln\left(\frac{k}{a_0 H_0}\right) - N_{rh} + \ln\left(\frac{a_{rh}}{a_0}\right) + \ln\left(\frac{H_k}{H_0}\right)$$
where $N_{rh}$ is the e-folds during reheating.

B. Parametrization of Reheating Dynamics

Instead of assuming a constant equation of state ($w_{rh}$) or a specific decay width, the authors assume the equation of state during reheating, $w_{rh}$, is a smooth function of the normalized e-folds $n = N/N_{rh}$ (where $n \in [0, 1]$).

• Boundary Conditions: $w_{rh}(n=0) = -1/3$ (end of inflation) and $w_{rh}(n=1) = 1/3$ (start of radiation domination).
• The Integral $W(\alpha)$: They define a parameter $W$ representing the integral of the equation of state profile:
  $$W(\alpha) = \int_0^1 w_{rh}(n) \, dn$$
  This parameter encapsulates the entire shape of the reheating history into a single number of order $\mathcal{O}(1)$.

C. Derivation of the Master Equation

By integrating the continuity equation with the general $w_{rh}(n)$, they express the energy density at the end of reheating ($\rho_{rh}$) and the reheating temperature ($T_{rh}$) in terms of $W$. Substituting these into the expression for $N_k$, they arrive at a master equation (Eq. 16):
$$N_k = \underbrace{\text{Observational Terms}}_{\text{Model Independent}} + \underbrace{\frac{1}{3(1+W)} \ln\left(\frac{\rho_e}{m_{Pl}^4}\right) + \frac{1-3W}{3(1+W)} \ln\left(\frac{T_{rh}}{m_{Pl}}\right)}_{\text{Model Dependent}}$$

• Model Independent: Terms involving $k, H_0, T_0, g_*$, and the scalar power spectrum $P_R$ and tensor-to-scalar ratio $r$.
• Model Dependent: Encapsulated entirely in the inflation endpoint energy density ($\rho_e$), the reheating temperature ($T_{rh}$), and the shape parameter $W$.

3. Key Contributions

1. Isolation of Model Dependence: The paper demonstrates that all model dependence regarding the dynamics of reheating can be sequestered into the single parameter $W$ (the average equation of state) and the boundary conditions ($\rho_e, T_{rh}$). This allows constraints on inflation models without specifying the microscopic reheating mechanism.
2. Refined Inflation Endpoint: The authors provide a more accurate determination of the inflaton field value at the end of inflation ($\phi_e$) by solving the equation of motion without relying solely on the slow-roll approximation $\epsilon_V=1$. They show that $\phi_e$ depends on the second slow-roll parameter $\eta_e$ and is robust against perturbative decay widths.
3. Analysis of $w_{rh}$ Profiles: The authors construct explicit, physically motivated examples of $w_{rh}(n)$ profiles (e.g., polynomial, hyperbolic tangent, and "bump" functions) to show how different shapes affect $W$ and consequently $N_k$.
4. Degeneracy and Breaking Mechanisms:
   • Degeneracy: They prove that if the universe is a single barotropic fluid with constant relativistic degrees of freedom ($g_*$), $N_k$ depends only on the average $W$. Different profiles with the same average are degenerate.
   • Breaking Degeneracy: They identify two ways to lift this degeneracy:
     1. Perturbative Observables: Observables sensitive to local time derivatives (e.g., the gravitational wave spectrum $\Omega_{GW}$) depend on the specific shape of $w_{rh}(n)$, not just the average.
     2. Varying $g_*$: If the effective number of relativistic species $g_*$ changes during reheating (e.g., due to phase transitions), the mapping between energy density and temperature becomes profile-dependent, breaking the degeneracy even at the background level.

4. Results

• Sensitivity to Shape: For a fixed reheating temperature $T_{rh}$, varying the shape parameter $\alpha$ (which controls the profile of $w_{rh}$) can change the total expansion $N_k$ by as much as $\Delta N_k \approx 10$.
  • Example: A "slower" reheating (where $w_{rh}$ stays closer to $-1/3$ for longer) results in a smaller $N_k$.
  • Example: If $w_{rh} > 1/3$ (e.g., kinetic domination), $N_k$ increases significantly.
• Instantaneous Reheating Limit: The formalism naturally recovers the instantaneous reheating limit when $N_{rh} \to 0$ (or $W=1/3$), where $T_{rh}$ becomes independent of the duration.
• MCMC Constraints: Using a toy model with a varying $g_*$, the authors show that precise measurements of $N_k$ can constrain the shape parameter $\alpha$ of the reheating profile, effectively breaking the degeneracy between different reheating histories that share the same average equation of state.

5. Significance

• Universal Applicability: The approach allows cosmologists to constrain inflationary models (via $P_R$ and $r$) without needing to commit to a specific, often speculative, reheating mechanism.
• Bridging Theory and Observation: It provides a rigorous framework to interpret CMB data (like Planck or future CMB-S4) in the context of the unknown reheating epoch.
• New Diagnostic Tool: By highlighting that $N_k$ is sensitive to the shape of reheating only when $g_*$ varies or when combined with gravitational wave data, the paper guides future observational strategies. It suggests that to fully reconstruct the reheating history, one must look beyond the background expansion (CMB) to perturbative observables (Gravitational Waves).
• Phenomenological Utility: The parametrization of $w_{rh}$ as a function of e-folds allows for numerical fitting of unknown dynamics, making the study of reheating accessible even without a microscopic derivation of the inflaton potential.

In summary, the paper successfully decouples the uncertainties of the reheating epoch from inflationary predictions, providing a robust, "weakly model-independent" tool to analyze the total expansion of the universe and offering clear pathways to eventually resolve the details of reheating.