Reheating as a variational probe of cosmological observables

This paper proposes a variational framework that treats reheating as a constrained optimization problem to identify equation-of-state histories that extremize specific cosmological observables, demonstrating that different signals like gravitational waves and primordial black holes select distinct regions of post-inflationary expansion histories without relying on microscopic models.

The Gist

The Big Picture: The Universe's "Missing Link"

Imagine the Big Bang as a massive explosion that created the universe. But there's a problem: right after that explosion, the universe was cold and empty, filled only with the energy of a mysterious field called "inflation." To get to the hot, bustling universe we see today (filled with stars, planets, and us), that energy had to be transferred to normal matter.

This transfer process is called Reheating. Think of it like a chef taking a giant block of ice (inflation energy) and trying to melt it into a perfect soup (the hot Big Bang).

The problem is, we don't know how the chef did it. Did they use a microwave? A slow stove? Did they stir it fast or slow? We only know the starting point (ice) and the ending point (soup). The paper asks: If we look at the final soup, can we figure out exactly how the chef cooked it?

The New Approach: "Reverse Engineering" the Recipe

Usually, scientists try to guess the recipe by making up a specific story about the chef's tools (microscopic physics). This paper takes a different approach. Instead of guessing the tools, the author asks: "What specific cooking style would produce the most delicious soup for a specific ingredient?"

The author treats the history of the universe's expansion (the "cooking style") as a variable that can be tweaked. They use a mathematical tool called a Variational Principle.

The Analogy of the Hiking Trail:
Imagine you are trying to find the best path up a mountain.

• Traditional method: You assume the mountain has a specific shape (a specific model of physics) and try to find the path on that shape.
• This paper's method: You don't assume the mountain's shape. Instead, you ask: "If I want to see the Sunset (Observable A), what path should I take? If I want to see the Moonrise (Observable B), what path should I take?"

The paper finds that the path to the Sunset looks completely different from the path to the Moonrise. This proves that different things we observe in the universe "select" different histories of how the universe expanded.

The Three "Observables" (The Different Views)

The paper tests three different "views" of the universe to see what kind of reheating history they prefer.

1. Prompt Gravitational Waves (The "Instant Flash")

• What it is: Ripples in space-time created immediately during the heating process.
• The Result: To maximize this signal, the universe needs to stay "cold" (slowly expanding) for as long as possible, only heating up at the very last second.
• The Metaphor: Imagine a car trying to get the most fuel efficiency. It should drive very slowly and steadily, only speeding up right before it reaches the finish line. The "Prompt GW" view prefers a history where the universe drags its feet until the very end.

2. Induced Gravitational Waves (The "Echo")

• What it is: Ripples created later, caused by the interaction of smaller waves.
• The Result: To maximize this, the universe needs a long, steady "mushy" phase (matter-like) in the middle, and the main action needs to happen late in the process.
• The Metaphor: Think of a drum. If you want a deep, resonant echo, you need to hit the drum and let it ring out for a long time. The "Induced GW" view prefers a history where the universe lingers in a specific state for a long time, letting the "echo" build up before it finishes.

3. Primordial Black Holes (The "Collapse")

• What it is: Tiny black holes formed from clumps of energy in the early universe.
• The Result: To make the most black holes, the universe needs that same long "mushy" phase, but the clumping needs to start happening as early as possible.
• The Metaphor: Imagine a snowball rolling down a hill. To make the biggest snowball, you want the hill to be long and flat (so it can gather snow), but you want to start rolling immediately. The "Black Hole" view prefers a history where the universe stays in a "clumping-friendly" state for a long time, but starts the process right away.

The Key Takeaway

The most important discovery in this paper is that there is no single "best" history for the universe.

• If you look at Prompt Gravitational Waves, the universe looks like it dragged its feet.
• If you look at Induced Gravitational Waves, it looks like it lingered in the middle.
• If you look at Black Holes, it looks like it lingered but started early.

The Conclusion:
Cosmological observables (like gravitational waves or black holes) act like different "lenses." Each lens focuses on a different part of the universe's history. By using this new mathematical method, scientists can systematically explore all the possible ways the universe could have expanded, without needing to know the tiny, invisible details of the physics that caused it. It turns the study of the early universe from "guessing the model" into "mapping the landscape of possibilities."

Technical Summary

Technical Summary: Reheating as a Variational Probe of Cosmological Observables

Problem Statement
The reheating epoch, which bridges cosmic inflation and the standard radiation-dominated Big Bang, is typically characterized by coarse-grained parameters such as an averaged equation of state ($w$) or a reheating duration. However, the actual evolution of the equation of state, $w(N)$, as a function of the number of e-folds $N$, remains largely unconstrained. While the boundary conditions are fixed by the end of inflation ($w = -1/3$) and the onset of radiation domination ($w = 1/3$), the functional form of $w(N)$ in the interior is unknown due to the lack of a first-principles description of the microphysics governing energy transfer. Traditional approaches attempt to reconstruct reheating dynamics from specific microscopic models (e.g., inflaton decay or scattering), but these are model-dependent and often wash out distinct features once thermal equilibrium is reached. This paper addresses the inverse problem: rather than predicting observables from a specific reheating model, it asks which reheating histories are preferred by specific cosmological observables under minimal physical assumptions.

Methodology
The authors formulate reheating as a constrained variational problem in the space of equation-of-state histories. The core methodology involves:

1. Regularized Functional Formulation: To avoid ill-posed extremization problems that yield unphysical or highly oscillatory solutions due to the vast freedom of $w(N)$, the authors introduce a regularized functional:
   $$J[w] \equiv O[w] - \lambda I_{\text{prior}}[w]$$
   where $O[w]$ is the cosmological observable to be extremized. The prior term $I_{\text{prior}}[w]$ suppresses pathological histories by penalizing rapid variations (via a derivative term $\int (w')^2 dN$) and anchoring the evolution around a fiducial value $w^*$ (via a quadratic term $\int (w-w^*)^2 dN$). This prior encodes minimal expectations of smoothness without committing to a specific microphysical model.

2. Parametrization: The reheating history is parametrized as a deformation around a linear reference profile $w_{\text{ref}}(N)$ that connects the boundary conditions:
   $$w(N) = w_{\text{ref}}(N) + \sum_{k=1}^{K} b_k k^{-p} \phi_k(N)$$
   where $\phi_k(N) = \sin(k\pi N / N_{\text{rh}})$ are basis functions vanishing at the endpoints, and $b_k$ are coefficients optimized numerically. The factor $k^{-p}$ provides mild spectral regularization.

3. Application to Observables: The framework is applied to three distinct observables:

   • Prompt Gravitational Waves (GWs): Generated by inflaton scattering, dependent on the instantaneous inflaton energy density.
   • Induced GWs: Sourced by scalar perturbations, dependent on the temporal structure of the source evolution and non-local in the reheating background.
   • Primordial Black Holes (PBHs): Abundance determined by the collapse probability, which is exponentially sensitive to the collapse exponent and the scales at which efficient collapse begins.

Key Results
The variational analysis reveals that different observables select qualitatively distinct regions of the reheating-history space, defining unique "extremal directions":

• Prompt GWs: The extremized histories remain close to the lower boundary $w \approx -1/3$ for the majority of the reheating epoch, approaching radiation domination only near the end. This is driven by the production rate scaling as $\rho_\phi^2$; minimizing the dilution of the inflaton energy density (by keeping $w$ low) maximizes the GW abundance. The source contribution is concentrated near the end of reheating.
• Induced GWs: In contrast to prompt GWs, the optimization favors an extended matter-like phase ($N_{\text{eMD}} \gtrsim 3$). Furthermore, the solution exhibits a "late source localization," where the dominant contribution to the GW signal is delayed to later times during reheating. This suggests that induced GWs are sensitive to the accumulation of scalar perturbations over an extended period and the delay of their dilution.
• Primordial Black Holes (PBHs): The extremized histories also favor a prolonged matter-dominated phase ($N_{\text{eMD}} \approx 3.15$) but differ significantly from the induced GW case in the timing of the relevant contribution. PBH formation is optimized by shifting the onset of efficient collapse to earlier scales ($\log_{10} k_{\text{turn}} \approx -2.35$). The variational preference is robust across different collapse prescriptions (HYK, EGS, PT) and prior parameters, indicating that the observable is controlled by the scales where the collapse exponent first becomes sufficiently small.

Significance and Claims
The paper claims that cosmological observables act as probes of the "geometry" of the reheating-history space. By treating the reheating history itself as the object of optimization, the framework allows for a systematic exploration of post-inflationary expansion histories without relying on specific microscopic models.

The authors emphasize that:

1. Distinct Probes: Different observables do not merely constrain the same parameter space; they select fundamentally different evolutionary paths (e.g., prompt GWs prefer $w \approx -1/3$ throughout, while PBHs and induced GWs prefer extended matter domination but with different temporal localizations).
2. Model Independence: The approach shifts the focus from "reconstructing the true dynamics" to "probing observable imprints." It identifies which classes of histories are naturally selected by the physics of the observables themselves.
3. Robustness: The results are shown to be insensitive to the specific choice of regularization parameters or collapse prescriptions, suggesting that the identified extremal directions are intrinsic features of the observables.

The work concludes that this variational perspective provides a controlled, model-independent tool to explore the space of reheating histories, with potential future applications to other early-universe probes such as spectral distortions and non-Gaussianity.