Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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Reheating temperature in a sudden-decay modulus model

Physical scope

An exact algebraic deduction inside the explicitly defined instantaneous-decay prescription: a homogeneous quadratic scalar and a comoving radiation fluid with p_r=rho_r/3 supply the total pre-decay stress, the flat-FLRW Einstein equation holds with zero cosmological term, and all pre-decay energy is instantaneously transferred to an ideal thermal bath at H=Gamma. It does not derive a modulus decay width, thermalization, a continuous-decay correction, a BBN threshold, or a string compactification.

Statement

Use Einstein-frame natural units $ℏ = c = k_{B} = 1$ and the inherited FLRW coordinate convention of FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter: $t$ and $a$ have length dimension, with dimensionless spatial coordinates and $γ$ . Thus $[H] = [m] = [M_{P}] = [Γ] = [T] = m a s s$ , $[ρ_{ϕ}] = [ρ_{r}] = [p_{r}] = {m a s s}^{4}$ , and $g_{*}$ is dimensionless. Let $J \subseteq R$ be a nonempty open cosmic-time interval and take a spatially flat FLRW model datum from the FLRW model-data definition def-gr-flrw-model-data, with $M = J \times Σ$ , $a \in C^{\infty} (J, (0, \infty))$ , metric $g = - d t^{2} + a (t)^{2} γ$ and $H = \dot{a} / a$ . Write $π_{J} : M \to J$ for the first-factor projection; in these units the smooth comoving unit timelike vector field is $n = \partial_{t}$ . Let $M_{P} = (8 π G_{N})^{- 1 / 2} > 0$ be the reduced four-dimensional Planck mass and let $Γ > 0$ be a constant decay rate of mass dimension one. Let $ϕ \in C^{\infty} (J, R)$ be a scalar time profile and set the spacetime scalar to $Φ : = ϕ \circ π_{J}$ , so homogeneity means that $Φ$ is independent of $x \in Σ$ . In the one-field convention of the scalar EFT and scalar-field matter definitions (def-r5-cosmology-4d-scalar-eft-data and def-gr-scalar-field-matter-model), take target metric $G_{11} = 1$ , scalar coupling $K = 1$ , and constant $m > 0$ of mass dimension one, with potential $V (Φ) = m^{2} Φ^{2} / 2$ , and comoving density profile $ρ_{ϕ} (t) = (\dot{ϕ} (t)^{2} + m^{2} ϕ (t)^{2}) / 2$ . For the pre-decay radiation component, choose smooth profiles ${\tilde{ρ}}_{r}, {\tilde{p}}_{r} \in C^{\infty} (J, R)$ and define the perfect-fluid fields on $M$ by $ρ_{r} = {\tilde{ρ}}_{r} \circ π_{J}$ and $p_{r} = {\tilde{p}}_{r} \circ π_{J}$ . These factorization identities specify homogeneity; use the same comoving velocity field $n = \partial_{t}$ in the perfect-fluid matter model, so $T_{r}^{a b} = (ρ_{r} + p_{r}) n^{a} n^{b} + p_{r} g^{a b}$ . Here radiation means $p_{r} = ρ_{r} / 3$ with $ρ_{r} \geq 0$ ; its rest energy density is $ρ_{r}$ . This equation of state is part of the model data, although the temporal stress calculation below depends only on the rest density. Let $ρ_{t o t} (t) : = ρ_{ϕ} (t) + {\tilde{ρ}}_{r} (t)$ be the total comoving energy density; the scalar and fluid stress tensors are the only sources in the adopted zero-cosmological-term Einstein equation. For any pre-decay quantity $f$ with a left limit at $t_{d}$ , write $f (t_{d}^{-}) : = \lim_{t ↑ t_{d}} f (t)$ . A sudden-decay model here supplies a time $t_{d} \in J$ at which $H (t_{d}^{-}) = Γ$ and instantaneously transfers all of $ρ_{t o t} (t_{d}^{-})$ into an ideal, immediately thermalized radiation bath with vanishing chemical potentials. Let $g_{*} : (0, \infty) \to (0, \infty)$ be the specified dimensionless effective number of relativistic degrees of freedom, so the adopted bath density is $ρ_{r a d} (T) = π^{2} g_{*} (T) T^{4} / 30$ . Any positive reheating temperature $T_{R H}$ satisfying $ρ_{r a d} (T_{R H}) = ρ_{t o t} (t_{d}^{-})$ then obeys

$T_{R H} = {(\frac{90}{π^{2} g_{*} (T_{R H})})}^{1 / 4} \sqrt{Γ M_{P}} .$

The equation is implicit if $g_{*}$ depends on temperature and asserts no existence or uniqueness for that implicit equation. It is exact for this sudden-decay algebra; it gives no continuous-decay or entropy-injection correction and no universal string prediction for $Γ$ .

Statement references

Four-dimensional Einstein-frame multi-scalar cosmology EFT data

Adopted two-derivative multi-scalar cosmology EFT

The Friedmann, acceleration and continuity equations of an FLRW model datum

FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter

Ricci, scalar and Einstein tensors of the FLRW metric in product charts

The scalar-field matter model of the classical gravitational framework

First variation, Hilbert tensor and conservation of the scalar-field model

The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy

Facts & Assumptions

Given: The four-dimensional Einstein-frame Planck convention, a spatially flat homogeneous scalar-plus-radiation model with zero cosmological term and additive comoving stress, the positive decay rate $Γ$ , the supplied sudden-decay/total-energy-matching prescription, and the adopted equilibrium radiation-density relation with $g_{*} (T) > 0$ .

[F1]

Adopted two-derivative multi-scalar cosmology EFT: adopt the direct Einstein equation $G_{μ ν} = M_{P}^{- 2} T_{μ ν}$ with zero cosmological term; any nonzero additional matter sector must have its stress tensor and couplings stated.

[F9]

Adopted two-derivative multi-scalar cosmology EFT: in the adopted natural-unit normalization, $M_{P}^{2} = (8 π G_{N})^{- 1}$ .

[F2]

Ricci, scalar and Einstein tensors of the FLRW metric in product charts: in the spatially flat FLRW case, the temporal Einstein component is $G_{00} = 3 {\dot{a}}^{2} / a^{2} = 3 H^{2}$ when $c = 1$ .

[F3]

Four-dimensional Einstein-frame multi-scalar cosmology EFT data: $M_{P} > 0$ is the reduced Planck mass of mass dimension one and the scalar EFT is in Einstein frame with $ℏ = c = 1$ ; this item additionally fixes $k_{B} = 1$ so temperature is measured in energy units.

[F4]

The Friedmann, acceleration and continuity equations of an FLRW model datum: the Hubble function is $H = \dot{a} / a$ on the cosmic-time interval; the definition of the Friedmann system alone does not assert its physical derivation.

[F5]

The scalar-field matter model of the classical gravitational framework: for the homogeneous canonical scalar with $K = 1$ , $c = 1$ , and $g_{00} = - 1$ , the covariant $00$ component of its candidate stress tensor is ${\dot{ϕ}}^{2} / 2 + V (ϕ)$ .

[F6]

First variation, Hilbert tensor and conservation of the scalar-field model: the stated scalar candidate tensor is the Hilbert tensor of the scalar action under the declared compact-variation convention.

[F7]

The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy defines a perfect-fluid datum by smooth rest-density and pressure fields $ε, p$ and a smooth future-directed unit timelike field $n$ , with $T^{a b} = (ε + p) n^{a} n^{b} + p g^{a b}$ ; here $ε = ρ_{r}$ and $p = p_{r}$ . FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter defines the FLRW comoving field as $c^{- 1} \partial_{t}$ ; since $c = 1$ here, $n = \partial_{t}$ , so $n^{0} = 1$ and $g^{00} = - 1$ . Substitution gives $T_{00} = ε = ρ_{r}$ , independently of the pressure.

[F10]

Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions: a model supplies its nongravitational fields or matter and their explicit equations and couplings as model input.

[F11]

FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter: in product coordinates for the FLRW metric, $g_{00} = - c^{2}$ , which is $- 1$ in the units fixed here.

[A1]

The field content is model input as in [F10]: a homogeneous canonical scalar and the smooth comoving radiation-fluid data specified in the Statement are the only pre-decay stress sources, their total tensor is the sum of the scalar and perfect-fluid tensors, and the adopted Einstein law applies to that total. The radiation equation of state is $p_{r} = ρ_{r} / 3$ ; no averaged-pressureless approximation or controlled averaging family is required for the scalar, and the residual radiation fraction need not be zero.

[A2]

At the supplied $t_{d}$ , impose $H (t_{d}^{-}) = Γ$ and instantaneously set $ρ_{r a d} (T_{R H}) = ρ_{t o t} (t_{d}^{-})$ ; the conversion is complete, with no unthermalized hidden products or additional entropy source.

[A3]

Adopt $ρ_{r a d} (T) = π^{2} g_{*} (T) T^{4} / 30$ for the specified ideal relativistic equilibrium bath, with $T > 0$ and $g_{*} (T) > 0$ . No value, uncertainty, or microscopic prediction for $g_{*}$ or $Γ$ is asserted.

Proof

1.1givenF5F6F7F10F11A1algebra

Put $V (ϕ) = m^{2} ϕ^{2} / 2$ on the pre-decay branch. The tensor formula in [F5], with $K = 1$ , $c = 1$ , the homogeneous field, and $g_{00} = - 1$ from [F11], gives $T_{00}^{ϕ} = {\dot{ϕ}}^{2} - g_{00} (- {\dot{ϕ}}^{2} / 2 + V) = {\dot{ϕ}}^{2} / 2 + V = ρ_{ϕ}$ ; [F6] confirms this candidate is the scalar action's Hilbert tensor. For the radiation fluid, [F7] gives $T_{r}^{00} = (ρ_{r} + p_{r}) (n^{0})^{2} + p_{r} g^{00} = (ρ_{r} + p_{r}) - p_{r} = ρ_{r}$ since $n^{0} = 1$ and $g^{00} = - 1$ ; lowering both indices leaves $T_{00}^{r} = ρ_{r}$ . By [A1] and [F10] the total source is the stated additive matter model, so $T_{00} = ρ_{ϕ} + ρ_{r} = ρ_{t o t}$ . This tensor computation does not require an averaged equation of state for the scalar and specifies no decay channel.

2.1F1F2F3F4F9step 1.1algebra

The direct Einstein equation [F1] and exact flat-FLRW tensor component [F2], with $k = 0$ and $c = 1$ , give $3 H^{2} = 8 π G_{N} ρ_{t o t}$ . By [F9], $8 π G_{N} = M_{P}^{- 2}$ , hence $ρ_{t o t} = 3 M_{P}^{2} H^{2}$ . This uses the spatially flat branch and zero cosmological term explicitly; it does not use an equation of state for the combined source.

3.1A2step 2.1algebra

Evaluate step 2.1 at the supplied pre-decay event. Since $H (t_{d}^{-}) = Γ$ by [A2], $ρ_{t o t} (t_{d}^{-}) = 3 M_{P}^{2} Γ^{2}$ .

4.1A2A3step 3.1algebra

By [A2], the entire pre-decay energy becomes the thermal bath, and [A3] gives $π^{2} g_{*} (T_{R H}) T_{R H}^{4} / 30 = 3 M_{P}^{2} Γ^{2}$ . Multiplication by $30 / (π^{2} g_{*} (T_{R H}))$ , permitted because $g_{*} (T_{R H}) > 0$ , yields $T_{R H}^{4} = 90 M_{P}^{2} Γ^{2} / (π^{2} g_{*} (T_{R H}))$ .

5.1A3F3step 4.1algebra∎

Since $T_{R H}$ , $M_{P}$ , $Γ$ and $g_{*} (T_{R H})$ are positive, taking the positive fourth root of step 4.1 gives the displayed equation. Its dimensions agree: $g_{*}$ is dimensionless and $\sqrt{Γ M_{P}}$ has mass dimension one, the dimension of $T_{R H}$ in the adopted $k_{B} = 1$ convention. No inversion of the possibly temperature-dependent function $g_{*}$ was used, so no uniqueness claim follows.

Reheating temperature in a sudden-decay modulus model

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources