Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-04 (gpt-6.1-sol)

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A quadratic modulus has leading averaged dust behavior

Physical scope

A leading-limit deduction for one homogeneous canonical real scalar on a prescribed expanding, spatially flat FLRW background. The result concerns only the averaged pressure and energy dilution in the stated quadratic, adiabatic, and uniform-window regime. It asserts no global string compactification, pointwise dust stress, finite-parameter O(H/m) correction, observation, decay law, or behavior for nonquadratic potentials or backgrounds dynamically determined by the scalar rather than prescribed.

Statement

Use units $ℏ = c = 1$ and assign length dimension to the FLRW coordinates $t$ and $a_{n}$ , and dimensionless coordinates $x^{i}$ and metric components $δ_{i j}$ as displayed below. Thus $[H_{n}] = [m_{n}] = m a s s$ , $[Δ_{n}] = {m a s s}^{- 1}$ , $[φ_{n}] = m a s s$ , and $[ρ_{n}] = [p_{n}] = {m a s s}^{4}$ . In this item, a modulus mode means a one-field scalar model with target $R$ , metric $G_{11} = 1$ , coupling $K = 1$ , positive mass constant on $I$ , and exact quadratic potential $V (φ) = m^{2} φ^{2} / 2$ on the field range; “modulus” is a model label and no compactification is assumed. Let $I = [t_{-}, t_{+}]$ be a nondegenerate compact interval inside an open cosmic-time interval $J$ , with $a_{n} \in C^{\infty} (J, (0, \infty))$ and $H_{n} = {\dot{a}}_{n} / a_{n} > 0$ on $I$ . For each $n \geq 1$ , let $φ_{n} \in C^{\infty} (J, R)$ be the time profile of the homogeneous spacetime scalar $(t, x) \mapsto φ_{n} (t)$ on $J \times R^{3}$ . Use the one-field target coordinate $T_{0} = R$ with metric $G_{11} = 1$ and scalar coupling $K = 1$ , as in the cited scalar EFT and scalar-field conventions. Let $m_{n} > 0$ be constant on $I$ and have mass dimension one and let $V_{n} (φ) = m_{n}^{2} φ^{2} / 2$ throughout the field range. The prescribed spatially flat FLRW metric is $d s^{2} = - d t^{2} + a_{n} (t)^{2} δ_{i j} d x^{i} d x^{j}$ on $J \times R^{3}$ . Assume the canonical homogeneous scalar equation ${\ddot{φ}}_{n} + 3 H_{n} {\dot{φ}}_{n} + m_{n}^{2} φ_{n} = 0$ and define $ρ_{n} = {\dot{φ}}_{n}^{2} / 2 + m_{n}^{2} φ_{n}^{2} / 2$ and $p_{n} = {\dot{φ}}_{n}^{2} / 2 - m_{n}^{2} φ_{n}^{2} / 2$ . For each $n$ , let $Δ_{n} > 0$ . For $Q \in {ρ_{n}, p_{n}}$ define the centered uniform window average ${\overset{ˉ}{Q}}_{n} (t) = Δ_{n}^{- 1} \int_{t - Δ_{n} / 2}^{t + Δ_{n} / 2} Q (s) d s$ whenever the window lies in $I$ . Write $W_{n} (t) = [t - Δ_{n} / 2, t + Δ_{n} / 2]$ and call $t$ an admissible center when $W_{n} (t) \subseteq I$ . Consider nonzero solutions and admissible centers with ${\overset{ˉ}{ρ}}_{n} (t) > 0$ , and suppose

$\sup_{I} H_{n} / m_{n} \to 0, \sup_{I} ∣ {\dot{H}}_{n} ∣ / m_{n}^{2} \to 0, \sup_{I} ∣ {\ddot{H}}_{n} ∣ / m_{n}^{3} \to 0, m_{n}^{- 1} \int_{I} ∣ {\dot{H}}_{n} (s) ∣ d s \to 0,$

$m_{n} Δ_{n} \to \infty, Δ_{n} \sup_{I} H_{n} \to 0, Δ_{n} \sup_{I} ∣ {\dot{H}}_{n} ∣ / m_{n} \to 0, Δ_{n} \sup_{I} ∣ {\ddot{H}}_{n} ∣ / m_{n}^{2} \to 0.$

Fix a reference center $t_{0} \in I$ such that $W_{n} (t_{0}) \subseteq I$ and ${\overset{ˉ}{ρ}}_{n} (t_{0}) > 0$ for all sufficiently large $n$ . For every such fixed reference center, the joint limit is ${\overset{ˉ}{p}}_{n} (t) / {\overset{ˉ}{ρ}}_{n} (t) \to 0$ and $a_{n} (t)^{3} {\overset{ˉ}{ρ}}_{n} (t) / (a_{n} (t_{0})^{3} {\overset{ˉ}{ρ}}_{n} (t_{0})) \to 1$ , uniformly over admissible centers $t$ with ${\overset{ˉ}{ρ}}_{n} (t) > 0$ . Define the local averaged envelope by $A_{n} (t) = (2 Δ_{n}^{- 1} \int_{t - Δ_{n} / 2}^{t + Δ_{n} / 2} φ_{n} (s)^{2} d s)^{1 / 2}$ . Then $A_{n} (t) / A_{n} (t_{0}) = (a_{n} (t_{0}) / a_{n} (t))^{3 / 2} (1 + o (1))$ and ${\overset{ˉ}{ρ}}_{n} (t) = m_{n}^{2} A_{n} (t)^{2} / 2 (1 + o (1))$ . The total-variation limit $m_{n}^{- 1} \int_{I} ∣ {\dot{H}}_{n} (s) ∣ d s \to 0$ controls the accumulated ${\dot{H}}_{n}$ term in the dilution estimate. This is an averaged leading-limit claim: it supplies no pointwise field-amplitude remainder or finite-parameter $O (H / m)$ correction, and makes no claim for nonquadratic potentials or for backgrounds dynamically determined by the scalar rather than prescribed as in the hypotheses.

Statement references

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

Conditional homogeneous multi-field FLRW equations

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

The scalar-field matter model of the classical gravitational framework

Facts & Assumptions

Given: The one-field unit-metric branch of the four-dimensional Einstein-frame scalar EFT, its homogeneous scalar equation on a spatially flat expanding FLRW background, the quadratic potential and positive constant mass, the stated smoothness and compact interval, and the joint pointwise, total-variation and window limits in the Statement.

[F1]

Four-dimensional Einstein-frame multi-scalar cosmology EFT data: the scalar EFT uses $ℏ = c = 1$ , a smooth scalar target with positive metric, and a smooth potential; for this one-field item the target patch is $R$ with metric $G_{11} = 1$ .

[F2]

Conditional homogeneous multi-field FLRW equations: under its smooth, homogeneous, Euclidean-chart hypotheses and its matter-free adopted model (no other stress tensor and no direct scalar source), scalar stationarity gives $D_{t} {\dot{ϕ}}^{i} + 3 H {\dot{ϕ}}^{i} + G^{i j} \partial_{j} V = 0$ , with $ρ = G_{i j} {\dot{ϕ}}^{i} {\dot{ϕ}}^{j} / 2 + V$ and $p = G_{i j} {\dot{ϕ}}^{i} {\dot{ϕ}}^{j} / 2 - V$ . Here this supplies the one-field convention for comparison; the scalar ODE and energy expressions are explicitly assumed in the Statement, rather than inferred by extending that theorem to a prescribed background.

[F3]

The Friedmann, acceleration and continuity equations of an FLRW model datum: the FLRW Hubble function is $H = \dot{a} / a$ and the Friedmann system is a named collection of pointwise scalar equations on the time interval; this mathematical definition itself adopts no equation of state.

[F4]

The scalar-field matter model of the classical gravitational framework: in the single-real-scalar convention the scalar equation and candidate stress tensor are specified separately; with $K = 1$ and $c = 1$ its homogeneous canonical density and pressure have the same kinetic-minus/plus-potential signs used here.

[A1]

For each family member the field is nonzero, $m_{n}$ is constant and positive on $I$ , the potential is exactly quadratic over the field range, $H_{n} > 0$ , the background is prescribed, the windows are centered uniform windows contained in $I$ , and all limits in the Statement hold. Backreaction is neglected in the prescribed-background calculation or is treated separately; no empirical input is used.

Proof

1.1givenF1F2F3F4algebra

Use the explicitly assumed equation ${\ddot{φ}}_{n} + 3 H_{n} {\dot{φ}}_{n} + m_{n}^{2} φ_{n} = 0$ and definitions $ρ_{n} = ({\dot{φ}}_{n}^{2} + m_{n}^{2} φ_{n}^{2}) / 2$ and $p_{n} = ({\dot{φ}}_{n}^{2} - m_{n}^{2} φ_{n}^{2}) / 2$ , with $H_{n} = {\dot{a}}_{n} / a_{n}$ as in [F3]. For comparison with [F2], the constant target metric $G_{11} = 1$ of [F1] gives $G^{11} = 1$ and $Γ^{1}_{11} = \frac{1}{2} G^{11} \partial_{1} G_{11} = 0$ , while $V_{n}^{'} (φ) = m_{n}^{2} φ$ ; hence the assumed expressions have exactly its canonical one-field form and the signs of [F4]. This comparison does not invoke [F2]'s matter-free model or its Einstein equations on the prescribed background.

2.1A1step 1.1algebra

Put $q_{n} = φ_{n} {\dot{φ}}_{n}$ . Differentiating the energy and using step 1.1 gives ${\dot{ρ}}_{n} = {\dot{φ}}_{n} ({\ddot{φ}}_{n} + m_{n}^{2} φ_{n}) = - 3 H_{n} {\dot{φ}}_{n}^{2}$ . Differentiating $q_{n}$ and substituting the same equation gives ${\dot{q}}_{n} = {\dot{φ}}_{n}^{2} - 3 H_{n} q_{n} - m_{n}^{2} φ_{n}^{2}$ , hence $p_{n} = ({\dot{q}}_{n} + 3 H_{n} q_{n}) / 2$ . Finally, $2 m_{n} ∣ q_{n} ∣ \leq {\dot{φ}}_{n}^{2} + m_{n}^{2} φ_{n}^{2} = 2 ρ_{n}$ by $(∣ {\dot{φ}}_{n} ∣ - m_{n} ∣ φ_{n} ∣)^{2} \geq 0$ , so $∣ q_{n} ∣ \leq ρ_{n} / m_{n}$ .

3.1A1step 2.1algebra

Write $J_{n} (t) = [t - Δ_{n} / 2, t + Δ_{n} / 2]$ . Since ${\dot{φ}}_{n}^{2} \leq 2 ρ_{n}$ and $H_{n} \leq H_{n, \max} : = \sup_{I} H_{n}$ , step 2.1 gives $∣ {\dot{ρ}}_{n} ∣ \leq 6 H_{n, \max} ρ_{n}$ . For $s \geq u$ , the derivative of $e^{- 6 H_{n, \max} (s - u)} ρ_{n} (s)$ is nonpositive, so $ρ_{n} (s) \leq e^{6 H_{n, \max} (s - u)} ρ_{n} (u)$ . For $s \leq u$ , the derivative of $e^{6 H_{n, \max} (s - u)} ρ_{n} (s)$ is nonnegative, giving $ρ_{n} (s) \leq e^{6 H_{n, \max} (u - s)} ρ_{n} (u)$ . Thus $ρ_{n} (s) \leq e^{6 H_{n, \max} Δ_{n}} ρ_{n} (u)$ for all $s, u \in J_{n} (t)$ ; averaging over $u$ bounds each endpoint energy by $e^{6 H_{n, \max} Δ_{n}} {\overset{ˉ}{ρ}}_{n} (t)$ . Averaging the identity for $p_{n}$ in step 2.1 and using $∣ q_{n} ∣ \leq ρ_{n} / m_{n}$ now yields $∣ {\overset{ˉ}{p}}_{n} (t) ∣ \leq {\overset{ˉ}{ρ}}_{n} (t) (e^{6 H_{n, \max} Δ_{n}} / (m_{n} Δ_{n}) + 3 H_{n, \max} / (2 m_{n}))$ .

3.2A1step 2.1F3algebra

Let $R_{n} = a_{n}^{3} ρ_{n}$ . step 2.1 and $H_{n} = {\dot{a}}_{n} / a_{n}$ give ${\dot{R}}_{n} = 3 H_{n} a_{n}^{3} ρ_{n} + a_{n}^{3} {\dot{ρ}}_{n} = - 3 H_{n} a_{n}^{3} p_{n}$ . Substituting $p_{n} = ({\dot{q}}_{n} + 3 H_{n} q_{n}) / 2$ and integrating by parts on $[t_{0}, t]$ gives $R_{n} (t) - R_{n} (t_{0}) = - \frac{3}{2} [H_{n} a_{n}^{3} q_{n}]_{t_{0}}^{t} + \frac{3}{2} \int_{t_{0}}^{t} (H_{n} a_{n}^{3})^{'} q_{n} d s - \frac{9}{2} \int_{t_{0}}^{t} H_{n}^{2} a_{n}^{3} q_{n} d s$ . Since $(H_{n} a_{n}^{3})^{'} = a_{n}^{3} {\dot{H}}_{n} + 3 H_{n}^{2} a_{n}^{3}$ , the two $H_{n}^{2}$ integral terms, with coefficients $9 / 2$ and $- 9 / 2$ , cancel exactly, leaving $R_{n} (t) - R_{n} (t_{0}) = - \frac{3}{2} [H_{n} a_{n}^{3} q_{n}]_{t_{0}}^{t} + \frac{3}{2} \int_{t_{0}}^{t} a_{n}^{3} {\dot{H}}_{n} q_{n} d s$ .

4.1A1step 3.1algebra

The first term in parentheses in step 3.1 tends to zero because $m_{n} Δ_{n} \to \infty$ and $H_{n, \max} Δ_{n} \to 0$ ; the second tends to zero because $H_{n, \max} / m_{n} \to 0$ . Division is permitted by the stated ${\overset{ˉ}{ρ}}_{n} (t) > 0$ . This proves ${\overset{ˉ}{p}}_{n} (t) / {\overset{ˉ}{ρ}}_{n} (t) \to 0$ uniformly for admissible centers.

4.2A1step 2.1step 3.1step 3.2algebra

The reference window is admissible and has ${\overset{ˉ}{ρ}}_{n} (t_{0}) > 0$ . If $R_{n} (t_{0}) = a_{n} (t_{0})^{3} ρ_{n} (t_{0}) = 0$ , then $ρ_{n} (t_{0}) = 0$ ; the comparison in step 3.1 with $u = t_{0}$ gives $0 \leq ρ_{n} (s) \leq e^{6 H_{n, \max} Δ_{n}} ρ_{n} (t_{0}) = 0$ throughout $W_{n} (t_{0})$ . This would force ${\overset{ˉ}{ρ}}_{n} (t_{0}) = 0$ , a contradiction, so $R_{n} (t_{0}) > 0$ . Because $R_{n}$ is continuous on compact $I$ , let $M_{n} = \sup_{s \in I} R_{n} (s) < \infty$ and $C_{n} = \frac{3}{2 m_{n}} (2 H_{n, \max} + \int_{I} ∣ {\dot{H}}_{n} (s) ∣ d s)$ . The bound $a_{n}^{3} ∣ q_{n} ∣ \leq R_{n} / m_{n}$ from step 2.1 and the exact identity in step 3.2 imply $∣ R_{n} (t) - R_{n} (t_{0}) ∣ \leq C_{n} M_{n}$ . Hence $R_{n} (t) \leq R_{n} (t_{0}) + C_{n} M_{n}$ for every $t$ , and taking the supremum yields $M_{n} \leq R_{n} (t_{0}) + C_{n} M_{n}$ . By the $H / m$ and total-variation limits, $C_{n} \to 0$ ; eventually $C_{n} < 1$ , so $M_{n} \leq R_{n} (t_{0}) / (1 - C_{n})$ and $\sup_{I} ∣ R_{n} (t) / R_{n} (t_{0}) - 1 ∣ \leq C_{n} / (1 - C_{n}) \to 0$ .

5.1A1step 4.2algebra

For $s \in J_{n} (t)$ , $a_{n} (s)^{3} / a_{n} (t)^{3} = \exp (3 \int_{t}^{s} H_{n} (u) d u) = 1 + o (1)$ uniformly because $H_{n, \max} Δ_{n} \to 0$ ; step 4.2 also gives $R_{n} (s) / R_{n} (t) = 1 + o (1)$ uniformly. Since $ρ_{n} = R_{n} / a_{n}^{3}$ , it follows that $ρ_{n} (s) = ρ_{n} (t) (1 + o (1))$ uniformly on the window and hence $a_{n} (t)^{3} {\overset{ˉ}{ρ}}_{n} (t) = R_{n} (t) (1 + o (1))$ . Combining this with step 4.2 proves $a_{n} (t)^{3} {\overset{ˉ}{ρ}}_{n} (t) / (a_{n} (t_{0})^{3} {\overset{ˉ}{ρ}}_{n} (t_{0})) \to 1$ .

6.1A1step 4.1step 5.1algebra∎

The definitions give the exact identity ${\overset{ˉ}{ρ}}_{n} (t) - {\overset{ˉ}{p}}_{n} (t) = m_{n}^{2} Δ_{n}^{- 1} \int_{J_{n} (t)} φ_{n} (s)^{2} d s = m_{n}^{2} A_{n} (t)^{2} / 2$ . By step 4.1, ${\overset{ˉ}{p}}_{n} / {\overset{ˉ}{ρ}}_{n} \to 0$ , so ${\overset{ˉ}{ρ}}_{n} = m_{n}^{2} A_{n}^{2} / 2 (1 + o (1))$ . step 5.1 then implies $A_{n} (t) / A_{n} (t_{0}) = (a_{n} (t_{0}) / a_{n} (t))^{3 / 2} (1 + o (1))$ , with the positive square root fixed by the definition of $A_{n}$ . These are averaged leading-limit statements, not pointwise dust or a pointwise sinusoidal approximation.

A quadratic modulus has leading averaged dust behavior

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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