arxiv: 2512.22531 · v2 · submitted 2025-12-27 · 🌀 gr-qc · astro-ph.CO

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Cosmological long-wavelength solutions in non-adiabatic multi-fluid systems

Hayami Iizuka , Tomohiro Harada

Authors on Pith no claims yet

Pith reviewed 2026-05-16 19:22 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.CO

keywords cosmological perturbationsmulti-fluid systemsnon-adiabatic perturbationssuperhorizon scalesnonlinear solutionsentropy modesgradient expansioncurvature perturbation

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The pith

Nonlinear long-wavelength solutions in multi-fluid cosmologies include both adiabatic and entropy modes at leading nonlinear order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a formulation for nonlinear cosmological perturbations on superhorizon scales in systems with multiple fluids. It uses the ADM formalism and a spatial gradient expansion to construct explicit solutions in a flat FLRW background. These solutions incorporate both adiabatic and entropy modes from the leading nonlinear order because multi-fluid systems are non-adiabatic. The authors examine how different definitions of pure entropy perturbations influence the evolution of the curvature perturbation and density perturbations in two-fluid examples.

Core claim

The central discovery is the explicit construction of nonlinear long-wavelength solutions for cosmological perturbations in multi-fluid systems. These solutions admit both adiabatic and entropy modes at leading nonlinear order due to the inherent non-adiabatic nature of such systems. The work defines adiabatic and entropy perturbations, notes the non-uniqueness in defining pure entropy perturbations, and analyzes the time evolution of physical quantities like the curvature perturbation in the geodesic slicing for two-fluid systems.

What carries the argument

The Arnowitt-Deser-Misner formalism combined with a spatial gradient expansion, with the small parameter being the ratio of the comoving wavenumber to the Hubble scale.

If this is right

Both adiabatic and entropy modes are present already at the leading nonlinear order in the solutions.
The time evolution of the curvature perturbation depends on the choice of pure entropy initial conditions.
Density perturbations in two-fluid systems can be tracked using these nonlinear solutions in geodesic slicing.
The framework applies to any multi-fluid non-adiabatic cosmological setup on superhorizon scales.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This construction could provide initial conditions for numerical simulations of structure formation that include entropy modes.
It may help model the effects of isocurvature perturbations in the cosmic microwave background at nonlinear level.
Extensions to include more fluids or interactions could reveal new behaviors in early universe cosmology.

Load-bearing premise

The background spacetime is a flat Friedmann-Lemaître-Robertson-Walker universe and the spatial gradient expansion is valid for small values of the comoving wavenumber divided by the Hubble scale.

What would settle it

A direct calculation or observation in a two-fluid cosmological model where the evolution of the curvature perturbation under specific entropy initial conditions does not match the constructed nonlinear solutions would falsify the central claim.

Figures

Figures reproduced from arXiv: 2512.22531 by Hayami Iizuka, Tomohiro Harada.

**Figure 2.** Figure 2: FIG. 2. The spatial dependence of the integral functions [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p041_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The snapshots of the time evolution of the spherically symmetric density perturbation for [PITH_FULL_IMAGE:figures/full_fig_p042_5.png] view at source ↗

read the original abstract

We develop a formulation of nonlinear cosmological perturbations on superhorizon scales in multi-fluid systems. It is based on the Arnowitt-Deser-Misner formalism combined with a spatial gradient expansion characterized by a small expansion parameter defined as the ratio of the comoving wavenumber to the Hubble scale. The background spacetime is assumed to be a flat Friedmann-Lemaitre-Robertson-Walker universe. Within this framework, we explicitly construct nonlinear long-wavelength solutions for cosmological perturbations. Since multi-fluid systems are inherently non-adiabatic, these solutions admit both adiabatic and entropy modes already at leading nonlinear order. We define adiabatic and entropy perturbations and discuss the non-uniqueness in defining pure entropy perturbations. Using different choices of pure entropy initial conditions, we analyze the time evolution of physical quantities such as the curvature perturbation and density perturbations in the geodesic slicing for two-fluid systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs explicit nonlinear long-wavelength solutions for superhorizon perturbations in multi-fluid cosmologies using ADM gradient expansion, a straightforward extension that handles entropy modes at leading order.

read the letter

The main contribution is the explicit construction of nonlinear solutions for cosmological perturbations on superhorizon scales in multi-fluid systems. They combine the ADM formalism with a spatial gradient expansion on a flat FLRW background, where the small parameter is the usual ratio of comoving wavenumber to Hubble scale. Because multi-fluid setups are non-adiabatic, the solutions include both adiabatic and entropy modes already at leading nonlinear order. They define these modes, note the ambiguity in choosing pure entropy perturbations, and track the evolution of curvature and density perturbations in geodesic slicing for two-fluid cases. The approach follows the standard separate-universe logic at zeroth order in gradients without adding new free parameters or circular definitions. This is useful for anyone needing concrete expressions beyond linear theory in multi-component early-universe models. The derivations appear internally consistent on the outline given, with no obvious contradictions in the setup or failure of the expansion. The discussion of non-uniqueness in entropy definitions is a reasonable clarification rather than a flaw. The main limitation is that the abstract and summary do not include explicit reductions to linear limits or single-fluid cases, so the practical size of the nonlinear corrections remains unclear without the full steps. This paper is for researchers modeling multi-fluid cosmologies, such as during reheating or with several scalar fields, where superhorizon entropy modes affect curvature perturbations. A reader who works with gradient expansions or needs ready-to-use nonlinear forms would get direct value from the solutions. I would send it for peer review because the framework is grounded in established techniques and the explicit constructions can be checked by referees.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a formulation of nonlinear cosmological perturbations on superhorizon scales in multi-fluid systems using the ADM formalism combined with a spatial gradient expansion (small parameter given by the ratio of comoving wavenumber to Hubble scale) in a flat FLRW background. It explicitly constructs nonlinear long-wavelength solutions that admit both adiabatic and entropy modes at leading nonlinear order due to inherent non-adiabaticity, discusses the non-uniqueness of pure entropy perturbation definitions, and analyzes the time evolution of the curvature perturbation and density perturbations under different entropy initial conditions in geodesic slicing for two-fluid systems.

Significance. If the explicit construction and evolution equations hold under the stated expansion, the work provides a concrete extension of separate-universe techniques to non-adiabatic multi-fluid cosmologies, allowing entropy modes to be tracked at nonlinear order without reduction to fitted parameters. This strengthens the toolkit for early-universe models involving multiple fluids or fields and supplies falsifiable predictions for the evolution of curvature and density contrasts in geodesic slicing.

minor comments (2)

[Abstract] The abstract and introduction would benefit from a short paragraph explicitly reducing the constructed solutions to the linear adiabatic limit (e.g., vanishing entropy mode) to confirm consistency with standard results.
Notation for the entropy perturbation variable and its non-uniqueness should be introduced with a clear equation reference when first defined, rather than only in the discussion section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment and recommendation of minor revision. The summary accurately captures the scope and contributions of the work.

read point-by-point responses

Referee: The manuscript develops a formulation of nonlinear cosmological perturbations on superhorizon scales in multi-fluid systems using the ADM formalism combined with a spatial gradient expansion (small parameter given by the ratio of comoving wavenumber to Hubble scale) in a flat FLRW background. It explicitly constructs nonlinear long-wavelength solutions that admit both adiabatic and entropy modes at leading nonlinear order due to inherent non-adiabaticity, discusses the non-uniqueness of pure entropy perturbation definitions, and analyzes the time evolution of the curvature perturbation and density perturbations under different entropy initial conditions in geodesic slicing for two-fluid systems.

Authors: We appreciate the referee's concise and accurate summary of the paper's content and results. The explicit construction of the nonlinear long-wavelength solutions, the admission of both adiabatic and entropy modes at leading order, the discussion of non-uniqueness in entropy definitions, and the analysis of curvature and density evolution in geodesic slicing for two-fluid systems are indeed the central elements, as presented in Sections 3–5. No changes are required in response to this summary. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper performs an explicit construction of nonlinear long-wavelength solutions via the standard ADM formalism plus spatial gradient expansion (with the small parameter being the ratio of comoving wavenumber to Hubble scale) on a flat FLRW background. The admission of both adiabatic and entropy modes at leading order follows directly from the non-adiabatic character of multi-fluid systems and the separate-universe logic at zeroth order in gradients; the discussion of non-uniqueness of pure entropy definitions is an analytical observation rather than a definitional loop. No step reduces a claimed prediction to a fitted input, imports uniqueness via self-citation chains, or renames a known result as a new derivation. The central result is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The formulation depends on the standard ADM decomposition and the validity of the gradient expansion approximation in a flat FLRW background.

free parameters (1)

small expansion parameter
Ratio of comoving wavenumber to Hubble scale, taken to be small to justify the long-wavelength approximation.

axioms (1)

domain assumption Background spacetime is a flat FLRW universe
Explicitly stated as the assumed background for the perturbation analysis.

pith-pipeline@v0.9.0 · 5447 in / 1181 out tokens · 26421 ms · 2026-05-16T19:22:36.064559+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · 12 internal anchors

[1]

Expansion scheme 11 C

Cosmological conformal decomposition 9 B. Expansion scheme 11 C. Background solution 12 D. Slicing conditions 13 III. Long-wavelength solutions in multi-fluid systems 16 A. Adiabatic and entropy perturbations 16

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Adiabatic perturbation 16

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Energy density at an equality time 17

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Zeroth-order adiabatic and entropy perturbations 18

Entropy perturbation 18 B. Zeroth-order adiabatic and entropy perturbations 18

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Zeroth-order solutions 18

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[6]

Adiabatic perturbation 19

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Defining pure entropy perturbations 22

Entropy perturbation 20 C. Defining pure entropy perturbations 22

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Primeval isocurvature condition 23

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Zero-average condition 23

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Leading-order solution with slicing conditions 25

Initially isocurvature condition 24 D. Leading-order solution with slicing conditions 25

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Uniform e-folding slice 25

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Long-wavelength solutions in two-fluid systems 27 A

Geodesic slice 26 IV. Long-wavelength solutions in two-fluid systems 27 A. Background solutions 28 B. Long-wavelength solutions 29 2

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Long-wavelength solutions with the geodesic slice 29

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Asymptotic behaviours of the curvature and density perturbations 30

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Adiabatic and isocurvature conditions 33

Curvature perturbations and density perturbations 32 C. Adiabatic and isocurvature conditions 33

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Adiabatic condition 33

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Primeval isocurvature condition 34

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separate universes

Initially isocurvature condition 35 V. Demonstration: matter - radiation systems 35 A. Background and leading-order solutions 35 B. Time evolution of the curvature perturbation 36 C. Time evolution of the density perturbations 38 VI. Conclusion 40 Acknowledgments 44 A. Linear limit of the nonlinear adiabatic condition 44 B. Evolution equation for ˜Aij 45 ...

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[19]

ADM formalism We employ the ADM formalism [25]. The line element in four-dimensional spacetime has the following form: ds2 =−α 2dt2 +γ ij βidt+ dx i βjdt+ dx j ,(2.1) whereα,γ ij, andβ i are the lapse function, the induced metric, and the shift vector, respec- tively, and the Latin uppercase indices take values from 1 to 3. We lower these indices using γi...

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[20]

The functiona(t) is the scale factor of a fiducial flat FLRW spacetime

Cosmological conformal decomposition We decompose the induced metricγ ij into the following form [9, 10]: γij =a 2(t)ψ4˜γij.(2.20) ˜γij is chosen so that its determinant ˜γis corresponding to that of the three-dimensional flat metricη ij, which is independent of time. The functiona(t) is the scale factor of a fiducial flat FLRW spacetime. The extrinsic cu...

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[21]

The background solution for the matter energy density is given by Eq

Adiabatic perturbation Let us consider an adiabatic perturbation and entropy perturbation conditions on a given time slice. The background solution for the matter energy density is given by Eq. (2.56): (b)ρ(α) =D (α)a−3Γ(α).(3.1) Introducing the e-folding number asN(t) = lna(t), the energy density is reduced to (b)ρ(α) =D (α)e−3Γ(α)N(t) .(3.2) If there ex...

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[22]

, D(N)),(3.7) ρtot = NX α=1 D(α)e−3Γ(α)(N(t)+δN(t,x k)) = (b)ρtot(N(t) +δN(t, x k);D (1),

Energy density at an equality time The total background energy density and the total energy density of the adiabatic per- turbation are written as (b)ρtot = NX α=1 D(α)e−3Γ(α)N(t) = (b)ρtot(N;D (1), . . . , D(N)),(3.7) ρtot = NX α=1 D(α)e−3Γ(α)(N(t)+δN(t,x k)) = (b)ρtot(N(t) +δN(t, x k);D (1), . . . , D(N)).(3.8) For anN-fluid background, we define theαβ–...

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[23]

Entropy perturbation Solving the adiabatic condition, Eq. (3.3), forN+δN, we find the following solution N(t) +δN(t, x k) =− 1 3Γ(α) ln ρ(α) D(α) ,orδN(t, x k) =− 1 3Γ(α) ρ(α) (b)ρ(α) .(3.17) From this solution, we conclude that−1/(3Γ (α)) ρ(α)/((b)ρ(α)) does not depend on a com- ponent of (α) for the adiabatic perturbation. This is rewritten as − 1 3Γ(α)...

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[24]

Zeroth-order solutions Integrating Eqs. (2.47) and (2.49), we find the energy density at the zeroth order under the general slice: (0)ρ(α)(t, xk) =C (α)(xk) a(t)ψ2 −3Γ(α) ,(3.21) whereC (α)(xk) is an integral function which comes from integration of the energy equation for each fluid. For later convenience, we decompose the integral function as follows: ¯...

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[25]

Substituting the zeroth-order solution Eq

Adiabatic perturbation We now discuss the necessary and sufficient condition for the zeroth-order solution (3.27) to be an adiabatic perturbation. Substituting the zeroth-order solution Eq. (3.27) into Eq. (3.3), we obtain ¯C(α)(xk)D(α)e−3Γ(α)N =D (α)e−3Γ(α)(N+δN) .(3.28) Here, assuming that the values ofNon both sides are identical, we find that the exis...

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[26]

Then, Eq

Entropy perturbation We introduce a functionδN (α)(xk) as ¯C(α)(xk) =e −3Γ(α)δN(α)(xk),(3.33) for later convenience. Then, Eq. (3.27) can be rewritten as (0)ρ(α) =D (α)e−3Γ(α)(N+δN (α)(xk)),(3.34) 20 for all (α). In this way, each pointx k can be interpreted as being described by a separate uni- verse on top of a common background, in which the evolution ...

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[27]

, Ncan then be chosen freely

Primeval isocurvature condition A simple way to fix the choice ofδN ad(xk) is, for example, δN ad =δN (1),or equivalently ¯C(1) =e −3Γ(1)δN ad .(3.46) The remaining functions ¯C(α)(x) forα= 2, . . . , Ncan then be chosen freely. In this choice, the total density fluctuation (0)δtot, which is defined by (0)δtot ≡ (0)ρtot − (b)ρtot (b)ρtot = P α (b)ρ(α)(0)δ...

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[28]

Zero-average condition An another choice forδN ad(xk) is given by δN ad = 1 N X α δN(α)(xk).(3.50) This can be interpreted as the reference adiabatic perturbation corresponding to a separate- universe picture obtained by averaging theδN (α) over all components (α) in the original non-adiabatic perturbation. In the two-fluid case, this gives eδN ad(xk) = ¯...

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[29]

The zeroth-order solution (0)ρ(α)(t, xk) is integrated as in Eq

Initially isocurvature condition Alternatively, one can use the freedom in choosingδN ad to fix ¯C(α)(xk) such that, under a certain gauge condition, ψ(t0, xk) = 1.(3.54) Here, we consider one example of this approach. The zeroth-order solution (0)ρ(α)(t, xk) is integrated as in Eq. (3.23), where the back- ground solution (b)ρ(α)(t) is multiplied by ¯C(α)...

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[30]

(2.47): ψ=ψ(x k).(3.58) Moreover, from Eq

Uniform e-folding slice The condition for the uniform e-folding slice (Nslice) is given by αK=−3 ˙a a .(3.57) Taking this slice, we find the following relation from Eq. (2.47): ψ=ψ(x k).(3.58) Moreover, from Eq. (2.49), we find that the energy densities for the fluid components are expressed by (0)ρ(α)(t, xk) =c (α)(xk)a−3Γ(α)(t),(3.59) wherec (α)(xk) is ...

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[31]

(2.47) and (2.49) gives (0)ρ(α)(t, xk) =c ′ (α)(xk) a(t)ψ2(t, xk) −3Γ(α) ,(3.64) wherec ′ (α)(xk) are integral function which comes from energy equation

CMC slice Taking CMC slice, Eqs. (2.47) and (2.49) gives (0)ρ(α)(t, xk) =c ′ (α)(xk) a(t)ψ2(t, xk) −3Γ(α) ,(3.64) wherec ′ (α)(xk) are integral function which comes from energy equation. Putting Eq. (3.64) into Eq. (2.46), Eq. (2.46) is reduced to ˙a a 2 = 8π 3 NX α=1 c′ (α)(xk) a(t)ψ2(t, xk) −3Γ(α) .(3.65) The left-hand side of the above equation corresp...

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[32]

Geodesic slice As discussed in Sec.II D, once we choose the geodesic slice, the proper time acts as the coordinate. Using the proper time as a time coordinate, the leading-order basic equations 26 are reduced to (0)K2 =24π(0)E,(3.69) (0)K=−3 ∂t(aψ2) aψ2 ,(3.70) ∂t (0)K= 1 3 (0)K2 + 4π(0) E+S i i ,(3.71) ∂t a3ψ6(0)ρ(α) =a3ψ6 Γ(α) −1 (0)ρ(α) (0)K.(3.72) We ...

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[33]

(4.18) shows (0)Ξ(t, xk) is monotonically increasing

Long-wavelength solutions with the geodesic slice For the perturbed system, if we choose the geodesic slice, the Hamiltonian constraint (3.74) reduces to ∂t(aΨ2) aΨ2 2 = 8π 3 h C(1)(xk) aΨ2 −3Γ(1) +C (2)(xk) aΨ2 −3Γ(2) i .(4.13) Introducing the following quantities, (0)Ξ(t, xk)≡a(t)Ψ 2(t, xk),(4.14) (0)ρeq(xk) 2 ≡C (1)(xk)Γ(2)/(Γ(2)−Γ(1))C(2)(xk)Γ(1)/(Γ(1...

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[34]

The limit corresponds to the conditionY≪1

Asymptotic behaviours of the curvature and density perturbations Let us consider the fluid (1) dominated limit of the functionq(Y). The limit corresponds to the conditionY≪1. Denoting the integrand in the left-hand side of Eq. (4.19) byF, we have q(Y) = Z Y 0 F( ˜Y)d ˜Y .(4.23) In the limit ofY≪1, the functionF(Y) is expanded as F(Y) = √ 2Y (3Γ(1)/2)−1 +....

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[35]

Curvature perturbations and density perturbations In two-fluid systems, we find the leading-order solution is given by Eq. (4.32). Rewriting the solution by using the functionsh(x k),δN (1) andδN (2), we obtain Ψ = s q−1(hHeqt) q−1(Heqt) exp −1 2 Γ(1) Γ(1) −Γ (2) δN(1) − Γ(2) Γ(1) −Γ (2) δN(2) ,(4.36) = s q−1(hHeqt) q−1(Heqt) exp −1 2 Γ(1) Γ(1) −Γ (2) δN ...

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(3.3) is satisfied at an equality timet0 = ˜teq(xk) in two fluid systems, the condition is expressed as (0)ρ(α)(˜teq(xk), xk) = (b)ρ(α)(˜teq(xk) + ∆t(xk)),(4.40) forα= 1,2

Adiabatic condition Assuming that the adiabatic condition Eq. (3.3) is satisfied at an equality timet0 = ˜teq(xk) in two fluid systems, the condition is expressed as (0)ρ(α)(˜teq(xk), xk) = (b)ρ(α)(˜teq(xk) + ∆t(xk)),(4.40) forα= 1,2. The energy densities of the two components are equal to each other at the equality time. Therefore, one can find that, fro...

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[37]

Primeval isocurvature condition This condition corresponds toδN (1) = 0 in two-fluids systems discussed in Sec. III C 1. Then, Eq. (4.29) with vanishingδN (1) implies Ψ→1,(4.54) in the limit oft→0. Therefore, in the early universe, the pure entropy condition constructed from the long-wavelength dynamics of inflation coincides with the standard isocurvatur...

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The pure entropy condition Eq

Initially isocurvature condition Following the normalization condition,a eq = 1, the Friedmann equation at the equality time is reduced to H2 eq = 8π 3 ρeq = 16π 3 D Γ(2)/(Γ(2)−Γ(1)) (1) D Γ(1)/(Γ(1)−Γ(2)) (2) = 16π 3 D(1).(4.55) The integral functions ¯C(α)(xk) are rewritten by using (0) ˜Heq(xk) and (0)Ξeq(xk) as follows: D(α) ¯C(α)(xk) = 3 16π (b)H2 eq...

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