Not R Kurvature: Beating Large-Scale White Noise

Wayne Hu

REVIEW 2 major objections 4 minor 22 references

Not R Kurvature: Beating Large-Scale White Noise

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Kurvature's large-scale white noise does not create an infrared-divergent curvature mode.

desk verdict This is a serious, careful second-order calculation that kills the BIS-II IR-divergence conjecture for the ideal radiation fluid it studies, and it deserves a real referee. read the letter →

arxiv 2608.09709 v1 pith:E4PDA2CP submitted 2026-08-10 astro-ph.CO gr-qc

Wayne Hu This is my paper · ORCID

classification astro-ph.COgr-qc

keywords kurvaturelarge-scalewhitenoisesecond-ordercosmologicalperturbationsradiationdominationcurvatureperturbationextrinsiccosmicmicrowavebackgroundinfrareddivergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Kurvature is a local curvature invariant built from the energy density and the expansion rate of the cosmic fluid in its center-of-momentum frame. Earlier work argued that beats between high-momentum, or hard-hard, modes generically give kurvature large-scale white noise, and that a Poisson-like relation would convert that noise into an infrared-divergent curvature perturbation visible in the cosmic microwave background. This paper tests that inference with an explicit second-order calculation for acoustic waves in a radiation-dominated, irrotational perfect fluid. It finds that the kurvature density $\delta_K$ does become white on large scales, but the Hamiltonian constraint splits the noise into intrinsic 3-curvature and extrinsic shear, equivalently density and expansion, and only the extrinsic terms grow like a density fluctuation above the horizon. The Laplacian of the second-order curvature perturbation has zero white contribution, so the supposed infrared-divergent relic in $\mathcal{R}$ does not arise, removing a proposed ultraviolet-sensitivity constraint on early-universe physics while keeping the direct hard-hard curvature power ultraviolet-convergent.

What carries the argument

The load-bearing object is the kurvature $K=8\pi G\rho/3-\theta^2/9$ in the center-of-momentum frame, together with its exact Hamiltonian-constraint decomposition on comoving slices, $K={}^{(3)}R/6-\sigma^2/3$. This identity splits the second-order kurvature density into intrinsic 3-curvature and the quadratic shear composite $(\sigma^{(1)})^2$, locating the growing white-noise contribution in the extrinsic part. The second workhorse is the sourced wave equation for the second-order curvature mode, $\ddot{\mathcal{R}}_2+2H\dot{\mathcal{R}}_2+(k_L^2/3)\mathcal{R}_2=q^2 S_{\mathcal{R}}$, whose source is regular as the soft wavenumber $k_L\to 0$; that regularity, verified at the level of the Laurent expansion of the source, is what prevents an inverse-Laplacian $k_L^{-2}$ response of the kind the BIS-II conjecture requires.

What would settle it

Perform the same second-order calculation, or a Boltzmann simulation, for the full photon-baryon plasma with viscosity and diffusion damping, and measure the superhorizon curvature power generated purely by hard-hard modes: if it grows like $q_{\max}/k$ and depends on the ultraviolet cutoff rather than vanishing as $(k\eta)^3$, the paper's central claim is false.

Watch

Extended reading notes

Core claim

The paper's central claim is that the BIS-II conjecture fails for the standard radiation-fluid system: kurvature does acquire large-scale white noise, but the white part comes from the extrinsic curvature of acoustic beats rather than from the intrinsic 3-curvature that a Poisson equation would connect to $\mathcal{R}$. Concretely, the second-order hard-hard response obeys $[\nabla^2\mathcal{R}^{(2)}]_{\rm LSWN}=0$ (Eq. 48), whereas the BIS-II Poisson construction $\nabla^2\mathcal{R}_{\rm BIS}=-4\pi G a^2\Delta\rho^{(2)}$ would give a nonzero, infrared-divergent, cutoff-sensitive result. The actual hard-hard curvature power $\Delta^2_{22}$ is white but strongly convergent, scaling as $(k\eta)^3$ on superhorizon scales and peaking for modes that cross the sound horizon near the evaluation epoch. Thus, beyond linear order, kurvature is not the potential for cosmological curvature perturbations, and no infrared relic in $\mathcal{R}$ follows from purely ultraviolet modes.

Load-bearing premise

The explicit counterexample assumes an idealized scalar, irrotational, perfect radiation fluid with no viscosity or diffusion damping; if the real photon-baryon plasma restores a Poisson-like link between kurvature and the curvature mode, the no-relic conclusion would not hold.

Editorial extensions

If this is right

Applying the BIS-II Poisson construction to the computed $\delta_K$ gives $\Delta^2_{\rm BIS}\propto q_{\max}/k$, whereas the direct hard-hard curvature power satisfies $\Delta^2_{22}\propto (k\eta)^3$ on superhorizon scales.
The Sachs-Wolfe contribution to the CMB temperature is not infrared-enhanced by hard-hard mode beats, so no ultraviolet-cutoff constraint on the primordial power of high-momentum modes follows from this mechanism.
The dominant hard modes are near the sound horizon at the evaluation epoch ($x\simeq 2.51$), so the second-order curvature power is ultraviolet-convergent rather than cutoff sensitive.
If a physical process switches off the hard-hard sources, the matched long-wavelength response is a finite white constant plus a decaying piece (Eq. 73), not a $k^{-2}$ curvature potential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A reader might push the same intrinsic-versus-extrinsic decomposition onto any local curvature invariant built from the fluid congruence; the general lesson would be that white noise in an invariant does not imply an infrared-divergent metric perturbation unless that invariant is genuinely the Poisson source of the perturbation.
The natural next test is a Boltzmann-level computation in the full photon-baryon plasma with viscosity and diffusion damping; the paper's damping argument suggests the matched remnant stays finite and white, but the multicomponent case is not computed here.
The logarithmically growing quadrupole in the curvature response suggests that even without a monopole infrared relic, hard-hard beats could imprint a tidal, direction-dependent signature on horizon-scale observables, which would show up in higher-order CMB statistics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The useful thing in this paper is the direct second-order calculation. BIS-II's claimed infrared-divergent curvature spectrum is built on a Poisson-like relation between kurvature and R. Hu shows that for an irrotational radiation fluid the relation fails beyond linear order: the kurvature white noise is carried by the extrinsic shear and expansion of acoustic beats, while the intrinsic 3-curvature piece has identically zero LSWN in the Laplacian of R. That is a clean, concrete counterexample, and I believe it is new relative to BIS [1,2] and the other cited work.

The paper also does several things well. The derivation is given in full length, with appendices that check the result independently: a synchronous-gauge construction pulled back to comoving gauge, a direct evaluation of the expansion contribution, and a reconciliation with the exact Langlois-Vernizzi conservation law. The source-level Laurent test is a nice analytic tool. There are no fitted parameters, no invented entities, and the paper is candid in listing what it has not done. The self-citation to ILH is used as a cross-check, not as the basis for the conclusion, so I do not read circularity into it.

The soft spots are real but proportional. The explicit no-relic conclusion, Eq. (48), is proven only for a single irrotational perfect radiation fluid with w=c_s^2=1/3 and no damping. The actual photon-baryon plasma has baryon loading, diffusion, and neutrino anisotropic stress, all of which change the first-order kernels. The stress-test concern is fair: Eq. (73) shows that when a damping envelope cuts off the sources, the matched remnant is finite and white, but it does not test whether modified kernels can regenerate a 1/epsilon term in R2. The paper also computes only the P22 hard-hard piece, not the full second-order power spectrum, and does not run a Boltzmann code. The geometric distinction between kurvature and R is argued more generally and is likely robust; the quantitative claim that no IR relic survives in CMB observables is not fully demonstrated for the realistic plasma. The paper itself says so, which makes the limitation easier to live with.

Who should read this: anyone working on second-order perturbation theory, CMB power spectra, or the BIS white-noise program. It deserves serious peer review, not a desk reject. My recommendation: send it out, and ask the referee to focus on whether the ideal-fluid counterexample can be promoted to the realistic plasma, or whether the no-relic claim should be scoped down accordingly.

Referee Report

2 major / 4 minor

Summary. This paper addresses the BIS-II conjecture that large-scale white noise in the curvature invariant "kurvature" implies an infrared-divergent, ultraviolet-sensitive contribution to the curvature perturbation R. Working in second-order perturbation theory for an irrotational perfect radiation fluid in comoving gauge, the author derives the sourced master equation for the long-wavelength curvature response R2 and shows that the hard-hard source is regular in the squeezed limit, so [∇²R^(2)]_LSWN = 0 (Eq. 48). The kurvature density δK nevertheless acquires white noise, but from extrinsic shear/expansion terms rather than from intrinsic 3-curvature. The paper then computes the hard-hard power spectrum Δ²₂₂ ∝ A_s² (kη)³, which is UV convergent and IR suppressed, and contrasts it with the BIS-II Poisson construction Δ²_BIS ∝ (q_max/k) A_s². Appendices provide a synchronous-gauge construction with pullback to comoving gauge and a check against the Langlois-Vernizzi conservation law.

Significance. The explicit calculation is a substantial and largely convincing counterexample to the BIS-II inference. Its strengths are that it is parameter-free (the only input is the initial spectrum normalization), the full second-order system is solved rather than assumed, and the result is cross-checked by an independent synchronous-gauge computation and by the exact LV conservation identity. If correct, the paper cleanly separates the geometric statement (kurvature is not related to R by a Poisson equation beyond linear order) from the model-dependent quantitative statement about the absence of an IR relic. The distinction between intrinsic and extrinsic contributions to δK is illuminating and likely robust. The main weakness is the scope of the quantitative no-relic claim, which is established only for the idealized single-fluid case.

major comments (2)

[Sec. I, Sec. IV, Eq. (73)] The abstract's no-relic statement ('leaving no IR relic in R from purely ultraviolet modes') and the CMB-relevance framing are stronger than what the calculation demonstrates. Equation (48) and the cancellations in Eqs. (42)-(43) are derived for a single irrotational perfect radiation fluid with c_s²=1/3 and no dissipation. The physical photon-baryon plasma has c_s²=1/[3(1+R_b)] < 1/3, photon diffusion, and neutrino anisotropic stress; these modify the first-order transfer functions (21)-(22) and hence the Laurent coefficients of S_R in Eq. (40). Equation (73) only multiplies the source by a damping envelope W(x); it does not verify that the modified kernels still have no 1/epsilon term, which is precisely the term that would produce a white component in ∇²R2 and restore the BIS-II IR divergence. Please either restrict the title and abstract claims to the ideal-fluid counterexample or perform a leading-order check (for example, recompute S_{R,0} with a finite baryon loading and with a damping transfer function) to show that the cancellation is not an artifact of w=1/3.
[Sec. IIIF, Eq. (65)] The quantitative UV-convergence statement is demonstrated for a scale-invariant initial spectrum, n_s=1, where each logarithmic hard-mode interval contributes with weight W_lnq ~ x^{-3} log²(2x). The text near Eq. (64) suggests a more general conclusion ('any initial power spectrum with support for such q'), but for a sufficiently blue primordial spectrum the integral in Eq. (65) would not be UV convergent and the hard-hard contribution could depend on the cutoff. Since the paper's central claim is that there is 'no UV cutoff sensitivity,' the range of spectral indices for which the conclusion holds should be stated explicitly, or the claim should be limited to the nearly scale-invariant spectra relevant to CMB anisotropies.

minor comments (4)

[Eqs. (51), (55), (59), (60), (B2), (B3)] The notation 'sin2x' is used for (sin x)^2 while 'cos2x' is used for cos(2x); this is confusing and should be typeset as sin^2 x (or with an explicit superscript) throughout.
[Appendix C, Eq. (C3)] The reference to a correction in 'an arXiv v3 update of ILH' should include the arXiv version number and date, since the numbering of equations (e.g., their Eq. B.27) may differ between versions.
[Fig. 2] The horizontal axis label 'large scale kη' combined with the logarithmic scale is unclear; please specify that the BIS curves are plotted for fixed q_max η and fixed evaluation epoch η, with kη varying.
[Sec. IIA, Eq. (3)] The distinction between the script K (kurvature) and the plain K (trace of extrinsic curvature) is introduced clearly, but it would help to add a one-sentence remark that the sign convention K=-θ on comoving slices is used consistently in Eq. (23) and Appendix A.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central no-IR-relic result follows from an explicit parameter-free second-order calculation; the only self-citation is a non-load-bearing cross-check.

full rationale

The paper's central claim, Eq. (48) [∇²R⁽²⁾]_LSWN = 0 and the UV-convergent P22 spectrum, is derived by an explicit, parameter-free second-order calculation in comoving gauge. The first-order transfer functions R1 = sin x/x, A1, δ1, b are solved from the linear Einstein-Euler system (Eqs. 15-22), not imported from the target. The second-order master equation (40) is obtained by eliminating constraints, and the source-level Laurent test (Eqs. 42-43) plus the integral solution (Eqs. 46-47) are algebraic consequences of the Einstein equations. Nothing is fitted to the BIS-II result; the BIS-II conjecture (Eq. 8) is quoted as the object to be tested and is shown to fail by explicit computation. The only self-citation is Ref. [7] (ILH), used in Appendix C as a synchronous-gauge cross-check of the direct comoving calculation; the author even identifies and corrects a typographical omission in ILH, and Fig. 4 shows agreement at finite soft momentum. Thus the self-citation is corroborative, not load-bearing. The paper itself flags modeling limitations—hard-hard modes only, irrotational perfect radiation fluid, no diffusion damping (Sec. IV and Appendix D)—but these are physical-idealization or completeness concerns, not circularity: the calculation does not assume the target result, and the exact Langlois-Vernizzi conservation law is checked as an external benchmark. No step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The calculation is parameter-free and relies on standard cosmological perturbation theory plus the stated perfect-fluid idealization. No new entities are introduced. No free parameters are fitted; As is only a normalization for the illustrative power-spectrum plots.

assumptions (4)

standard math Standard second-order cosmological perturbation theory in comoving and synchronous gauge is a valid expansion around FLRW.
The entire calculation uses the standard iterative Einstein and fluid equations (Sec. IIIA-IIID, Appendix C).
domain assumption The matter is a scalar, irrotational, perfect radiation fluid (w=1/3) with no viscosity or heat conduction.
Stated in Sec. I and Sec. IV; this restricts the scope and matches the BIS example mechanism.
domain assumption The LSWN limit is epsilon=k_L/q -> 0 at fixed acoustic time x; this is the relevant superhorizon soft limit.
Defined in Eq. (11) and used throughout Sec. III; it is the regime where BIS predicts IR divergence.
domain assumption The primordial spectrum is scale-invariant for the illustrative power spectra in Fig. 2.
Eq. (13) with n_s=1; not load-bearing, used for visualization.

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Cite this review

Pith. "Pith review of Not R Kurvature: Beating Large-Scale White Noise." pith.science (2026). https://pith.science/paper/E4PDA2CP

@misc{pith2026260809709,
  author       = {Pith},
  title        = {Pith review of: Not R Kurvature: Beating Large-Scale White Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4PDA2CP}},
  note         = {Machine review of arXiv:2608.09709}
}

read the original abstract

Kurvature, a recently identified curvature invariant, has been argued to acquire superhorizon, or large-scale, white noise from hard-hard momentum coupling even when matter nonlinearities are small. If kurvature were related directly to cosmological curvature perturbations R through a Poisson equation, this white noise would cause an infrared-divergent variance sensitive to ultraviolet hard-mode physics. However, this relation does not generically hold. Kurvature is not intrinsic 3-curvature: on comoving slices it contains extrinsic-curvature terms, and intrinsic curvature is not related to R by a Poisson equation beyond linear order. We test this inference with second-order perturbation theory in radiation domination, relevant to CMB observables. Quadratic hard-hard composites do generate large-scale white noise in the kurvature density, but the Hamiltonian constraint separates it into intrinsic curvature and extrinsic shear, or equivalently density and expansion. Only the extrinsic terms carry the growing dimensionless kurvature density that mimics an ordinary density fluctuation above the horizon. The direct hard-hard curvature power is ultraviolet convergent, dominated by horizon-scale modes at evaluation, and leaves no IR relic in R from purely ultraviolet modes. By contrast, the Poisson construction of a curvature potential from kurvature is infrared divergent and cutoff sensitive; its white noise arises from extrinsic curvature associated with nonlinear acoustic beat modes in a radiation fluid.

Figures

Figures reproduced from arXiv: 2608.09709 by the authors.

**Figure 2.** FIG. 2. LSWN curvature power spectra for the hard-hard contributions at second order [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. Evolution of the curvature and density hard-hard mode response. Top: the monopole [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Direct comoving-gauge curvature response (solid) compared with the synchronous-evolution-plus-pullback [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 16 canonical work pages

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For a wave alongˆzthe velocity isv(z,η)ˆz, so the deformation-rate tensor has a single non-vanishing eigendirection, Dij =λˆziˆzj.(A1) 14 The transverse directions are untouched

Longitudinal acoustic compression and rarefaction A plane acoustic wave is a longitudinal, alternating compression and rarefaction of the fluid. For a wave alongˆzthe velocity isv(z,η)ˆz, so the deformation-rate tensor has a single non-vanishing eigendirection, Dij =λˆziˆzj.(A1) 14 The transverse directions are untouched. This one-dimensional longitudinal...

work page
[2]

In the ADM foliation used in Sec

DeWitt form of the hard-leg cancellation The preceding discussion phrases the cancellation in the fluid picture while our extrinsic curvature is a property of the embedded spatial slice. In the ADM foliation used in Sec. IIA, the spatial projector is hµν =g µν +nµnν. The induced metric on a constant-time hypersurface is its spatial pullback,γij =h µνeµ ie...

work page
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5 Note also that √ 3x=q/H

When it is important to keep the temporal dependence of both modes we explicitly writexp≡pη/ √ 3 = (p/q)xbut generally usexthroughout. 5 Note also that √ 3x=q/H. The limit of interest isϵ→0at fixedx. At late timesϵ −1≫x≫1, this means the hard modes are acoustically oscillating while the soft mode is outside the horizonkLη≪1appropriate for the LSWN calcula...

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Stebbins, Generation of Isocurvature from Curvature Inhomogeneities on Super-Horizon Scales, (2023), arXiv:2311.17379 [astro-ph.CO]

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In the main text, we took the short cut of using the Hamiltonian constraint to infer it from the shear term

Direct computation ofδ K,expansion The direct computation of the expansion term in the kurvature is operationally more cumbersome since unlike the shear, it enters with a background contribution and requires a higher order expansion. In the main text, we took the short cut of using the Hamiltonian constraint to infer it from the shear term. To avoid the i...

work page
[7]

(B3)), with the dashed guide line−4 ln(2x) + 17 2 −4γ E showing the late-time cycle mean

Middle: the quadrupoleR ℓ=2 2 (Eq. (B3)), with the dashed guide line−4 ln(2x) + 17 2 −4γ E showing the late-time cycle mean. Bottom: the comoving densityδ2 (Eq. (59)), with oscillations that decay toward its asymptotic value−4. takes the free constant-plus-1/xdecaying form of Eq. (73), so the logarithmic growth ceases and fixes a finite constant component...

work page
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Synchronous variables and evolution For a hard leg of comoving wavevectorb(b∈{q,p}) with unit primordial curvature, writexb≡|b|η/ √ 3, so thatxb =xfor theqleg. We follow the synchronous-gauge notation and results of Inomata, Lee, and Hu (ILH) [7] rather than reproducing them here:Ψb and ˆEb are respectively the isotropic and scalar trace-free parts of the...

work page

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Second, we follow BIS in assuming an irrotational fluid and so do not address the effect of the rotation termω2 on kurvature. Finally, our finding of a small white-noisek0 term in the second-order comoving densityδ2 and curvature fluctuationsR 2 might seem to contradict naive ...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

For a wave alongˆzthe velocity isv(z,η)ˆz, so the deformation-rate tensor has a single non-vanishing eigendirection, Dij =λˆziˆzj.(A1) 14 The transverse directions are untouched

Longitudinal acoustic compression and rarefaction A plane acoustic wave is a longitudinal, alternating compression and rarefaction of the fluid. For a wave alongˆzthe velocity isv(z,η)ˆz, so the deformation-rate tensor has a single non-vanishing eigendirection, Dij =λˆziˆzj.(A1) 14 The transverse directions are untouched. This one-dimensional longitudinal...

work page

[2] [2]

In the ADM foliation used in Sec

DeWitt form of the hard-leg cancellation The preceding discussion phrases the cancellation in the fluid picture while our extrinsic curvature is a property of the embedded spatial slice. In the ADM foliation used in Sec. IIA, the spatial projector is hµν =g µν +nµnν. The induced metric on a constant-time hypersurface is its spatial pullback,γij =h µνeµ ie...

work page

[3] [3]

5 Note also that √ 3x=q/H

When it is important to keep the temporal dependence of both modes we explicitly writexp≡pη/ √ 3 = (p/q)xbut generally usexthroughout. 5 Note also that √ 3x=q/H. The limit of interest isϵ→0at fixedx. At late timesϵ −1≫x≫1, this means the hard modes are acoustically oscillating while the soft mode is outside the horizonkLη≪1appropriate for the LSWN calcula...

work page

[4] [4]

Stebbins, Generation of Isocurvature from Curvature Inhomogeneities on Super-Horizon Scales, (2023), arXiv:2311.17379 [astro-ph.CO]

A. Stebbins, Generation of Isocurvature from Curvature Inhomogeneities on Super-Horizon Scales, (2023), arXiv:2311.17379 [astro-ph.CO]

arXiv 2023

[5] [5]

Stebbins, A Space-Time Fluid (Unabridged), (2026), arXiv:2601.16996 [physics.gen-ph]

A. Stebbins, A Space-Time Fluid (Unabridged), (2026), arXiv:2601.16996 [physics.gen-ph]

work page arXiv 2026

[6] [6]

In the main text, we took the short cut of using the Hamiltonian constraint to infer it from the shear term

Direct computation ofδ K,expansion The direct computation of the expansion term in the kurvature is operationally more cumbersome since unlike the shear, it enters with a background contribution and requires a higher order expansion. In the main text, we took the short cut of using the Hamiltonian constraint to infer it from the shear term. To avoid the i...

work page

[7] [7]

(B3)), with the dashed guide line−4 ln(2x) + 17 2 −4γ E showing the late-time cycle mean

Middle: the quadrupoleR ℓ=2 2 (Eq. (B3)), with the dashed guide line−4 ln(2x) + 17 2 −4γ E showing the late-time cycle mean. Bottom: the comoving densityδ2 (Eq. (59)), with oscillations that decay toward its asymptotic value−4. takes the free constant-plus-1/xdecaying form of Eq. (73), so the logarithmic growth ceases and fixes a finite constant component...

work page

[8] [8]

Synchronous variables and evolution For a hard leg of comoving wavevectorb(b∈{q,p}) with unit primordial curvature, writexb≡|b|η/ √ 3, so thatxb =xfor theqleg. We follow the synchronous-gauge notation and results of Inomata, Lee, and Hu (ILH) [7] rather than reproducing them here:Ψb and ˆEb are respectively the isotropic and scalar trace-free parts of the...

work page