One-point holographic correlator in the expanding universe

Gopinath Guin; Souvik Paul; Sunandan Gangopadhyay

REVIEW 3 major objections 3 minor 63 references

One-point holographic correlator in the expanding universe

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A single ratio of brane position to horizon controls the one-point function in expanding holographic universes.

desk verdict First one-point function in the moving-brane braneworld, with clean power laws, but the GM-formula transplant is not justified and the paper has internal normalization slips. read the letter →

arxiv 2607.21107 v1 pith:JNU2XWHU submitted 2026-07-23 hep-th

Souvik Paul , Gopinath Guin , Sunandan Gangopadhyay This is my paper · ORCID

classification hep-th

keywords holographicone-pointfunctionbraneworldcosmologyRandall-SundrumIIAdSblackbranethermalgeodesicapproximationp-branegasIsraeljunctioncondition

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

The paper aims to show that thermal one-point functions of heavy operators in an expanding braneworld universe are governed by one number: the ratio of the moving brane's radial position to the black-brane horizon. Using the geodesic formula for large-mass operators, it finds that the correlator is essentially that ratio raised to the conformal dimension, up to a constant p-dependent phase, so all time dependence enters through the brane's trajectory. Applied to radiation-, matter-, and exotic-matter-dominated universes, this gives late-time decays scaling as τ^{-Δ/2}, τ^{-2Δ/3}, and τ^{-Δ}. A sympathetic reader would care because it connects the expansion history of the universe to the decay of a holographic condensate through a simple power law, and it brings the interior-horizon one-point function machinery into a cosmological setting.

What carries the argument

The central object is the ratio z̄(τ)/z_h, where z̄ = 1/r is the inverse radial position of the brane and z_h is the black-brane horizon. The geodesic formula expresses the one-point function as e^{-Δ(l_h + iT_s)}, with l_h the renormalized spacelike length from the brane to the horizon and T_s the proper time from horizon to singularity. For the p-brane metric these lengths evaluate to log(z_h/z̄) + log(4/(4-p)) and π/(4-p), producing the power law above. The formula does the work of converting a geometric distance-to-horizon into an operator expectation value.

What would settle it

One concrete check: evaluate the one-point function by solving the bulk scalar equation on the time-dependent brane geometry without the fixed-brane approximation and see whether the late-time scaling remains τ^{-Δ/2}, τ^{-2Δ/3}, τ^{-Δ}; a different power law would falsify the central claim.

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Extended reading notes

Core claim

The central claim is eqs. (4.24)-(4.25): for a bulk p-brane gas, the thermal one-point function of an operator of conformal dimension Δ is ⟨O⟩_p(τ) ≈ (z̄(τ)/z_h)^Δ 2^{-2Δ/(4-p)} e^{-iΔπ/(4-p)}, so its modulus is just the current position-to-horizon ratio raised to Δ. Substituting the brane trajectories obtained from the Israel junction condition yields explicit time dependence: radiation gives τ^{-Δ/2}, matter gives τ^{-2Δ/3}, and exotic matter gives τ^{-Δ} at late times, with radiation dominating the early-time behavior and matter or exotic matter governing the late-time decay in multi-component universes.

Load-bearing premise

The load-bearing premise, flagged in the paper's own footnote, is that the geodesic formula derived for operators on the boundary of an AdS black hole remains valid when the boundary is replaced by a moving brane treated as momentarily fixed; if that quasi-static step fails, the predicted power-law decay does not follow.

Editorial extensions

If this is right

In single-component universes the modulus of the one-point function decays monotonically in time, with exponents Δ/2, 2Δ/3, and Δ for radiation, matter, and exotic matter; the phase is constant.
In multi-component universes the early-time behavior is set by radiation and the late-time behavior by the dominant pressure component, matching the standard cosmological timeline.
The time dependence factorizes from the operator: all heavy operators with the same conformal dimension share the same decay exponent, only the normalization differs.
For the multi-component cases T_s is independent of the brane position, so the phase never contributes to the time evolution and only the modulus carries cosmological information.

Reading between the lines

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

If the single-ratio rule survives the quasi-static approximation, it suggests that in these braneworlds every geodesically controlled heavy-operator observable is a function of z̄(τ)/z_h; one could test that by computing two-point or higher-point functions on the same brane trajectories.
The late-time decay implies ⟨O⟩ behaves like a^{-Δ} in terms of the scale factor, since z̄ ∝ a^{-1} in these solutions; this hints at a general statement about how vacuum condensates redshift in holographic cosmologies.
A direct numerical check would be to solve the bulk scalar equation on the actual time-dependent brane geometry—including brane motion in the saddle-point integral—rather than freezing the brane position, and compare the late-time scaling.
Because the phase in the multi-component cases is purely constant, one can look for observables sensitive to that phase alone as a clean signature of the horizon-to-singularity geodesic length.

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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

Short version: this is the first computation of the thermal one-point function in the RS-II braneworld with p-brane gas. The late-time scalings — τ^{-Δ/2} for radiation, τ^{-2Δ/3} for matter, τ^{-Δ} for exotic matter — are clean and will probably get quoted. The brane trajectories come from prior parameter-free Israel-junction work, not from fitting, and the key integrals (e.g. T_s = π/(4-p)) check out by substitution. That said, the central move is an unexamined transplant of the Grinberg–Maldacena formula to a finite, moving brane, and there are smaller consistency issues that look like an unfinished draft. I would send it to a referee, but I would not take it as is.

Novelty is real: [36] did two-point functions, [46,50] did entanglement measures, and no one has done the one-point function in this braneworld cosmology. The structure — plug the time-dependent brane position zbar(τ) into (zbar/z_h)^Δ — produces concrete predictions for single- and multi-component universes, and the early/late dominance pattern (radiation at early times, matter/exotic matter at late times) is physically sensible. The geodesic-length integrals in Sec. 4 are straightforward and mostly sound.

Soft spots, in proportion. The load-bearing one is the GM-formula transplant. Eq. (4.14) is derived for large-mass operators on a static AdS black hole, using a WKB saddle at complex radius and analytic continuation to the singularity. The paper replaces the AdS boundary with a finite RS-II brane and substitutes the instantaneous zbar(τ); footnote 3 asserts the quasi-static approximation by fiat, with no estimate of the neglected time-derivative terms. On top of that, the plots use Δ = 1.5–2.1, corresponding to m²R² near the BF bound, far outside the mR≫1 regime. If the saddle argument fails at the cutoff — or if the large-mass saddle does not continue to these Δ values — the power laws do not follow from the calculation as presented. The scalings may well be robust because they are inherited from the brane dynamics, but 'may well' is not a proof, and the paper currently offers no argument.

Then there are the fixable slips: the normalization in (4.24) does not match (4.26)/(4.29)/(4.33), and the text repeatedly calls the early-time behavior 'growth' when the equations (4.27), (4.30), (4.34) show the correlator is decreasing in τ. These are the sort of things a referee can get cleaned up, but they make the paper harder to trust.

Bottom line. This is a paper for people working on holographic cosmology and braneworld correlators. It deserves a serious referee because the first computation of a quantity is useful even when the prefactor and phase are later corrected; the referee should focus on the cutoff/adiabatic justification and the Δ range, and ask for consistency fixes. Reader's CONDITIONAL verdict seems right to me.

Referee Report

3 major / 3 minor

Summary. The manuscript computes the time-dependent thermal one-point function of a massive scalar operator in an expanding Randall–Sundrum II braneworld, with the matter content realized by p-brane gas in the bulk. The brane trajectory z̄(τ) is obtained from the second Israel junction condition for radiation (p=0), matter (p=1), and exotic matter (p=2), as well as for radiation–matter and radiation–exotic-matter universes. The one-point function is then obtained by inserting these trajectories into the Grinberg–Maldacena geodesic formula ⟨O⟩ ∼ e^{-Δ(l_h + i T_s)}, with l_h computed from the brane to the horizon and T_s from the horizon to the singularity. The paper claims clean late-time power laws: τ^{-Δ/2}, τ^{-2Δ/3}, τ^{-Δ} for the three single-component cases, with radiation dominating early-time and matter/exotic matter dominating late-time behavior in the multi-component cases.

Significance. If the calculation were fully justified, the paper would provide a simple holographic prediction for how one-point correlators decay in an expanding braneworld, with the time dependence entering purely through the ratio z̄(τ)/z_h and no fitted parameters. The algebraic skeleton is largely sound: the integrals (4.19)–(4.22) are correct and the substitution of z̄(τ) into the exponential is transparent. However, the central result is conditional on two unproven steps: the transplant of the static, boundary-at-infinity Grinberg–Maldacena formula to a moving finite brane, and the use of that WKB/large-mass formula for conformal dimensions near the BF bound. Both are load-bearing for every displayed power law. The paper is therefore not yet in publishable form, but the issues are of a kind that could be addressed by restricting the claims, adding estimates, or supplying the missing derivation.

major comments (3)

[§4, Eq. (4.14)–(4.25), footnote 3] The central step is the replacement of the static AdS-boundary setup of [1] by a moving finite brane. Eq. (4.14) is derived for a static AdS black hole with the boundary at infinity, using a WKB saddle at complex radius and analytic continuation to the singularity. Here the same exponential is used with l_h measured from the instantaneous brane position z̄(τ), and z̄(τ) is then substituted from the Israel junction condition. Footnote 3 asserts that this is justified because the calculation is performed 'at a fixed cosmological time', but no estimate of the neglected z̄̇ terms or of the effect of the finite moving cutoff on the saddle/contour argument is supplied. This is load-bearing: without a derivation or a quantitative adiabaticity bound, Eq. (4.25) is an ansatz rather than a consequence. Citing [36] for the same practice does not remove the need for the estimate.
[§4.1, Eqs. (4.10)–(4.14), Figs. 3–5] The geodesic formula is derived under the WKB condition mR ≫ 1, with Δ ≈ mR (see the text between Eqs. (4.9) and (4.14)). Yet the paper plots and discusses Δ = 1.5, 1.7, 1.9, 2.1. Using Eq. (4.2) with the minus branch, these correspond to m²R² between -3.75 and -3.99, i.e. very close to the Breitenlohner–Freedman bound, far outside the mR ≫ 1 regime. The paper itself notes the BF bound but does not explain why the large-mass saddle-point result should continue to these Δ values. Either the claims must be restricted to Δ ≫ 1 and the small-Δ plots removed or explicitly labeled as unjustified extrapolations, or an analytic-continuation argument extending (4.14) to near-BF-bound dimensions must be supplied.
[§4.1, Eq. (4.24) versus Eqs. (4.26), (4.29), (4.33)] The advertised central result, Eq. (4.24), contains the prefactor 2^{-2Δ/(4-p)}. Setting p = 0, 1, 2 gives 2^{-Δ/2}, 2^{-2Δ/3}, and 2^{-Δ}, i.e. e^{-(2 ln 2)Δ/(4-p)}. The specialized single-component formulas (4.26), (4.29), and (4.33) instead contain e^{-Δ/2}, e^{-2Δ/3}, and e^{-Δ}, respectively, missing the factor ln2 in the exponent. The same incorrect normalization enters the plots. Although this error does not change the late-time powers, it is an inconsistency in the central formula itself and must be corrected.

minor comments (3)

[§3.1/§4.1, Eq. (4.27) and Conclusion] The text says the early-time one-point function 'grows as' τ^{1/2}, τ^{2/3}, and τ, and the conclusion repeats 'polynomial growth'. However Eqs. (4.27), (4.30), and (4.34) are of the form A(1 - c τ^α), so the correlator decreases with time. Please rephrase to say that the time-dependent correction grows in magnitude, or that the one-point function decays, to avoid contradicting the figures.
[§4.2.2, around Eq. (4.58)] In the paragraph after Eq. (4.57), the text refers to 'a universe with coexisting radiation and matter', but the calculation in this subsection is for radiation plus exotic matter. This wording appears several times and should be corrected.
[§4.1, Eqs. (4.18)–(4.21)] Notation and presentation: the statement in Eq. (4.18) that only leading-order terms in z̄ are written is imprecise, since the sum is subsequently evaluated exactly and the neglected terms are higher powers of z̄ in the l_h expansion. In Eq. (4.21), the first line appears to have a measure typo (dz/z_h instead of dz/z). Also, define r̃, m̃, δ̃, z_t, and r_t consistently and state their dimensions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the time-dependent one-point function follows from the external Grinberg–Maldacena formula plus the Israel-junction brane motion, with no fitted parameter renamed as a prediction.

full rationale

Walking the claimed derivation chain, the central result (4.24)–(4.25) is assembled from the external Grinberg–Maldacena geodesic formula (4.14), the direct integrations l_h = log(z_h/z̄) + log 4/(4−p) in (4.19) and T_s = π/(4−p) in (4.22), and the brane trajectories z̄(τ) obtained by integrating the Israel junction condition in Section 3. No parameter is fitted to the one-point function; z_i, z_h, and Δ are independent inputs, and the late-time powers τ^{−Δ/2}, τ^{−2Δ/3}, τ^{−Δ} are simply the composition of these explicit formulas. The self-citations [45,46] accompany brane-motion and lapse-function results that are re-derived in Section 3, so they are not load-bearing; deleting them would not change the calculation. The fixed-brane-position/quasi-static assumption in footnote 3 and the use of a large-mass saddle for Δ near the BF bound are validity or regime concerns, not circular reductions: they do not make the output equal to an input by construction. No specific circular step can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The derivation rests on the holographic dictionary, the Grinberg-Maldacena geodesic formula (used at a finite cutoff without re-derivation), the RS-II junction-condition dynamics, the p-brane-gas dictionary for the lapse function, and the quasi-static instant-evaluation ansatz. Free parameters are the integration/plot choices (z_i, z_h), the critical-tension tuning σ_c, and the density-ratio parameters; no new entities are introduced.

free parameters (5)

Initial brane position z_i = 0.5 (in plots)
Integration constant from the Israel junction condition; used in early-time expansions and Figs. 3-5; chosen by hand, not fitted to data.
Horizon position z_h = 2 (in plots)
Sets the black-brane horizon scale; chosen for the plot curves; appears in constants of all power-law results.
Critical brane tension σ_c = set by σ = σ_c
Condition used in eqs. (3.6)-(3.7), (3.12), (3.17) to make the brane FLRW-like; standard RS tuning, but a model input chosen rather than derived.
Density ratios (r̃, m̃, δ̃, z_t) = unspecified
Parameters of p-brane gas / black-brane mass in lapse function (3.5); treated as small perturbations in multi-component expansions; not fitted here.
Conformal dimension Δ = 1.5, 1.7, 1.9, 2.1 in plots
Used to draw curves; chosen within BF bound but outside the large-mass regime where the geodesic formula is derived.

assumptions (7)

domain assumption AdS/CFT correspondence with GKPW dictionary
Used throughout (§1, §4) to equate bulk heavy-scalar one-point functions with boundary operator one-point functions.
domain assumption Grinberg-Maldacena geodesic formula ⟨O⟩ ~ e^{-Δ(l_h + iT_s)}
Imported from [1]; the paper relies on the WKB saddle point, analytic continuation to the singularity, and W^2 source balance without re-derivation (eqs. (4.13)-(4.14)).
domain assumption Large-mass WKB limit: Δ ≈ mR for mR ≫ 1
Stated in §1 and §4 before eq. (4.10); contradicts the Δ ~ 1.5-2.1 values used in plots.
domain assumption RS-II braneworld with second Israel junction condition determines brane motion
Basic framework of §2-(2.18); time-dependent brane position equals scale factor.
domain assumption p-brane gas backreaction yields lapse function f(z)=1-(z/z_h)^{4-p}-m̃z^4
Result of §3 eq. (3.5), drawn from prior p-brane gas literature [46,49]; used for all subsequent integrals.
ad hoc to paper Quasi-static evaluation of correlator at fixed brane position
Footnote 3: 'the whole calculation is performed at a fixed cosmological time that is at a fixed brane position'; the instantaneous approximation is asserted, not derived.
standard math BF bound and stability of scalar in AdS
Eq. (4.3) used to justify plotting Δ values; standard stability criterion.

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Pith. "Pith review of One-point holographic correlator in the expanding universe." pith.science (2026). https://pith.science/paper/JNU2XWHU

@misc{pith2026260721107,
  author       = {Pith},
  title        = {Pith review of: One-point holographic correlator in the expanding universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNU2XWHU}},
  note         = {Machine review of arXiv:2607.21107}
}

abstract

In this article, we have calculated the time-dependent thermal one-point function of massive operators within an expanding universe. We employ the Randall-Sundrum II braneworld model combined with a $p$-brane gas in the bulk, enabling us to represent various matter-dominated cosmological scenarios localised on the brane. Applying the geodesic formula introduced by Grinberg and Maldacena in \cite{Grinberg:2020fdj}, we have calculated the thermal one-point functions of massive operators within the universe. The time-dependent one-point functions for different matter-dominated universes, both single-component and multi-component, are derived from the brane's evolving radial position. The time-dependent positions of the branes have been obtained using the second Israel junction condition. Additionally, we have analyzed the early and late-time behaviors of the thermal one-point function across various matter-dominated universes.

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