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arxiv: 2507.14358 · v1 · submitted 2025-07-18 · 🌀 gr-qc

Perturbative emergent modified gravity on cosmological backgrounds: Kinematics

Martin Bojowald , Manuel Diaz , Erick I. Duque This is my paper

Pith reviewed 2026-05-19 03:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords emergent modified gravityperturbative inhomogeneitycosmological backgroundscovariance conditionsmetric emergencegeneral relativity modificationsearly universe modelsinhomogeneous perturbations
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The pith

Emergent modified gravity allows new modifications in perturbative cosmological models while keeping the classical derivative order.

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

The paper extends emergent modified gravity from symmetry-reduced models to perturbative inhomogeneities on a spatially flat cosmological background. It separates the space-time metric that appears on solutions from the basic field variables used to write the equations, then shows that covariance conditions can still be met with new modifications that do not raise the derivative order. This keeps the setup at the same classical order as general relativity while allowing the metric to emerge only after the equations and conditions are solved. The result prepares the ground for later dynamical studies of early-universe models.

Core claim

In the perturbative treatment of emergent modified gravity on spatially flat cosmological backgrounds, new modifications are possible while maintaining the classical derivative order. The canonical formulation of general relativity yields a larger class of covariant modifications than action-based approaches because the metric is no longer fundamental but emerges after field equations and covariance conditions are solved. This distinction between the metric on a given solution and the basic field degrees of freedom carries over consistently to the perturbative setting with inhomogeneities.

What carries the argument

The distinction between the space-time metric on a given solution and the basic field degrees of freedom, together with covariance conditions that let the metric emerge after the equations are solved.

If this is right

  • New modifications become available in the perturbative regime on cosmological backgrounds.
  • The equations retain their classical second-order derivative structure.
  • The kinematic setup enables derivation of dynamical equations suitable for early-universe models.
  • Symmetry-reduced models extend to include small inhomogeneities while preserving covariance.

Where Pith is reading between the lines

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

  • These modifications could supply new parameters for fitting cosmological data without introducing higher-derivative instabilities.
  • The same separation of metric and fields might apply to other symmetry-reduced systems beyond cosmology.
  • Numerical evolution of the resulting dynamical equations could test whether emergent modifications affect structure formation.

Load-bearing premise

The perturbative expansion around the homogeneous background can be consistently combined with the covariance conditions so that the metric still emerges after the equations are solved.

What would settle it

A explicit check showing that every non-trivial modification at perturbative order either violates the covariance conditions or forces higher-than-second derivatives would disprove the possibility of new modifications.

read the original abstract

Emergent modified gravity has shown that the canonical formulation of general relativity gives rise to a larger class of covariant modifications than action-based approaches, so far in symmetry-reduced models. This outcome is made possible by distinguishing between the space-time metric on a given solution, and the basic field degrees of freedom in which equations of motion are formulated. In this general treatment, the metric is no longer fundamental but emerges after field equations and covariance conditions are solved. Here, the results are extended to perturbative inhomogeneity on a spatially flat cosmological background, showing that new modifications are possible while maintaining the classical derivative order and setting the stage for dynamical equations suitable for detailed studies of early-universe models.

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.

Referee Report

2 major / 2 minor

Summary. The paper extends the emergent modified gravity framework from symmetry-reduced models to linear perturbations around a spatially flat FLRW cosmological background. It shows that covariance conditions imposed at first order in inhomogeneities still permit non-trivial modification functions while preserving second-order derivative structure and ensuring that a space-time metric emerges from the basic canonical variables after the equations are solved.

Significance. If the kinematic construction holds, the work provides a controlled perturbative setting in which emergent modified gravity can be applied to inhomogeneous cosmologies. This is a necessary step toward dynamical studies of early-universe models that retain the framework’s key feature: a larger class of covariant modifications than those obtainable from metric actions, without raising the derivative order.

major comments (2)
  1. [§3.2] §3.2 (linear-order covariance conditions): the mixing of background quantities with first-order perturbations in the covariance constraints is not shown to leave room for non-GR modification functions. The text states that solutions exist, but no explicit solution or counting argument demonstrates that the system remains under-determined enough to admit new functions while keeping the emergent metric second-order.
  2. [§4] §4 (emergent metric reconstruction): the claim that the metric emerges at the same derivative order as in GR after solving the perturbed constraints relies on an implicit assumption that the linear-order conditions do not force higher derivatives or destroy metric emergence. An explicit check reducing to the homogeneous limit or a concrete non-trivial example would be required to substantiate the central claim.
minor comments (2)
  1. [Notation] Notation for the emergent metric versus the fundamental triad and connection variables should be made uniform between the background and perturbative sections to avoid confusion when the two are combined.
  2. [Table 1] A brief table comparing the allowed modification functions in the homogeneous case versus the perturbative case would clarify the new content introduced at linear order.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript extending emergent modified gravity to perturbative cosmological backgrounds. We address each major comment below and have made revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (linear-order covariance conditions): the mixing of background quantities with first-order perturbations in the covariance constraints is not shown to leave room for non-GR modification functions. The text states that solutions exist, but no explicit solution or counting argument demonstrates that the system remains under-determined enough to admit new functions while keeping the emergent metric second-order.

    Authors: We appreciate this observation. Upon re-examination, the linear-order covariance conditions in §3.2 do allow for non-trivial modification functions beyond GR, as the equations are under-determined after imposing the background conditions. However, to make this explicit, we will include a detailed counting of the independent equations versus free functions and provide a concrete example of a non-GR solution in the revised manuscript. This addition will clarify that the system admits new functions without increasing the derivative order of the emergent metric. revision: yes

  2. Referee: [§4] §4 (emergent metric reconstruction): the claim that the metric emerges at the same derivative order as in GR after solving the perturbed constraints relies on an implicit assumption that the linear-order conditions do not force higher derivatives or destroy metric emergence. An explicit check reducing to the homogeneous limit or a concrete non-trivial example would be required to substantiate the central claim.

    Authors: We agree that an explicit demonstration is beneficial. In the revised version, we will add to §4 an explicit reduction to the homogeneous limit, recovering the standard emergent metric from the symmetry-reduced case. Furthermore, we will present a specific non-trivial example of modification functions satisfying the first-order covariance conditions and explicitly solve for the emergent metric, verifying that it remains second-order in derivatives and consistent with the framework. revision: yes

Circularity Check

0 steps flagged

Self-citation to prior emergent modified gravity framework; perturbative extension adds independent content on covariance conditions

full rationale

The paper extends the authors' earlier emergent modified gravity results (cited in the abstract as having been shown in symmetry-reduced models) to linear perturbations on a flat FLRW background. The central derivation concerns imposing covariance conditions at first order in perturbations while preserving second-derivative order and metric emergence. This step introduces new analysis of how background and perturbation terms mix under the covariance conditions, rather than redefining prior parameters or reducing to a self-definitional identity. No equation is shown to equal its input by construction, and the self-citation supports the background framework without bearing the full load of the perturbative claim. The derivation remains self-contained against the stated assumptions about consistent combination of expansion and covariance.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior distinction between basic fields and the emergent metric, plus the assumption that perturbative inhomogeneity can be introduced while preserving covariance.

axioms (2)
  • domain assumption The canonical formulation of general relativity permits a larger class of covariant modifications than action-based approaches.
    Stated as the foundational premise of emergent modified gravity in the abstract.
  • domain assumption The metric emerges only after field equations and covariance conditions are solved.
    Core distinction introduced in the abstract and carried into the perturbative setting.

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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unclear
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Reference graph

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