Modified Friedmann equations and non-singular cosmologies in $d=4$ non-polynomial quasi-topological gravities

Jo\~ao Magueijo; Johanna Borissova

REVIEW 4 major objections 3 minor 9 cited by

Four-dimensional non-polynomial quasi-topological gravity yields three families of cosmologies without a Big Bang, all recovering the usual late-time expansion.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-15 11:55 UTC pith:WH27BZNP

load-bearing objection Abstract-only readable claim of three non-singular cosmologies in d=4 quasi-topological gravity; bounce and de Sitter-origin cases rest on constructions that need hard scrutiny. the 4 major comments →

arxiv 2603.17654 v2 pith:WH27BZNP submitted 2026-03-18 gr-qc astro-ph.CO

Modified Friedmann equations and non-singular cosmologies in d=4 non-polynomial quasi-topological gravities

Johanna Borissova , Jo\~ao Magueijo This is my paper

classification gr-qc astro-ph.CO PACS 04.50.Kd98.80.Jk04.20.Dw

keywords quasi-topological gravitynon-polynomial curvaturemodified Friedmann equationsnon-singular cosmologyBig Bang singularitybouncing universede Sitter originloitering universe

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The same higher-curvature quasi-topological theories already known to smooth black-hole interiors can also eliminate the Big Bang. In four dimensions, with non-polynomial curvature terms, the modified Friedmann equations admit three generic early-universe solutions for ordinary matter (including pure radiation). One universe emerges from a de Sitter phase whose curvature stays finite, even though the matter density formally diverges only at infinite affine distance. A second solution bounces, but needs a multi-valued Lagrangian. The third begins asymptotically from Minkowski space, with a nearly constant, sub-Planckian density that never becomes singular. All three recover ordinary Einstein gravity at late times, so the singularity resolution is an ultraviolet modification that leaves the observed cosmology intact.

Core claim

In d=4 non-polynomial quasi-topological gravities the modified Friedmann equations possess three families of solutions that match the standard infrared cosmology yet never encounter a Big-Bang singularity for radiation or other ordinary matter: (i) emergence from de Sitter with finite curvature invariants, (ii) a bounce that requires a multi-valued Lagrangian, and (iii) an eternally loitering, asymptotically Minkowski origin whose density remains finite and sub-Planckian for the entire past.

What carries the argument

The modified Friedmann equations obtained by reducing non-polynomial quasi-topological curvature densities to a second-order equation for the scale factor; these equations retain the correct Einstein infrared limit while opening new ultraviolet branches that avoid a=0.

Load-bearing premise

That the non-polynomial quasi-topological Lagrangians remain physically acceptable when they are multi-valued or when they allow a density that diverges only at infinite past affine distance.

What would settle it

Derive the full modified Friedmann equation for a concrete non-polynomial quasi-topological series and check whether any of the three claimed early-time branches (de Sitter origin, bounce, Minkowski loitering) actually solves it for radiation while recovering the standard late-time expansion.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Abstract-only readable claim of three non-singular cosmologies in d=4 quasi-topological gravity; bounce and de Sitter-origin cases rest on constructions that need hard scrutiny.

read the letter

The punchline is simple: they take the quasi-topological machinery that already resolves black-hole singularities and ask whether the same higher-curvature setup can kill the Big Bang for ordinary matter in four dimensions. From the abstract they get three families with the right late-time Friedmann limit—emergence from de Sitter, a bounce, and an asymptotically Minkowski loitering phase.

What is actually new is the cosmological classification itself, not the gravity theory. Quasi-topological gravities and their BH regularity are established; cataloguing the three IR-safe, singularity-free cosmologies for radiation (and standard matter) is a legitimate extension. The Minkowski/loitering branch looks cleanest: density stays finite, sub-Planckian, and roughly constant into the past. That is worth having on the table.

Soft spots are real and uneven. The bounce explicitly needs a multi-valued Lagrangian. That is not a minor technicality—variational principle, branch choice, and well-posedness of the equations through the bounce are all open until they show a continuous physical branch with continuous H and Ḣ. The de Sitter-origin case keeps curvature invariants finite but lets matter density diverge at infinite affine distance; geodesic completeness may hold, yet infinite energy density is still a matter-sector pathology, and calling it “non-singular” needs a clear criterion. The full manuscript we were given is encoding-corrupted, so the modified Friedmann equations, the explicit non-polynomial series, and the regularity proofs cannot be checked. That is a verification problem, not a proof of error.

This is for people already working on higher-curvature gravity, singularity theorems, and early-universe alternatives to inflation. A serious referee should see it: the claim shape is falsifiable and sits inside an active program. I would not cite it yet until the multi-valued and infinite-density issues are cleaned up and a readable text is available. Send it to peer review; do not desk-reject.

Referee Report

4 major / 3 minor

Summary. The manuscript studies whether non-polynomial quasi-topological gravities in four dimensions, already known to resolve black-hole singularities, can also remove cosmological Big-Bang singularities. From the abstract and stated setup, the authors derive modified Friedmann equations for these theories and classify three generic families of solutions that recover the standard infrared (late-time) Friedmann limit for pure radiation or other standard matter: (i) emergence from a de Sitter phase with finite curvature invariants but a matter density that diverges only at infinite affine distance; (ii) a bouncing cosmology that requires a multi-valued Lagrangian; and (iii) an asymptotically Minkowski (eternally loitering) origin whose matter density remains non-singular, approximately constant, and sub-Planckian for the entire past. The central claim is that the same quasi-topological mechanism that regularizes black holes yields non-singular cosmologies with correct IR behavior.

Significance. If the derivations hold, the work would extend the singularity-resolution programme of quasi-topological gravity from static black holes to dynamical FLRW cosmologies in d=4, a setting of direct observational interest. The three-scenario classification, the emphasis on infrared recovery of Einstein gravity, and the identification of a clean loitering branch with sub-Planckian density are potentially useful contributions to modified-gravity cosmology. The result would be of interest to the community working on higher-curvature and quasi-topological theories, provided the constructions are shown to be well-defined classical field theories and the modified Friedmann equations are fully specified and checked.

major comments (4)

The bouncing scenario is stated to require a multi-valued Lagrangian. A multi-valued action density is not a standard classical field theory: the variational principle, the map from curvature to the effective Friedmann function, and branch selection through the bounce are not automatically well-defined. The manuscript must either (a) exhibit a continuous single-valued physical branch with continuous H and Ḣ across the bounce and a well-posed initial-value problem, or (b) justify why a multi-valued Lagrangian remains an acceptable classical description. Without that, the bounce cannot be counted as a demonstrated non-singular cosmology in a well-defined theory.
For the de Sitter-origin scenario the paper counts the solution as non-singular because curvature invariants remain finite while the matter density diverges only at infinite affine distance. Infinite energy density is still a matter-sector pathology. The manuscript needs an explicit comparison with standard singularity criteria (geodesic incompleteness, curvature blow-up, and matter-sector regularity) and a clear statement of which criterion is being adopted and why a density divergence at infinite affine distance is acceptable. Otherwise the non-singularity claim for this branch rests on a narrow geometric criterion that is not yet justified.
The infrared recovery of Einstein gravity and the second-order character of the modified Friedmann equations are asserted for the full non-polynomial curvature series. These properties are load-bearing for all three scenarios. The manuscript should display the explicit form of the modified Friedmann equations (or the effective algebraic relation between H^{2} and ρ), state the conditions under which the series reduces to GR at low curvature, and confirm that no higher-than-second-order derivatives of the scale factor appear for the FLRW ansatz used. Without that verification the classification of solutions cannot be trusted.
The third (asymptotically Minkowski / loitering) branch is the cleanest of the three: density non-singular and sub-Planckian for the past eternity. The paper should quantify how generic this branch is within the space of non-polynomial quasi-topological couplings, and whether it requires fine-tuning relative to the de Sitter-origin and bounce branches. A parameter-space sketch or existence theorem would strengthen the claim that three generic families exist.

minor comments (3)

The abstract is clear, but the body should define notation for the non-polynomial quasi-topological densities and the precise sense of “correct infrared limit” (e.g., leading Einstein–Hilbert term plus controlled higher-curvature corrections) before the cosmological analysis.
When discussing affine distance and geodesic completeness for the de Sitter-origin case, a short appendix computing the affine parameter for radial null and timelike geodesics would make the “infinite affine distance” claim self-contained.
References to the black-hole singularity-resolution results in quasi-topological gravity should be cited with equation or theorem numbers so that the analogy to the cosmological case is easy to check.

Circularity Check

0 steps flagged

No circularity: standard Lagrangian-to-equations-to-solutions construction; no fitted inputs renamed as predictions and no load-bearing self-definitional loop visible.

full rationale

The paper’s claimed chain is ordinary modified-gravity theory construction: adopt a class of non-polynomial quasi-topological curvature Lagrangians in d=4, obtain the corresponding modified Friedmann equations, then classify homogeneous cosmologies (radiation or standard matter) that recover the infrared Einstein limit while avoiding a Big-Bang singularity. The three scenarios (de Sitter origin, multi-valued bounce, asymptotically Minkowski/loitering origin) are solution classes of those equations, not quantities fitted to data and re-labeled as predictions, nor quantities defined in terms of themselves. There is no empirical fit, no uniqueness theorem imported solely from the authors’ prior work that forces the target result by construction, and no renaming of a known empirical pattern. Physical caveats (multi-valued Lagrangian for the bounce; density divergence only at infinite affine distance for the de Sitter-origin case) are assumptions or limitations of the setup, not circular reductions of outputs to inputs. The supplied full-text encoding is corrupted, so individual equations cannot be re-derived here, but nothing in the readable abstract or stated claim structure exhibits a self-definitional, fitted-as-prediction, or self-citation-forced loop. Score 0 with empty steps is therefore the warranted finding.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

Abstract-only review. The central claim rests on the existence and consistency of a class of non-polynomial quasi-topological Lagrangians in four dimensions that (i) yield second-order equations for FLRW, (ii) recover Einstein gravity in the infrared, and (iii) admit the three listed regular cosmologies. No free parameters are numerically fitted in the abstract; the multi-valued Lagrangian for the bounce and the acceptance of infinite-affine-distance density divergence are the main ad-hoc structural choices.

axioms (4)

domain assumption Non-polynomial quasi-topological curvature terms in d=4 produce second-order modified Friedmann equations with a healthy infrared Einstein limit.
Stated as the theoretical setup of the paper; required for all three scenarios.
ad hoc to paper A multi-valued Lagrangian is an acceptable classical description for the bouncing branch.
Explicitly required by the second scenario in the abstract; not standard single-valued GR.
ad hoc to paper A matter-density divergence that occurs only at infinite affine distance does not count as a physical Big-Bang singularity when curvature invariants remain finite.
Used to classify the de Sitter-origin solution as non-singular.
domain assumption Universes filled with pure radiation or other standard matter (no exotic fluids) are the relevant matter content.
Stated in the abstract as the setting in which the three scenarios appear.

invented entities (1)

Multi-valued non-polynomial quasi-topological Lagrangian (bounce branch) no independent evidence
purpose: Enable a classical bounce without exotic matter while remaining inside the quasi-topological class.
Introduced as necessary for the second scenario; no independent experimental handle is given in the abstract.

pith-pipeline@v1.1.0-grok45 · 7848 in / 2725 out tokens · 24124 ms · 2026-07-15T11:55:53.430600+00:00 · methodology

0 comments

read the original abstract

Quasi-topological theories of gravity are known to resolve black-hole singularities. We investigate whether the same mechanism can remove cosmological singularities. Focusing on non-polynomial curvature quasi-topological gravities in $d=4$ dimensions, we find three generic scenarios with the correct infrared limit but without a Big-Bang singularity, for universes filled with pure radiation or other standard matter. The first scenario yields a universe emerging from a de Sitter phase, a case for which the curvature invariants remain finite but the matter density diverges, albeit only at infinite affine distance. The second one corresponds to a bouncing universe, which requires a multi-valued Lagrangian. The third possibility is an asymptotically Minkowski origin, reminiscent of an eternally loitering universe. The matter energy density for this solution is non-singular even at infinite affine distance and does not enter a super-Planckian regime, but is instead approximately constant for the past eternity.

discussion (0)

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