arxiv: 2512.08972 · v2 · submitted 2025-12-02 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Degenerate higher-order scalar-tensor theories in metric-affine gravity

Hamed Bouzari Nezhad

Authors on Pith no claims yet

Pith reviewed 2026-05-17 02:05 UTC · model grok-4.3

classification 🌀 gr-qc

keywords metric-affine gravityDHOST theoriesPalatini formalismdegenerate higher-order scalar-tensorgravitational wavesscalar-tensor theoriesmodified gravityconnection equation

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The pith

Metric-affine quadratic DHOST theories reduce to a one-function family when gravitational waves must travel at light speed.

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

The paper builds the metric-affine version of quadratic degenerate higher-order scalar-tensor theories by writing an action linear in curvature and at most quadratic in the covariant second derivatives of the scalar field. Solving the algebraic equation for the independent connection produces a closed effective metric theory. Standard metric DHOST degeneracy conditions then pick out a Palatini Class Ia branch controlled by two free functions of the scalar field. Requiring that tensor perturbations propagate exactly at the speed of light further collapses the family to a single free function. A reader would care because the result supplies an explicit, observationally filtered extension of scalar-tensor gravity inside the metric-affine setting.

Core claim

Starting from the metric-affine completion of the quadratic DHOST action, full decomposition of the distortion tensor yields an effective metric theory. Imposition of the metric DHOST degeneracy conditions isolates the Palatini Class Ia sector, which is parameterized by two arbitrary functions. The additional demand that the tensor speed equals the speed of light reduces the theory to a one-function family within the chosen operator basis.

What carries the argument

Full decomposition of the distortion tensor, which converts the metric-affine connection equation into a closed-form effective metric theory to which standard DHOST degeneracy conditions can be applied directly.

If this is right

The quadratic metric-affine Class Ia sector is completely characterized by two functions before the gravitational-wave condition is imposed.
Requiring light-speed gravitational waves selects a unique one-function subfamily.
The construction supplies a self-contained effective metric description for this branch of metric-affine DHOST theories.

Where Pith is reading between the lines

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

The same reduction procedure could be applied to cubic or higher-order extensions of the same operator basis.
The resulting one-function family offers a concrete target for cosmological tests that combine scalar-field dynamics with gravitational-wave propagation.
Explicit solutions for specific choices of the remaining function could be compared with existing metric DHOST black-hole or cosmological solutions.

Load-bearing premise

The connection enters the action only through curvature and the covariant second derivatives of the scalar field.

What would settle it

An observation or calculation showing that tensor modes propagate at a speed different from light, inside the regime where the effective metric theory applies, would rule out the one-function family.

read the original abstract

We construct the metric-affine analogue of the quadratic degenerate higher-order scalar-tensor (DHOST) theories. We begin with the metric-affine completion of the quadratic DHOST scalar-tensor action, which is linear in curvature and contains all operators that are at most quadratic in the covariant second derivatives of the scalar field, ensuring that the connection enters only through curvature and these second derivatives. Solving the connection equation by performing a full decomposition of the distortion tensor gives a closed-form effective metric theory. Imposing the standard metric DHOST degeneracy conditions then selects a Palatini Class Ia branch that is fully determined by two free functions in the original action. Analyzing the tensor sector shows that requiring gravitational waves to propagate at the speed of light further restricts the theory to a one-function family. These results provide a detailed and self-contained characterization of the quadratic metric-affine Class Ia sector within this operator basis and identify the theoretical conditions implied by gravitational wave observations.

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

This paper builds the metric-affine version of quadratic DHOST by solving the connection explicitly and restricts it to a one-function family with the GW speed condition, though the exact match to the metric DHOST basis after decomposition needs checking.

read the letter

This paper constructs the metric-affine analogue of quadratic degenerate higher-order scalar-tensor theories and reduces the free functions to one after imposing the gravitational wave speed condition. They start from a general quadratic action in metric-affine gravity that is linear in curvature and quadratic in the covariant second derivatives of the scalar field. The connection is solved by decomposing the distortion tensor completely, producing a closed-form effective metric theory. Imposing the standard metric DHOST degeneracy conditions isolates a Palatini Class Ia branch determined by two free functions. The tensor sector analysis then shows that requiring gravitational waves to propagate at the speed of light cuts this down to a one-function family. The new part is the explicit construction of this metric-affine completion and the demonstration that the usual degeneracy conditions apply after the connection is integrated out. The closed-form solution for the connection is useful because it avoids leaving auxiliary fields in the final theory. This gives a practical way to build models that satisfy both degeneracy and observational constraints from gravitational waves. One soft spot is the assumption that the effective metric action falls squarely inside the standard DHOST operator basis. The distortion decomposition might introduce relations among derivatives that are not present in the pure metric case, which could mean the degeneracy conditions need extra checks or that the function counting changes. The paper claims the conditions carry over directly, but the details of how the tensor perturbations are handled would be important to verify. This work is for people studying modified gravity in the metric-affine framework, particularly those looking at scalar-tensor theories with higher derivatives and their cosmological implications. Anyone working on ensuring consistency with gravitational wave observations would find the restriction to one function relevant. The paper shows clear thinking in extending previous metric results and provides enough structure to be worth refereeing.

Referee Report

1 major / 2 minor

Summary. The paper constructs the metric-affine analogue of quadratic degenerate higher-order scalar-tensor (DHOST) theories. Starting from a metric-affine completion of the quadratic DHOST action that is linear in curvature and quadratic in the covariant second derivatives of the scalar field, the authors solve the connection equation via a full decomposition of the distortion tensor. This yields a closed-form effective metric theory. Standard metric DHOST degeneracy conditions are then imposed to isolate a Palatini Class Ia branch determined by two free functions in the original action. Requiring the tensor sector to propagate gravitational waves at the speed of light further restricts the theory to a one-function family.

Significance. If the central mapping to an unmodified quadratic DHOST theory holds, the work provides a self-contained characterization of the quadratic metric-affine Class Ia sector and links it to observational constraints from gravitational-wave speed. The explicit reduction from two to one free function via the c_T=1 condition is a concrete phenomenological output. The approach of completing the action, decomposing the distortion, and transferring degeneracy conditions could be useful for other metric-affine extensions, provided the effective-action equivalence is verified in detail.

major comments (1)

[derivation of effective metric action after distortion decomposition] The load-bearing step is the assertion that the closed-form effective metric action obtained after distortion decomposition lies exactly within the standard quadratic DHOST operator basis, so that the usual degeneracy conditions apply without modification or extra constraints. This is stated in the abstract and the paragraph following the connection-equation solution, but the manuscript does not exhibit the explicit expanded form of the effective Lagrangian or verify term-by-term that no implicit relations among metric and scalar derivatives appear. Without this check, the counting of two free functions (and the subsequent reduction to one) is not yet secured.

minor comments (2)

[Introduction] The abstract and introduction refer to 'Palatini Class Ia branch' without a brief reminder of the classification used in the metric DHOST literature; adding one sentence would improve readability for readers outside the immediate subfield.
[connection equation and distortion decomposition] Notation for the distortion tensor and its irreducible pieces is introduced but not summarized in a single table or equation block; a compact reference list would help when the decomposition is used later in the tensor-sector analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in the derivation of the effective metric action. We address the major comment below and have revised the manuscript to incorporate the requested verification.

read point-by-point responses

Referee: The load-bearing step is the assertion that the closed-form effective metric action obtained after distortion decomposition lies exactly within the standard quadratic DHOST operator basis, so that the usual degeneracy conditions apply without modification or extra constraints. This is stated in the abstract and the paragraph following the connection-equation solution, but the manuscript does not exhibit the explicit expanded form of the effective Lagrangian or verify term-by-term that no implicit relations among metric and scalar derivatives appear. Without this check, the counting of two free functions (and the subsequent reduction to one) is not yet secured.

Authors: We agree that an explicit term-by-term verification strengthens the central claim. In the revised manuscript we have added the full expanded expression for the effective metric Lagrangian immediately after the solution of the connection equation (new subsection following Eq. (connection solution)). The resulting action is shown to contain precisely the standard quadratic DHOST operators—those built from the metric, the scalar field, and its first and second covariant derivatives—with no additional implicit relations or cross terms generated by the distortion decomposition. Because the effective theory therefore lies exactly inside the usual quadratic DHOST operator basis, the standard degeneracy conditions can be imposed without modification, confirming the reduction to a two-function family before the further restriction from c_T = 1. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation proceeds from general metric-affine action via independent connection solution to external DHOST conditions

full rationale

The paper begins with a general metric-affine completion of the quadratic DHOST action that is linear in curvature and quadratic in covariant second derivatives of the scalar field. It then solves the connection equation independently by full distortion-tensor decomposition to produce a closed-form effective metric theory. Standard metric DHOST degeneracy conditions (defined externally in the literature) are imposed on this derived effective action to select the Palatini Class Ia branch with two free functions, followed by an analysis of the tensor sector for c_T=1. No step defines a quantity in terms of itself, renames a fitted input as a prediction, or reduces the central claim to a self-citation chain; the construction remains self-contained against the stated operator basis and external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the specific operator basis limited to curvature and quadratic second derivatives of the scalar, plus the direct transferability of metric DHOST degeneracy conditions to the effective theory obtained after solving for the connection.

free parameters (2)

two free functions in the original action
These determine the Palatini Class Ia branch before the gravitational-wave speed condition is imposed.
one free function after restriction
The final family is controlled by a single arbitrary function once light-speed propagation of tensor modes is required.

axioms (2)

domain assumption The connection enters only through curvature and the covariant second derivatives of the scalar field.
Explicitly stated as the basis for constructing the metric-affine completion of the quadratic DHOST action.
domain assumption Standard metric DHOST degeneracy conditions apply directly to the effective metric theory after connection solution.
Used to select the Palatini Class Ia branch.

pith-pipeline@v0.9.0 · 5457 in / 1476 out tokens · 46073 ms · 2026-05-17T02:05:18.405525+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Solving the connection equation by performing a full decomposition of the distortion tensor gives a closed-form effective metric theory. Imposing the standard metric DHOST degeneracy conditions then selects a Palatini Class Ia branch that is fully determined by two free functions
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

c^2_T = f̃ / (f̃ - X ã1) ... imposing exact luminal propagation c^2_T=1 ... reduces the two-function Palatini Class Ia family to a one-function subclass

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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