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arxiv: 2402.07716 · v3 · submitted 2024-02-12 · 🌌 astro-ph.CO · gr-qc· hep-ph· hep-th

Λ_(rm s)CDM cosmology from a type-II minimally modified gravity

\"Ozg\"ur Akarsu , Antonio De Felice , Eleonora Di Valentino , Suresh Kumar , Rafael C. Nunes , Emre \"Oz\"ulker , J. Alberto Vazquez , Anita Yadav This is my paper

Pith reviewed 2026-05-24 03:48 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-phhep-th
keywords Λ_s CDMminimally modified gravityVCDMAdS-to-dS transitionscalar field potentialtransient accelerationcosmological tensions
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The pith

Embedding Λ_s CDM into VCDM gravity produces a predictive model for the AdS-to-dS transition and transient cosmic accelerations.

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

The paper integrates the Λ_s CDM scenario, which aims to ease cosmological tensions, into VCDM, a type of minimally modified gravity. This creates Λ_s VCDM, a model that determines the universe's expansion history on its own, even during the switch from anti-de Sitter to de Sitter behavior at late times. An auxiliary scalar field drives an effective cosmological constant whose potential can jump in slope, but the model smooths this jump with a sigmoid function to avoid instabilities. Two kinds of smooth transitions emerge: one with a bump in the effective constant and one without, each potentially causing brief extra periods of accelerated expansion around redshift 1.5 to 2.

Core claim

We integrate Λ_s CDM into VCDM to obtain the fully predictive Λ_s VCDM model that specifies the cosmological evolution self-consistently, including through the late-time AdS-to-dS transition epoch. An auxiliary scalar field generates an effective cosmological constant with a potential that permits an abrupt mirror AdS-to-dS transition via a piecewise-linear form with sudden slope change; smoothing the junction with a blended sigmoid interpolant removes the sudden singularity and yields two distinct realisations, agitated and quiescent, that can induce transient accelerated expansion around z∼1.5-2 in addition to present-day acceleration.

What carries the argument

The auxiliary scalar field whose potential is smoothed by a blended sigmoid interpolant, which generates the effective cosmological constant and permits continuous mirror AdS-to-dS transitions without singularities.

If this is right

  • A finite-width transition can induce a transient accelerated-expansion interval around z∼1.5-2 in addition to present-day acceleration.
  • If the background enters a region where V,ϕϕ>2/3, a nested super-acceleration episode appears.
  • Distinct transition types imprint different signatures on background and perturbation evolution.
  • The construction enables a self-consistent observational assessment of smooth Λ_s CDM realisations.

Where Pith is reading between the lines

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

  • Multi-probe data could distinguish agitated from quiescent transitions through their differing effects on the expansion rate near z=2.
  • The requirement that the transition remain stable under perturbations may constrain the allowed sharpness of the sigmoid smoothing in extensions that include matter coupling.
  • If the quiescent case keeps Λ_s(a) monotone, it could produce expansion histories closer to standard ΛCDM while still allowing late-time adjustments testable with distance-ladder measurements.

Load-bearing premise

Smoothing the junction using a blended sigmoid interpolant removes the associated sudden singularity and ensures stable evolution of the background and perturbations without introducing instabilities.

What would settle it

Detection or non-detection of a transient interval of accelerated expansion (ddot a > 0) at redshifts 1.5-2, together with any associated signatures in perturbation evolution arising from the transition layer.

Figures

Figures reproduced from arXiv: 2402.07716 by Anita Yadav, Antonio De Felice, Eleonora Di Valentino, Emre \"Oz\"ulker, J. Alberto Vazquez, \"Ozg\"ur Akarsu, Rafael C. Nunes, Suresh Kumar.

Figure 1
Figure 1. Figure 1: FIG. 1. Abrupt mirror AdS-dS transition. The upper panel [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Quiescent mirror AdS-dS transition. The upper [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Agitated mirror AdS-dS transition. The upper panel [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

We integrate $\Lambda_{\rm s}$CDM, a promising scenario for alleviating cosmological tensions, into VCDM, a type-II minimally modified gravity. This promotes the scenario to a fully predictive model (dubbed $\Lambda_{\rm s}$VCDM) that specifies the cosmological evolution self-consistently, including through the late-time AdS-to-dS transition epoch. In this theory, an auxiliary scalar field generates an effective cosmological constant with either a constant or a linear potential. This allows an abrupt mirror AdS-to-dS transition via a piecewise-linear potential with a sudden slope change. To remove the associated sudden singularity and ensure stable evolution, we smooth the junction using a blended sigmoid interpolant, obtaining rapid but continuous transitions. We identify two qualitatively distinct smooth mirror AdS-to-dS realisations of $\Lambda_{\rm s}$: (i) an agitated transition, in which the potential interpolates between equal-magnitude AdS and dS plateaus and $\Lambda_{\rm s}$ develops a central bump; and (ii) a quiescent transition, in which the potential remains continuous but changes slope across the transition layer, so that $\Lambda_{\rm s}(a)$ can remain monotone, with possible shallow shoulders, and a central bump is not automatic. Depending on type and sharpness, a finite-width transition can induce a transient accelerated-expansion interval ($\ddot a>0$) around $z\sim 1.5-2$, in addition to present-day acceleration, and, if the background enters a region where $V_{,\phi\phi}>2/3$, a nested super-acceleration episode. These distinct transient histories can imprint signatures on background and perturbation evolution. Our construction enables a self-consistent observational assessment of smooth $\Lambda_{\rm s}$CDM realisations and motivates multi-probe analyses to test transition dynamics and reassess cosmological tensions.

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 / 1 minor

Summary. The manuscript integrates the Λ_s CDM scenario into VCDM (a type-II minimally modified gravity) by replacing the piecewise-linear potential with a sigmoid-smoothed interpolant. This produces a predictive Λ_s VCDM model with two classes of smooth mirror AdS-to-dS transitions (agitated, with a central bump in Λ_s, and quiescent, with possible monotone Λ_s(a) and shallow shoulders). The smoothed construction is claimed to remove sudden singularities while permitting transient accelerated expansion (ä>0) around z∼1.5–2 in addition to late-time acceleration, and possible nested super-acceleration when V,φφ>2/3, with distinct imprints on background and perturbations.

Significance. If the stability of perturbations is confirmed, the work supplies a fully predictive, self-consistent realization of Λ_s CDM inside modified gravity, enabling observational tests of transition dynamics and their bearing on cosmological tensions. The explicit distinction between agitated and quiescent realizations, together with the identification of possible transient histories, constitutes a concrete advance over prior piecewise constructions.

major comments (2)
  1. [Abstract and section describing the smoothing procedure and realizations] The central claim that sigmoid smoothing yields stable background and perturbation evolution (abstract; discussion of the two realizations) is load-bearing, yet the quadratic action for scalar/tensor perturbations is not recomputed. VCDM perturbation equations depend on V,φφ and the auxiliary-field kinetic structure; without explicit evaluation of the signs of the kinetic and gradient coefficients across the finite-width transition layer, it remains possible that a transient region with c_s²<0 or V,φφ>2/3 appears even in the quiescent case.
  2. [Paragraph on transient accelerated-expansion interval] The statement that a finite-width transition can induce transient acceleration (ä>0) around z∼1.5–2 is presented as a qualitative outcome of the smoothed potential, but no explicit background integration or parameter scan is shown to confirm that the transition sharpness and slope values actually produce this interval without violating the quiescent monotonicity assumption.
minor comments (1)
  1. Notation for the auxiliary scalar field and the effective potential V(φ) should be introduced with a single consistent equation early in the text rather than piecemeal across the abstract and realizations section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments correctly identify that the manuscript's claims on perturbation stability and transient acceleration rest on qualitative arguments rather than explicit calculations. We address each point below and will incorporate the required computations in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and section describing the smoothing procedure and realizations] The central claim that sigmoid smoothing yields stable background and perturbation evolution (abstract; discussion of the two realizations) is load-bearing, yet the quadratic action for scalar/tensor perturbations is not recomputed. VCDM perturbation equations depend on V,φφ and the auxiliary-field kinetic structure; without explicit evaluation of the signs of the kinetic and gradient coefficients across the finite-width transition layer, it remains possible that a transient region with c_s²<0 or V,φφ>2/3 appears even in the quiescent case.

    Authors: We agree that the absence of an explicit recomputation of the quadratic action for perturbations across the finite-width transition constitutes a gap. Although the sigmoid smoothing preserves continuity and differentiability of V(φ) and the original VCDM kinetic structure away from the layer, the referee is correct that the signs of the kinetic and gradient coefficients must be verified inside the transition region for both realizations. In the revised manuscript we will derive the relevant coefficients from the perturbed action, evaluate them numerically through the transition layer, and demonstrate that c_s² remains positive and V,φφ does not exceed 2/3 in the quiescent case (while noting the agitated case may require additional parameter restrictions). revision: yes

  2. Referee: [Paragraph on transient accelerated-expansion interval] The statement that a finite-width transition can induce transient acceleration (ä>0) around z∼1.5–2 is presented as a qualitative outcome of the smoothed potential, but no explicit background integration or parameter scan is shown to confirm that the transition sharpness and slope values actually produce this interval without violating the quiescent monotonicity assumption.

    Authors: The referee correctly notes that the transient ä>0 interval is asserted without supporting numerical evidence. The smoothed potential permits a temporary excursion in the effective equation-of-state parameter that can produce ä>0 at intermediate redshifts while the late-time behavior remains accelerating. In the revision we will add explicit numerical integrations of the background equations for representative values of the transition sharpness and slope parameters in the quiescent realization, together with a short scan confirming that the ä>0 window around z∼1.5–2 can be realized without violating monotonicity of Λ_s(a). revision: yes

Circularity Check

0 steps flagged

No significant circularity; model construction is explicit and self-contained

full rationale

The paper defines Λ_s VCDM by embedding the Λ_s CDM scenario into VCDM via an explicit choice of smoothed sigmoid interpolant for the auxiliary-field potential. This is presented as a modeling decision to remove the sudden singularity, not as a derivation that reduces to prior fitted inputs or self-citations. No equations are shown reducing predictions to the transition parameters by construction, and the distinction between agitated and quiescent transitions follows directly from the chosen potential forms. The provided text contains no load-bearing self-citation chains or uniqueness theorems imported from the authors' prior work that would force the result. The construction is therefore independent of its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claim rests on postulating an auxiliary scalar field and a custom smoothed potential to realize the desired transitions; these are not derived from more fundamental principles but introduced to embed the Λ_s CDM scenario.

free parameters (2)
  • transition sharpness
    Parameter controlling the width and rapidity of the sigmoid interpolant used to smooth the potential junction.
  • potential slope values
    Magnitudes of the linear slopes on either side of the transition, chosen to achieve AdS and dS plateaus.
axioms (1)
  • domain assumption VCDM provides a consistent type-II minimally modified gravity framework with an auxiliary scalar field
    The integration assumes the prior VCDM theory is valid and can host the effective cosmological constant.
invented entities (1)
  • auxiliary scalar field no independent evidence
    purpose: Generates the effective cosmological constant with constant or linear potential allowing AdS-to-dS transition
    New field introduced to make the model predictive and self-consistent through the transition epoch.

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