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Reproducing $\Lambda$CDM-like Solutions in $f(Q)$ Gravity: A Comprehensive Study Across All Connection Branches

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read f(Q) gravity can reproduce the ΛCDM expansion exactly for all three connection branches

desk verdict Solid reconstruction study: the new Gamma2 analytic result is the clear contribution; the Gamma3 numerical branch is plausible but under-reported and needs code/data before the 'all three branches' claim is fully sealed. read the letter →

arxiv 2501.15159 v2 pith:KZMJMUZZ submitted 2025-01-25 gr-qc

Saikat Chakraborty , Jibitesh Dutta , Daniele Gregoris , Khamphee Karwan , Wompherdeiki Khyllep This is my paper · ORCID
classification gr-qc MSC 83F0583D05
keywords f(Q)gravityLambdaCDMmimicrysymmetricteleparallelcosmographicjerkparameterconnectionbranchescosmologicalreconstructionnonmetricityscalareffectivegravitationalcoupling
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 asks whether a modified theory of gravity based on the nonmetricity scalar $Q$ can reproduce the exact background expansion history of the standard $\Lambda$CDM model. The answer it argues for is yes: for each of the three allowed symmetric teleparallel connection branches in a flat, homogeneous, isotropic universe, there exists an $f(Q)$ theory whose Friedmann dynamics give $H^2(z)=H_0^2[c_1(1+z)^3+1-c_1]$ with $c_1=\tfrac{2}{3}(1+q_0)$. For two branches the reconstruction is analytic; for the third it is numerical. If correct, the result shows that $f(Q)$ gravity is not distinguished from $\Lambda$CDM by the expansion history alone, so any discriminating test must come from perturbations or the behavior of the connection.

What carries the argument

The reconstruction is driven by the cosmographic condition $j(z)=1$ on the jerk parameter, which is equivalent to demanding the $\Lambda$CDM-like Hubble rate $H^2(z)=H_0^2[c_1(1+z)^3+1-c_1]$. The field equations of $f(Q)$ gravity, together with the connection equation $\nabla_\mu\nabla_\nu(\sqrt{-g}f_Q P^{\mu\nu}{}_{\sigma})=0$, are then integrated to determine $f(Q)$ for each connection branch. For $\Gamma_2$ this integration gives an analytic quadratic form in $Q$; for $\Gamma_3$ the key step is rewriting the connection evolution as a decoupled first-order equation for $x=\gamma/H$, Eq. (117), which permits numerical solution. The physical-viability condition $f_Q>0$ is used throughout to constrain the free parameters, and the effective gravitational coupling $\kappa_{\rm eff}=1/f_Q$ is tracked to identify where the reconstructed theory coincides with (STE)GR.

What would settle it

A direct detection that standard matter has a non-minimal coupling to the symmetric teleparallel connection, giving non-vanishing hypermomentum, would invalidate the connection field equation used here and hence all three reconstructed $f(Q)$ forms; alternatively, a precise measurement showing that the dynamical connection function for $\Gamma_2$ does not grow linearly with the Hubble rate as in Eq. (79) would falsify the analytic reconstruction.

Watch

Extended reading notes

Core claim

The paper proves that $\Lambda$CDM-mimicking $f(Q)$ models exist for all three homogeneous, isotropic, spatially flat symmetric teleparallel connection branches. Branch $\Gamma_1$, the coincident gauge where $Q=-6H^2$, yields the previously known two-parameter family $f(Q)=-2\Lambda+\left(\Lambda/H_0^2/(1-2q_0)\right)Q+\beta\sqrt{-Q}$, with $\beta$ and $\Lambda$ the free parameters and with STEGR as a past attractor when $\Lambda=H_0^2(1-2q_0)$. Branch $\Gamma_2$ is reconstructed analytically as $f(Q)=-2\Lambda+\alpha Q-\beta Q^2$, a three-parameter family constrained so that the effective gravitational coupling stays positive, and with a parameter choice that yields a one-parameter model $f(Q)=-2H_0^2(1-2q_0)+Q-\frac{1}{4H_0^2}\left(\frac{1-D}{1-2q_0}\right)Q^2$. For branch $\Gamma_3$, the evolution equation for the dynamical connection function decouples from $Q$, allowing a numerical reconstruction. The central conclusion is that the background kinematics alone cannot distinguish $f(Q)$ gravity from $\Lambda$CDM in any of the three branches.

Load-bearing premise

The reconstruction assumes that ordinary matter feels only the metric, so the matter Lagrangian does not couple to the connection, hypermomentum vanishes, and dust density redshifts as $(1+z)^3$.

Editorial extensions

If this is right

  • If the paper is correct, the $\Lambda$CDM expansion history is compatible with infinitely many $f(Q)$ theories, one family for each connection branch, so background data alone cannot select among them.
  • For the $\Gamma_2$ branch the reconstructed theory can reproduce $\Lambda$CDM-like evolution with a vanishing cosmological constant, offering a concrete route to address the cosmological constant problem within symmetric teleparallel gravity.
  • The reconstructed models reduce to (STE)GR in the appropriate asymptotic regime when parameters are chosen within the stated bounds, so they can inherit standard early-universe behavior while modifying late-time dynamics.
  • The same cosmographic reconstruction procedure can be applied to other constant-jerk values or to a redshift-dependent jerk, providing a systematic way to generate 'almost-$\Lambda$CDM' $f(Q)$ models.
  • Because the background is indistinguishable from $\Lambda$CDM, the models are expected to differ only at the perturbation level, through the time-dependent effective gravitational coupling $\kappa_{\rm eff}$.

Reading between the lines

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

  • A natural extension is to confront the reconstructed $\Gamma_2$ form $f(Q)=-2\Lambda+\alpha Q-\beta Q^2$ with growth-rate and weak-lensing data, since its quadratic correction may produce a distinctive scale dependence in structure formation.
  • The decoupled evolution equation for $\Gamma_3$ suggests that the same numerical method could be applied to other background histories, such as an evolving-dark-energy parametrization, without needing an analytic form for $f(Q)$.
  • If future measurements find $j(z)\neq 1$ at high significance, the reconstruction procedure remains valid but the resulting $f(Q)$ would no longer be the exact $\Lambda$CDM mimic, illustrating how cosmographic data directly map onto the gravitational Lagrangian.
  • The authors' assumption of vanishing hypermomentum is the point most likely to be revisited; coupling matter to the connection would change the field equations and invalidate all three reconstructed forms.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper reconstructs f(Q) theories whose flat FLRW background reproduces exactly H²(z)=H0²[c1(1+z)^3+1-c1] with c1=2(1+q0)/3, for each of the three symmetric teleparallel connection branches. Branch Γ1 yields f(Q)=-2Λ+[Λ/H0²/(1-2q0)]Q+β√(-Q), recovering previous results; branch Γ2 is reconstructed analytically as a quadratic f(Q), with a consistency condition, parameter counting, and viability bounds; branch Γ3 is treated numerically after decoupling an ODE for the dynamical connection function. The paper also analyzes the effective gravitational coupling, the stability of the Γ1 solution, robustness to small deviations of the jerk parameter, and the model-dependence of the inferred Ωm0.

Significance. If correct, the paper gives a useful completeness statement: the ΛCDM background does not select a unique connection branch in f(Q) gravity. The analytic parts for Γ1 and Γ2 are the main strengths; the derivations are mostly explicit, the parameter counting is careful, and the fQ>0 constraints are applied systematically. The paper also explicitly flags that the vanishing-hypermomentum assumption is a physical limitation, which is appropriate and does not undermine the background-level claims. However, the Γ3 branch is the load-bearing piece of the 'all three branches' claim, and as presented it is not reproducible: it consists of plots without data, code, or a check of the inversion Q(z) and of the Friedmann constraint at z=0. The stress-test concern about Γ3 therefore lands, and it requires a substantive revision.

major comments (4)
  1. [Section VIII D, Figs. 3-7] The numerical reconstruction for Γ3 is not independently reproducible from the text. The paper reports only plots, with no tabulated (z, Q, fQ, f) values and no code or integration details for Eqs. (116)-(118). Since this branch is the only support for the abstract's 'all three branches' claim, please provide a reproducibility supplement (data files or code) and state the integration scheme, tolerances, and the exact parameter values used for each plotted curve.
  2. [Section VIII (after Eq. (114); Figs. 4 and 7)] The construction requires inverting Q=Q(z) to obtain fQ(Q)=fQ(z(Q)), but the paper never proves that Q(z) is monotone or single-valued on (0,(1+4z*)/3). The text itself notes that ∣Q changes sign near x=-4, so this is not a formality; if Q(z) is not one-to-one, fQ(Q) is multivalued and the 'reconstructed f(Q)' is not a function. The authors should prove monotonicity on the integration interval, restrict to monotonic subintervals, or give a different well-defined elimination argument.
  3. [Sections VIII B and VIII D, Eq. (123)] The plotted Γ3 curves are not validated against the z=0 constraint. Eq. (123), evaluated at z=0, is supposed to determine Ωm0 (or equivalently relate Λ, c, z*, Ωm0), but in the numerical section Ωm0=c1=0.3 is assumed and no value of Λ (or λ) is reported. Without this, one cannot check whether the displayed f(Q) satisfies the Friedmann equation at z=0, let alone on the whole interval. Please report the full parameter sets for the plotted solutions and demonstrate that Eq. (123) holds at z=0 and at representative redshifts.
  4. [Abstract and Section IX (Γ3 bullet)] The word 'proves' in the abstract is stronger than the Γ3 analysis supports. For Γ3 the result is a numerical reconstruction that depends on the ad hoc condition fQ(z*)=1 (i.e., x=-2 at z*), on the chosen z*, and on the auxiliary parameter c; no existence or uniqueness argument for the full f(Q) is given. Please either soften the claim to 'provides evidence for' or 'presents a numerical construction of', or supply a rigorous existence argument, and correspondingly adjust the summary section.
minor comments (4)
  1. [Figs. 3-7] Most axes lack labels and the horizontal tick labels are garbled (e.g., '1 5 1 0 5 0 1 0 0'), making it impossible to read the plotted ranges without guessing.
  2. [Eq. (115) and Section VII] The same symbol c is used with different dimensions for Γ2 (c≡C/H0) and Γ3 (c≡C/H0³); although a footnote warns the reader, the notation is easy to confuse in the adjacent sections and should be changed or emphasized more strongly.
  3. [Eqs. (127)-(135)] The approximate analytic reconstruction for Γ3 is presented as valid near z*, but the text does not quantify the order of the expansion or its radius of validity; please state the error order and the expected range of applicability.
  4. [Section II] The statement that c1 is 'completely kinematical' should be reconciled with the later, correct observation that its relation to Ωm0 is model-dependent; the current wording may confuse readers about the status of c1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the f(Q) forms are derived by integrating the f(Q) field equations with the assumed H(z) as input and then verified by substitution; the reconstruction is not a fit renamed as a prediction.

full rationale

The paper's central claim is an existence construction: impose the LambdaCDM-like H(z) of Eqs. (7)/(11) and solve the metric and connection field equations for f(Q) on each connection branch. For Gamma1, Eq. (36) is integrated to obtain Eq. (41), and substituting this f(Q) into the Friedmann equation reproduces Eq. (11) identically; the parameters beta and Lambda are identified as free parameters, not fitted to the target. For Gamma2, Eq. (29)/(70) is integrated for fQ(z), Eq. (77) determines gamma(z), Eq. (79) gives Q(z), and Eqs. (80)-(81) integrate to the quadratic f(Q); the redshift-independent constraint Eq. (82) is a consistency condition relating the parameters, not a recycled input. For Gamma3, the decoupled equation (113) is solved numerically and fQ(Q) is obtained by inverting Q(z); this is a genuinely independent numerical construction rather than a renaming or a fit to the target H(z). The self-citation [41] is used only to define 'LambdaCDM-like' kinematically, and the Gamma1 result is explicitly compared with prior independent work [18,33]; neither citation carries the derivation. The vanishing-hypermomentum assumption is explicitly flagged in Section IX as a limitation and does not make the reconstruction circular. A separate concern, not circularity, is that the Gamma3 numerical data are not tabulated and Q(z) monotonicity is not demonstrated, so the third-branch result is not independently reproducible from the paper alone; this weakens verifiability, not the logical independence of the reconstruction.

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

The central claim rests on the standard f(Q) field equations, the three-branch classification of symmetric teleparallel connections, the vanishing-hypermomentum assumption, and the cosmographic definition of LambdaCDM-like evolution. The integration constants beta, Lambda, c, and D are the free parameters of the reconstructed models. No new particles, forces, or geometrical entities are invented; the reconstructed f(Q) functions are outputs, not inputs.

free parameters (7)
  • beta (coefficient of sqrt(-Q), Gamma1) = Free; constrained by beta/H0 < 2*sqrt(6)*h(z) and by the bound on kappa_eff variation
    Integration constant A1 in Section VI, redefined as beta; cannot be fixed by the background reconstruction.
  • Lambda (cosmological constant term) = Free in general; Gamma1 bound 0<Lambda/H0^2<3/2*(1-2q0)/(1+q0); setting Lambda=H0^2(1-2q0) makes STEGR a past attractor
    Appears as integration constant A2 or -2Lambda in all branches; determines Omega_m0 via Eq. (42), (83), or (123).
  • c (Gamma2 dimensionless integration constant) = Negative; bounds -3c1/(2h(z_star)) <= c < 0
    Integration constant from Eq. (71); controls the evolution of fQ and the dynamical connection function.
  • D (Gamma2 integration constant) = 0 <= D < 1 - 2|c|/(3c1)
    Integration constant in fQ; can be traded for z_star, the redshift at which fQ=1 and the reconstructed theory coincides with STEGR.
  • c (Gamma3 dimensionless integration constant) = c=0.1 used in the numerical plots; bound c < 12*sqrt(3)*c1^(3/2)*sqrt(1+z_star)
    Integration constant from Eq. (106); chosen to keep fQ positive over the integration range.
  • z_star (Gamma3, and equivalently Gamma2) = z_star=20 and z_star=100 used in the numerical examples
    Redshift where the reconstructed f(Q) coincides with STEGR; sets the initial condition x=-2 in the Gamma3 numerical integration.
  • q0 (present-day deceleration parameter) = Approximately -0.55 used in figures; c1=2/3(1+q0)
    External cosmographic input defining the LambdaCDM-like trajectory; not a model parameter but a boundary condition.
assumptions (5)
  • domain assumption The connection branches Gamma1, Gamma2, and Gamma3 exhaust the curvature-free, torsion-free, FLRW-compatible symmetric teleparallel connections.
    Used in Section IV; the completeness result is taken from Hohmann (2021), cited as Ref. [34].
  • domain assumption Matter is a pressureless perfect fluid with vanishing hypermomentum, so the connection equation is Eq. (23) and dust energy is conserved with respect to the Levi-Civita connection.
    Used throughout Section V; the authors note in Section IX that neglecting hypermomentum has no inherent physical motivation.
  • domain assumption Physical viability requires fQ>0.
    Used to constrain parameters in Sections VI B, VII B, and VIII C, following Ref. [65].
  • domain assumption The LambdaCDM-like evolution is defined by j=1, i.e. H^2=H0^2[c1(1+z)^3+1-c1] with c1=2/3(1+q0).
    Section II defines the target background via cosmography rather than via the GR field equations.
  • ad hoc to paper For the Gamma3 numerical reconstruction, f(Q) is assumed to match STEGR at z=z_star in the matter-dominated era (fQ(z_star)=1, x=-2), and z is restricted to (0,(1+4z_star)/3) to avoid a singularity.
    Sections VIII A and VIII C; this assumption is not derived from first principles and restricts the domain of validity of the Gamma3 reconstruction.

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Pith. "Pith review of Reproducing $\Lambda$CDM-like Solutions in $f(Q)$ Gravity: A Comprehensive Study Across All Connection Branches." pith.science (2026). https://pith.science/paper/KZMJMUZZ

@misc{pith2026250115159,
  author       = {Pith},
  title        = {Pith review of: Reproducing $\Lambda$CDM-like Solutions in $f(Q)$ Gravity: A Comprehensive Study Across All Connection Branches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZMJMUZZ}},
  note         = {Machine review of arXiv:2501.15159}
}
abstract

Given the remarkable success of the $\Lambda$CDM model in fitting various cosmological observations, a pertinent question in assessing the phenomenological viability of modified gravity theories is whether they can reproduce an exactly $\Lambda$CDM-like cosmic background evolution. In this paper, we address this question in the context of $f(Q)$ gravity, where $Q$ denotes the nonmetricity scalar. It is known that there are three possible symmetric teleparallel connection branches that respect the cosmological principles of spatial homogeneity, isotropy, and global spatial flatness. By enforcing a $\Lambda$CDM-like background evolution via the cosmographic condition $j(z)=1$, where $j$ is the jerk parameter, we reconstruct the $\Lambda$CDM-mimicking $f(Q)$ theory for each of the three possible connection branches. For the first connection branch, also known as the ``coincident gauge'' in cosmology, we recover the previously known result that a theory of the form $f(Q)=-2\Lambda+\alpha Q+\beta\sqrt{-Q}$ can exactly reproduce a $\Lambda$CDM-like cosmic evolution. Furthermore, we establish that the stability of the $\Lambda$CDM-like cosmic solution within this reconstructed $f(Q)$, as well as the robustness of the reconstructed $f(Q)$ form with respect to small errors in the astrophysical measurements of the jerk parameter. For the second connection branch, we analytically reconstruct the $\Lambda$CDM-mimicking $f(Q)$ to be of the form $f(Q)=-2\Lambda+\alpha Q-\beta Q^2$. For the third connection branch, we could decouple the evolution equation for the dynamical connection function, which enabled us to perform a numerical reconstruction. Our analysis proves that, at least at the background level, it is possible to obtain $\Lambda$CDM-mimicking $f(Q)$ models for all the three possible connection branches.

Figures

Figures reproduced from arXiv: 2501.15159 by the authors.

Figure 1
Figure 1. FIG. 1: Evolution of [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) Evolution of [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Plots (a) and (b) shows the evolution of [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Plots (a) and (b) shows the evolution of [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The plots for the [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The plots of [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The plots of [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

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