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Geometric formulation of $k$-essence and late-time acceleration

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A linear gravitational action built from integrable vectorial nonmetricity is exactly equivalent to quadratic purely kinetic k-essence, and the model is statistically indistinguishable from ΛCDM on late-time CC, Pantheon+, and DESI BAO…

desk verdict A clean geometric derivation of a known k-essence model, with an honest but unproven stability transfer; worth refereeing. read the letter →

arxiv 2505.15975 v2 pith:AN36JSY7 submitted 2025-05-21 gr-qc astro-ph.COmath-phmath.MP

Lehel Csillag , Erik Jensko This is my paper
classification gr-qcastro-ph.COmath-phmath.MP MSC 83D0583F05 PACS 04.50.Kd95.36.+x98.80.-k
keywords nonmetricityk-essencemetric-affinegravitydarkenergydynamicalsystemscosmologycosmicchronometersbaryonacousticoscillationsWeylgeometry
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 tries to establish that a geometry in which parallel transport changes vector lengths in a very constrained way—nonmetricity proportional to a gradient one-form with three coefficients—is not a separate dark-energy theory but the same thing as a familiar scalar-field model. With the integrability condition imposed by Lagrange multipliers, the linear Ricci-scalar action reduces exactly to the purely kinetic quadratic k-essence Lagrangian $P(X) = -b_1 X + b_2 X^2$. The authors show that $\Lambda$CDM's expansion history is embedded in this model, that a dynamical-systems analysis supplies a stable late-time de Sitter attractor, and that the physical stability conditions (non-negative energy density and non-negative squared sound speed) select the parameter region that survives comparison with cosmic chronometer, Pantheon+, and DESI BAO data. Their fit is statistically indistinguishable from $\Lambda$CDM ($\Delta\mathrm{AIC}=-0.03$), which they read as evidence that integrable vectorial nonmetricity is a viable geometric origin for dark energy and explains why the quadratic k-essence model was dismissed too quickly in the past.

What carries the argument

The load-bearing identity is the reduction of the affine Ricci scalar with integrable vectorial nonmetricity to a k-essence Lagrangian: after the Lagrange multiplier imposes $\pi_\mu=\partial_\mu\phi$, the action collapses to $L=R+b_1(\partial\phi)^2+\tfrac12 b_2(\partial\phi)^4$, i.e. $P(X)=-b_1X+b_2X^2$ with $b_1,b_2$ given by Eq. (19). The cosmological analysis is carried by the two-dimensional autonomous system in $x=-b_1\dot{\phi}^2/(6H^2)$ and $y=b_2\dot{\phi}^4/(4H^2)$; its fixed points $A=(0,1)$ (radiation tracking), $B=(1,0)$ (stiff matter), $C=(-2,3)$ (de Sitter), organized around the divergent line $x=-2y$, define the physical triangle $OAC$ that generates the parameter priors. The analytic Hubble solution $h(z)^2=\Omega_{m0}(1+z)^3-\tilde{X}(z)/3+\tfrac43 B\tilde{X}(z)^2$, whose $k_0=0$ branch is $\Lambda$CDM, carries the late-time observational claim.

What would settle it

Run a full linear perturbation analysis of action (16) with the independent connection and Lagrange-multiplier fields: a ghost or gradient instability appearing in those sectors while $\rho_\phi\ge 0$ and $c_s^2\ge 0$ would break the perturbative equivalence and invalidate the priors. Observationally, fitting the model to CMB-anchored early-time data would test the claim directly, since the analytic $h(z)$ deviates substantially from $\Lambda$CDM at $z\gtrsim 2$ once $k_0$ is non-zero.

Watch

Extended reading notes

Core claim

The central claim is that the linear action $$S=\frac{1}{2\kappa}\int $d^{4}$x\sqrt{-g}\left(R+\xi\nabla_\mu\pi^\mu\right)+S_\$\lambda$$$ with $S_\lambda$ enforcing $\pi_\mu=\partial_\mu\phi$, and with vectorial nonmetricity $$Q_{\mu\nu\rho}=c_1\pi_\mu g_{\nu\rho}+c_2(\pi_\rho g_{\mu\nu}+\pi_\nu g_{\rho\mu})+2c_3\pi_\mu\pi_\nu\pi_\rho,$$ is exactly equivalent, after eliminating the Lagrange multiplier, to the purely kinetic quadratic k-essence model $L_k=P(X)=-b_1 X+b_2 X^2$, where $X=-\partial_\mu\phi\,\partial^\mu\phi/2$ and $b_1,b_2$ are the parameter combinations in Eq. (19). In cosmology the model contains $\Lambda$CDM as the $k_0=0$ branch of the analytic Hubble solution $h(z)^2$, and its dynamical system has a stable de Sitter attractor that exists only for $b_1>0$, $b_2>0$, i.e. only for the completely symmetric geometry $c_1=c_2$. Imposing $\rho_\phi\ge 0$ and $c_s^2\ge 0$ restricts the physical phase space to the triangle with vertices $O,A,C$ and produces the bounded priors used in the MCMC. An MCMC fit to CC, Pantheon+, and DESI BAO data gives $\chi^2_{\rm red}\simeq 1.032$ and $\Delta\mathrm{AIC}=-0.03$ relative to $\Lambda$CDM, which the paper states as statistical indistinguishability at late times.

Load-bearing premise

The load-bearing premise is that the stability conditions derived in the equivalent scalar-field description—non-negative energy density and non-negative squared sound speed—also govern the underlying geometric theory, even though the equivalence is demonstrated at the background level and the full connection may carry extra degrees of freedom.

Editorial extensions

If this is right

  • $\Lambda$CDM is contained as the $k_0=0$ branch of the analytic Hubble solution, so the model inherits the late-time $\Lambda$CDM expansion history by construction.
  • An MCMC fit to CC, Pantheon+, and DESI BAO gives $\chi^2_{\rm red}\approx 1.032$ and $\Delta\mathrm{AIC}=-0.03$ relative to $\Lambda$CDM, making the two models statistically indistinguishable at late times.
  • The stable de Sitter attractor requires $b_1>0$ and $b_2>0$, selecting the completely symmetric geometry ($c_1=c_2$) as the only special geometry that can host late-time acceleration.
  • Imposing $\rho_\phi\ge 0$ and $c_s^2\ge 0$ removes the divergent line $x=-2y$ from the physical phase space and yields bounded priors on $B$; without these priors the MCMC samples unstable regions.
  • The late-time Hubble function is governed almost entirely by $B$, leaving $\Omega_{m0}$ nearly unconstrained; varying the fixed $\Omega_{m0}$ from $0$ to $0.303$ leaves $H_0$, $r_d$, $M$, and the goodness of fit unchanged within $2\sigma$.

Reading between the lines

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

  • The equivalence is proven at the field-equation level, but the full metric-affine theory contains the Lagrange multiplier and connection as independent fields; a linear perturbation analysis of action (16) could reveal extra degrees of freedom whose stability is not captured by $c_s^2\ge 0$.
  • Because $\Omega_{m0}$ decouples from late-time $h(z)$, combining CMB or other early-time data with this model may break the degeneracy and change the early-time fit relative to $\Lambda$CDM; the paper identifies this as the next step but does not perform it.
  • The construction is a hierarchy: adding higher self-interactions of $\pi_\mu$ in the nonmetricity ansatz would produce generic power-law $P(X)$ k-essence models, making the quadratic model the first nontrivial member of a geometric family.
  • The triangular physical phase space suggests a diagnostic for other k-essence models: stability boundaries, not just best-fit statistics, may explain why purely kinetic models have been prematurely ruled out by late-time data.
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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

2 major / 5 minor

Summary. The paper constructs a metric-affine theory in which nonmetricity is of the integrable vectorial form Q_μνρ = c1 π_μ g_νρ + c2(π_ρ g_μν + π_ν g_μρ) + 2c3 π_μ π_ν π_ρ, imposes the integrability condition π = dφ via a Lagrange multiplier, and shows that the linear-in-Ricci-scalar action reduces to the purely kinetic quadratic k-essence model P(X) = -b1 X + b2 X^2. The authors perform a dynamical-systems analysis of the cosmological background, identify fixed points including a stable de Sitter point, derive analytic solutions for h(z) that contain ΛCDM as the k0=0 limit, and use these stability conditions to set priors for an MCMC fit to CC, Pantheon+, and DESI BAO data. They report that the model is statistically indistinguishable from ΛCDM at late times, with ΔAIC = -0.03, and discuss implications for the Hubble tension and for why this k-essence model has been overlooked.

Significance. If the central claims hold, this is a significant result: it gives a geometric, non-Riemannian origin for a well-studied k-essence model, embeds ΛCDM as a special limit, and shows that late-time background observations cannot distinguish the model from ΛCDM. The paper's strengths include the careful derivation of the field equations in Appendices A and B, the explicit analytic solution (D5) with its manifest ΛCDM limit, the honest discussion of singularities in the dynamical-systems formulation, and the use of public data with standard MCMC methods. The statistical comparison and the viability claim, however, rest on two load-bearing assumptions that need to be made explicit and checked: the transfer of scalar-tensor stability conditions to the geometric variables, and the treatment of Ωm0 as a fixed rather than a free parameter in the model comparison.

major comments (2)
  1. [Sec. III B, Eq. (28); Sec. V B, Eq. (82)] The paper's viability conclusion and the MCMC priors rest on the conditions ρφ ≥ 0 and cs² ≥ 0 being transferable from the scalar-tensor representation to the original geometric variables. The equivalence in Eqs. (24)-(25) and the field equations (17)-(18) is established by eliminating the Lagrange multiplier λμ and imposing π = ∂φ; this is an on-shell reduction. Section III B itself states that "the constraints one imposes from the scalar-tensor representation may not necessarily hold the same weight in the geometric setting," but no linear perturbation analysis in the original variables (g, π, φ, λ), and no Hamiltonian/constraint count demonstrating the absence of extra modes, is provided. Because Eq. (65) and the physical phase space in Fig. 4 are used as priors for the MCMC and to support the indistinguishability claim in Table IV, this missing check is load-bearing rather than cosmetic. I request either a direct perturbative stability calculation in the geometric theory or an explicit argument that the on-shell equivalence preserves the fluctuation dynamics.
  2. [Sec. V C, Table IV; Sec. V B, Eq. (82)] The model comparison credits the geometric k-essence model with four free parameters by fixing Ωm0, while ΛCDM has four free parameters including Ωm0. If Ωm0 is instead counted as a parameter of the k-essence model, the reported ΔAIC = -0.03 becomes approximately +2.0, and ΔBIC = -0.03 becomes approximately +7.4 (using N = 1728), reversing the claimed mild preference for k-essence into a preference for ΛCDM under BIC. Appendix E demonstrates insensitivity to the fixed value of Ωm0, but it does not marginalize over Ωm0 or justify treating it as known a priori. The text should either include Ωm0 in the parameter count, perform a marginalization over Ωm0, or explicitly frame the comparison as conditional on an externally fixed Ωm0.
minor comments (5)
  1. [Sec. III B, Eq. (29); Sec. IV B, Table II] The statement that ρφ ≥ 0 and cs² ≥ 0 imply b1 ≤ 0 and b2 ≥ 0 for all X is not consistent with the stable de Sitter point C, which requires b1 > 0 and b2 > 0. Please clarify that Eq. (29) is a sufficient global condition, whereas the analysis actually imposes Eq. (28) pointwise along physical trajectories.
  2. [Figures 3, 6, and 9] Several axis labels and legends are garbled or missing in the compiled text (e.g., "Ωm Ωm ΛCDM" in Fig. 3 and the legend entries in Fig. 6). Please redraw these figures with clean, complete labels.
  3. [Section IV A, Eq. (49)-(50)] The phrase "all other points on the x = -2y line represent genuine pathological behaviour" should be qualified, since the divergence is a property of the chosen dynamical variables and the text later explains that the physical phase space excludes this line; as written it could be read as a statement about the underlying theory rather than the formulation.
  4. [Appendix A, Eq. (A16)] The angle-preservation condition in Eq. (A16) is written in a non-standard form that mixes g(X,Y) with ∇_Z g; please add a clear definition of the norm being used or replace this with the standard condition (∇_Z g)(X,Y) = λ(Z) g(X,Y).
  5. [Throughout] The terminology "cubic nonmetricity term" (for the 2c3 π_μ π_ν π_ρ term in Eq. (9)) and "quartic kinetic terms" (for the b2 X² term in the action) is potentially confusing; consider adding a sentence clarifying that the nonmetricity is cubic in π while the corresponding action term is quartic in ∂φ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geometric/k-essence equivalence is derived from the action, the ΛCDM limit is a parameter choice, and the observational constraints are genuine fits to external data.

full rationale

The paper's central chain is self-contained rather than circular. The equivalence between the linear nonmetricity action (16) and the quadratic k-essence Lagrangian (25) is obtained by direct substitution of Eq. (9) into the Ricci scalar, contraction, and the algebraic definitions (19); the target result is not assumed in the inputs. The claim that ΛCDM is a special case is supported by an explicit analytic branch, k0 = 0 in Eq. (D4), which reduces h(z)^2 to the ΛCDM form (D5), again a parameter specialization rather than a fit renamed as a prediction. The MCMC analysis uses external CC, Pantheon+, and DESI BAO likelihoods and reports ΔAIC = -0.03, which is a statistical comparison, not a forced equality. The one potentially self-referential point is the use of scalar-tensor stability conditions ρφ ≥ 0 and c_s^2 ≥ 0 (Eq. 28) to define the physical phase space and the priors (65). However, the authors explicitly flag in Section III B that 'the constraints one imposes from the scalar-tensor representation may not necessarily hold the same weight in the geometric setting' and do not claim a proof of transfer; they present the viability result as conditional on these standard k-essence stability criteria, which is a transparent modeling choice rather than a hidden circular reduction. Self-citations to the authors' prior work on Schrödinger and completely symmetric geometries support the geometric classification but are not load-bearing for the derivation of the field equations, the fixed point analysis, or the likelihood evaluation. No equation is shown to be identical to its input by construction, and no fitted parameter is relabeled as a prediction. The appropriate finding is therefore no significant circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model rests on a specific nonmetricity ansatz, the integrability condition, a linear gravitational action, and the transfer of k-essence stability conditions to the geometric theory. These are explicit or implicit assumptions; none are derived from deeper principles.

free parameters (2)
  • B = 0.0925+0.0015-0.0018 (JOINT, Ωm0=0.29)
    Dimensionless parameter controlling the quartic kinetic term in the k-essence Lagrangian (Eq. 54); fitted to CC, Pantheon+ and DESI BAO data.
  • Ωm0 = Fixed to 0, 0.1, 0.2, 0.29, 0.303 in separate runs
    Present-day matter density parameter; fixed rather than sampled because it is degenerate with B at late times (Appendix E).
assumptions (5)
  • domain assumption Nonmetricity has the restricted vectorial form Q_{μνρ} = c1 π_μ g_{νρ} + c2(π_ρ g_{μν} + π_ν g_{ρμ}) + 2c3 π_μ π_ν π_ρ (Eq. 9).
    This defines the class of geometries studied and is assumed, not derived.
  • domain assumption The nonmetricity vector is integrable, π_μ = ∂_μ φ (Section III A).
    Enforced by a Lagrange multiplier; the scalar-tensor equivalence depends on this condition.
  • ad hoc to paper The gravitational action is linear in the Ricci scalar with an added ξ∇_μ π^μ term (Eq. 16).
    The simplest linear action is chosen; higher-order terms are excluded by construction.
  • domain assumption The k-essence stability conditions ρ_φ ≥ 0 and c_s² ≥ 0 apply to the geometric theory (Section III B).
    Assumed even though the equivalence is shown at background level and the full perturbation theory of the geometric model is not computed.
  • standard math FLRW background with pressureless matter (Eq. 30, Section V A).
    Standard cosmological symmetry and matter assumption.

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Cite this review

Pith. "Pith review of Geometric formulation of $k$-essence and late-time acceleration." pith.science (2026). https://pith.science/paper/AN36JSY7

@misc{pith2026250515975,
  author       = {Pith},
  title        = {Pith review of: Geometric formulation of $k$-essence and late-time acceleration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AN36JSY7}},
  note         = {Machine review of arXiv:2505.15975}
}
abstract

We study a class of geometries in which nonmetricity is fully determined by a vectorial degree of freedom and three independent coefficients. Formulating the simplest linear action in this geometry, implemented through Lagrange multipliers, naturally leads to an equivalence with the purely kinetic $k$-essence models with quadratic kinetic terms. A detailed dynamical systems analysis reveals that the $\Lambda$CDM phenomenology is embedded within the model. Crucially, we find that if stability conditions such as a positive sound speed squared and energy density are not enforced, the model generically exhibits instabilities and divergent behaviour in the phase space. These physical viability criteria allow us to isolate stable regions of the parameter space and derive well-motivated priors for parameter inference. Using Markov Chain Monte Carlo methods and late-time observational data, including cosmic chronometers, Pantheon$^{+}$ Type Ia supernovae, and DESI baryon acoustic oscillations, we constrain the degrees of freedom associated with nonmetricity and demonstrate the viability of the model. We discuss the implications of these results in light of the recent cosmic tensions, and give a possible explanation as to why the equivalent $k$-essence models have been missed as serious competitors to $\Lambda$CDM in the past. Finally, we review the geometric foundations of the theory and show that the integrable Weyl, Schr\"{o}dinger and completely symmetric geometries are embedded within our framework as special cases.

Figures

Figures reproduced from arXiv: 2505.15975 by the authors.

Figure 1
Figure 1. FIG. 1: Illustration of vectorial nonmetricity and its special cases. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Phase portraits with pressureless matter equation of state [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Evolution of the matter density parameter Ω [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Phase plot showing the physical phase space (black border) and accelerating region (green) used to define [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Corner plots showing the posterior constraints on the model parameters for the geometric [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Plot of the Hubble function with Moresco CC data points (left panel) and the Hubble function difference [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Redshift evolution of the equation of state parameter [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Plots of [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Corner plots showing the posterior constraints on the model parameters for the geometric [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]

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Reference graph

Works this paper leans on

167 extracted references · 29 canonical work pages · cited by 3 Pith papers

  1. [1]

    The likelihood function , which quantifies the agreement between the model’s theoretical predictions and the observational data

  2. [2]

    The form of the likelihood function depends strongly on the specific datasets employed

    The prior distributions, which encode assumptions or constraints on the model parameters and are supplied to the emcee sampler. The form of the likelihood function depends strongly on the specific datasets employed. In this study, we make use of three datasets, detailed below. The explicit form of the likelihoods corresponding to each dataset is presented...

  3. [3]

    A. G. Riess et al. (Supernova Search Team), Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201

    arXiv 1998

  4. [4]

    D. J. Eisenstein et al. (SDSS), Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies, Astrophys. J. 633, 560 (2005), arXiv:astro-ph/0501171

    arXiv 2005

  5. [5]

    Aghanim et al

    N. Aghanim et al. (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    arXiv 2020

  6. [6]

    Joyce, B

    A. Joyce, B. Jain, J. Khoury, and M. Trodden, Beyond the Cosmological Standard Model, Phys. Rept. 568, 1 (2015), arXiv:1407.0059 [astro-ph.CO]

    arXiv 2015

  7. [7]

    P. J. E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev. Mod. Phys.75, 559 (2003), arXiv:astro- ph/0207347

    arXiv 2003

  8. [8]

    Weinberg, The Cosmological Constant Problem, Rev

    S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61, 1 (1989)

    1989

Show all 167 references
  1. [9]

    Ratra and P

    B. Ratra and P. J. E. Peebles, Cosmological Consequences of a Rolling Homogeneous Scalar Field, Phys. Rev. D 37, 3406 (1988). 34 60 70 80 H0 140 160 180 rd 0.090 0.094 0.098 B −19.8 −19.6 −19.4 −19.2 M −19.8 −19.5 −19.2 M 0.090 0.095 B 140 160 180 rd (JOINT) Geometric k-essenc...

    1988

  2. [10]

    R. R. Caldwell, R. Dave, and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett. 80, 1582 (1998), arXiv:astro-ph/9708069

    1998 arXiv

  3. [11]

    E. J. Copeland, M. Sami, and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006), arXiv:hep- th/0603057

    2006

  4. [12]

    Wetterich, The Cosmon model for an asymptotically vanishing time dependent cosmological ’constant’, Astron

    C. Wetterich, The Cosmon model for an asymptotically vanishing time dependent cosmological ’constant’, Astron. As- trophys. 301, 321 (1995), arXiv:hep-th/9408025

    1995 arXiv

  5. [13]

    Armendariz-Picon, T

    C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, k - inflation, Phys. Lett. B 458, 209 (1999), arXiv:hep- th/9904075

    1999

  6. [14]

    Armendariz-Picon, V

    C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, Essentials of k essence, Phys. Rev. D 63, 103510 (2001), arXiv:astro-ph/0006373

    2001 arXiv

  7. [15]

    Armendariz-Picon, V

    C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, A Dynamical solution to the problem of a small cosmological constant and late time cosmic acceleration, Phys. Rev. Lett. 85, 4438 (2000), arXiv:astro-ph/0004134

    2000 arXiv

  8. [16]

    Abdalla et al

    E. Abdalla et al. , Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies, JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO]

    2022 arXiv

  9. [17]

    E. Di Valentino et al., The cosmoverse white paper: Addressing observational tensions in cosmology with systematics and fundamental physics, arXiv:2504.01669 [astro-ph.CO] (2025), white paper, unpublished

    2025 arXiv

  10. [18]

    A. G. Riess et al. , A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s −1 Mpc−1 Uncertainty from the Hubble Space Telescope and the SH0ES Team, Astrophys. J. Lett.934, L7 (2022), arXiv:2112.04510 [astro-ph.CO]. 35

    2022 arXiv

  11. [19]

    Verde, T

    L. Verde, T. Treu, and A. G. Riess, Tensions between the Early and the Late Universe, Nature Astron. 3, 891 (2019), arXiv:1907.10625 [astro-ph.CO]

    2019 arXiv

  12. [20]

    Di Valentino et al., Snowmass2021 - Letter of interest cosmology intertwined II: The hubble constant tension, Astropart

    E. Di Valentino et al., Snowmass2021 - Letter of interest cosmology intertwined II: The hubble constant tension, Astropart. Phys. 131, 102605 (2021), arXiv:2008.11284 [astro-ph.CO]

    2021 arXiv

  13. [21]

    Abdul Karim et al

    M. Abdul Karim et al. (DESI), DESI DR2 Results I: Baryon Acoustic Oscillations from the Lyman Alpha Forest (2025), arXiv:2503.14739 [astro-ph.CO]

    2025 arXiv

  14. [22]

    Abdul Karim et al

    M. Abdul Karim et al. (DESI), DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints (2025), arXiv:2503.14738 [astro-ph.CO]

    2025 arXiv

  15. [23]

    H. F. M. Goenner, On the history of unified field theories, Living Rev. Rel. 7, 2 (2004)

    2004

  16. [24]

    Clifton, P

    T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Modified Gravity and Cosmology, Phys. Rept. 513, 1 (2012), arXiv:1106.2476 [astro-ph.CO]

    2012 arXiv

  17. [25]

    Blagojevi´ c and F

    M. Blagojevi´ c and F. W. Hehl, eds., Gauge Theories of Gravitation: A Reader with Commentaries (World Scientific, Singapore, 2013)

    2013

  18. [26]

    F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rept. 258, 1 (1995), arXiv:gr-qc/9402012

    1995 arXiv

  19. [27]

    T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2, 212 (1961)

    1961

  20. [28]

    F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester, General Relativity with Spin and Torsion: Foundations and Prospects, Rev. Mod. Phys. 48, 393 (1976)

    1976

  21. [29]

    Trautman, Einstein-Cartan theory (2006), arXiv:gr-qc/0606062

    A. Trautman, Einstein-Cartan theory (2006), arXiv:gr-qc/0606062

    2006 arXiv

  22. [30]

    I. L. Shapiro, Physical aspects of the space-time torsion, Phys. Rept. 357, 113 (2002), arXiv:hep-th/0103093

    2002 arXiv

  23. [31]

    Luz and J

    P. Luz and J. P. S. Lemos, Relativistic cosmology and intrinsic spin of matter: Results and theorems in Einstein-Cartan theory, Phys. Rev. D 107, 084004 (2023), arXiv:2303.03104 [gr-qc]

    2023 arXiv

  24. [32]

    Weyl, Gravitation and electricity, Sitzungsber

    H. Weyl, Gravitation and electricity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1918, 465 (1918)

    1918

  25. [33]

    D. M. Calderbank and H. Pederson, Einstein-Weyl geometry (Odense Universitet. Institut for Matematik og Datalogi, 1997)

    1997

  26. [34]

    J. T. Wheeler, Weyl geometry, Gen. Rel. Grav. 50, 80 (2018), arXiv:1801.03178 [gr-qc]

    2018 arXiv

  27. [35]

    Hayashi and T

    K. Hayashi and T. Nakano, Extended translation invariance and associated gauge fields, Prog. Theor. Phys. 38, 491 (1967)

    1967

  28. [36]

    J. M. Nester and H.-J. Yo, Symmetric teleparallel general relativity, Chin. J. Phys. 37, 113 (1999), arXiv:gr-qc/9809049

    1999 arXiv

  29. [37]

    Y. N. Obukhov and J. G. Pereira, Metric affine approach to teleparallel gravity, Phys. Rev. D67, 044016 (2003), arXiv:gr- qc/0212080

    2003

  30. [38]

    Aldrovandi and J

    R. Aldrovandi and J. G. Pereira, Teleparallel Gravity: An Introduction (Springer, 2013)

    2013

  31. [39]

    Puetzfeld, Status of non-Riemannian cosmology, New Astron

    D. Puetzfeld, Status of non-Riemannian cosmology, New Astron. Rev. 49, 59 (2005), arXiv:gr-qc/0404119

    2005 arXiv

  32. [40]

    T. P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82, 451 (2010), arXiv:0805.1726 [gr-qc]

    2010 arXiv

  33. [41]

    G. R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79, 124019 (2009), arXiv:0812.1205 [astro-ph]

    2009 arXiv

  34. [42]

    E. V. Linder, Einstein’s Other Gravity and the Acceleration of the Universe, Phys. Rev. D 81, 127301 (2010), [Erratum: Phys.Rev.D 82, 109902 (2010)], arXiv:1005.3039 [astro-ph.CO]

    2010 arXiv

  35. [43]

    Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. 79, 106901 (2016), arXiv:1511.07586 [gr-qc]

    2016 arXiv

  36. [44]

    Beltr´ an Jim´ enez, L

    J. Beltr´ an Jim´ enez, L. Heisenberg, and T. Koivisto, Coincident General Relativity, Phys. Rev. D 98, 044048 (2018), arXiv:1710.03116 [gr-qc]

    2018 arXiv

  37. [45]

    Beltr´ an Jim´ enez, L

    J. Beltr´ an Jim´ enez, L. Heisenberg, and T. S. Koivisto, The Geometrical Trinity of Gravity, Universe 5, 173 (2019), arXiv:1903.06830 [hep-th]

    2019 arXiv

  38. [46]

    Beltr´ an Jim´ enez, L

    J. Beltr´ an Jim´ enez, L. Heisenberg, T. S. Koivisto, and S. Pekar, Cosmology inf (Q) geometry, Phys. Rev. D 101, 103507 (2020), arXiv:1906.10027 [gr-qc]

    2020 arXiv

  39. [47]

    Bahamonde, K

    S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud, and E. Di Valentino, Teleparallel gravity: from theory to cosmology, Rept. Prog. Phys. 86, 026901 (2023), arXiv:2106.13793 [gr-qc]

    2023 arXiv

  40. [48]

    C. G. Boehmer and E. Jensko, Modified gravity: A unified approach, Phys. Rev. D 104, 024010 (2021), arXiv:2103.15906 [gr-qc]

    2021 arXiv

  41. [49]

    C. G. Boehmer and E. Jensko, Modified gravity: A unified approach to metric-affine models, J. Math. Phys. 64, 082505 (2023), arXiv:2301.11051 [gr-qc]

    2023 arXiv

  42. [50]

    Jensko, A unified approach to geometric modifications of gravity, Ph.D

    E. Jensko, A unified approach to geometric modifications of gravity, Ph.D. thesis, U. Coll. London (2023), arXiv:2401.12567 [gr-qc]

    2023 arXiv

  43. [51]

    Csillag, A

    L. Csillag, A. Agashe, and D. Iosifidis, Schr¨ odinger connections: from mathematical foundations towards Yano–Schr¨ odinger cosmology, Class. Quant. Grav.41, 235005 (2024), arXiv:2402.06167 [gr-qc]

    2024 arXiv

  44. [52]

    Csillag and T

    L. Csillag and T. Harko, Semi-symmetric metric gravity: From the Friedmann–Schouten geometry with torsion to dy- namical dark energy models, Phys. Dark Univ. 46, 101596 (2024), arXiv:2402.06114 [gr-qc]

    2024 arXiv

  45. [53]

    Aringazin and A

    A. Aringazin and A. Mikhailov, Matter fields in spacetime with vector nonmetricity, Classical and Quantum Gravity 8, 1685 (1991)

    1991

  46. [54]

    Beltran Jimenez and T

    J. Beltran Jimenez and T. S. Koivisto, Spacetimes with vector distortion: Inflation from generalised Weyl geometry, Phys. Lett. B 756, 400 (2016), arXiv:1509.02476 [gr-qc]

    2016 arXiv

  47. [55]

    Beltran Jimenez, L

    J. Beltran Jimenez, L. Heisenberg, and T. S. Koivisto, Cosmology for quadratic gravity in generalized weyl geometry, 36 JCAP 2016 (04), 046, arXiv:1602.07287 [hep-th]

    2016 arXiv

  48. [56]

    Guendelman, E

    E. Guendelman, E. Nissimov, and S. Pacheva, Unified Dark Energy and Dust Dark Matter Dual to Quadratic Purely Kinetic K-Essence, Eur. Phys. J. C 76, 90 (2016), arXiv:1511.07071 [gr-qc]

    2016 arXiv

  49. [57]

    Y. Yang, X. Ren, Q. Wang, Z. Lu, D. Zhang, Y.-F. Cai, and E. N. Saridakis, Quintom cosmology and modified gravity after DESI 2024, Sci. Bull. 69, 2698 (2024), arXiv:2404.19437 [astro-ph.CO]

    2024 arXiv

  50. [58]

    R. J. Scherrer, Purely kinetic k-essence as unified dark matter, Phys. Rev. Lett. 93, 011301 (2004), arXiv:astro- ph/0402316

    2004

  51. [59]

    Bertacca, S

    D. Bertacca, S. Matarrese, and M. Pietroni, Unified Dark Matter in Scalar Field Cosmologies, Mod. Phys. Lett. A 22, 2893 (2007), arXiv:astro-ph/0703259

    2007 arXiv

  52. [60]

    Bertacca, N

    D. Bertacca, N. Bartolo, and S. Matarrese, Unified Dark Matter Scalar Field Models, Adv. Astron. 2010, 904379 (2010), arXiv:1008.0614 [astro-ph.CO]

    2010 arXiv

  53. [61]

    Quiros, T

    I. Quiros, T. Gonzalez, U. Nucamendi, R. De Arcia, and F. A. H. Rangel, Revisiting purely kinetic k-essence, arXiv:2501.14177 [gr-qc] (2025), unpublished preprint

    2025 arXiv

  54. [62]

    J. A. Schouten, Ricci-calculus: an introduction to tensor analysis and its geometrical applications , Vol. 10 (Springer Science & Business Media, 2013)

    2013

  55. [63]

    F. W. Hehl, G. D. Kerlick, and P. Von Der Heyde, On Hypermomentum in General Relativity. 2. The Geometry of Space-Time, Z. Naturforsch. A 31, 524 (1976)

    1976

  56. [64]

    F. W. Hehl, G. D. Kerlick, and P. Von Der Heyde, On Hypermomentum in General Relativity. 3. Coupling Hypermo- mentum to Geometry, Z. Naturforsch. A 31, 823 (1976)

    1976

  57. [65]

    Ne’eman and F

    Y. Ne’eman and F. W. Hehl, Test matter in a space-time with nonmetricity, Class. Quant. Grav. 14, A251 (1997), arXiv:gr-qc/9604047

    1997 arXiv

  58. [66]

    Iosifidis, Exactly Solvable Connections in Metric-Affine Gravity, Class

    D. Iosifidis, Exactly Solvable Connections in Metric-Affine Gravity, Class. Quant. Grav. 36, 085001 (2019), arXiv:1812.04031 [gr-qc]

    2019 arXiv

  59. [67]

    Iosifidis, Cosmological Hyperfluids, Torsion and Non-metricity, Eur

    D. Iosifidis, Cosmological Hyperfluids, Torsion and Non-metricity, Eur. Phys. J. C 80, 1042 (2020), arXiv:2003.07384 [gr-qc]

    2020 arXiv

  60. [68]

    Andrei, D

    I. Andrei, D. Iosifidis, L. J¨ arv, and M. Saal, Friedmann cosmology with hyperfluids, Phys. Rev. D 111, 064063 (2025), arXiv:2411.19127 [gr-qc]

    2025 arXiv

  61. [69]

    Schr¨ odinger,Space-time structure (Cambridge University Press, 1985)

    E. Schr¨ odinger,Space-time structure (Cambridge University Press, 1985)

    1985

  62. [70]

    Csillag, R

    L. Csillag, R. Hama, M. J´ ozsa, T. Harko, and S. V. Sab˘ au, Length-preserving biconnection gravity and its cosmological implications, Journal of Cosmology and Astroparticle Physics 2024 (12), 034

    2024

  63. [71]

    F. W. Hehl, E. A. Lord, and Y. Ne’eman, Hadron Dilation, Shear and Spin as Components of the Intrinsic Hypermomen- tum Current and Metric Affine Theory of Gravitation, Phys. Lett. B 71, 432 (1977)

    1977

  64. [72]

    F. W. Hehl, E. A. Lord, and Y. Ne’eman, Hypermomentum in Hadron Dynamics and in Gravitation, Phys. Rev. D 17, 428 (1978)

    1978

  65. [73]

    Iosifidis, Non-Riemannian cosmology: The role of shear hypermomentum, Int

    D. Iosifidis, Non-Riemannian cosmology: The role of shear hypermomentum, Int. J. Geom. Meth. Mod. Phys. 18, 2150129 (2021), arXiv:2010.00875 [gr-qc]

    2021 arXiv

  66. [74]

    Klemm and L

    S. Klemm and L. Ravera, An action principle for the Einstein–Weyl equations, J. Geom. Phys. 158, 103958 (2020), arXiv:2006.15890 [hep-th]

    2020 arXiv

  67. [75]

    Iosifidis, The full quadratic metric-affine gravity (including parity odd terms): exact solutions for the affine-connection, Class

    D. Iosifidis, The full quadratic metric-affine gravity (including parity odd terms): exact solutions for the affine-connection, Class. Quant. Grav. 39, 095002 (2022), arXiv:2112.09154 [gr-qc]

    2022 arXiv

  68. [76]

    T. P. Sotiriou, f(R) gravity and scalar-tensor theory, Class. Quant. Grav. 23, 5117 (2006), arXiv:gr-qc/0604028

    2006 arXiv

  69. [77]

    T. P. Sotiriou and S. Liberati, Metric-affine f(R) theories of gravity, Annals Phys. 322, 935 (2007), arXiv:gr-qc/0604006

    2007 arXiv

  70. [78]

    De Felice and S

    A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13, 3 (2010), arXiv:1002.4928 [gr-qc]

    2010 arXiv

  71. [79]

    J. Wu, G. Li, T. Harko, and S.-D. Liang, Palatini formulation of f (R, T) gravity theory, and its cosmological implications, Eur. Phys. J. C 78, 430 (2018), arXiv:1805.07419 [gr-qc]

    2018 arXiv

  72. [80]

    Amarzguioui, O

    M. Amarzguioui, O. Elgaroy, D. F. Mota, and T. Multamaki, Cosmological constraints on f(r) gravity theories within the palatini approach, Astron. Astrophys. 454, 707 (2006), arXiv:astro-ph/0510519

    2006 arXiv

  73. [81]

    D. A. Gomes, R. Briffa, A. Kozak, J. Levi Said, M. Saal, and A. Wojnar, Cosmological constraints of palatini f(r) gravity, JCAP 2024 (01), 011, arXiv:2310.17339 [gr-qc]

    2024 arXiv

  74. [82]

    K. Aoki, S. Bahamonde, J. Gigante Valcarcel, and M. A. Gorji, Cosmological perturbation theory in metric-affine gravity, Phys. Rev. D 110, 024017 (2024), arXiv:2310.16007 [gr-qc]

    2024 arXiv

  75. [83]

    Bahamonde and J

    S. Bahamonde and J. Gigante Valcarcel, Stability in cubic metric-affine gravity, Phys. Rev. D 111, 084058 (2025), arXiv:2411.12954 [gr-qc]

    2025 arXiv

  76. [84]

    Bahamonde, A

    S. Bahamonde, A. Hell, D. Blixt, and K. F. Dialektopoulos, Revisiting stability in new general relativity, Phys. Rev. D 111, 064080 (2025), arXiv:2404.02972 [gr-qc]

    2025 arXiv

  77. [85]

    Velten and T

    H. Velten and T. R. P. Caramˆ es, To conserve, or not to conserve: A review of nonconservative theories of gravity, Universe 7, 38 (2021), arXiv:2102.03457 [gr-qc]

    2021 arXiv

  78. [86]

    Bahamonde and J

    S. Bahamonde and J. Gigante Valcarcel, Observational constraints in metric-affine gravity, Eur. Phys. J. C81, 495 (2021), arXiv:2103.12036 [gr-qc]

    2021 arXiv

  79. [87]

    Iosifidis, The perfect hyperfluid of metric-affine gravity: The foundation, JCAP 2021 (04), 072, arXiv:2101.07289 [gr-qc]

    D. Iosifidis, The perfect hyperfluid of metric-affine gravity: The foundation, JCAP 2021 (04), 072, arXiv:2101.07289 [gr-qc]

    2021 arXiv

  80. [88]

    Iosifidis and T

    D. Iosifidis and T. S. Koivisto, Hyperhydrodynamics: relativistic viscous fluids from hypermomentum, JCAP 2024 (05), 001, arXiv:2312.06780 [gr-qc]

    2024 arXiv

  81. [89]

    Iosifidis, E

    D. Iosifidis, E. Jensko, and T. S. Koivisto, Relativistic interacting fluids in cosmology, JCAP 2024 (11), 043, 37 arXiv:2406.01412 [gr-qc]

    2024 arXiv

  82. [90]

    Miritzis, Acceleration in Weyl integrable spacetime, Int

    J. Miritzis, Acceleration in Weyl integrable spacetime, Int. J. Mod. Phys. D 22, 1350019 (2013), arXiv:1301.5696 [gr-qc]

    2013 arXiv

  83. [91]

    Paliathanasis, G

    A. Paliathanasis, G. Leon, and J. D. Barrow, Einstein-aether theory in Weyl integrable geometry, Eur. Phys. J. C 80, 1099 (2020), arXiv:2007.06435 [gr-qc]

    2020 arXiv

  84. [92]

    Paliathanasis and G

    A. Paliathanasis and G. Leon, Integrability and cosmological solutions in Einstein-æther-Weyl theory, Eur. Phys. J. C 81, 255 (2021), arXiv:2101.01606 [gr-qc]

    2021 arXiv

  85. [93]

    J. M. Salim and S. L. Sautu, Gravitational theory in Weyl integrable space-time, Class. Quant. Grav. 13, 353 (1996)

    1996

  86. [94]

    Quiros, Phenomenological signatures of gauge invariant theories of gravity with vectorial nonmetricity, Phys

    I. Quiros, Phenomenological signatures of gauge invariant theories of gravity with vectorial nonmetricity, Phys. Rev. D 107, 104028 (2023), arXiv:2208.10048 [gr-qc]

    2023 arXiv

  87. [95]

    Novello and S

    M. Novello and S. E. P. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463, 127 (2008), arXiv:0802.1634 [astro-ph]

    2008 arXiv

  88. [96]

    Koivisto, Covariant conservation of energy momentum in modified gravities, Class

    T. Koivisto, Covariant conservation of energy momentum in modified gravities, Class. Quant. Grav. 23, 4289 (2006), arXiv:gr-qc/0505128

    2006 arXiv

  89. [97]

    M. L. Pucheu, F. A. P. Alves Junior, A. B. Barreto, and C. Romero, Cosmological models in Weyl geometrical scalar-tensor theory, Phys. Rev. D 94, 064010 (2016), arXiv:1602.06966 [gr-qc]

    2016 arXiv

  90. [98]

    Chatzidakis, A

    S. Chatzidakis, A. Giacomini, P. G. L. Leach, G. Leon, A. Paliathanasis, and S. Pan, Interacting dark energy in curved FLR W spacetime from Weyl Integrable Spacetime, JHEAp 36, 141 (2022), arXiv:2206.06639 [gr-qc]

    2022 arXiv

  91. [99]

    Chiba, T

    T. Chiba, T. Okabe, and M. Yamaguchi, Kinetically driven quintessence, Phys. Rev. D 62, 023511 (2000), arXiv:astro- ph/9912463

    2000

  92. [100]

    de Putter and E

    R. de Putter and E. V. Linder, Kinetic k-essence and Quintessence, Astropart. Phys. 28, 263 (2007), arXiv:0705.0400 [astro-ph]

    2007 arXiv

  93. [101]

    Bouhmadi-L´ opez, K

    M. Bouhmadi-L´ opez, K. S. Kumar, J. ao Marto, J. ao Morais, and A. Zhuk, K-essence model from the mechanical approach point of view: coupled scalar field and the late cosmic acceleration, JCAP 2016 (07), 050, arXiv:1605.03212 [gr-qc]

    2016 arXiv

  94. [102]

    Garriga and V

    J. Garriga and V. F. Mukhanov, Perturbations in k-inflation, Phys. Lett. B 458, 219 (1999), arXiv:hep-th/9904176

    1999 arXiv

  95. [103]

    S. X. Tian and Z.-H. Zhu, Early dark energy in k-essence, Phys. Rev. D 103, 043518 (2021), arXiv:2102.06399 [gr-qc]

    2021 arXiv

  96. [104]

    A. L. Ferreira Junior, N. Pinto-Neto, and J. Zanelli, Inflation and late-time accelerated expansion driven by k-essence degenerate dynamics, Phys. Rev. D 109, 023515 (2024), arXiv:2311.01456 [hep-th]

    2024 arXiv

  97. [105]

    Huang, K

    Q. Huang, K. Zhang, H. Huang, B. Xu, and F. Tu, CMB Power Spectrum in the Emergent Universe with K-Essence, Universe 9, 221 (2023), arXiv:2210.16166 [gr-qc]

    2023 arXiv

  98. [106]

    S. Das, A. Panda, G. Manna, and S. Ray, Raychaudhuri Equation in K-essence Geometry: Conditional Singular and Non-Singular Cosmological Models, Fortsch. Phys. 71, 2200193 (2023), arXiv:2303.03118 [gr-qc]

    2023 arXiv

  99. [107]

    Huang, Statistics of thawing k-essence dark energy models, Phys

    Z. Huang, Statistics of thawing k-essence dark energy models, Phys. Rev. D 104, 103533 (2021), arXiv:2108.06089 [astro- ph.CO]

    2021 arXiv

  100. [108]

    E. J. Copeland, A. Moss, S. S. M. noz, and J. M. M. White, Scaling solutions as Early Dark Energy resolutions to the Hubble tension, JCAP 2024 (05), 078, arXiv:2309.15295 [astro-ph.CO]

    2024 arXiv

  101. [109]

    S. A. Hosseini Mansoori and H. Moshafi, Alleviating H 0 and S 8 Tensions Simultaneously in K-essence Cosmology, Astrophys. J. 975, 275 (2024), arXiv:2405.05843 [astro-ph.CO]

    2024 arXiv

  102. [110]

    Hussain, S

    S. Hussain, S. Nelleri, and K. Bhattacharya, Comprehensive study of k-essence model: dynamical system analysis and observational constraints from latest type ia supernova and bao observations, JCAP 2025 (03), 025, arXiv:2406.07179 [astro-ph.CO]

    2025 arXiv

  103. [111]

    A. V. Minkevich and A. S. Garkun, Isotropic cosmology in metric-affine gauge theory of gravity, arXiv:gr-qc/9805007 (1998), unpublished preprint

    1998 arXiv

  104. [112]

    Iosifidis, I

    D. Iosifidis, I. G. Vogiatzis, and C. G. Tsagas, Friedmann-like universes with non-metricity, Eur. Phys. J. C 83, 216 (2023), arXiv:2205.03139 [gr-qc]

    2023 arXiv

  105. [113]

    Bahamonde, C

    S. Bahamonde, C. G. B¨ ohmer, S. Carloni, E. J. Copeland, W. Fang, and N. Tamanini, Dynamical systems applied to cosmology: dark energy and modified gravity, Phys. Rept. 775-777, 1 (2018), arXiv:1712.03107 [gr-qc]

    2018 arXiv

  106. [114]

    A. A. Coley, Dynamical systems and cosmology (Kluwer, Dordrecht, Netherlands, 2003)

    2003

  107. [115]

    C. G. Boehmer, E. Jensko, and R. Lazkoz, Cosmological dynamical systems in modified gravity, Eur. Phys. J. C 82, 500 (2022), arXiv:2201.09588 [gr-qc]

    2022 arXiv

  108. [116]

    W. Fang, H. Tu, Y. Li, J. Huang, and C. Shu, Full Investigation on the Dynamics of Power-Law Kinetic Quintessence, Phys. Rev. D 89, 123514 (2014), arXiv:1406.0128 [gr-qc]

    2014 arXiv

  109. [117]

    Chakraborty, A

    A. Chakraborty, A. Ghosh, and N. Banerjee, Dynamical systems analysis of a k -essence model, Phys. Rev. D 99, 103513 (2019), arXiv:1904.10149 [gr-qc]

    2019 arXiv

  110. [118]

    Amendola and S

    L. Amendola and S. Tsujikawa, Scaling solutions and weak gravity in dark energy with energy and momentum couplings, JCAP 2020 (06), 020, arXiv:2003.02686 [gr-qc]

    2020 arXiv

  111. [119]

    Fritis, D

    A. Fritis, D. Villalobos-Silva, Y. Vasquez, C. H. Lopez-Caraballo, and J. C. Helo, Dynamical system analysis and obser- vational constraints of cosmological models in mimetic gravity, Phys. Dark Univ. 46, 101650 (2024), arXiv:2312.07765 [gr-qc]

    2024 arXiv

  112. [120]

    Perko, Differential equations and dynamical systems , Vol

    L. Perko, Differential equations and dynamical systems , Vol. 7 (Springer Science & Business Media, 2013)

    2013

  113. [121]

    E. J. Copeland, A. R. Liddle, and D. Wands, Exponential potentials and cosmological scaling solutions, Phys. Rev. D 57, 4686 (1998), arXiv:gr-qc/9711068

    1998 arXiv

  114. [122]

    E. V. Linder, Exploring the expansion history of the universe, Phys. Rev. Lett.90, 091301 (2003), arXiv:astro-ph/0208512

    2003 arXiv

  115. [123]

    P. S. Corasaniti and E. J. Copeland, A Model independent approach to the dark energy equation of state, Phys. Rev. D 67, 063521 (2003), arXiv:astro-ph/0205544. 38

    2003 arXiv

  116. [124]

    Li, Z.-K

    H. Li, Z.-K. Guo, and Y.-Z. Zhang, Parametrization of k-essence and its kinetic term, Mod. Phys. Lett. A 21, 1683 (2006), arXiv:astro-ph/0601007

    2006 arXiv

  117. [125]

    V. H. C´ ardenas, N. Cruz, S. Mu˜ noz, and J. R. Villanueva, Reconstruction of a kinetic k-essence Lagrangian from a modified of dark energy equation of state, Eur. Phys. J. C 78, 591 (2018), arXiv:1804.02762 [gr-qc]

    2018 arXiv

  118. [126]

    Foreman-Mackey, D

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The mcmc hammer, Publications of the Astronomical Society of the Pacific 125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

    2013 arXiv

  119. [127]

    P. C. Gregory, Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica ® Support (Cambridge University Press, 2005)

    2005

  120. [128]

    Lewis and S

    A. Lewis and S. Bridle, Cosmological parameters from cmb and other data: a monte carlo approach, Physical Review D 66, 103511 (2002), astro-ph/0205436

    2002 arXiv

  121. [129]

    Gelman, J

    A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis , 3rd ed. (Chapman and Hall/CRC, 2013)

    2013

  122. [130]

    Jimenez and A

    R. Jimenez and A. Loeb, Constraining cosmological parameters based on relative galaxy ages, The Astrophysical Journal 573, 37 (2002)

    2002

  123. [131]

    Vagnozzi, A

    S. Vagnozzi, A. Loeb, and M. Moresco, Eppur ` e piatto? the cosmic chronometers take on spatial curvature and cosmic concordance, The Astrophysical Journal 908, 84 (2021)

    2021

  124. [132]

    Moresco, Cccovariance: Data - hztable mm bc03.dat, https://gitlab.com/mmoresco/CCcovariance/-/blob/ master/data/HzTable_MM_BC03.dat (2020)

    M. Moresco, Cccovariance: Data - hztable mm bc03.dat, https://gitlab.com/mmoresco/CCcovariance/-/blob/ master/data/HzTable_MM_BC03.dat (2020)

    2020

  125. [133]

    M. Moresco and the Cosmic Chronometers Collaboration, Improved constraints on the expansion rate of the universe up to z ∼ 1.1 from the spectroscopic evolution of cosmic chronometers, JCAP 2012 (08), 006, arXiv:1201.3609 [astro-ph.CO]

    2012 arXiv

  126. [134]

    M. Moresco, Raising the bar: new constraints on the hubble parameter with cosmic chronometers at z ∼ 2, Monthly Notices of the Royal Astronomical Society: Letters 450, L16 (2015), https://academic.oup.com/mnrasl/article- pdf/450/1/L16/54653923/mnrasl 450 1 l16.pdf

    2015

  127. [135]

    Moresco, L

    M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro, and D. Wilkin- son, A 6% measurement of the hubble parameter at z ∼ 0.45: direct evidence of the epoch of cosmic re-acceleration, Journal of Cosmology and Astroparticle Phy...

    2016

  128. [136]

    Brout et al

    D. Brout et al. , The Pantheon+ Analysis: Cosmological Constraints, Astrophys. J. 938, 110 (2022), arXiv:2202.04077 [astro-ph.CO]

    2022 arXiv

  129. [137]

    Pantheon+ Collaboration, Pantheon+ Data Release: Distances and Covariance Matrices, https://github.com/ PantheonPlusSH0ES/DataRelease/tree/main/Pantheon%2B_Data/4_DISTANCES_AND_COVAR (2022)

    2022

  130. [138]

    Conley, R

    A. Conley, R. G. Carlberg, J. Guy, D. A. Howell, M. Sullivan, et al. , Supernova constraints and systematic uncertainties from the first three years of the supernova legacy survey, The Astrophysical Journal Supplement Series 192, 1 (2011)

    2011

  131. [139]

    A. G. Adame and the DESI Collaboration, Desi 2024 vi: Cosmological constraints from the measurements of baryon acoustic oscillations, JCAP 2025 (02), 021, arXiv:2404.03002 [astro-ph.CO]

    2024 arXiv

  132. [140]

    Moresco, Cosmic chronometers covariance estimate, https://gitlab.com/mmoresco/CCcovariance (2021)

    M. Moresco, Cosmic chronometers covariance estimate, https://gitlab.com/mmoresco/CCcovariance (2021)

    2021

  133. [141]

    D. Collaboration, Desi 2024 gaussian bao covariance matrices, https://github.com/CobayaSampler/bao_data/blob/ master/desi_2024_gaussian_bao_ALL_GCcomb_cov.txt (2024), covariance matrices for DESI 2024 BAO analysis

    2024

  134. [142]

    Vagnozzi, Seven hints that early-time new physics alone is not sufficient to solve the hubble tension, Universe 9, 10.3390/universe9090393 (2023)

    S. Vagnozzi, Seven hints that early-time new physics alone is not sufficient to solve the hubble tension, Universe 9, 10.3390/universe9090393 (2023)

    2023 doi

  135. [143]

    W. Lin, X. Chen, and K. J. Mack, Early universe physics insensitive and uncalibrated cosmic standards: Constraints on ωm and implications for the hubble tension, The Astrophysical Journal 920, 159 (2021)

    2021

  136. [144]

    Jedamzik, L

    K. Jedamzik, L. Pogosian, and G.-B. Zhao, Why reducing the cosmic sound horizon alone can not fully resolve the hubble tension, Communications Physics 4, 123 (2021)

    2021

  137. [145]

    L. P. Chimento, Extended tachyon field, Chaplygin gas and solvable k-essence cosmologies, Phys. Rev. D 69, 123517 (2004), arXiv:astro-ph/0311613

    2004 arXiv

  138. [146]

    Yang and X.-T

    R.-J. Yang and X.-T. Gao, Observational constraints on purely kinetic k-essence dark energy models, Chin. Phys. Lett. 26, 089501 (2009)

    2009

  139. [147]

    Tauscher, D

    K. Tauscher, D. Rapetti, and J. Burns, A new goodness-of-fit statistic and its application to 21-cm cosmology, Journal of Cosmology and Astroparticle Physics 2018 (12), 015

    2018

  140. [148]

    A. R. Liddle, Information criteria for astrophysical model selection, Monthly Notices of the Royal Astronomical Society: Letters 377, L74 (2007)

    2007

  141. [149]

    M. Y. J. Tan and R. Biswas, The reliability of the akaike information criterion method in cosmological model selection, Monthly Notices of the Royal Astronomical Society 419, 3292 (2012)

    2012

  142. [150]

    K. P. Burnham and D. R. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. (Springer, New York, 2002)

    2002

  143. [151]

    S. I. Vrieze, Model selection and psychological theory: A discussion of the differences between the akaike information criterion (aic) and the bayesian information criterion (bic), Psychological Methods 17, 228 (2012)

    2012

  144. [152]

    Rezaei and M

    M. Rezaei and M. Malekjani, Comparison between different methods of model selection in cosmology, The European Physical Journal Plus 136, 219 (2021)

    2021

  145. [153]

    Arevalo, A

    F. Arevalo, A. Cid, and J. Moya, Aic and bic for cosmological interacting scenarios, The European Physical Journal C 77, 565 (2017)

    2017

  146. [154]

    J. A. Kable, G. Benevento, N. Frusciante, A. D. Felice, and S. Tsujikawa, Probing modified gravity with integrated Sachs-Wolfe CMB and galaxy cross-correlations, JCAP 2022 (09), 002, arXiv:2111.10432 [astro-ph.CO]

    2022 arXiv

  147. [155]

    Malquarti, E

    M. Malquarti, E. J. Copeland, A. R. Liddle, and M. Trodden, A New view of k-essence, Phys. Rev. D 67, 123503 (2003), 39 arXiv:astro-ph/0302279

    2003 arXiv

  148. [156]

    Bamba, J

    K. Bamba, J. Matsumoto, and S. Nojiri, Cosmological perturbations in k-essence model, Phys. Rev. D 85, 084026 (2012), arXiv:1109.1308 [hep-th]

    2012 arXiv

  149. [157]

    Iosifidis and K

    D. Iosifidis and K. Pallikaris, Biconnection gravity as a statistical manifold, Phys. Rev. D 108, 044026 (2023)

    2023

  150. [158]

    ichi Amari and H

    S. ichi Amari and H. Nagaoka, Methods of Information Geometry , Translations of Mathematical Monographs, Vol. 191 (American Mathematical Society, Providence, RI, 2000) translated by Daishi Harada

    2000

  151. [159]

    ichi Amari, Information geometry, in Contemporary Mathematics, Vol

    S. ichi Amari, Information geometry, in Contemporary Mathematics, Vol. 203 (American Mathematical Society, 1997) pp. 81–96

    1997

  152. [160]

    Matsuzoe et al

    H. Matsuzoe et al. , Statistical manifolds and affine differential geometry, in Advances in Mathematics: Studies in Pure Mathematics, Vol. 57 (Mathematical Society of Japan, 2010) pp. 303–321

    2010

  153. [161]

    S. L. Lauritzen, Statistical manifolds, in Differential Geometry in Statistical Inference , IMS Lecture Notes – Monograph Series, Vol. 10 (Institute of Mathematical Statistics, Hayward, CA, 1987) pp. 163–216

    1987

  154. [162]

    Iosifidis, On a torsion/curvature analogue of dual connections and statistical manifolds, J

    D. Iosifidis, On a torsion/curvature analogue of dual connections and statistical manifolds, J. Geom. Phys. 196, 105064 (2024), arXiv:2303.13259 [math.DG]

    2024 arXiv

  155. [163]

    Csillag and T

    L. Csillag and T. Harko, Semi-symmetric metric gravity: From the friedmann–schouten geometry with torsion to dynam- ical dark energy models, Physics of the Dark Universe 46, 101596 (2024)

    2024

  156. [164]

    Chaudhary, L

    H. Chaudhary, L. Csillag, and T. Harko, Semi-symmetric metric gravity: A brief overview, Universe 10, 10.3390/uni- verse10110419 (2024)

    2024 doi

  157. [165]

    M. E. Aydin, R. L´ opez, and A. Mihai, Constant sectional curvature surfaces with a semi-symmetric non-metric connection, Journal of Mathematical Analysis and Applications 542, 128795 (2025)

    2025

  158. [166]

    Matsuzoe, J

    H. Matsuzoe, J. ichi Takeuchi, and S. ichi Amari, Equiaffine structures on statistical manifolds and bayesian statistics, Differential Geometry and its Applications 24, 567 (2006)

    2006

  159. [167]

    Aghanim et al

    N. Aghanim et al. (Planck), Planck 2018 results. VIII. Gravitational lensing, Astron. Astrophys. 641, A8 (2020), arXiv:1807.06210 [astro-ph.CO]

    2020 arXiv

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