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REVIEW 4 major objections 4 minor 9 references

The Study on Modified Theories of General Relativity: A Differential Geometric Approach

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The thesis claims that modified gravity built from non-metricity, torsion, and matter-coupled curvature can reproduce ΛCDM expansion, support traversable wormholes, and pass BBN plus late-time tests.

desk verdict A PhD thesis compiled from published papers, with a load-bearing flaw in the f(R,L_m) chapters: the action is complex-valued for the fitted parameters, and the ΛCDM embedding is by construction rather than an independent test. read the letter →

arxiv 2507.04031 v1 pith:QVQUOQHX submitted 2025-07-05 gr-qc astro-ph.CO

N.S. Kavya This is my paper · ORCID
classification gr-qcastro-ph.CO MSC 83D0583C1583F05 PACS 04.20.-q04.50.Kd98.80.-k
keywords modifiedgravitynon-metricityteleparallelcosmologicalreconstructiontraversablewormholesenergyconditionsBigBangnucleosynthesisdecelerationparameter
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

This thesis tries to establish that modified gravity theories obtained by replacing the Ricci scalar of Einstein's action with other geometric invariants, notably non-metricity Q, torsion T, and matter-coupled curvature f(R,L_m), are capable of describing the Universe's late-time acceleration and of hosting traversable wormholes. It introduces a two-parameter deceleration law that, fitted to cosmic chronometer, BAO, and supernova data, yields a present deceleration parameter near $q_0=-0.7$ with a transition redshift around $0.79$, and uses those numbers to constrain $f(Q)$ models that shadow ΛCDM. It then embeds the ΛCDM expansion analytically in non-minimally coupled $f(Q,L_m)$ gravity, fits an anisotropic Bianchi-I model in $f(R,L_m)$ gravity, constructs wormhole shape functions in $f(Q,T)$ and $f(R,L_m)$ gravity, and constrains hybrid $f(T)$ models with Big Bang nucleosynthesis and late-time data. A sympathetic reader would care because the thesis offers a menu of geometric alternatives that fit much of the same expansion data as ΛCDM, and in one wormhole case claims to do so without exotic matter.

What carries the argument

The load-bearing construction is the replacement of the linear Ricci scalar in the Einstein-Hilbert action by arbitrary functions of geometric invariants: $f(Q)$ for non-metricity, $f(T)$ for torsion, $f(R,L_m)$ and $f(Q,L_m)$ for curvature or non-metricity coupled to the matter Lagrangian, and $f(Q,T)$ for non-metricity coupled to the trace of the stress-energy tensor. Varying these actions with respect to metric and connection yields modified Friedmann and wormhole equations. In the wormhole chapters the decisive mechanism is the conformal Killing vector condition, $\mathcal{L}_\eta g_{ab}=\psi(r)g_{ab}$, which turns the non-linear field equations into ordinary differential equations whose solution gives the wormhole shape function $S_f(r)$; in the $f(R,L_m)$ wormhole chapter, non-commutative Gaussian and Lorentzian smearing of the energy density plays the same role. In the reconstruction chapters the decisive object is the first-order differential equation obtained by inserting the ΛCDM form of the non-metricity scalar into the modified Friedmann equation, whose closed-form solution yields the geometric function $f_1(Q)$ for a prescribed coupling $f_2(Q)$.

What would settle it

Compute $(-\rho)^\alpha$ at the best-fit value $\alpha=4.76$ and a positive cosmological density; unless the paper specifies which branch of the complex power is meant, the field equations (4.18)-(4.20) do not follow from a real action, and the Chapter 4 and Chapter 6 results are undefined before comparison with data.

Watch

Extended reading notes

Core claim

The central claim is that the geometric trinity of general relativity, built on curvature, torsion, and non-metricity, admits modified extensions whose actions produce the observed cosmic history and support wormhole geometries. In the $f(Q)$ chapter, the thesis derives the deceleration parameter $q(z)=-1+a(1+z)^3/(z^3+5z^2+b)$ with best-fit values $a=1.513$, $b=5.04$, $H_0=74.43\, \mathrm{km\,s^{-1}Mpc^{-1}}$, and reconstructs power-law and logarithmic $f(Q)$ forms whose effective evolution tracks ΛCDM. In the non-minimally coupled $f(Q,L_m)$ chapter, it solves the reconstruction equation analytically for power-law and logarithmic couplings, obtaining the geometric function $f_1(Q)$ that makes the background expansion exactly ΛCDM, and validates the coefficients cosmographically with Pantheon+ data. In the $f(R,L_m)$ Bianchi-I chapter, the equation of state is fitted to Hubble and Pantheon samples and found to be phantom-like, with a small anisotropy parameter. The wormhole chapters show that in $f(Q,T)=\alpha Q+\beta T$ with conformal symmetry, three equations of state yield shape functions satisfying the throat and flare-out conditions, with the anisotropic case satisfying both null energy conditions, while in $f(R,L_m)$ Gaussian and Lorentzian non-commutative profiles yield wormhole shape functions and energy-condition analyses. The final chapter constrains two hybrid $f(T)$ models so that their BBN predictions and late-time cosmic-chronometer plus gamma-ray-burst data leave overlapping parameter ranges, with intermediate epochs checked by cosmography.

Load-bearing premise

The load-bearing premise is that $L_m=-\rho$ may be used in $f(R,L_m)=R/2+L_m^\alpha$ with non-integer fitted exponents $\alpha$, so that $(-\rho)^\alpha$ is interpreted as a real number even though ordinary exponent rules give a complex value for positive $\rho$.

Editorial extensions

If this is right

  • If the reconstructed $f(Q)$ forms are correct, the same expansion data that support ΛCDM are reproduced by non-metricity-based gravity without a cosmological constant term in the action.
  • If the $f(Q,L_m)$ embedding holds, the ΛCDM background can be realized exactly by a non-minimal geometry-matter coupling, so late-time acceleration need not require a vacuum-energy cosmological constant.
  • If the conformally symmetric $f(Q,T)$ wormholes are genuine solutions, traversable wormholes can exist with ordinary non-exotic matter in the anisotropic-pressure case, making the exotic-matter obstacle model-dependent.
  • If the hybrid $f(T)$ models pass BBN and late-time constraints, teleparallel gravity offers an early-to-late cosmological viability with model parameters confined by overlapping early- and late-time bounds.
  • If the Bianchi-I $f(R,L_m)$ fit is reliable, a small but measurable anisotropy is compatible with the same Hubble and Pantheon samples used to support an accelerating Universe.

Reading between the lines

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

  • A natural next test the thesis does not perform is perturbation theory: background equivalence to ΛCDM in $f(Q,L_m)$ does not guarantee the same growth of structure, so cosmic-shear and redshift-space-distortion data could discriminate the models.
  • The $f(Q)$ reconstruction in Chapter 2 is anchored to $H_0\approx 74.4\, \mathrm{km\,s^{-1}Mpc^{-1}}$; if the Hubble tension resolves toward the lower Planck value, the fitted $\alpha,\beta$ values change and the models' apparent closeness to ΛCDM may weaken.
  • For the $f(R,L_m)$ wormholes, the non-commutative profiles are assumed in the form of Gaussian and Lorentzian smearing; a decisive extension would derive such profiles from the theory's own minimal-length structure rather than impose them.
  • The BBN constraint pipeline could be applied to the non-metricity theories as well: the same $\Delta T_F/T_F$ bound plus cosmography would give an early-time test for $f(Q)$ and $f(Q,T)$, not only for $f(T)$.
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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. The manuscript is a PhD thesis compilation, arXiv:2507.04031, presenting seven chapters on modified theories of gravity. It introduces a two-parameter deceleration parameter and uses it to reconstruct f(Q) models (Chapter 2), embeds the ΛCDM expansion history into non-minimally coupled f(Q,L_m) gravity by solving for the functions f1 and f2 (Chapter 3), studies LRS Bianchi-I cosmological models in f(R,L_m)=R/2+L_m^alpha (Chapter 4), constructs wormhole solutions in f(Q,T) gravity with conformal symmetries (Chapter 5), repeats the f(R,L_m) wormhole analysis with non-commutative geometry (Chapter 6), and constrains hybrid f(T) models using Big Bang Nucleosynthesis and late-time data (Chapter 7). The abstract and preface claim that these modified gravity theories are viable frameworks for cosmic acceleration, traversable wormholes (in one case without exotic matter), and BBN-compatible late-time cosmology.

Significance. If the collection is correct, it would provide a menu of modified gravity models that all reproduce the background expansion history and yield explicit wormhole solutions, with the f(T) chapter providing cross-checks between early- and late-time constraints. The strengths of the manuscript include the use of current observational datasets (Pantheon+, CC, BAO), MCMC parameter estimation, explicit analytic reconstruction formulas in Chapters 2 and 3, and a reasonably complete energy-condition analysis in the wormhole chapters. The thesis is also grounded in a series of already-published papers, which lends some confidence that the individual calculations were externally reviewed. However, the load-bearing issues identified below—especially the complex-valued f(R,L_m) action—mean that the central viability claims are not currently supported as stated.

major comments (4)
  1. [Chapter 4, Eq. (4.13) and Table 4.1; Chapter 6, Eqs. (6.1)-(6.4)] The action f(R,L_m)=R/2+L_m^alpha with L_m=-rho and positive rho is not real-valued for the fitted non-integer alpha values (4.76, 5.57, 6.4). In Eq. (4.18)-(4.20) the term (-rho)^alpha appears explicitly; for alpha not an integer this is complex, so the field equations do not follow from a real stationary action. No branch convention or real-part prescription is stated anywhere in Chapters 4 or 6. Since the MCMC constraints, the reconstructed H(z), and the wormhole energy-condition results all depend on Eqs. (4.18)-(4.20), the f(R,L_m)-based conclusions are invalid as stated. The authors should either restrict alpha to integers (or odd integers, e.g. alpha=5), adopt and justify a specific branch of the complex power, or replace the model with a manifestly real Lagrangian, and then redo the fits and wormhole analysis.
  2. [Chapter 3, Eq. (3.13); Chapter 2, Eqs. (2.22)-(2.23)] The 'embedding' of ΛCDM into f(Q,L_m) is a reconstruction by construction: Eq. (3.13) is the ΛCDM Friedmann equation rewritten in terms of Q, and f1 and f2 are then solved so that the resulting model reproduces the assumed H(z). Consequently the analytic solutions in Sections 3.5.1-3.5.3 do not independently test against ΛCDM; agreement is guaranteed by the construction. Similarly, in Chapter 2 the parameters (alpha, beta) of the f(Q) models are computed from the fitted q0, H0, and an externally fixed Omega_m0 via Eqs. (2.22)-(2.23), so the reconstructed f(Q) curves are consistency checks rather than falsifiable predictions. The chapters should state this limitation explicitly and temper the viability language accordingly.
  3. [Chapter 5, Sections 5.5.1-5.5.3 and Eqs. (5.10)-(5.24)] The wormhole solutions are not asymptotically flat, as the authors themselves note after Eq. (5.24): S_f/r tends to a non-zero constant (e.g. (8 beta + 6)/(10 beta + 9) in Case 1), so the spatial metric approaches a conical or constant-deficit geometry rather than Minkowski space at infinity. For standard Morris-Thorne traversable wormholes, asymptotic flatness is one of the defining boundary conditions. The claims that these are physically viable traversable wormhole solutions therefore require either a matching to an exterior flat region or a clear statement that these are wormhole-like configurations in a non-asymptotically-flat background, which changes their physical interpretation.
  4. [Chapter 4, Table 4.1 and Eq. (4.25)] The fitted anisotropy parameter Delta is of order 1.6, and the directional Hubble ratio n is of order 27-31. This means the directional expansion rates differ by a factor of about 27, which is difficult to reconcile with the chapter's stated motivation that the Universe has only very small large-scale anisotropy. The text claims the model predicts an anisotropy 'in agreement with the dataset used' (Section 4.6), but no comparison is made with observational bounds on cosmological anisotropy (e.g. CMB quadrupole or shear constraints). The conclusion that the model is consistent with observed isotropy is therefore unsupported; the authors should either compare Delta with actual bounds or refrain from that claim.
minor comments (4)
  1. [Front matter] The manuscript contains university declaration pages, a plagiarism report, and submission metadata. These are inappropriate for a journal article and should be removed or moved to a supplementary file.
  2. [Throughout] The text contains typographical errors and inconsistent notation, e.g. 'Covarient' (Section 1.2), 'Christeffol' (Section 1.5.2.2), and the use of both 'article' and 'chapter' for self-reference. A thorough proofread is needed.
  3. [Chapter 5, Eq. (5.1)] The action in Eq. (5.1) has a formatting error: 'SM = R Lm sqrt(-g) d4x f(Q,T)' is not a well-formed expression. The intended definition of the matter action and the f(Q,T) model should be written explicitly.
  4. [Chapter 3, Sections 3.5-3.6] The hypergeometric and regularized hypergeometric function expressions (e.g. Eqs. (3.18), (3.23), (3.28)) are written without specifying their domains of convergence or the allowed parameter ranges; since these formulas are central to the reconstruction, a brief statement on parameter validity would help the reader.

Circularity Check

2 steps flagged · score 7.0 of 10

Chapters 2 and 3 reduce by construction: f(Q) parameters are solved from the fitted q0/H0/Omega_m0, and f1/f2 are solved from the LambdaCDM Hubble rate; the rest of the thesis is mostly self-constrained model fitting rather than circular.

  1. self definitional [Chapter 3, Sections 3.4-3.5, Eq. (3.13) and Eq. (3.18)/(3.19)]
    "To derive fi’s that replicate the ΛCDM expansion, it is necessary to incorporate all of the relevant parameters expressed as functions of the non-metricity scalar into the Friedmann equation (3.4). Now, substituting all the parameters in terms of Q into (3.4), we obtain a first-order inhomogeneous differential equation, expressed as follows Q/2 = ... (3.13)."

    Equation (3.13) is the Friedmann equation (3.4) evaluated with the ΛCDM Hubble function inserted via (3.8)-(3.12). The functions f1 and f2 are then obtained as solutions of this very differential equation, so the later statement that the model "would accurately replicate the expansion history of the ΛCDM model" is the input ansatz returned after solving an equation that already contains ΛCDM. No independent ΛCDM-like expansion is predicted; it is enforced by construction.

  2. fitted input called prediction [Chapter 2, Section 2.7, Eqs. (2.19)-(2.26), Table 2.3, Figs. 2.8-2.9]
    "By substituting the values q0 = −0.7, H0 = 74.43 obtained in this study along with Ωm0 = 0.315 [44], into Equations (2.22) and (2.23), we find the corresponding parameter values for the power-law f (Q) gravity model as α = 0.255708 and β = −0.839416."

    The parameters α and β are not constrained by an independent prediction of the f(Q) model; they are algebraically solved from the two present-day Friedmann equations (2.22)-(2.23) after inserting the MCMC-fitted q0 and H0 together with an adopted Ωm0. The reconstructed f(Q) curves in Figs. 2.8-2.9 are therefore a repackaging of those fitted inputs, and the comparison with ΛCDM is a consistency check rather than a test. The logarithmic model is treated identically via (2.25)-(2.26).

full rationale

The clearest circularity is in Chapter 3. Equation (3.13) is the f(Q,Lm) Friedmann equation with the ΛCDM H(Q) substituted; the paper then solves that differential equation for f1 or f2. The claim that the resulting theory 'replicates' ΛCDM is therefore true by construction, not by empirical test. Chapter 2 has a milder but real version: α and β for the two f(Q) ansatze are obtained by solving the present-day Friedmann equations with the MCMC-fitted q0, H0, and an adopted Ωm0, so the reconstructed f(Q) curves are functions of those inputs; comparing them with ΛCDM is a self-consistency check. The other chapters — Bianchi-I f(R,Lm), wormholes in f(Q,T) and f(R,Lm), BBN in f(T) — fit their parameters to the data they then display, which is normal likelihood-based model validation rather than circularity in the sense of this review. No load-bearing self-citation chain was found. A separate correctness issue, not a circularity, is that Chapter 4's model f = R/2 + L_m^α with L_m = -ρ and fitted non-integer α makes (-ρ)^α complex, so Eqs. (4.18)-(4.20) require an unstated branch choice; this affects viability but does not by itself make the derivation circular.

Assumptions & free parameters 15 free parameters · 10 assumptions · 0 invented entities

The central claims rest on many fitted or hand-chosen parameters: the q(z) shape parameters, the coefficients of the f1(Q) Taylor expansions, the f(Q) and f(R,Lm) model parameters, and the wormhole profile parameters. The axioms include standard differential geometry, FLRW and Bianchi-I backgrounds, perfect-fluid and barotropic assumptions, matter-Lagrangian choices, and the unstated assumption that fractional powers of negative densities are real. No genuinely new particles, forces, or dimensions are introduced.

free parameters (15)
  • a in q(z) parametrization = 1.513+0.073/-0.073
    Fitted to CC+BAO+SNeIa in Chapter 2, Eq. (2.15).
  • b in q(z) parametrization = 5.04+0.37/-0.44
    Fitted with a and H0 in Chapter 2; positivity of b is imposed to avoid divergences.
  • H0 in Chapter 2 = 74.43+0.18/-0.18 km/s/Mpc
    Fitted jointly with a and b in the MCMC analysis.
  • alpha in power-law f(Q) model = 0.255708
    Computed from fitted q0, H0, and Omega_m0 in Eq. (2.22)-(2.23); no uncertainty propagated.
  • beta in power-law f(Q) model = -0.839416
    Same consistency computation as alpha in Chapter 2.
  • alpha in logarithmic f(Q) model = 1.155
    Computed from Eq. (2.25)-(2.26) using fitted q0, H0, and Omega_m0.
  • beta in logarithmic f(Q) model = -0.42
    Same consistency computation as alpha in Chapter 2.
  • f(Q,Lm) power-law case parameters = xi=-1.28, nu=0.801, F1=2.23849e4, F2=0.073, F3=0.85, F4=0.01, H0=78.3, Omega_m0=0.316
    MCMC constraints from 1701 Pantheon+SH0ES points in Chapter 3.
  • f(Q,Lm) logarithmic case parameters = xi=-1.19, F1=-4.0411e3, F2=-0.073, F3=0.85, F4=0.012, H0=79.35, Omega_m0=0.299
    MCMC constraints for the logarithmic f2 case in Chapter 3.
  • w (EoS parameter) in f(R,Lm) = -1.245 (H(z)), -1.175 (Pantheon), -1.281 (H(z)+Pantheon)
    Constrained by MCMC in Chapter 4; indicates phantom behavior.
  • alpha in f(R,Lm) model = 5.57 (H(z)), 6.4 (Pantheon), 4.76 (H(z)+Pantheon)
    Model parameter in f(R,Lm)=R/2+L_m^alpha; fitted non-integer values create the fractional-power issue.
  • n in Bianchi-I anisotropy relation = 26.73 (H(z)), 31.09 (Pantheon), 28.65 (H(z)+Pantheon)
    Defines A=B^n in Eq. (4.14); yields anisotropy measure Delta near 1.6.
  • Chapter 5 wormhole parameters = alpha=1.5, beta=0.9, C2=2, r0=1, w=0.4 (Case 3); alpha=1.4, C2=1.26, r0=0.35, A=2.25 (Case 2)
    Chosen by hand to satisfy the flare-out condition; not fitted to observational data.
  • Chapter 6 non-commutative wormhole parameters = M=1.2, theta=4, r0=1, alpha varied
    Chosen by hand for Gaussian and Lorentzian smearing profiles.
  • Chapter 7 f(T) hybrid model parameters = n (hybrid exponential) and a (hybrid tanh), ranges in Table 7.2
    Constrained using BBN bounds and late-time CC/GRB data.
assumptions (10)
  • standard math Standard differential geometry definitions and tensor calculus are assumed.
    Chapter 1 provides definitions of manifolds, connections, curvature, torsion, and non-metricity as background.
  • standard math Einstein field equations and the LambdaCDM expansion history are taken as baseline.
    Used for comparison and as the target history in Chapter 3, Eq. (3.8).
  • domain assumption Cosmological chapters use a flat FLRW or LRS Bianchi-I background.
    Assumed in Chapters 2, 3, 4, and 7.
  • domain assumption Matter is modeled as a perfect fluid with barotropic equation of state p=w rho.
    Used throughout the cosmological reconstructions and explicitly in Eq. (4.17).
  • domain assumption The matter Lagrangian is chosen as Lm=p in f(Q,Lm) and Lm=-rho in f(R,Lm).
    Choice affects the continuity equation and the extra force; discussed in Chapter 3 and used in Chapter 4.
  • ad hoc to paper The deceleration parameter has the assumed form q(z)=-1+a(1+z)^3/(z^3+5z^2+b).
    This is the central parametrization of Chapter 2; all subsequent constraints depend on it.
  • ad hoc to paper Specific Lagrangian forms are assumed: power/log f(Q), power/log f2 in f(Q,Lm), f(R,Lm)=R/2+L_m^alpha, f(Q,T)=alpha Q+beta T, and hybrid f(T) models.
    These choices define the models being tested; they are not derived from a deeper principle.
  • ad hoc to paper Wormhole chapters impose conformal Killing vectors and non-commutative Gaussian or Lorentzian profiles.
    Used to close the wormhole equations in Chapters 5 and 6.
  • ad hoc to paper The expression (-rho)^alpha is treated as real for positive rho and non-integer alpha.
    Unstated assumption in Chapter 4 (Eqs. (4.18)-(4.20)) and Chapter 6; it is false for general non-integer alpha.
  • domain assumption Standard Big Bang nucleosynthesis abundance predictions are valid and f(T) corrections are small perturbations.
    Used to constrain f(T) models in Chapter 7.

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Cite this review

Pith. "Pith review of The Study on Modified Theories of General Relativity: A Differential Geometric Approach." pith.science (2026). https://pith.science/paper/QVQUOQHX

@misc{pith2026250704031,
  author       = {Pith},
  title        = {Pith review of: The Study on Modified Theories of General Relativity: A Differential Geometric Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVQUOQHX}},
  note         = {Machine review of arXiv:2507.04031}
}
read the original abstract

The introduction of General Relativity (GR) in 1915 revolutionized our understanding of gravity, but over time, its limitations in explaining phenomena like dark energy, dark matter, and quantum gravity have motivated alternative theories. Early modifications, such as Weyl's 1919 proposal, focused on adding higher-order terms to the Einstein-Hilbert action. GR's non-renormalizability further strengthened the case for extending it. A central theme of modern gravity research is modifying the geometric structure, often by changing the gravitational Lagrangian. This leads to theories such as teleparallel and symmetric teleparallel gravity, utilizing torsion or non-Levi-Civita connections, with differential geometry providing the essential framework. This thesis explores several modified gravity models. Chapter 1 introduces necessary mathematical tools. Chapter 2 develops a novel parametrization of the deceleration parameter, constrained using MCMC and observational data, and applies it to f(Q) gravity. Chapter 3 embeds the LambdaCDM model into f(Q, L\_m) gravity with non-minimal coupling, producing analytic solutions and matching observations through cosmographic analysis. Chapter 4 considers Bianchi-I spacetime in f(R, L\_m) gravity with observational constraints to measure anisotropy. Chapter 5 presents wormhole solutions in f(Q, T) gravity with conformal symmetries. Chapter 6 explores wormholes in f(R, L\_m) with non-commutative geometry, analyzing shape functions, energy conditions, and stability. Chapter 7 studies Big Bang Nucleosynthesis in f(T) gravity, constraining hybrid models using early- and late-time data, and validating intermediate epochs via cosmography.

Figures

Figures reproduced from arXiv: 2507.04031 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p040_1.png] view at source ↗
Figure 1.1
Figure 1.1. Schematic representation of the proposed modified gravity theories by [PITH_FULL_IMAGE:figures/full_fig_p041_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Schematic representation of the proposed modified gravity theories by [PITH_FULL_IMAGE:figures/full_fig_p041_1_2.png] view at source ↗
Figures from the paper (72 more)
Figure 1.3
Figure 1.3. Figure 1.3: Schematic representation of the proposed modified gravity theories by [PITH_FULL_IMAGE:figures/full_fig_p042_1_3.png]
Figure 1.4
Figure 1.4. Figure 1.4: Schematic representation of the proposed modified gravity theories by [PITH_FULL_IMAGE:figures/full_fig_p043_1_4.png]
Figure 1.5
Figure 1.5. Figure 1.5: A potential categorization of gravity theories arising within the framework [PITH_FULL_IMAGE:figures/full_fig_p044_1_5.png]
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p045_1.png]
Figure 1.6
Figure 1.6. Figure 1.6: Schematic representation of equivalent theories and their extensions [PITH_FULL_IMAGE:figures/full_fig_p046_1_6.png]
Figure 1.7
Figure 1.7. Figure 1.7: Pictorial representation of the tetrad formalism [PITH_FULL_IMAGE:figures/full_fig_p046_1_7.png]
Figure 1.8
Figure 1.8. Figure 1.8: Pictorial representation of the behavior of tetrads under the absence and [PITH_FULL_IMAGE:figures/full_fig_p047_1_8.png]
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p047_1.png]
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Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p049_1.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p063_2.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p064_2.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p065_2.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p066_2.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p067_2.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p069_2.png]
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Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p071_2.png]
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p072_2.png]
Figure 2.1
Figure 2.1. Figure 2.1: 2D-contour plot of the model parameters a, b, and H0, indicating the most likely values and confidence regions up to 3σ obtained from the combined analysis of CC, BAO and SNeIa datasets. 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 z 0.2 0.3 0.4 0.5 0.6 0.7 O m(z) Mean CDM model …
Figure 2.2
Figure 2.2. Figure 2.2: The behavior of the Om(z) diagnostic vs. redshift z. 51 [PITH_FULL_IMAGE:figures/full_fig_p074_2_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: Comparison of the obtained best-fit theoretical curves (red line) of the Hub [PITH_FULL_IMAGE:figures/full_fig_p075_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: A plot illustrating the current values of [PITH_FULL_IMAGE:figures/full_fig_p076_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: A graph displaying the present values of deceleration parameter, accompa [PITH_FULL_IMAGE:figures/full_fig_p076_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: Deceleration parameters as a function of redshift for proposed parametric [PITH_FULL_IMAGE:figures/full_fig_p077_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: Jerk parameters as a function of redshift for the proposed model, obtained [PITH_FULL_IMAGE:figures/full_fig_p077_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: Comparison between the reconstructed f(Q) = Q + αQ0  Q Q0 β and ΛCDM model. α=1.155, β=-0.42 α=0.772, β=-0.228 α=0.774, β=-0.224 ΛCDM 0 10 20 30 40 0 10 20 30 40 Q/(100 km/s/Mpc) 2 f(Q)/(100 km / s /Mpc) 2 [PITH_FULL_IMAGE:figures/full_fig_p078_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Graph of f(Q) = αQ + βQ0 log(Q/Q0) against ΛCDM model. 55 [PITH_FULL_IMAGE:figures/full_fig_p078_2_9.png]
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p081_1.png]
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p095_3.png]
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p097_3.png]
Figure 3.1
Figure 3.1. Figure 3.1: 2D likelihood contours obtained from MCMC with power-law form [PITH_FULL_IMAGE:figures/full_fig_p102_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: 2D likelihood contours obtained from MCMC with logarithmic form [PITH_FULL_IMAGE:figures/full_fig_p103_3_2.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p113_4.png]
Figure 4.1
Figure 4.1. Figure 4.1: Contour plot with 1 − σ and 2 − σ errors for the parameters w, α and n along with the constraint values for Hubble dataset. 92 [PITH_FULL_IMAGE:figures/full_fig_p115_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: The profile of Hubble parameter versus redshift [PITH_FULL_IMAGE:figures/full_fig_p116_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Contour plot with 1 − σ and 2 − σ errors for the parameters w, α and n along with the constraint values for pantheon dataset. 93 [PITH_FULL_IMAGE:figures/full_fig_p116_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: The profile of distance modulus versus redshift [PITH_FULL_IMAGE:figures/full_fig_p117_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Contour plot with 1 − σ and 2 − σ errors for the parameters w, α and n along with the constraint values for H(z)+Pantheon dataset. 94 [PITH_FULL_IMAGE:figures/full_fig_p117_4_5.png]
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p132_5.png]
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p133_5.png]
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p134_5.png]
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p136_5.png]
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p137_5.png]
Figure 5.1
Figure 5.1. Figure 5.1: Case 1: Plot showing the characteristics of [PITH_FULL_IMAGE:figures/full_fig_p138_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Case 1: Plot showing the profile of energy density and various ECs for [PITH_FULL_IMAGE:figures/full_fig_p139_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Case 2: Plot showing the characteristics of [PITH_FULL_IMAGE:figures/full_fig_p140_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Case 2: Plot showing the profile of energy density and various ECs for [PITH_FULL_IMAGE:figures/full_fig_p141_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: Case 3: Plot showing the characteristics of [PITH_FULL_IMAGE:figures/full_fig_p142_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: Case 3: Plot showing the profile of energy density and various ECs for [PITH_FULL_IMAGE:figures/full_fig_p143_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Plot showing the three-dimensional embedding diagram of wormhole so [PITH_FULL_IMAGE:figures/full_fig_p144_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: A plot showing the equilibrium state of different wormhole solutions [PITH_FULL_IMAGE:figures/full_fig_p145_5_8.png]
Figure 5.9
Figure 5.9. Figure 5.9: Plot showing the two-dimensional embedding diagram of wormhole solu [PITH_FULL_IMAGE:figures/full_fig_p146_5_9.png]
Figure 6.1
Figure 6.1. Figure 6.1: Gaussian Distribution: The influence of model parameter [PITH_FULL_IMAGE:figures/full_fig_p162_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: Gaussian Distribution: plot showing the profile of (a) energy density vary [PITH_FULL_IMAGE:figures/full_fig_p163_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: Lorentzian Distribution: The influence of model parameter [PITH_FULL_IMAGE:figures/full_fig_p164_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: Lorentzian Distribution: plot showing the profile of (a) energy density vary [PITH_FULL_IMAGE:figures/full_fig_p165_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: The profile of hydro-static and anisotropic forces for different values of [PITH_FULL_IMAGE:figures/full_fig_p166_6_5.png]
Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p170_1.png]
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p176_7.png]
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p177_7.png]
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p178_7.png]
Figure 7.1
Figure 7.1. Figure 7.1: n vs ∆TF/TF for the Hybrid exponential model. The blue dashed line rep￾resents the upper bound of ∆TF/TF. 1.80 1.85 1.90 1.95 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 a Δ T F/TF [PITH_FULL_IMAGE:figures/full_fig_p181_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: a vs ∆TF/TF for the Hybrid tangent hyperbolic model. The blue dashed line represents the upper bound of ∆TF/TF. 158 [PITH_FULL_IMAGE:figures/full_fig_p181_7_2.png]
Figure 7.3
Figure 7.3. Figure 7.3: Likelihood probability distributions and 2D contours for the Hybrid ex [PITH_FULL_IMAGE:figures/full_fig_p182_7_3.png]
Figure 7.4
Figure 7.4. Figure 7.4: Likelihood probability distributions and 2D contours for the Hybrid ex [PITH_FULL_IMAGE:figures/full_fig_p182_7_4.png]
Figure 7.5
Figure 7.5. Figure 7.5: Likelihood probability distributions and 2D contours for the Hybrid tan [PITH_FULL_IMAGE:figures/full_fig_p183_7_5.png]
Figure 7.6
Figure 7.6. Figure 7.6: Likelihood probability distributions and 2D contours for the Hybrid tan [PITH_FULL_IMAGE:figures/full_fig_p183_7_6.png]
Figure 7.7
Figure 7.7. Figure 7.7: The Hubble function (constrained from CC and CC+GRB) against redshift [PITH_FULL_IMAGE:figures/full_fig_p184_7_7.png]
Figure 7.8
Figure 7.8. Figure 7.8: The Hubble function (constrained from CC and CC+GRB) against redshift [PITH_FULL_IMAGE:figures/full_fig_p184_7_8.png]
Figure 7.9
Figure 7.9. Figure 7.9: The distance modulus function (constrained from CC and CC+GRB) against [PITH_FULL_IMAGE:figures/full_fig_p185_7_9.png]
Figure 7.10
Figure 7.10. Figure 7.10: The distance modulus function (constrained from CC and CC+GRB) [PITH_FULL_IMAGE:figures/full_fig_p185_7_10.png]
Figure 7.11
Figure 7.11. Figure 7.11: Deceleration parameter (constrained from CC and CC+GRB) vs redshift [PITH_FULL_IMAGE:figures/full_fig_p186_7_11.png]
Figure 7.12
Figure 7.12. Figure 7.12: Deceleration parameter (constrained from CC and CC+GRB) vs redshift [PITH_FULL_IMAGE:figures/full_fig_p186_7_12.png]

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