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Observational analysis of bulk viscous modified Chaplygin gas in (2+1)-dimensional universe using MCMC

T0 review · 6 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A bulk-viscous modified Chaplygin gas in (2+1)-dimensional spacetime is claimed to reproduce the observed expansion history, yielding $H_0 = 67.90$ km s$^{-1}$ Mpc$^{-1}$, and to damp the structure-formation oscillations that have…

desk verdict The Friedmann equations are mutually inconsistent and not derivable from the stated field equations, so the solutions and the MCMC H0 rest on broken foundations; this should be desk-rejected. read the letter →

arxiv 2504.18612 v1 pith:SK2BCETL submitted 2025-04-25 gr-qc

Praveen Kumar Dhankar , Aritra Sanyal , Albert Munyeshyaka , Saibal Ray , Behnam Pourhassan This is my paper · ORCID
classification gr-qc
keywords bulkviscositymodifiedChaplygingas(2+1)-dimensionalcosmologyFLRWuniverseMarkovchainMonteCarloHubbleconstantPantheonsupernovaestructureformationperturbations
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 sets out to show that a single cosmic fluid—a modified Chaplygin gas with bulk viscosity, placed in a (2+1)-dimensional FLRW spacetime—can reproduce the broad features of the observed expansion history. It derives analytical solutions for the energy density and Hubble parameter in both the inviscid and constant-viscosity cases, then fits the model with MCMC to 30 cosmic-chronometer Hubble measurements and the 1048-point Pantheon supernova sample. The fit gives $H_0 = 67.90$ km s$^{-1}$ Mpc$^{-1}$, close to the Planck $\Lambda$CDM estimate, and the perturbation analysis finds that bulk viscosity inserts a wave-number-dependent damping term that suppresses the oscillation problems of non-viscous Chaplygin models. If these results stand, a lower-dimensional viscous fluid would be a tractable laboratory for dark-energy mechanisms, and bulk viscosity would offer a concrete way to ease Chaplygin gas's structure-formation tension. The authors are careful to frame the constraints as mathematical projections onto a lower-dimensional framework rather than a literal model of the observable universe.

What carries the argument

The load-bearing object is the bulk-viscous modified Chaplygin gas: the equation of state $p = \gamma\rho - A/\rho^\beta$ with $A > 0$ and $0 < \beta \le 1$, modified by bulk viscosity through $\bar p = p - 2\xi H$, and inserted into the assumed (2+1)-dimensional Friedmann equations $H^2 = \rho/2$ and $\ddot a/a = -\bar p$. The Chaplygin term supplies late-time negative pressure; the viscous term shifts the deceleration parameter and, in perturbations, contributes the damping term $2\xi k^2\delta$ to the density-contrast equation. The MCMC stage enters through a $\chi^2$ built from the theoretical $H(z)$ and distance modulus against the Hubble and Pantheon datasets. All downstream results—the analytical solutions, the fitted $H_0$, and the claimed damping of structure oscillations—depend on this fluid prescription and on the 2+1 field equations.

What would settle it

Recompute the Einstein tensor for the FLRW metric (1) in 2+1 dimensions and compare $G_{00}$ and $G_{ab}$ with Eqs. (6)-(7); under the paper's own normalization $G_{ij}=2\pi G T_{ij}$, if $G_{00}$ is not $\rho/2$, the derived $\rho(z)$ and $H(z)$ used in the MCMC fit are invalid. A second check is to re-run the MCMC with the corrected Friedmann equations and see whether $H_0$ remains $67.90$ km s$^{-1}$ Mpc$^{-1}$.

Watch

Extended reading notes

Core claim

On its own terms, the paper's finding is that a (2+1)-dimensional FLRW universe filled with a modified Chaplygin gas $p = \gamma\rho - A/\rho^\beta$ and a bulk-viscous pressure $\bar p = p - 2\xi H$ obeys $H^2 = \rho/2$ and $\ddot a/a = -\bar p$, and that this system admits closed-form solutions for the energy density in both the $\xi = 0$ and constant-$\xi$ cases. The inviscid solution produces a smooth deceleration-to-acceleration transition; the viscous solution gives a consistently negative deceleration parameter. Fitting the resulting $H(z)$ and distance modulus to cosmic-chronometer and Pantheon data by MCMC yields $H_0 = 67.90$ km s$^{-1}$ Mpc$^{-1}$, which the authors take as consistency with Planck. In the perturbation sector, the viscous term $2\xi k^2\delta$ damps oscillations in the density contrast, presented as a way around the structure-formation problem that has disfavored Chaplygin cosmologies. The paper's own framing is that the whole construction is a theoretical laboratory rather than a direct model of the observable universe.

Load-bearing premise

The whole derivation rests on the assumed (2+1)-dimensional Friedmann equations $H^2 = \rho/2$ and $\ddot a/a = -\bar p$; if the standard reduction of the stated Einstein equations yields different equations, the analytical solutions, fitted parameters, and claimed Planck consistency do not follow.

Editorial extensions

If this is right

  • A single bulk-viscous Chaplygin fluid would reproduce the observed deceleration-to-acceleration transition and sustain late-time accelerated expansion without a separate dark-energy component.
  • The fitted $H_0 = 67.90$ km s$^{-1}$ Mpc$^{-1}$ would place the model in agreement with the Planck $\Lambda$CDM value, suggesting that low-redshift expansion data alone do not demand (3+1)-dimensional dynamics.
  • The viscous damping term $2\xi k^2\delta$ would suppress small-scale oscillations in the density contrast, potentially easing the structure-formation tension that has ruled out non-viscous Chaplygin models.
  • Because the viscous model approaches a de Sitter-like phase at late times, its late-time predictions become degenerate with $\Lambda$CDM, so future late-time observations would discriminate poorly between them.

Reading between the lines

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

  • If the (2+1)-dimensional Friedmann equations were derived from the stated Einstein equation rather than assumed, the solutions and the fitted $H_0$ would likely change; re-running the MCMC with the standard reduction would test whether the Planck agreement survives.
  • The structure-damping result is established in a 2+1 setting with no matter power spectrum; translating the same viscosity prescription to (3+1) perturbations would show whether oscillations disappear without over-suppressing structure growth.
  • The authors' 'mathematical projection' caveat implies an implicit holographic or brane-world reading of the fits; making that mapping explicit would turn the Planck consistency from a numerical coincidence into a physical statement.
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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

6 major / 6 minor

Summary. This paper studies a bulk-viscous modified Chaplygin gas (MCG) in a (2+1)-dimensional spatially flat FLRW spacetime. The authors claim to derive analytic solutions for the energy density, Hubble parameter, and deceleration parameter in non-viscous and viscous cases; to show that bulk viscosity dampens structure-growth oscillations; and to constrain the model by a Markov chain Monte Carlo fit to 30 cosmic-chronometer H(z) points and 1048 Pantheon supernova distances, obtaining H0 = 67.90 km s−1 Mpc−1, which the abstract presents as 'remarkable consistency with Planck LCDM estimations despite the dimensional reduction.' The derivation rests on the Friedmann pair H² = ρ/2 and ä/a = −p̄ (Eqs. (6)–(7)), a reduced conservation equation (Eq. (9)), and a viscous energy-density ansatz taken from Ref. [53]. The manuscript repeatedly cautions that the model is a theoretical laboratory rather than a direct alternative to the standard cosmological model.

Significance. If the derivations were sound, the paper would offer a tractable toy model in which a viscous Chaplygin gas in 2+1 dimensions reproduces the main features of cosmic expansion, a mechanism (bulk viscosity) for curing the matter-power-spectrum oscillations that afflict Chaplygin models, and a conventional two-dataset MCMC pipeline. The manuscript deserves credit for unusually candid limitation statements: it concedes that the fitted constraints are 'mathematical consistency checks rather than physically meaningful constraints' and that Chaplygin models are disfavored by current DESI-era data. These strengths do not carry the paper's central claims, because the equations do not support them: the foundational pair (6)–(7) is internally inconsistent, the claimed solution (11) does not solve (10), and the headline H0 agreement restates a fitted quantity rather than constituting a prediction. No machine-checked proofs or reproducible chains accompany the analysis, and the perturbation claim in the abstract is not backed by any displayed numerical result.

major comments (6)
  1. [§2, Eqs. (6)–(7)] The two Friedmann equations used throughout the paper are mutually inconsistent, independently of any coupling convention. Differentiating Eq. (6) with respect to cosmic time and inserting the paper's own conservation equation (8) gives 2Ḣ = −(ρ + p̄), whereas combining Eqs. (6) and (7) gives 2Ḣ = −(ρ + 2p̄) (using ä/a = Ḣ + H²); the two expressions agree only when p̄ = 0, which the modified Chaplygin gas is not. Additionally, Eqs. (6)–(7) do not follow from the stated field equations: a direct computation of G_ij for metric (1) yields G_00 = H² and G_ij = −(ä/a) g_ij (spatial), so that Eq. (2) with T_00 = ρ + 2p̄ from Eq. (3) cannot produce the coefficient 1/2 in (6) and the coefficient 1 in (7) simultaneously. The cited Ref. [19] is a (3+1)-dimensional paper and does not support Eqs. (6)–(7). Since every subsequent result, including Eqs. (9)–(27) and the fitted H0, rests on these equations, this is a load-bearing error.
  2. [§2, Eq. (9)] Eq. (9) is an incorrect reduction of Eq. (8). Substituting H = √(ρ/2) from Eq. (6) and p̄ = γρ − A ρ^{−β} − 2ξH from Eqs. (4)–(5) into Eq. (8) gives ρ̇ + √2(γ+1)ρ^{3/2} − 2ξρ − √2 A ρ^{1/2−β} = 0. The paper's Eq. (9) replaces √2 A ρ^{1/2−β} by √2 A, which is valid only for β = 1/2. Because β is a free parameter in the equation of state (5) and does not appear anywhere in the fitting formulas (25)–(27), the paper silently restricts the model to β = 1/2 without stating or justifying this. The MCMC constraints are therefore not for the model defined by Eq. (5).
  3. [§2, Eqs. (10)–(14)] Eq. (11) does not solve Eq. (10). Direct substitution of Eq. (11) into Eq. (10), using ȧ/a = √(ρ/2) from Eq. (6), leaves a non-vanishing residual for generic A, γ, c; the equality would hold only for special values such as A = ±1 and c = 0. The correct solution of Eq. (10) is ρ(a) = [A/(γ+1) + c a^{−3(γ+1)}]^{2/3}, which diverges as a → 0, whereas Eq. (11) tends to zero in that limit. Equation (13) is also not real-valued at high redshift for c > 0 because A − c(1+z)^{3(γ+1)} becomes negative, so the plotted curves in Figs. 1–2 are complex-valued over a substantial part of the displayed range. Equations (12)–(14) and the corresponding analysis therefore rest on an algebraic error.
  4. [§2, Eqs. (15)–(27)] The viscous solution is not derived or verified. The form ρ = E/t² + F/t + ht + De^{bt} is imported as an ansatz from Ref. [53], a paper co-authored by one of the current authors, and 'comparing like coefficients' in Eq. (16) is not a legitimate procedure because ρ^{3/2} of the ansatz is not a linear combination of 1/t², 1/t, t, and e^{bt}; the resulting expressions (17)–(23) are never checked by substitution into Eq. (9). Independently, the asserted redshift relation t(z) = (1/(nα)) ln(1 + (1+z)^{−n}) corresponds to a(t) = (e^{nαt} − 1)^{1/n}, for which H = αe^{nαt}/(e^{nαt} − 1) → α as t → ∞; Eq. (6) then requires ρ = 2H² → 2α², whereas Eq. (23) gives ρ ~ √2 A t at late times. Equations (6), (23), and the t(z) ansatz are mutually incompatible, so the model H(z) in Eq. (27) that feeds the MCMC fit is unsupported.
  5. [§4, §§4.1–4.3] The headline result is a fitted value, not a prediction, and the fit is under-documented. H0 is determined by the fit itself, through H(z) at z → 0 and through the distance-scale normalization in Eq. (39), so the abstract's 'remarkable consistency with Planck' restates the fit outcome rather than providing an independent check; the manuscript itself concedes that the constraints 'should be interpreted primarily as mathematical consistency checks rather than physically meaningful constraints.' Of the parameters entering Eq. (27) (n, α, γ, A, ξ, and the implicit unit normalization), only H0 and n are reported with uncertainties; no χ² minimum, priors, chain lengths, or convergence diagnostics are given. The analysis cannot be reproduced or checked as presented.
  6. [§3, Eq. (36) and §3.3] The perturbation analysis is asserted rather than demonstrated. Eq. (36) is presented without derivation from the perturbed Einstein equations in 2+1 dimensions, the viscous damping term 2ξk²δ is not derived and no dimensional analysis is given, and §3.3 states that the perturbation equations were 'numerically solved' without showing a single plot, parameter value, or table. The abstract's claim that 'bulk viscosity dampens the structure growth oscillations' is therefore unsupported by any displayed evidence, and §3.4 itself acknowledges that the analysis cannot be compared with observational data.
minor comments (6)
  1. [§2, Case (i)–(ii)] The cross-references are unreliable: 'After solving Eq. (24)' (before Eq. (30)) should refer to the equation just solved, Eq. (29); 'Eq. (25), becomes as ρ = 2/((γ+1)²t²)' refers to the same A = ξ = 0 result rather than to Eq. (25); 'From Eq. (27), it is observed that energy density ρ decreases' appears in a paragraph about ρ(t); and in Case (ii), 'using the value of ρ from Eq. (13) in Eq. (9)' should refer to the ansatz Eq. (15).
  2. [§1, §4] There are numerous typos and grammatical errors, including 'regardred' and 'severel' in Section 1, 'stranded error' for 'standard error' in §4.1, 'depreciate' for 'marginalized' in §4.2, and the abstract's opening 'This paper investigates regarding cosmological implications of...'. A thorough language edit is needed.
  3. [§2, Eqs. (21)–(26)] The notation O(γ^n) is defined in Eq. (22) as an explicit polynomial while n is simultaneously one of the fitted model parameters; the nested definitions X1–X5 and Δ1, Δ2 in Eqs. (25)–(26) are very difficult to track; and no statement is given about the units of α, ξ, or t needed to evaluate Eq. (27).
  4. [§4, Figs. 6–12] The model curves in Figs. 6–12 are computed for fixed illustrative values (γ = 0.3, A = 3.4, c = 1) rather than the MCMC best-fit values, and no χ², reduced χ², or other goodness-of-fit statistic is reported, so the claimed 'excellent agreement with observational data' cannot be assessed quantitatively.
  5. [§4.3, Figs. 13–14] Figures 13–14 show contours for 'H0 and q' and for 'x, y, z', but q is not a parameter of the model H(z) in Eq. (27) and x, y, z are never defined; moreover the fitted q = 0.322 is positive, in apparent contradiction with the text's statement that q(z) is negative throughout the viscous model (Figs. 8–9).
  6. [§4.2, Eqs. (38)–(40)] The units are not specified: Eq. (27) evaluates H in natural units, yet the fit reports H0 in km s−1 Mpc−1, and Eq. (39) reintroduces c/H0 without stating the conversion convention; this needs to be specified for the quoted constraints to be reproducible.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline Planck agreement is the fitted H0 itself, and the viscous solution is a same-author ansatz imported from Ref. [53].

  1. fitted input called prediction [Abstract and Section 4.3, MCMC analysis.]
    "Using Markov chain Monte Carlo (MCMC) techniques with Hubble parameter and Pantheon supernova datasets, we impose constraints on our model parameters, obtaining H0 = 67.90 km/s/Mpc, showing remarkable consistency with Planck ΛCDM estimations despite the dimensional reduction. ... The outcome of this computational operation is that we can determine the best fit values of the model parameters as H0 = 67.90 and n = 3.036 due to effective application of the MCMC procedure."

    H0 is one of the parameters being fitted by the MCMC procedure, not a quantity predicted before fitting. The 'remarkable consistency with Planck' is therefore a restatement of the best-fit value: the same data were used to set H0, and then the fitted H0 is presented as a successful consequence of the model. No independent prediction of H0 is made, so the agreement is forced by the fit rather than by the (2+1)-dimensional theory itself.

  2. ansatz smuggled in via citation [Section 2, Case (ii), Eq. (15).]
    "In this case we follow the particular form of ρ given earlier by Saadat and Pourhassan [53] as ρ = E/t^2 + F/t + ht + De^{bt}."

    The central viscous density profile that generates H(z) for the MCMC fit is imported verbatim from a prior paper coauthored by one of the current authors (Pourhassan). The subsequent derivation only matches coefficients of this pre-chosen functional form, so the output ρ in Eq. (23) has the same functional structure as the input ansatz by construction. Thus the 'analytical solution' for the viscous case is not independently derived from the Friedmann equations; it is a self-cited ansatz that determines the expansion history used later.

full rationale

The clearest circular element is the H0/Planck consistency claim: the abstract presents the fitted best-fit H0 as though the model had reproduced the Planck value, but H0 is a free parameter of the MCMC fit. The second circularity is the viscous solution: Eq. (15) is explicitly taken from Saadat and Pourhassan [53], a paper overlapping with the present authors, and the coefficient matching that follows does not derive the functional form independently. I did not count the inconsistency between Eqs. (6)-(7) and the stated field equation (2) as a circularity step: that is a correctness/foundational defect rather than a prediction reducing to an input. Similarly, the perturbation claim that viscosity damps oscillations follows from a damping term inserted by hand into Eq. (36); it is a model property, not an independent test, so it does not add a separate circular step. No machine-checked or externally falsifiable calibration is provided to break the self-citation chain for the viscous ansatz. Overall score 6 reflects that one headline prediction reduces to a fitted parameter and the central viscous solution is a load-bearing same-author ansatz.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The model depends on two fitted parameters (H0, n), two arbitrary parameters (γ, A), an uncontrolled viscosity ξ, an implied β=1/2, and an ad hoc t(z) relation. The analytical backbone is taken from prior work by the same group, so the ledger is heavy and the independent contribution is small.

free parameters (8)
  • H0 = 67.90 km/s/Mpc (best fit; 67.905+0.174-0.163 in Fig. 13)
    Fitted with MCMC to CC+Pantheon data; the 'Planck consistency' is a consequence of the fit, not an independent prediction.
  • n = 3.036
    Fitted with MCMC; shape parameter in the redshift-time relation.
  • alpha
    Appears in the ad hoc t(z) = (1/(n α)) log(1+(1+z)^{-n}) relation; no best-fit value or prior is reported.
  • gamma = 0.3 (chosen)
    MCG equation-of-state parameter; fixed to 0.3 in the plots, not marginalized.
  • A = 3.4 (chosen)
    Chaplygin parameter; fixed to 3.4 in the plots, not marginalized.
  • xi
    Bulk viscosity coefficient; treated as a constant but no fitted value is reported.
  • beta = 1/2 (implied)
    The conservation equation Eq. (9) contains -√2A instead of -√2A ρ^{1/2-β}, silently setting β=1/2; the paper never declares this.
  • c = 1 (chosen)
    Integration constant in Eq. (11); fixed to 1 in plots.
assumptions (4)
  • domain assumption The Friedmann equations H² = ρ/2 and ä/a = -p̄ (Eqs. 6-7) are assumed to follow from G_ij = 2πG T_ij (Eq. 2) for the FLRW metric (Eq. 1).
    This is not verified; standard reduction gives different equations. The paper cites Ref. [19] but no derivation is shown.
  • ad hoc to paper The conservation equation Eq. (8) reduces to Eq. (9) with -√2A rather than -√2A ρ^{1/2-β}.
    Holds only if β=1/2, which is never stated; inconsistent with the general β used in Eq. (35).
  • ad hoc to paper The viscous energy density has the form ρ = E/t² + F/t + ht + D e^{bt} from Ref. [53].
    An ansatz imported from the authors' prior work; no derivation, and the computed coefficients diverge as ξ→0.
  • ad hoc to paper The redshift-time relation t(z) = (1/(n α)) log(1+(1+z)^{-n}).
    Introduced without derivation; used to convert the time-domain solution into the observable H(z) and distance modulus.

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Cite this review

Pith. "Pith review of Observational analysis of bulk viscous modified Chaplygin gas in (2+1)-dimensional universe using MCMC." pith.science (2026). https://pith.science/paper/SK2BCETL

@misc{pith2026250418612,
  author       = {Pith},
  title        = {Pith review of: Observational analysis of bulk viscous modified Chaplygin gas in (2+1)-dimensional universe using MCMC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SK2BCETL}},
  note         = {Machine review of arXiv:2504.18612}
}
abstract

This paper investigates regarding cosmological implications of a bulk viscous modified Chaplygin gas (MCG) in (2+1)-dimensional Friedmann-Robertson-Walker spacetime, incorporating both theoretical analysis and observational constraints. We derive analytical solutions for both viscous and non-viscous cases, revealing distinct behavior in energy density evolution, Hubble parameter dynamics, and deceleration parameter transitions. A comprehensive perturbation analysis illustrates how bulk viscosity dampens the structure growth oscillations, addressing a key challenge faced by Chaplygin gas models in higher dimensions. Using Markov chain Monte Carlo (MCMC) techniques with Hubble parameter and Pantheon supernova datasets, we impose constraints on our model parameters, obtaining $H_0 = 67.90$ km s$^{-1}$ Mpc$^{-1}$, showing remarkable consistency with Planck $\Lambda$CDM estimations despite the dimensional reduction. Our findings suggest that lower-dimensional viscous cosmology captures essential features of cosmic evolution while providing valuable theoretical insights into the interplay between dissipative effects and exotic equations of state.

Figures

Figures reproduced from arXiv: 2504.18612 by the authors.

Figure 1
Figure 1. The energy density ρ(z) is shown against redshift z for different values of the constants γ = 0.3, A = 3.4, c = 1 for the Case (i) [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The energy density ρ(z) is shown against redshift z in the contour plot for the different values of γ = 0.3, A = 3.4, c = 1 for the Case (i) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The Hubble parameter H(z) is shown against redshift z for different values of the constants γ = 0.3, A = 3.4, c = 1 for the Case (i). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: The Hubble parameter H(z) vs redshift z in the contour plot for the different values of γ = 0.3, A = 3.4, c = 1 for the Case (i) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: The deceleration parameter q(z) is shown against redshift z for different values of the constants γ = 0.3, A = 3.4, c = 1 for the Case (i). 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The energy density ρ is shown against redshift z for different values of the constants γ = 0.3, A = 3.4, c = 1 for the Case (ii) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: The Hubble parameter H(z) is shown against redshift z for different values of the constants γ = 0.3, A = 3.4, c = 1 for the Case (ii). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: The deceleration parameter q(z) is shown against redshift z for different values of the constants γ = 0.3, A = 3.4, c = 1 for the Case (ii) [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: The deceleration parameter q(z) vs redshift z in the contour plot for different values of γ = 0.3, A = 3.4, c = 1 for the Case (ii). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: The error bar plot of Hubble H versus z for the theoretical model (blue curve) and ΛCDM model (red dotted curve) where the blue dots depict the 30 points of the Hubble data [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: The figure shows the error bar for the Hubble parameter with the standard [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: The figure shows the distance modulus versus redshift [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]
Figure 13
Figure 13. Figure 13: Contour plot with 1-σ and 2-σ errors for the model parameters H0 and q along with the constraint values for CC datasets. The outcome of this computational operation is that we can determine the best fit values of the model parameters as H0 = 67.90 and n = 3.036 due to…
Figure 14
Figure 14. Figure 14: Contour plot with 1-σ and 2-σ errors for the model parameters H0 and q along with the constraint values for combined CC + SC datasets [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]

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