A Nonsingular Logarithmic Bouncing Cosmology in $f(R,T)$ Gravity with Thermodynamic Viability

Anjani; Milan Srivastava; Pankaj Kumar; S.H. Shekh

REVIEW 4 major objections 4 minor 88 references

A Nonsingular Logarithmic Bouncing Cosmology in $f(R,T)$ Gravity with Thermodynamic Viability

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The logarithmic scale factor $a(t)=m\log(\alpha+\beta t^2)$ in $f(R,T)=R+\lambda T$ gravity produces a nonsingular bouncing universe that is classically stable and thermodynamically viable for $-\frac12<\lambda<0$.

desk verdict A clean ansatz-based f(R,T) bounce whose own late-time equations contradict the claimed viability window -1/2<λ<0; worth a quick referee, but not a viable cosmology as written. read the letter →

arxiv 2608.11824 v1 pith:CEUHA5JV submitted 2026-08-12 gr-qc

Anjani , Pankaj Kumar , S.H. Shekh , Milan Srivastava This is my paper · ORCID

classification gr-qc MSC 83F0583D05 PACS 04.50.Kd98.80.-k

keywords bouncingcosmologyf(RT)gravitynonsingularlogarithmicscalefactorenergyconditionsgeneralizedsecondlawofthermodynamicsmatter-geometrycouplingsquaredspeedsound

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 claims that the scale factor ansatz $a(t)=m\log(\alpha+\beta t^2)$ in $f(R,T)=R+\lambda T$ gravity produces a nonsingular cosmological bounce: the scale factor stays finite at $t=0$, the Hubble parameter vanishes there with $\dot H(0)=2\beta/(\alpha\log\alpha)>0$, and the universe contracts for $t<0$ and expands for $t>0$. Within the coupling range $-\frac12<\lambda<0$, the paper reports positive energy density and negative pressure at the bounce, violation of the null and strong energy conditions only near the bounce with restoration away from it, a squared speed of sound that stays positive, and a generalized second law that holds in the expanding phase. The significance, if the claims hold, is that a concrete bouncing solution replaces the initial singularity in an established modified theory of gravity while keeping the matter–geometry coupling compatible with independent compact-object bounds.

What carries the argument

The engine of the model is the logarithmic scale-factor ansatz $a(t)=m\log(\alpha+\beta t^2)$ with $m,\beta>0$ and $\alpha>1$, together with the linear matter–geometry coupling $f(R,T)=R+\lambda T$. This ansatz does the geometric work: it forces $\dot a<0$ before $t=0$, $\dot a=0$ and $\ddot a=2m\beta/\alpha>0$ at the bounce, and $\dot a>0$ afterward, so the Hubble rate $H=2\beta t/[ (\alpha+\beta t^2)\log(\alpha+\beta t^2)]$ changes sign with positive slope at the transition. The coupling parameter $\lambda$ then controls every physical viability condition through the explicit density and pressure formulas, dictating where energy conditions are violated, where $V_s^2$ is positive, and where the generalized second law is satisfied.

What would settle it

Evaluate Eqs. (17) and (18) at a large positive time for any $\lambda$ in $(-\frac12,0)$, for instance $\lambda=-0.3$, $\alpha=2$, $\beta=1$: the energy density turns negative and the pressure positive once $\log(\alpha+\beta t^2)>2(4\lambda+3)/(-\lambda)$. This directly contradicts the claimed positive-density, negative-pressure expanding phase unless the density is reinterpreted as an effective quantity.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the simple analytic choice $a(t)=m\log(\alpha+\beta t^2)$ solves the Friedmann equations of linear $f(R,T)$ gravity with a finite minimum scale factor $a_{\min}=m\log\alpha$, a bounce at $t=0$ where $H=0$ and $\dot H>0$, and no singularity anywhere. Substituting the ansatz into the modified field equations yields closed-form energy density and pressure, Eqs. (17) and (18), whose signs at the bounce select the allowed coupling $-\frac12<\lambda<0$: this window gives $\rho(t_b)>0$, $p(t_b)<0$, and the required NEC and SEC violations, with positive squared sound speed $V_s^2$ near the bounce. The effective equation of state crosses the phantom divide around the bounce and tends toward a stiff-fluid value at late times. The entropy production rate is negative during contraction, positive during expansion, and singular at $t=0$, which the paper interprets as the breakdown of near-equilibrium thermodynamics during the transition. In the compact-object normalization $f(R,T)=R+2\lambda_p T$, the cosmological window becomes $-\frac14<\lambda_p<0$, which overlaps the published white-dwarf and neutron-star limits.

Load-bearing premise

The model's viability is certified by checking the energy density and pressure only at the instant of the bounce, on the unspoken assumption that their signs stay the same at all later times; the paper's own formulas give the opposite signs at late times for $-\frac12<\lambda<0$.

Editorial extensions

If this is right

The big-bang singularity of standard cosmology is replaced by a finite, smooth bounce in this model.
The matter–geometry coupling must be negative and small ($-\frac12<\lambda<0$) for a physically acceptable bounce; positive $\lambda$ is excluded because it makes the bounce density negative.
Null and strong energy conditions are violated only in a neighborhood of the bounce and are restored outside it, matching the standard picture of a successful bounce.
The model is classically stable in the allowed parameter region, since the squared speed of sound remains positive and can be kept subluminal.
The generalized second law of thermodynamics holds in the expanding phase, while the entropy production rate is negative during contraction and diverges at the bounce.

Reading between the lines

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

The viability window $-\frac12<\lambda<0$ is selected by evaluating $\rho$ and $p$ only at $t=0$; reading Eqs. (17)–(18) at arbitrary late times shows that for this same window $\rho$ becomes negative and $p$ positive once $\log(\alpha+\beta t^2)$ exceeds the threshold $2(4\lambda+3)/(-\lambda)$. A full-time positivity check is therefore a natural next test, and it appears to fail.
The divergence of $\dot S_{\rm total}$ at the bounce means the generalized second law is not defined at the transition itself; calling this a breakdown of near-equilibrium thermodynamics is an interpretation rather than a derivation, and a non-equilibrium entropy formulation would be needed to decide whether the bounce is thermodynamically well posed.
The comparison with compact-object bounds uses the normalization $\lambda=2\lambda_p$; the cosmological window $-\frac14<\lambda_p<0$ is much wider than the astrophysical intervals, so the stated compatibility rests on overlap, not on astrophysics singling out the bounce range.
A direct extension would impose $\rho\ge0$ and $p\le0$ for all $t$ and ask whether any coupling survives; if none does, the model would have to be reinterpreted as an effective-geometry description rather than a literal perfect-fluid cosmology.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read this one if you need a concrete example of why checking a bounce only at t=0 is not enough. The paper is a textbook f(R,T)=R+λT reconstruction: assume a(t)=m log(α+βt²), solve the modified Friedmann equations, then test energy conditions and sound speed. The logarithmic ansatz is new to the bouncing-cosmology literature as far as I can tell, and the algebra leading to rho and p is clean and repeatable. The paper also honestly flags the GSL divergence at the bounce. That part is fine.

What is actually new is the scale factor itself, plus the derived H, q, rho, p. The framework and the viability tests are standard for the subfield, which is okay; this is a reconstruction paper, not a derivation from microscopic physics.

The soft spots are not minor, though. The central viability claim collapses. The allowed window -1/2<λ<0 is fixed by requiring rho(t_b)>0 and p(t_b)<0 at the bounce. But for λ in that same window, Eqs. (17)-(18) at large t give rho<0 and p>0: in (17) the bracket is dominated by λβt² log(α+βt²)<0, and in (18) the bracket is dominated by -(3λ+2)βt² log(...)<0 with an outer minus sign. So the signs needed for a physically viable expanding phase do not persist. The paper's claims that w approaches 1 and that V_s² is positive at late times are also wrong; both (19) and (24) tend to (3λ+2)/λ, which is negative in this window. The plots look safe only because the chosen times are not large enough for the log terms to dominate. Since the viability constraints are derived from bounce-point values and then applied to all t, the reasoning is unsound.

The stress-test note is correct; I checked the signs independently. This is not a marginal issue, it attacks the paper's main conclusion. On circularity: the bounce is put in by hand, which is standard for this kind of reconstruction, so I would not count that against the paper by itself. The problem is the unstated assumption that bounce-point signs persist.

Who is this for? Someone cataloguing ansatz-based bounces in f(R,T), or teaching why late-time asymptotics need checking before declaring a model viable. For a serious journal, I would not desk-reject—the calculation is coherent and the claim is crisp enough that a referee can settle it quickly. But the expected outcome is reject unless the authors restrict the time domain, allow negative effective density with explicit justification, or revise the viability claims. Worth reading as a cautionary example, not worth citing as a viable model.

Referee Report

4 major / 4 minor

Summary. The paper constructs a spatially flat FLRW bouncing model in f(R,T)=R+λT gravity using an assumed logarithmic scale factor a(t)=m log(α+βt^2). It derives H, q, ρ, p, the effective equation-of-state parameter, energy conditions, the squared sound speed, and the total entropy production rate. Imposing positive energy density and negative pressure at the bounce restricts the coupling to -1/2<λ<0, and on this basis the paper claims a classically stable, thermodynamically viable bounce, with the coupling range compared to compact-object constraints.

Significance. The construction is a direct ansatz-based exercise, and the algebra leading from the field equations to Eqs. (17)-(18) is internally consistent. The paper also correctly identifies the need for NEC violation near the bounce and attempts to test the model against energy conditions, stability, the generalized second law, and astrophysical bounds. However, the central viability claim of a physically acceptable expanding phase is not supported by the derived expressions: the bounce-point sign analysis is local, and the same equations produce opposite signs at late times in the claimed allowed window. If the model were viable, it would be a concrete new bouncing solution in a well-studied modified-gravity framework, but the presented evidence does not establish that.

major comments (4)

[Section 5.1, Eqs. (17)-(18)] The allowed window -1/2<λ<0 is fixed by requiring ρ(t_b)>0 and p(t_b)<0 at the bounce, but the text also asserts that the energy density must remain positive throughout the evolution. This assertion is false: with L=log(α+βt^2), as t→∞ Eq. (17) gives ρ ~ 2λ/[(λ+1)(2λ+1)t^2L] < 0, and Eq. (18) gives p ~ 2(3λ+2)/[(λ+1)(2λ+1)t^2L] > 0 for -1/2<λ<0. The post-bounce expanding phase therefore has negative energy density and positive pressure, directly contradicting the abstract's and Section 9's claim of a viable expanding phase.
[Section 5.2, Eq. (19) and following sentence] The statement that the EoS parameter eventually approaches w≈1 is not correct in the constrained window. The late-time limit of Eq. (19) is (3λ+2)/λ, which is negative whenever -1/2<λ<0 because λ<0 and 3λ+2>0. Thus the claimed stiff-matter late-time phase is not realized for the model's own allowed parameters.
[Section 5.4, Eq. (24)] The stability claim that V_s^2>0 during late-time evolution is contradicted by the formula. For t→∞, Eq. (24) tends to (3λ+2)/λ < 0 in the allowed window, so the model is classically unstable at late times by the paper's own stability criterion. The bounce-point expression Eq. (25) is local and does not establish global stability.
[Sections 5.1 and 5.3] The manuscript does not state whether ρ and p in Eqs. (17)-(18) are physical fluid quantities or effective quantities arising from the matter-geometry coupling. If they are physical, the late-time sign changes violate the weak energy condition and make the fluid unacceptable; if they are effective, the energy-condition analysis and the comparison with compact-object constraints in Section 7 require a different interpretation and should be stated explicitly. Either way, the summary in Section 5.3 that all required conditions are satisfied in -1/2<λ<0 is inaccurate.

minor comments (4)

[Throughout] There are numerous typographical and grammatical errors that should be corrected, including 'Type la supernova' for 'Type Ia supernova', 'anstaz' for 'ansatz', and 'comic fluid' for 'cosmic fluid'.
[Figure 3] Figure 3 includes λ=-0.55, which lies outside the claimed allowed interval -1/2<λ<0 and would give ρ(t_b)<0; the figure and the parameter constraints in the text should be reconciled.
[Section 6, Eqs. (27)-(30)] The quantities S_in and S_prod are introduced but never separately defined, so the decomposition in Eq. (27) and the use of the Gibbs equation in Eq. (30) are not fully transparent.
[References] References [35] and [37] duplicate the same source, and several citations in the introduction are not clearly tied to the specific claims they support; the bibliography should be checked for duplicates and relevance.

Circularity Check

0 steps flagged · score 0.0 of 10

No constructional circularity: the scale factor is an explicit ansatz and the lambda window is a consistency constraint, not a fitted prediction.

full rationale

No load-bearing circular step is established. The paper is transparent that the logarithmic scale factor is an assumption: it says "A logarithmic time-dependent scale factor is assumed to realize a smooth transition" and "we introduce a logarithmic form of scale factor as, a(t)=m log(alpha+beta t^2)" (Eq. 14). The subsequent Hubble, deceleration, density, pressure, energy-condition, sound-speed, and entropy expressions are algebraic consequences of this ansatz together with the f(R,T) field equations (Eqs. 11-13). That is standard reconstruction from an explicitly stated ansatz, not a hidden identification of output with input. The window -1/2<lambda<0 is derived in Sec. 5.1 by imposing rho(t_b)>0 and p(t_b)<0 at the bounce; the paper presents this as a parameter constraint ("the conditions ... are required to observe rho(t_b)>0"), not as a prediction, so it is not a fitted input called a prediction. There is no load-bearing self-citation, no imported uniqueness theorem, and no ansatz smuggled in via citation; the compact-object comparison is external and independent, with an explicit normalization conversion. The serious problems in the paper are correctness and internal-consistency failures rather than circularity: the bounce-point sign analysis is extrapolated to the whole evolution, while in the late-time limit Eqs. (17)-(18) give rho<0 and p>0 for -1/2<lambda<0 and Eq. (24) gives V_s^2<0; also the paper itself notes that the entropy production rate diverges at the bounce and that the GSL cannot be strictly defined there. These are substantive scientific objections, but they do not make the derivation equivalent to its inputs by construction. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

Everything in the paper follows from the chosen scale factor and the linear f(R,T) model. The free parameters alpha, beta, and m set the shape of the bounce; lambda is fixed by demanding the desired signs of rho and p and the energy conditions, which makes the 'constraint' an output of the requirements rather than a prediction. No new particles, fields, or forces are introduced.

free parameters (4)

m (scale factor amplitude) = >0, unconstrained (cancels in H)
Appears in the ansatz (14) but cancels in H(t); no observational constraint is derived.
alpha (bounce depth) = >1, chosen per figure (1.1 to 6.2)
Controls depth and position of the bounce; values are selected to make plots, with no data fit.
beta (bounce sharpness) = >0, chosen per figure (0.001 to 5)
Controls sharpness; selected for plots and to keep the sound speed positive.
lambda (matter-geometry coupling) = -1/2 < lambda < 0
Constraint obtained by requiring rho>0, p<0, and NEC/SEC violation at the bounce, not from independent data; the same range makes rho negative at late times.

assumptions (6)

domain assumption Linear form f(R,T)=R+lambda T
Assumed in Sec. 2 (Eqs. 6 and 11); all field equations depend on it.
domain assumption Spatially flat FLRW metric
Line element Eq. (8); homogeneity and isotropy are assumed.
domain assumption Perfect fluid with matter Lagrangian L_m=-p
Used in Sec. 2 to compute Theta_mu nu and T=rho-3p; affects all density and pressure expressions.
ad hoc to paper Logarithmic scale factor ansatz
Eq. (14) is chosen so that dot a(0)=0 and ddot a(0)>0, so the bounce is an input rather than a consequence of the dynamics.
domain assumption Sound speed V_s^2 = dot p / dot rho is a valid stability diagnostic
Eq. (24); the paper does not perform a full cosmological perturbation analysis.
domain assumption Hayward-Kodama temperature and the GSL entropy formulas hold in f(R,T) gravity
Eqs. (28)-(33) are taken from the literature; their validity near H=0 is not established because the entropy rate diverges at the bounce.

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Pith. "Pith review of A Nonsingular Logarithmic Bouncing Cosmology in $f(R,T)$ Gravity with Thermodynamic Viability." pith.science (2026). https://pith.science/paper/CEUHA5JV

@misc{pith2026260811824,
  author       = {Pith},
  title        = {Pith review of: A Nonsingular Logarithmic Bouncing Cosmology in $f(R,T)$ Gravity with Thermodynamic Viability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CEUHA5JV}},
  note         = {Machine review of arXiv:2608.11824}
}

abstract

We present a nonsingular bouncing cosmological model in the framework of modified $f(R,T)$ gravity within a spatially flat Friedmann--Robertson--Walker universe. A logarithmic time-dependent scale factor is assumed to realize a smooth transition from a contracting phase to an expanding phase without encountering an initial singularity. Based on this assumption, the dynamical evolution of the Hubble parameter, deceleration parameter, energy density, and pressure is obtained for various choices of the model and the matter--geometry coupling parameter to confirm the occurrence of a successful bounce. The effective equation of state parameter is examined to characterize the cosmic fluid during different evolutionary phases. The violation of energy conditions, necessary for the realization of the bouncing behavior, is also discussed. The stability of the model is investigated using the squared speed of sound and is found to remain positive within the allowed parameter space, indicating classical stability. Furthermore, constraints on the matter--geometry coupling parameter are obtained by demanding positive energy density, negative pressure, and a viable cosmological evolution. The obtained cosmological constraint on the coupling parameter is also shown to be compatible with the currently available compact-object constraints. The thermodynamic behavior of the model is examined by testing the generalized second law of thermodynamics. The total entropy production rate remains negative during the contracting phase and changes its sign to positive during the expanding phase. However, it becomes singular at the bouncing point, reflecting the breakdown of the standard thermodynamic description during the transition phase.

Figures

Figures reproduced from arXiv: 2608.11824 by the authors.

**Figure 2.** FIG. 2. The plot of evolution of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. The plot of evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. The plot of evolution of pressure with cosmic time for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. The plot of evolution of the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 8.** Figure 8: FIG. 8. The plot of stability analysis as speed of sound [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 7.** Figure 7: FIG. 7. The plot of SEC ( [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 9.** Figure 9: FIG. 9. The plot of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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Works this paper leans on

88 extracted references · 69 canonical work pages

[1]

A Nonsingular Logarithmic Bouncing Cosmology in $f(R,T)$ Gravity with Thermodynamic Viability

INTRODUCTION General Relativity (GR) is widely regarded as the most successful theory of gravitation, providing a fun- damental geometrical description of the gravitational phenomena in spacetime [1, 2]. However, observations at both galactic and cosmological scales reveal signifi- cant discrepancies with GR predictions. In particular, the flat rotation c...

work page Pith review arXiv 2026
[2]

The EoS parameterw=p/ρexhibits a phantom crossing near the bouncing epoch, allowing a tempo- rary violation of the null energy condition

Positivity of the ED and negative pressure necessary to avoid gravitational collapse at the bounce commonly requires− 1 2 <λ<0. The EoS parameterw=p/ρexhibits a phantom crossing near the bouncing epoch, allowing a tempo- rary violation of the null energy condition. At the bouncew b = 3λ+2 λ , showing explicit dependence on the coupling parameter. After th...

work page
[3]

According to GR, the big-bang corresponds to the singular point where the scale factor vanishes, and physical quantities such as energy density, pres- sure and curvature diverge

BOUNCING COSMOLOGY The fascinating questions, what existed before the big- bang? and what preceded time itself, if big-bang marks the beginning of time?, lie at the heart of modern cos- mology. According to GR, the big-bang corresponds to the singular point where the scale factor vanishes, and physical quantities such as energy density, pres- sure and cur...
[4]

Each of these is characterized by distinct phys- ical and mathematical conditions

PARAMETERIZATION OF SCALE FACTOR Introduced through Friedmann field equations in Ein- stein’s GR equation, the scale factora(t)plays a pivotal role in distinguishing among various cosmological sce- narios. Each of these is characterized by distinct phys- ical and mathematical conditions. Its time-dependent nature encodes different cosmic phases of the uni...
[5]

Energy Density (ED) and Pressure TheEDand pressure of the universe are regulated by the matter, radiation and dark energy

DYNAMICAL ANALYSIS OF THE PARAMETERS AND THEIR PHYSICAL INTERPRETATION 5.1. Energy Density (ED) and Pressure TheEDand pressure of the universe are regulated by the matter, radiation and dark energy. From eq (13), we calculated the expressions ofEDand pressure as ρ= 2β[2βt 2(4λ+3) +λ(−α+βt 2)log(α+βt 2)] (λ+1)(2λ+1)[(α+βt 2)log(α+βt 2)]2 (17) p=− 2β[2βt 2 ...
[6]

The positivity ofEDand the requirement of negative pressure impose constraints on coupling parameterλ

It is important to note that whenλ falls in− 1 2 <λ<0, both the requirementρ(t b)>0 and p(tb)<0 are satisfied. The positivity ofEDand the requirement of negative pressure impose constraints on coupling parameterλ. Inf(R,T)gravity, the known feature of matter and ge- ometry coupling modifies the effectiveEDand pressure differently because both contain diff...
[7]

At the bouncing pointt=0, the null energy condition reduces to ρ+p=− 4β α(λ+1)logα

are satisfied outside the bouncing regime which indi- cates the restoration of energy conditions. At the bouncing pointt=0, the null energy condition reduces to ρ+p=− 4β α(λ+1)logα . (22) Sinceβ>0 andα>1, the sign of(ρ+p)is deter- mined entirely by the factor(λ+1). TheNECviolation (ρ+p<0), therefore requires 1+λ>0⇒λ> −1. This shows that the coupling param...
[8]

These perturbations can be gener- ated by gravitational effects, compression of baryonic matter and quantum fluctuations in the comic fluid

which makes the cosmological solutions sensitive to small peturbations. These perturbations can be gener- ated by gravitational effects, compression of baryonic matter and quantum fluctuations in the comic fluid. In this work, we focus on the speed of sound as a diag- nostic for stability [47]. The sound speed determines how small perturbations in pressur...

Show all 88 references

[9]

A deep connection be- tween gravity and thermodynamics was first revealed in by suggesting that gravity may have a thermodynamic origin [71]

THERMODYNAMICS In this section, we discuss the thermodynamic be- haviour of our bouncing model. A deep connection be- tween gravity and thermodynamics was first revealed in by suggesting that gravity may have a thermodynamic origin [71]. This idea is further strengthened by Pa...
[10]

This indicates that the GSL cannot be strictly de- fined at the exact moment of the bounce. This behav- ior can be physically understood as a consequence of the highly non-equilibrium nature of the bounce, where rapid changes in the Hubble parameter and the effective thermodyn...
[11]

CONSISTENCY WITH COMPACT-OBJECT CONSTRAINTS The matter–geometry coupling parameter plays a fundamental role in the linearf(R,T)gravity model and therefore its allowed range should be compatible with both cosmological and astrophysical observations. While the present work const...
[12]

DISCUSSION OF RESULTS In this section, we present a comprehensive discus- sion of the cosmological and thermodynamic implica- tions of the logarithmic bouncing model constructed within the framework off(R,T) =R+λTgravity. The analysis is organized by examining the behavior of ...
[13]

CONCLUSION In this work, a non-singular bouncing cosmologi- cal model has been developed within the framework off(R,T) =R+λTgravity by adopting a logarith- mic form of the scale factor. The proposed model suc- cessfully replaces the initial big-bang singularity with a smooth a...
[14]

Einstein, Ann

A. Einstein, Ann. Phys. 49 (7) (1916) 769–822

1916
[15]

C. M. Will, Living Rev. Relativity 17, 4 (2014)

2014
[16]

A. G. Riess et al., Astron. J. 116, 1009 (1998)

1998
[17]

Perlmutter et al., Astrophys

S. Perlmutter et al., Astrophys. J.517, 565 (1999)

1999
[18]

D. J. Eisenstein et al., Astrophys. J.633, 560 (2005)

2005
[19]

Tegmark et al., Phys

M. Tegmark et al., Phys. Rev. D69, 103501 (2004)

2004
[20]

D. N. Spergel et al., Astrophys. J. Suppl.170, 377 (2007)

2007
[21]

N. A. et al., Astronomy and Astrophysics 641 (2020)

2020
[22]

DESI Collaboration, Abbott et al., Astrophys. J. Lett.973, L14 (2024)

2024
[23]

Abdul-Karim, J et al., arXiv e- prints (2025)

DESI Collaboration, M. Abdul-Karim, J et al., arXiv e- prints (2025)

2025
[24]

S. M. Carroll, Living Rev. Relativity 4, 1 (2001)

2001
[25]

Weinberg, Springer, Berlin, p

S. Weinberg, Springer, Berlin, p. 18 (2001); astro- ph/0005265

2001
[26]

Sahni, A

V . Sahni, A. Shafieloo and A. A. Starobinsky, Astrophys. J. Lett.793, L40 (2014)

2014
[27]

H. A. Buchdahl, Mon. Not. Roy. Astron. Soc.150, 1 (1970)

1970
[28]

J. P . Duruisseau, R. Kerner and P . Eysseric,Gen. Rel. Grav. 15, 797 (1983)

1983
[29]

Kleinert and H.-J

H. Kleinert and H.-J. Schmidt, Gen. Rel. Grav.34, 1295 (2002)

2002
[30]

Bamba, S

K. Bamba, S. Nojiri, S. D. Odintsov and M. Sasaki,Phys. Rev. D77, 123532 (2008)

2008
[31]

Bamba, S

K. Bamba, S. D. Odintsov, L. Sebastiani and S. Zerbini, Eur. Phys. J. C67, 295 (2010)

2010
[32]

S. D. Odintsov, V . K. Oikonomou and K. Bamba, Eur. Phys. J. C79, 708 (2019)

2019
[33]

B. Li, T. P . Sotiriou and J. D. Barrow, Phys. Rev. D83, 104017 (2011)

2011
[34]

Jensko, Class

E. Jensko, Class. Quantum Grav.42, 055011 (2025)

2025
[35]

Heisenberg, Phys

L. Heisenberg, Phys. Rep.1066, 1 (2024)

2024
[36]

Cai, S.-H

Y.-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta and E. N. Sari- dakis, Class. Quantum Grav.28, 215011 (2011)

2011
[37]

F. Xu, Y. Guo and Z. H. Zhu, Eur. Phys. J. C79, 708 (2019)

2019
[38]

Agrawal, L

A. Agrawal, L. Pati, S. Tripathy and B. Mishra,Phys. Dark Univ.33, 100863 (2021)

2021
[39]

C. P . Singh and P . Kumar, Eur. Phys. J. C74, 3070 (2014)

2014
[40]

S. W. Hawking, R. Penrose, Proc. R. Soc. Lond. A314, 529 (1970)

1970
[41]

Borde, A

A. Borde, A. H. Guth, and A. Vilenkin, Phys. Rev. Lett.90, 151301 (2003)

2003
[42]

A. R. Liddle, P . Parsons, and J. D. Barrow, Phys. Rev. D50, 7222 (1994)

1994
[43]

P . J. Steinhardt and N. Turok, Phys. Rev. D65, 126003 (2002)

2002
[44]

Lohakare, S

S. Lohakare, S. D. Odintsov and S. K. Maurya, Universe 8, 636 (2022)

2022
[45]

Ijjas and P

A. Ijjas and P . J. Steinhardt, Phys. Lett. B747, 191 (2015)

2015
[46]

Haro et al.,Phys

J. Haro et al.,Phys. Rev. D95, 123507 (2017)

2017
[47]

S. D. Odintsov and V . K. Oikonomou, Int. J. Geom. Meth. Mod. Phys.15, 1850165 (2018)

2018
[48]

Novello and S

M. Novello and S. E. Perez Bergliaffa,Phys. Rept.463, 127 (2008)

2008
[49]

Raychaudhuri, Phys

A. Raychaudhuri, Phys. Rev.98, 1123 (1955)

1955
[50]

Novello and S

M. Novello and S. E. Perez Bergliaffa, Bouncing Cosmolo- gies, Phys. Rept.463, 127 (2008)

2008
[51]

P . J. E. Peebles and B. Ratra, Astrophys. J. Lett.325, L17 (1988)

1988
[52]

Ratra and P

B. Ratra and P . J. E. Peebles, Phys. Rev. D37, 3406 (1988)

1988
[53]

Armendariz-Picon, V

C. Armendariz-Picon, V . F. Mukhanov, and P . J. Stein- hardt, Phys. Rev. Lett.85, 4438 (2000)

2000
[54]

R. R. Caldwell, Phys. Lett. B545, 23 (2002)

2002
[55]

Nojiri, S

S. Nojiri, S. D. Odintsov and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005)

2005
[56]

Kamenshchik et al.,Phys. Lett. B511, 265 (2001)

2001
[57]

Copeland, M

E.J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D15, 1753 (2006)

2006
[58]

Perivolaropoulos et al., New Astron

L. Perivolaropoulos et al., New Astron. Rev.95, 101659 (2022); arXiv:2105.05208

2022 arXiv
[59]

Nojiri, S

S. Nojiri, S. D. Odintsov and V . K. Oikonomou, Phys. Rept.692, 1 (2017)

2017
[60]

Gsponer and J

S. Gsponer and J. Noller, Phys. Rev. D105, 064002 (2022)

2022
[61]

Clifton, P

T. Clifton, P . G. Ferreira, A. Padilla and C. Skordis, Phys. Rept.513, 1 (2012)

2012
[62]

Harko, F

T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, Phys. Rev. D84, 024020 (2011)

2011
[63]

F. G. Alvarenga et al., Phys. Rev. D87, 103526 (2013)

2013
[64]

Bouli, et al., Mon

A. Bouli, et al., Mon. Not. R. Astron. Soc.526, 4192 (2023)

2023
[65]

Shabani and F

H. Shabani and F. Ziaie, Phys. Rev. D95, 123541 (2017)

2017
[66]

M. J. S. Houndjo et al.,Int. J. Mod. Phys. D26, 1750106 (2017)

2017
[67]

M. J. S. Houndjo, O. F. Piattella, and A. V . Kpadonou,Eur. Phys. J. C78, 368 (2018)

2018
[68]

Sharif and M

M. Sharif and M. Zubair, J. Cosmol. Astropart. Phys.03, 028 (2012)

2012
[69]

J. K. Singh, K. Bamba, R. Nagpal, and S. K. J. Pacif, Phys. Rev. D97, 123536 (2018) 16

2018
[70]

Agrawal, L

A. Agrawal, L. Pati, S. Tripathy, and B. Mishra, Phys. Dark Univ.33, 100863 (2021)

2021
[71]

Shabani and F

H. Shabani and F. Ziaie, Phys. Rev. D99, 123525 (2019)

2019
[72]

P . K. Sahoo, S. Bhattacharjee, S. Tripathy, and P . Sahoo, Eur. Phys. J. C80, 1123 (2020)

2020
[73]

Bhattacharjee and P

S. Bhattacharjee and P . K. Sahoo, Int. J. Geom. Meth. Mod. Phys.17, 2050181 (2020)

2020
[74]

J. K. Singh, K. Bamba, et al., Eur. Phys. J. C83, 215 (2023)

2023
[75]

Devi and P

K. Devi and P . Kumar, Eur. Phys. J. C81, 631 (2021)

2021
[76]

Pradhan, G

A. Pradhan, G. Goswami, R. Rani, and A. Bee- sham,arXiv:2210.15433

arXiv
[77]

Singh, V

C.P . Singh, V . Singh, Gen. Relativ. Gravit. 46, 1696 (2014)

2014
[78]

Nagpal, J.K

R. Nagpal, J.K. Singh, S.K.J. Pacif, Int. J. Geom. Meth. Mod. Phys. 16, 1950046 (2019)

2019
[79]

Harko and F

T. Harko and F. S. N. Lobo,Generalized curvature-matter couplings in modified gravity,Universe, MDPI (2017)

2017
[80]

Singh, S

V . Singh, S. Jokweni, A. Beesham, Universe 10, 272 (2024)

2024
[81]

Tiwari, A

R.K. Tiwari, A. Beesham, A. Pradhan, Gravitation and Cosmology 23, 176114 (2017)

2017
[82]

Y.-F. Cai, D. A. Easson and R. Brandenberger, JCAP08, 020 (2012)

2012
[83]

Momeni, P .H.R.S

D. Momeni, P .H.R.S. Moraes, R. Myrzakulov, Astrophys. Space Sci. 361, 228 (2016)

2016
[84]

Jacobson, Phys

T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995)

1995
[85]

Padmanabhan, Class

T. Padmanabhan, Class. Quantum Grav. 19, 5387 (2002)

2002
[86]

J. D. Barrow and A. C. Ottewill, J. Phys. A: Math. Gen.16, 2757 (1983)

1983
[87]

Carvalho et al., Eur

G.A. Carvalho et al., Eur. Phys. J. C 77, 871 (2017)

2017
[88]

Lobato et al., J

R. Lobato et al., J. Cosmol. Astropart. Phys.12, 039 (2020)

2020

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

A Nonsingular Logarithmic Bouncing Cosmology in $f(R,T)$ Gravity with Thermodynamic Viability

INTRODUCTION General Relativity (GR) is widely regarded as the most successful theory of gravitation, providing a fun- damental geometrical description of the gravitational phenomena in spacetime [1, 2]. However, observations at both galactic and cosmological scales reveal signifi- cant discrepancies with GR predictions. In particular, the flat rotation c...

work page Pith review arXiv 2026

[2] [2]

The EoS parameterw=p/ρexhibits a phantom crossing near the bouncing epoch, allowing a tempo- rary violation of the null energy condition

Positivity of the ED and negative pressure necessary to avoid gravitational collapse at the bounce commonly requires− 1 2 <λ<0. The EoS parameterw=p/ρexhibits a phantom crossing near the bouncing epoch, allowing a tempo- rary violation of the null energy condition. At the bouncew b = 3λ+2 λ , showing explicit dependence on the coupling parameter. After th...

work page

[3] [3]

According to GR, the big-bang corresponds to the singular point where the scale factor vanishes, and physical quantities such as energy density, pres- sure and curvature diverge

BOUNCING COSMOLOGY The fascinating questions, what existed before the big- bang? and what preceded time itself, if big-bang marks the beginning of time?, lie at the heart of modern cos- mology. According to GR, the big-bang corresponds to the singular point where the scale factor vanishes, and physical quantities such as energy density, pres- sure and cur...

[4] [4]

Each of these is characterized by distinct phys- ical and mathematical conditions

PARAMETERIZATION OF SCALE FACTOR Introduced through Friedmann field equations in Ein- stein’s GR equation, the scale factora(t)plays a pivotal role in distinguishing among various cosmological sce- narios. Each of these is characterized by distinct phys- ical and mathematical conditions. Its time-dependent nature encodes different cosmic phases of the uni...

[5] [5]

Energy Density (ED) and Pressure TheEDand pressure of the universe are regulated by the matter, radiation and dark energy

DYNAMICAL ANALYSIS OF THE PARAMETERS AND THEIR PHYSICAL INTERPRETATION 5.1. Energy Density (ED) and Pressure TheEDand pressure of the universe are regulated by the matter, radiation and dark energy. From eq (13), we calculated the expressions ofEDand pressure as ρ= 2β[2βt 2(4λ+3) +λ(−α+βt 2)log(α+βt 2)] (λ+1)(2λ+1)[(α+βt 2)log(α+βt 2)]2 (17) p=− 2β[2βt 2 ...

[6] [6]

The positivity ofEDand the requirement of negative pressure impose constraints on coupling parameterλ

It is important to note that whenλ falls in− 1 2 <λ<0, both the requirementρ(t b)>0 and p(tb)<0 are satisfied. The positivity ofEDand the requirement of negative pressure impose constraints on coupling parameterλ. Inf(R,T)gravity, the known feature of matter and ge- ometry coupling modifies the effectiveEDand pressure differently because both contain diff...

[7] [7]

At the bouncing pointt=0, the null energy condition reduces to ρ+p=− 4β α(λ+1)logα

are satisfied outside the bouncing regime which indi- cates the restoration of energy conditions. At the bouncing pointt=0, the null energy condition reduces to ρ+p=− 4β α(λ+1)logα . (22) Sinceβ>0 andα>1, the sign of(ρ+p)is deter- mined entirely by the factor(λ+1). TheNECviolation (ρ+p<0), therefore requires 1+λ>0⇒λ> −1. This shows that the coupling param...

[8] [8]

These perturbations can be gener- ated by gravitational effects, compression of baryonic matter and quantum fluctuations in the comic fluid

which makes the cosmological solutions sensitive to small peturbations. These perturbations can be gener- ated by gravitational effects, compression of baryonic matter and quantum fluctuations in the comic fluid. In this work, we focus on the speed of sound as a diag- nostic for stability [47]. The sound speed determines how small perturbations in pressur...