Bounce, Turnaround, and the Anisotropy Problem in Cyclic Cosmology on a Brane with a Timelike Extra Dimension

Arkajit Aich; Kaushik Bhattacharya; Rikpratik Sengupta

REVIEW 3 major objections 4 minor 55 references

Bounce, Turnaround, and the Anisotropy Problem in Cyclic Cosmology on a Brane with a Timelike Extra Dimension

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A single scalar rolling at a constant rate on a timelike-extra-dimension brane can produce smooth bounces, cyclic evolution, and CMB-compatible inflation.

desk verdict A cyclic braneworld whose bounce, periodicity, and shear suppression are all fixed by two undereived ansatze; the CMB 'agreement' is parameter fitting dressed as prediction. read the letter →

arxiv 2602.08974 v2 pith:KKM565TP submitted 2026-02-09 gr-qc

Rikpratik Sengupta , Arkajit Aich , Kaushik Bhattacharya This is my paper · ORCID

classification gr-qc MSC 83F0583E15 PACS 98.80.Cq04.50.-h

keywords braneworldcosmologytimelikeextradimensioncyclicuniversecosmologicalbounceBianchiIanisotropyuniformrateinflationprimordialperturbationsShtanov-Sahnibrane

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 aims to show that one scalar field rolling at a constant rate can drive inflation on an anisotropic braneworld whose extra dimension is timelike, and that the same setup replaces the Big Bang singularity with a smooth, repeated bounce. In the Shtanov-Sahni version of this idea, the effective Friedmann equation acquires a negative energy-density correction that caps the density at a critical value, so a contracting universe can turn around into expansion. The authors treat shear as a stiff fluid, derive closed-form background solutions in which the shear field shrinks near the bounce, and then use the δN formalism to compute the perturbation spectra. Their CMB-normalized parameters reproduce the observed scalar tilt (n_s = 0.965900) and a very small tensor-to-scalar ratio (r ≈ 10⁻⁶). If the construction holds, cyclic cosmology and single-field inflation can coexist without exotic matter fields.

What carries the argument

The machinery is the uniform-rate condition φ̇ = −λ, which turns the Klein-Gordon equation into V′(φ) = 3Hλ, together with the timelike-extra-dimension Friedmann relation H² = (ρ/3)(1 − ρ/ρ_c), where ρ_c = 2|σ| is the brane critical density. Bianchi-I shear enters as a stiff fluid with σ_{αβ}σ^{αβ} = 6Σ²/a⁶, represented by a massless scalar φ_a; the system is closed by two posited relations between φ̇_a² and the inflaton potential. These choices make V(φ) sinusoidal, ρ(φ) = ρ_c sin²(...), and H(φ) = √(ρ_c/12) sin(...), so every background quantity is periodic and known in closed form. The δN formalism then converts that background into predictions for the scalar power spectrum P_R, the tilt

What would settle it

Derive the evolution of φ_a from the full Bianchi-I junction conditions on the timelike-extra-dimension brane without imposing the two ansätze; if the resulting φ̇_a² does not match either assumed form, the periodic solution is an artifact rather than a prediction. Observationally, a CMB measurement of n_s far from 0.9659 or r above 10⁻⁶ at pivot scales would rule out the stated parameter point, as would detecting a blue-tilted scalar spectrum sourced by the contracting phase.

Watch

Extended reading notes

Core claim

On the flat, dark-radiation-free branch of the anisotropic Shtanov-Sahni brane, the uniform-rate condition φ̇ = −λ plus two assumed forms for the shear-field kinetic term yields a periodic potential and a periodic Hubble parameter. The energy density caps at ρ_c = 2|σ|, and the scale factor oscillates between finite minima: smooth non-singular bounces, with low-density turnarounds returning the universe to contraction. Shear, a stiff fluid with σ_{αβ}σ^{αβ} = 6Σ²/a⁶, would grow in naive contraction, but the timelike-extra-dimension corrections suppress it near t = 0, avoiding the Mixmaster/BKL instability. A general turning-point analysis says bounce requires the negative high-energy correct

Load-bearing premise

The periodic potential, bounded density, and smooth bounce all follow from two posited relations (Eqs. 23–25 and 58–60) between the shear field's kinetic term and the inflaton potential; if those relations are not consequences of the Bianchi-I brane dynamics, the central cyclic-bounce claim does not stand.

Editorial extensions

If this is right

If correct, a non-singular early universe can be obtained without phantom or ekpyrotic matter; the timelike extra dimension itself supplies the bounce and the shear damping.
The closed-form periodic background makes the model highly tractable: the potential, density, Hubble rate, and scale factor are all known exactly, easing perturbation-theory calculations.
The CMB-normalized parameter set gives n_s = 0.965900 and r ≈ 10⁻⁶, consistent with current bounds, so the model is observationally viable at the two-point level.
Because the shear field freezes near the bounce and is then diluted by inflation, the model avoids the Mixmaster/BKL instability that otherwise tends to destroy bounces in anisotropic contraction.
The two cases quantify the anisotropy problem: sustaining a long weak-shear cyclic phase compatible with observations forces the shear amplitude far below H_*², with the required suppression depending on the pivot density fraction.

Reading between the lines

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

Because the two ansätze are posited rather than derived, the most direct extension of this work is to derive φ̇_a² from the Bianchi-I junction conditions; that derivation would turn the conditional bounce into a theorem.
The δN computation counts only modes that exit during inflation; a full treatment of perturbations through the contracting phase and the bounce could alter the predicted spectra, so a transfer-function calculation is a natural testable extension.
The near-identical values of λ and ρ_c in Cases 1 and 2 suggest a degeneracy: two-point CMB data fix the background dynamics but leave the anisotropy sector weakly constrained, so higher-order statistics such as non-Gaussianity could discriminate between the two cases.
The predicted tensor-to-scalar ratio is tiny (≈10⁻⁶); a future CMB experiment that measures r at the 10⁻³ level or above would falsify this particular parameter point.

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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

Short version: the central mechanism is an assumption, not a result. The two ansatze in Sections 2.1 and 2.2 — Eqs. (24) and (59) — are posited, not derived. They are what produce the sinusoidal potential, the bounded density, the periodic Hubble parameter, and the claimed suppression of the effective shear field near the bounce. Absent an independent derivation from the Bianchi-I brane equations or bulk dynamics, the timelike extra dimension isn't doing the work; the authors are. The paper gives some credit where it's due: the analytic background treatment of uniform-rate inflation on a Bianchi-I Shtanov–Sahni brane is genuinely new, and the δN computation is worked through in enough detail to check. The prose is readable and the relevant brane literature is cited. But the soft spots are load-bearing, not cosmetic. The two ansatze are asserted with "we assume," and everything that makes the model cyclic — the bounce, the periodicity, the shear freeze-out — follows from those choices. Then the CMB normalization is imposed: ζ is set to 4.9×10^-5 Mpc^-1, λ and ρ_c are hand-picked, and Σ is tuned to match. The resulting n_s and r are then presented as agreement with Planck. That is fitting, not predicting. There are also internal consistency problems: ζ is assigned units Mpc^-1 even though the curvature perturbation should be dimensionless; the Hubble parameter in Eq. (27) misses the λ factor that should come from φdot = -λ, and also appears to miss a cos factor from the H^2 law; consequently P_R has the wrong dimensions. The paper's equations don't all talk to each other. On top of that, the arXiv abstract promises a general turning-point classification and a discussion of the anisotropy problem, but the full text contains neither. That mismatch inflates the apparent scope. Bottom line: this is a nice example of how easy it is to get a bounce by choosing the anisotropy by hand. It deserves a serious referee because the algebra is extensive and the flaws are checkable — a referee can point to the exact equations. But the referee letter should say plainly that the central physical claim is not established. I wouldn't cite it as a working model; I might mention it as a cautionary example in a footnote.

Referee Report

3 major / 4 minor

Summary. The manuscript constructs a cyclic, non-singular cosmological model on an anisotropic Bianchi-I Shtanov–Sahni braneworld with a timelike extra dimension. A scalar inflaton rolls at constant rate dotphi=-lambda; anisotropy is encoded in a second scalar phi_a with vanishing potential. Two ansatze for dotphi_a^2 (Cases 1 and 2, Eqs. 24 and 59) lead to sinusoidal inflaton potentials, periodic H(t), repeated bounces at rho=rho_c, and the claimed suppression of phi_a at the bounce. The delta-N formalism is used to compute P_R, P_T, n_s, r, and a CMB-normalized parameter set is claimed to give n_s=0.965900, r=1e-6.

Significance. If the construction were derived from the brane/bulk dynamics, the model would be a useful analytic example of a cyclic braneworld with controlled anisotropies and CMB-compatible perturbations. The algebra is explicit and the two cases are worked in detail; the use of the uniform-rate condition and the delta-N formalism is transparent. However, as discussed below, the key physical conclusions are not extracted from the underlying dynamics but are encoded in the two assumed shear-field relations, so the significance as stated is not established.

major comments (3)

[§2.1–§2.2, Eqs. (24) and (59)] The two relations for dotphi_a^2 are introduced as assumptions ('we assume', §2.1). They are the only origin of the sinusoidal V(phi), the bounded density, the periodic H(t), the bounce, and the freeze-out of phi_a near the bounce. No derivation from the Bianchi-I brane equations or from the Maartens–Sahni–Saini closure is given, and no consistency check is offered. Since these relations are equivalent to the phenomenology the paper claims to derive, the central claim that the timelike extra dimension suppresses shear is imposed by construction rather than demonstrated.
[§2.1, Eqs. (24)–(25) and text after Eq. (25)] The physical shear in Bianchi I is Sigma^2/a^6, a positive energy density that grows during contraction. The paper replaces it by a scalar phi_a and asserts that phi_a 'vanishes at the bounce'. But from Eq. (24), at t=0 dotphi_a^2 is generally non-zero, so the shear energy density rho_a=dotphi_a^2/2 does not vanish; it is instead balanced by a negative inflaton potential V(phi_0) so that the total density equals rho_c. The claimed suppression of anisotropy is therefore a cancellation between tuned sectors, not the dynamical dilution by the timelike extra dimension. This undermines the resolution of the anisotropy problem.
[§2.1, text after Eq. (57); §2.2, text after Eq. (87)] The quoted observational agreement is a fit, not a prediction. The amplitude zeta is fixed to 4.9e-5 Mpc^-1, lambda and rho_c are chosen by hand, and Sigma is then determined; n_s and r follow. Moreover, zeta has units Mpc^-1 although in the delta-N formalism the curvature perturbation is dimensionless. The two cases produce Sigma values differing by 11 orders of magnitude (-7.1e16 vs -2.62e5 Mpc^-1), illustrating that the 'observational consistency' is not robust but the result of parameter selection.

minor comments (4)

[Eq. (25)] The incomplete elliptic integral parameter is not defined, and Fig. 2 does not specify the parameter values used in the plot.
[Fig. 7 caption] Typo: 'fore case 2' should be 'for case 2'. Similar grammatical slips appear in a few other figure captions.
[Title/abstract] The arXiv title and abstract differ substantively from the manuscript's title and abstract; the body also does not contain the 'general turning-point classification' promised in the arXiv abstract. Please unify these.
[§2, Eqs. (4)–(5)] The definitions of beta, Lambda_eff, and G_eff are not used later; consider removing or clarifying their role.

Circularity Check

3 steps flagged · score 7.0 of 10

Bounce, cyclicity, and shear suppression are built into the two posited φ̇a–V ansatze; Case 2 then enforces the CMB observables it cites as agreement.

self definitional [Sections 2.1–2.2 (Eqs. 22–27 and 58–62); text after Eqs. (25), (26), (27)]
"“To proceed further, we assume two different relations between the effective scalar field describing anisotropic effects and the inflaton potential... the total energy density can be written as ρ(φ)=ρc sin2[...]... The oscillatory behaviour of the Hubble parameter clearly indicates a cyclic evolution of the universe... the effective scalar field describing the anisotropic effects decreases during the contracting phase and vanishes at the bounce... clearly a consequence of the higher dimensional braneworld effects stemming from the timelike extra dimension.”"

With either closure, ρa=φ̇a²/2 is a prescribed linear function of V (Eqs. 24, 59), so Eq. (22) forces a sinusoidal solution: ρ(φ)=ρc sin²(...) and H(φ)∝sin(...) (Eqs. 26–27, 61–62). Bounded density, H=0 bounce/turnaround, periodic V, and infinite cyclicity are mathematical consequences of the assumed φ̇a–V relation — output already in the input. Yet the paper reports them as braneworld findings: “the effective scalar field describing the anisotropic effects decreases during the contracting phase and vanishes at the bounce... clearly a consequence of the higher dimensional braneworld effects.” Moreover, the physical shear (Eq. 12) grows as a⁻⁶ toward small a; the claimed suppression is read off from the ansatz-determined φa(t), not from the Bianchi-I shear dynamics. The central phenomenolog
fitted input called prediction [Section 2.2, closing parameter-determination paragraph (after Eq. 87)]
"“By enforcing the cosmic microwave background normalization ζ = 4.9×10⁻⁵ Mpc⁻¹ and adopting the observationally consistent values of the scalar spectral index n_s = 0.965900 and the tensor-to-scalar ratio r = 10⁻⁶ obtained in Case 1, we determine the corresponding background parameters... Notably, the inferred values of the fundamental parameters λ and ρc... are nearly identical to those found in Case 1. This demonstrates the robustness of the inflationary background against changes in the anisotropic sector.”"

In Case 2, n_s=0.965900 and r=10⁻⁶ are not outputs: they are imposed along with ζ, and the three parameters (λ, ρc, Σ) are then solved for (three equations, three unknowns). The model therefore matches the CMB by construction. The conclusion that the parameters are “nearly identical... This demonstrates the robustness of the inflationary background” is a statement about the enforced fit, not an independent consistency check; the “observational consistency” advertised for the scenario is the input written as an output.

1 more flagged steps

fitted input called prediction [Section 2.1, CMB-normalization paragraph after Eq. (57)]
"“We impose the CMB normalization ζ = 0.000049 Mpc⁻¹, consider a realistic value of the critical density ρc = 1.44×10⁻¹³ Mpc⁻² and the uniform rate of rolling of the uniform as λ = 3.85×10⁻¹¹ Mpc⁻¹... Imposing these values for the constant model parameters gives us the scalar spectral index as n_s = 0.965900, the tensor to scalar ratio as r = 0.000001 and the initial constant anisotropy or shear parameter as Σ = −7.1×10¹⁶ Mpc⁻¹. The values of both the scalar spectral index and tensor to scalar ratio are in excellent agreement with the Planck CMB observations.”"

n_s (Eq. 54) and r (Eq. 57) are functions of (λ, ρc, Σ) only, and Σ is fixed by the CMB-normalization equation (Eq. 50) once λ and ρc are chosen. The paper chooses λ=3.85×10⁻¹¹ and ρc=1.44×10⁻¹³ by hand (“we consider a realistic value...”, “not unreasonable to assume a small rate”), with no independent constraint or fitting procedure shown, and then reports n_s=0.965900, r≈10⁻⁶ as “excellent agreement with the Planck CMB observations.” The agreement is therefore a function of the same hand-picked inputs the test is supposed to check; no independent determination of λ and ρc is exhibited that would make the “prediction” nontrivial.

full rationale

The central circularity is structural: the two undereived closures φ̇a²=Σ(Σ−V/λ) (Case 1, Eq. 24) and φ̇a²=(2Σ/λ)(e^{Σ/λ}−1)V (Case 2, Eq. 59) make the total density ρ linear in V, so Eq. (22) forces the sinusoidal solution ρ(φ)=ρc sin²(...), H∝sin(...). Consequently the bounded density, non-singular bounce, cyclic evolution, and the claimed φa→0 shear suppression are consequences of the assumed relation, not of the Bianchi-I braneworld dynamics per se; the text admits the relations are assumed (“we assume two different relations... ansatz”) and then attributes the bounce to “higher dimensional braneworld effects.” This is the definition of the model containing its results. The CMB-facing claims are weakened in the same way. In Case 2 the observed n_s and r are imposed (“adopting the observationally consistent values... we determine the corresponding background parameters”), so agreement with Planck is enforced, then reported as robustness. In Case 1, n_s and r are functions of (λ, ρc, Σ) with Σ fixed by ζ and λ, ρc hand-picked as “realistic”; no independent determination or parameter scan is shown, so the “excellent agreement” is not an independent test of the model. Not every part is circular. The turning-point classification for NEC fluids and the explicit bounce/turnaround conditions follow from the brane Friedmann equation (Eq. 9) and are independent content; the δN spectral computations are formal consequences of the background. The braneworld input (Eq. 9, from Shtanov–Sahni [41]) and the dark-radiation-free Bianchi-I closure (Maartens–Sahni–Saini [50]) are external and checkable, and none of the authors' self-citations is load-bearing for the ansatz or the bounce. Because the distinctive results (bounce + shear suppression + CMB agreement) reduce by construction to posited closures and imposed observables, while the mathematical machinery around them is genuine, the appropriate score is 7: partial but decisive circularity of the central claim.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The paper contributes essentially no free dynamics: the bounce and cyclic behavior are forced by ad hoc relations between the shear field and the inflaton potential, and the CMB agreement is obtained by tuning lambda, rho_c, and Sigma. The only independent input from prior literature is the Shtanov-Sahni effective Friedmann equation; everything else is either an ansatz or a fitted number.

free parameters (4)

lambda (uniform rolling rate) = 3.85e-11 Mpc^-1 (both cases)
Constant scalar rolling rate; chosen by hand together with rho_c to land n_s and r near observed values.
rho_c (critical brane density / brane tension sigma) = 1.44e-13 Mpc^-2 (case 1), 1.42e-13 Mpc^-2 (case 2)
Hand-picked so that the model's n_s and r match Planck; parametrizes the brane tension.
Sigma (shear/anisotropy amplitude) = -7.1e16 Mpc^-1 (case 1), -2.62e5 Mpc^-1 (case 2)
Solved from the CMB normalization zeta = 4.9e-5; not predicted by the theory.
phi0 (integration constant / initial field offset) = cos(k phi0) = -1; arbitrary integer n
Chosen to cancel alpha(phi0) and set the bounce at t = 0.

assumptions (6)

domain assumption The Shtanov-Sahni effective Friedmann equation H^2 = (rho/3)(1 - rho/rho_c) with C = 0 applies to the Bianchi-I brane with a timelike extra dimension
Taken from the cited brane literature; necessary for the bounce and for defining rho_c.
domain assumption Anisotropy is representable as a stiff perfect fluid rho_a proportional to a^-6 with a canonical scalar phi_a having zero potential
Converts shear into a matter component; depends on the claimed closure of the Bianchi-I brane equations.
domain assumption The uniform-rate condition dot_phi = -lambda is imposed exactly, with motivation from de Broglie-Bohm quantum cosmology
No slow-roll or dynamical derivation; the entire potential V(phi) is then solved from this condition.
ad hoc to paper Case-1 ansatz dot_phi_a^2 = Sigma(Sigma - V/lambda) and Case-2 variant dot_phi_a^2 = (2Sigma/lambda)(e^{Sigma/lambda} - 1)V
No derivation from brane or bulk dynamics; chosen to make V, rho, and H sinusoidal and to force shear to vanish at the bounce.
domain assumption The delta-N formalism with delta_phi = H/(2pi) gives the curvature power spectrum on this bouncing brane
Imported from uniform-rate inflation literature; no derivation of its validity across the bounce is given.
domain assumption CMB normalization zeta = 4.9e-5 and 60 e-folds of inflation are assumed
These observational inputs convert parameter choice into the claimed consistency.

invented entities (1)

Effective anisotropy scalar field phi_a
purpose: Represents Bianchi-I shear as a stiff fluid whose kinetic energy is fixed by an ad hoc function of the inflaton potential, making the total density sinusoidal.
No independent dynamics or measurement; its constructed evolution is what produces shear suppression at the bounce.

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Cite this review

Pith. "Pith review of Bounce, Turnaround, and the Anisotropy Problem in Cyclic Cosmology on a Brane with a Timelike Extra Dimension." pith.science (2026). https://pith.science/paper/KKM565TP

@misc{pith2026260208974,
  author       = {Pith},
  title        = {Pith review of: Bounce, Turnaround, and the Anisotropy Problem in Cyclic Cosmology on a Brane with a Timelike Extra Dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKM565TP}},
  note         = {Machine review of arXiv:2602.08974}
}

abstract

We study cosmological bounces, turnarounds, and cyclic evolution on an anisotropic Bianchi-I brane embedded in a five-dimensional bulk with a \emph{timelike} extra dimension, within the Shtanov--Sahni braneworld framework. Restricting to the flat, dark-radiation-free, effective-$\Lambda$-free branch of the general anisotropic brane Friedmann equation, we drive the dynamics with a single canonical scalar field obeying the uniform-rate condition $\dot\phi=-\lambda=\mathrm{const}$, with shear anisotropy encoded through a geometric term $\Omega_\sigma(a)\propto a^{-6}$. We derive a general turning-point classification valid for any fluid obeying the null energy condition: turnarounds at negative energy density occur unconditionally, while bounces at $\rho>\rho_c$ occur only when the negative high-energy brane correction dominates the decelerating shear term. Specializing to the uniform-rate scalar, we obtain closed-form bounce and turnaround conditions, the leading-order excess of the bounce density above critical, and a matching condition for a finite cyclic branch connecting a bounce at $N_B$ to a turnaround at $N_T$. We identify post-bounce superinflationary and post-shear-dilution ordinary-inflationary regimes, compute the single-field curvature power spectrum, and derive parameter relations fixing $H_*$, $\lambda$, $\rho_c$, and the shear amplitude $\Sigma_g^2$ in terms of the observed amplitude $A_s$ and tilt $n_{s*}$. An explicit CMB-normalized parameter point shows that sustaining a long, weak-shear cyclic phase compatible with observations requires the shear amplitude suppressed by $10^{2}$--$10^{3}$ orders of magnitude below $H_*^2$, depending sensitively on the pivot density fraction $x_*=\rho_{\phi*}/\rho_c$. We discuss the physical origin of this anisotropy problem, its parametric dependence, and the status of the periodicity condition required for a genuinely cyclic $V(\phi)$.

Figures

Figures reproduced from arXiv: 2602.08974 by the authors.

**Figure 2.** Variation of anisotropy field ϕa with cosmic time t. In [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Variation of total energy density with cosmic time. The energy density is found to vary periodically over time, depending on essential model parameters like the anisotropy parameter, and the brane tension σ. One can immediately notice that the energy density has a finite maximum at the bounce, thereby justifying the energy bound of the brane universe with a timelike extra dimension. We obtain the Hubble parameter as… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: The oscillatory behaviour of the Hubble parameter clearly indicates a [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 4.** Figure 4: Variation of the Hubble parameter H with cosmic time t. of smooth non-singular contraction to expansion transitions occur at regular intervals, giving rise to an eternal periodic universe. At this point, it is convenient for our analysis to introduce the parameter α, w…

**Figure 5.** Figure 5: Variation of the scale factor with cosmic time [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Variation of the inflation potential V (ϕ) vs the inflaton field ϕ for case 2 ϕ˙ a(t) = ± vuuut 2 Σ λ e Σ λ − 1 1 + Σ λ e Σ λ − 1 r ρc cos2(ωt) − λ2 2 (59) where ω = λ 2 2 h 1 + Σ λ e Σ λ − 1 i q 3 ρc . For ρc ≫ λ 2 , we have ρc cos2 (ωt) − λ 2 2 ≈ ρc cos2 (…

**Figure 7.** Figure 7: Variation of anisotropy field ϕa with cosmic time t fore case 2 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: Variation of total energy density ρ with cosmic time t for case 2. The variation of the energy density with cosmic time is plotted in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Variation of the Hubble parameter H with cosmic time t for case 2. The variation of the Hubble parameter H(t) with cosmic time is plotted in [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Variation of the scale factor with cosmic time [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

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Pith tools

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