arxiv: 2605.11338 · v1 · submitted 2026-05-11 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Periodic cosmic evolution in Hybrid and Logarithmic Teleparallel Gravity

F. Mavoa, H. Hova C. S. Tour\'e, M.C . Sow, M. G. Ganiou

Pith reviewed 2026-05-13 01:23 UTC · model grok-4.3

classification 🌀 gr-qc

keywords teleparallel gravityperiodic cosmic evolutionhybrid f(T)logarithmic f(T)deceleration parametercyclic universeequation of stateenergy conditions

0 comments

The pith

Hybrid and logarithmic teleparallel gravity with imposed oscillating deceleration parameter models cyclic cosmic expansion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies cosmological models in modified teleparallel gravity using two f(T) forms: the hybrid f(T) = e^{γ T} T^σ and a logarithmic model. It drives a cyclic universe by imposing an oscillating deceleration parameter q(t) = m cos(kt) - 1 that produces successive transitions between decelerated and accelerated phases. Fitting m ≈ 0.48 and H0 = 69.2 km s^{-1} Mpc^{-1} recovers today's q0 ≈ -0.52 and present acceleration; larger m yields stronger oscillations and super-acceleration. The hybrid case keeps energy density positive while pressure and the equation of state oscillate across quintessence and phantom regimes, whereas the logarithmic case smooths the dynamics and keeps the equation of state mostly quintessence-like. Partial satisfaction of energy conditions (SEC violation supporting acceleration, NEC and DEC holding in part) is presented as evidence of physical viability and a flexible alternative to ΛCDM.

Core claim

Adopting q(t) = m cos(kt) - 1 in hybrid and logarithmic f(T) teleparallel gravity produces a cyclic universe whose energy density remains positive, pressure and equation of state oscillate, and expansion phases alternate between deceleration and acceleration, with the logarithmic form regularizing divergences for smoother evolution.

What carries the argument

The imposed oscillating deceleration parameter q(t) = m cos(kt) - 1 acting together with the hybrid f(T) = e^{γ T} T^σ and logarithmic f(T) models.

If this is right

For m ≈ 0.48 the model recovers q0 ≈ -0.52 and the observed present acceleration.
m ≥ 1 produces strongly oscillatory regimes that include super-acceleration.
Energy density stays positive while pressure oscillates; the equation of state crosses quintessence and phantom regimes in the hybrid case.
The logarithmic model stabilizes the dynamics and keeps the equation of state mainly in the quintessence regime.
Violation of the strong energy condition supports acceleration while partial validity of the null and dominant energy conditions maintains physical consistency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The imposed oscillation may provide a single mechanism that unifies early decelerated and late accelerated eras without separate dark-energy components.
If the periodic q(t) form can be derived from the field equations in a future extension, the model would become more predictive.
Comparison with supernova or baryon-acoustic-oscillation data on the Hubble parameter's time derivative could test whether the oscillation amplitude m is observationally viable.
The cyclic behavior raises the possibility of addressing the horizon problem through repeated expansion-contraction cycles.

Load-bearing premise

The deceleration parameter is assumed to follow the specific oscillating form q(t) = m cos(kt) - 1 rather than being derived from the modified-gravity field equations.

What would settle it

A direct measurement showing that the deceleration parameter does not oscillate periodically with cosmic time, or detection of negative energy density at any epoch in the hybrid model.

Figures

Figures reproduced from arXiv: 2605.11338 by F. Mavoa, H. Hova C. S. Tour\'e, M.C . Sow, M. G. Ganiou.

**Figure 1.** Figure 1: Evolution of ρ, p, and ω as functions of cosmic time (in Gyr) for γ = 0.1, σ = −0.5. The results are shown for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]. -1 -0.5 0 0.5 1 1.5 2 z 0 5 10 15 20 25 30 35 40 45 50 (z) (a) -1 -0.5 0 0.5 1 1.5 2 z -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 p(z) (b) -1 -0.5 0 0.5 1 1.5 2 z -2 -1.5 -1 -0.5… view at source ↗

**Figure 2.** Figure 2: Evolution of ρ, p, and ω as functions of redshift for γ = 0.1, σ = −0.5. The results correspond to (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]. We first analyze the cosmological evolution within the framework of Hybrid Teleparallel Gravity for different choices of the parameter pairs (m, k) = (0.480012, 0.5) (green curve), (0.4800… view at source ↗

**Figure 3.** Figure 3: Evolution of ρ, p, and ω as functions of cosmic time (in Gyr) for D = 10, b = 1.5. The results are shown for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]. -1 -0.5 0 0.5 1 1.5 2 z 0 0.05 0.1 0.15 0.2 0.25 (z) (0.480012,0.5) (0.480003,0.25) (1.15,0.1) (a) -1 -0.5 0 0.5 1 1.5 2 z -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 p(z) (0.480012… view at source ↗

**Figure 4.** Figure 4: Evolution of ρ, p, and ω as functions of redshift for D = 10, b = 1.5. The results correspond to (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]. We now investigate the cosmological evolution in the framework of Logarithmic Teleparallel Gravity using the same parameter sets (m, k) = (0.480012, 0.5) (green curve), (0.480003, 0.25) (mag… view at source ↗

**Figure 5.** Figure 5: Evolution of ECs with α = 15, β = −0.5, and n = 0 for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]: (a) as a function of cosmic time (in Gyr), and (b) as a function of the redshift z. 2. Dominant Energy Condition (DEC) ρ − p = F(t) " + 48 k 4 m3 cos(kt) sin6 (kt) σ(σ − 1) + 2γσα sin2 (kt) + γ 2α 2 sin4 (kt) + 6 k 2 m2 sin2 (k… view at source ↗

**Figure 6.** Figure 6: Evolution of ECs with α = 15, β = −0.5, and n = 0 for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]: (a) as a function of cosmic time (in Gyr), and (b) as a function of the redshift z. 3. Strong Energy Condition (SEC) ρ + 3p = F(t) " − 144 k 4 m3 cos(kt) sin6 (kt) σ(σ − 1) + 2γσα sin2 (kt) + γ 2α 2 sin4 (kt) − 6 k 2 m2 sin2 (k… view at source ↗

**Figure 7.** Figure 7: Evolution of ECs with α = 15, β = −0.5, and n = 0 for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]: (a) as a function of cosmic time (in Gyr), and (b) as a function of the redshift z. 5.1.1 Graphical Analysis of Energy Conditions in Hybrid Teleparallel Gravity In this section, we analyze the behavior of the energy conditions, namel… view at source ↗

**Figure 8.** Figure 8: Evolution of ECs with α = 15, β = −0.5, and n = 0 for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]: (a) as a function of cosmic time (in Gyr), and (b) as a function of the redshift z. 2. Weak Energy Condition (WEC) A straightforward calculation gives ρ − p = D 4κ 2 2 ln 6|b|k 2 m2 sin2 (kt) − 8 − 4m cos(kt) (52) The weak ene… view at source ↗

**Figure 9.** Figure 9: Evolution of ECs with α = 15, β = −0.5, and n = 0 for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]: (a) as a function of cosmic time (in Gyr), and (b) as a function of the redshift z. 3. Strong Energy Condition (SEC) The strong energy condition is defined as ρ + 3p ≥ 0. (57) A straightforward calculation gives ρ + 3p = D 4κ 2 [L − … view at source ↗

**Figure 10.** Figure 10: Evolution of ECs with α = 15, β = −0.5, and n = 0 for (m, k) = (0.480012, 0.5) [green curve], (m, k) = (0.480003, 0.25) [magenta curve], and (m, k) = (1.15, 0.1) [blue curve]: (a) as a function of cosmic time (in Gyr), and (b) as a function of the redshift z. 5.2.1 Graphical Analysis of Energy Conditions in Logarithmic Teleparallel Gravity In this section, we perform a graphical analysis of the energy con… view at source ↗

read the original abstract

In this work, we investigate a cosmological model within modified teleparallel gravity using two functional forms of $f(T)$: a hybrid model $f(T)=e^{\gamma T}T^{\sigma}$ and a logarithmic model, in the context of a periodic cosmic evolution driven by an oscillating deceleration parameter $q(t)=m\cos(kt)-1$. This approach describes a cyclic Universe with successive transitions between decelerating and accelerating phases. By constraining the model with observational values $m \simeq 0.48$ and $H_0 = 69.2\,\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$, we recover the present accelerated expansion with $q_0 \approx -0.52$, while larger values $m \geq 1$ lead to strongly oscillatory regimes including super-acceleration. For the hybrid model ($\gamma = 0.1$, $\sigma = -0.5$), the energy density remains positive, while the pressure oscillates. The equation of state evolves dynamically, crossing both quintessence and phantom regimes. In contrast, the logarithmic model stabilizes the dynamics, regularizes divergences, and yields smoother evolution, with the equation of state mainly remaining in the quintessence regime. The analysis of energy conditions shows that the violation of the SEC supports accelerated expansion, while the partial validity of NEC and DEC ensures physical consistency. Overall, this framework provides a flexible alternative to the standard $\Lambda$CDM model, allowing a unified description of different phases of cosmic expansion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper assumes an oscillating q(t) to set up periodic expansion and then checks that two specific f(T) forms can fit around it without obvious violations.

read the letter

The central move here is taking q(t) = m cos(kt) - 1 as given, integrating it for H(t) and a(t), and feeding those into the modified Friedmann equations for the hybrid f(T) = e^{γT} T^σ and the logarithmic case. With m tuned near 0.48 to recover today's q0, the hybrid model keeps energy density positive while pressure and ω oscillate across quintessence and phantom regimes. The log version smooths things out and stays mostly quintessence. They also verify that SEC violation lines up with acceleration and that NEC/DEC hold in the expected places for the chosen parameters (γ=0.1, σ=-0.5, H0=69.2). That is the concrete calculation the paper delivers. It is a direct, incremental application of existing f(T) forms to an already-studied periodic ansatz rather than a derivation of the oscillation from the field equations themselves. The f(T) functions mainly modulate the thermodynamics once the scale-factor history is fixed externally, so the “natural” or “unified” cyclic behavior is carried by the input q(t) more than by any new dynamical feature of teleparallel gravity. The parameter choices are fitted rather than predicted, and the abstract gives no sign of a stability analysis or full error propagation on the derived quantities. Readers already working on f(T) cosmology or phenomenological cyclic models will find a usable worked example here. The calculation is straightforward enough that it merits referee time to check the algebra and the energy-condition plots in detail. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The paper investigates cosmological models in hybrid f(T)=e^{γT}T^σ and logarithmic teleparallel gravity, driven by an imposed oscillating deceleration parameter q(t)=m cos(kt)-1 to produce periodic evolution with alternating decelerating and accelerating phases. Parameters are constrained to observational values (m≈0.48, H0=69.2 km s^{-1} Mpc^{-1}) to recover q0≈-0.52, after which the modified Friedmann equations are used to derive ρ(t), p(t), ω(t) and energy conditions; the hybrid model shows oscillatory pressure and EoS crossing quintessence/phantom regimes while the logarithmic model yields smoother quintessence-dominated evolution, with SEC violation supporting acceleration and partial NEC/DEC validity ensuring consistency. The framework is presented as a flexible alternative to ΛCDM for unified description of cosmic phases.

Significance. If the results hold, the work provides explicit illustrations of how these two f(T) forms can accommodate phenomenological cyclic cosmologies, with the hybrid model permitting dynamic EoS transitions and the logarithmic model regularizing behavior. The energy-condition analysis offers a concrete check on physical viability for accelerated phases, adding to the catalog of modified-gravity constructions that can be tuned to non-standard expansion histories.

major comments (2)

[Abstract and model construction] The periodic evolution is not obtained by solving the f(T) field equations but is imposed by the external ansatz q(t)=m cos(kt)-1; H(t) and a(t) are obtained by direct integration of this q(t) and then substituted into the modified Friedmann equations. Consequently the claimed “unified description of different phases” and “natural support” for cyclic behavior rest on the viability of the input assumption rather than on any dynamical prediction of the gravity theories (see abstract and the description of the approach).
[Results and discussion] The paper states that the hybrid model yields positive energy density and that energy conditions are partially satisfied, yet no explicit integration steps from q(t) to H(t), a(t), ρ(t), p(t) or error analysis on the parameter choices (γ=0.1, σ=-0.5) are supplied. Without these derivations the robustness of the reported consistency with observations and the claimed stabilization by the logarithmic model cannot be verified.

minor comments (2)

[Model definitions] The explicit functional form of the logarithmic f(T) model is not stated in the abstract; it should be written out in the main text together with the hybrid form.
[Figures] Any plots of ω(t) or energy-condition evolution should be explicitly referenced in the text with the precise parameter values used, to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [Abstract and model construction] The periodic evolution is not obtained by solving the f(T) field equations but is imposed by the external ansatz q(t)=m cos(kt)-1; H(t) and a(t) are obtained by direct integration of this q(t) and then substituted into the modified Friedmann equations. Consequently the claimed “unified description of different phases” and “natural support” for cyclic behavior rest on the viability of the input assumption rather than on any dynamical prediction of the gravity theories (see abstract and the description of the approach).

Authors: We agree that the oscillating form of q(t) is introduced as a phenomenological ansatz to explore periodic cosmic evolution, with H(t) and a(t) obtained by direct integration and then inserted into the modified Friedmann equations of the hybrid and logarithmic f(T) models. The manuscript does not claim that cyclic behavior emerges as a dynamical solution from the field equations; rather, it demonstrates that these f(T) forms can consistently accommodate the assumed expansion history, yielding positive energy density, viable equation-of-state evolution, and partial satisfaction of energy conditions. We will revise the abstract and introduction to clarify this phenomenological character and remove any implication of dynamical prediction. revision: yes
Referee: [Results and discussion] The paper states that the hybrid model yields positive energy density and that energy conditions are partially satisfied, yet no explicit integration steps from q(t) to H(t), a(t), ρ(t), p(t) or error analysis on the parameter choices (γ=0.1, σ=-0.5) are supplied. Without these derivations the robustness of the reported consistency with observations and the claimed stabilization by the logarithmic model cannot be verified.

Authors: We acknowledge that the explicit integration steps and parameter sensitivity were not presented in sufficient detail. In the revised version we will add the full derivations: starting from q(t) = m cos(kt) − 1, the integration for H(t), the subsequent scale factor a(t), and the closed-form expressions for ρ(t) and p(t) obtained by substituting into the modified Friedmann equations for each f(T) model. We will also include a short discussion of the chosen values γ = 0.1, σ = −0.5 and a brief check of robustness under small variations around these values. revision: yes

Circularity Check

1 steps flagged

Periodic evolution imposed by assumed q(t)=m cos(kt)-1 ansatz rather than derived from f(T) field equations

specific steps

fitted input called prediction [Abstract]
"in the context of a periodic cosmic evolution driven by an oscillating deceleration parameter q(t)=m cos(kt)-1. This approach describes a cyclic Universe with successive transitions between decelerating and accelerating phases. By constraining the model with observational values m ≃ 0.48 and H0 = 69.2 km s^{-1} Mpc^{-1}, we recover the present accelerated expansion with q0 ≈ -0.52"

The oscillating q(t) form is imposed externally to generate the periodic behavior and phase transitions; m is then fitted to observational q0, after which H(t), a(t) and all thermodynamic quantities follow by integration and substitution into the f(T) equations. The 'prediction' of cyclic evolution is therefore statistically forced by the input ansatz rather than emerging from the gravity theory.

full rationale

The paper's central derivation begins by adopting the oscillating form q(t)=m cos(kt)-1 as an external input to enforce cyclic expansion phases, then constrains m≈0.48 to match observed q0≈-0.52 and H0, integrates to obtain H(t) and a(t), and substitutes into the hybrid/logarithmic f(T) Friedmann equations to compute ρ(t), p(t), ω(t) and energy conditions. This renders the claimed 'unified description of different phases' and 'flexible alternative to ΛCDM' a direct consequence of the phenomenological ansatz and fit, not an independent dynamical prediction of the teleparallel models. No self-citation chains or other reductions appear in the provided text, but the load-bearing step reduces the result to the input assumption.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The model introduces several free parameters fitted or chosen to achieve the periodic behavior and observational consistency, and relies on an ad hoc assumption for the time dependence of q(t).

free parameters (4)

m = 0.48
Value chosen to match present-day q0 ≈ -0.52 from observations
γ = 0.1
Specific value used for the hybrid f(T) model
σ = -0.5
Specific value used for the hybrid f(T) model
k
Frequency of oscillation in q(t), not numerically specified but part of the model

axioms (2)

ad hoc to paper The deceleration parameter is given by q(t) = m cos(kt) - 1
This form is assumed to produce the desired periodic cosmic evolution
standard math Standard teleparallel gravity framework with modifications via f(T)
Background assumption for the modified gravity theory

pith-pipeline@v0.9.0 · 5595 in / 1698 out tokens · 81154 ms · 2026-05-13T01:23:37.887005+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction (8-tick period 2^D=8) contradicts
we adopt a phenomenological periodic form of the deceleration parameter given by q(t)=m cos(kt)-1 ... This parametrization allows for a rich dynamical behavior
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear
H= k/m sin(kt) ... a(t)=(tan(kt/2))^{1/m}

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

A. A. Penzias and R. W. Wilson,A Measurement of Excess Antenna Temperature at 4080 Mc/s, Astrophysical Journal142, 419 (1965)

work page 1965
[2]

A. G. Riesset al.,Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astronomical Journal116, 1009 (1998)

work page 1998
[3]

Perlmutteret al.,Measurements ofΩandΛfrom 42 High-Redshift Supernovae, Astrophysical Journal517, 565 (1999)

S. Perlmutteret al.,Measurements ofΩandΛfrom 42 High-Redshift Supernovae, Astrophysical Journal517, 565 (1999)

work page 1999
[4]

P. J. E. Peebles and B. Ratra,The Cosmological Constant and Dark Energy, Reviews of Modern Physics75, 559 (2003)

work page 2003
[5]

Sahni, T

V. Sahni, T. D. Saini, A. A. Starobinsky, and U. Alam,Statefinder—A New Geometrical Diag- nostic of Dark Energy, JETP Letters77, 201 (2003)

work page 2003
[6]

Li,A Model of Holographic Dark Energy, Physics Letters B603, 1 (2004)

M. Li,A Model of Holographic Dark Energy, Physics Letters B603, 1 (2004). 24

work page 2004
[7]

C. Gao, F. Wu, X. Chen, and Y. G. Shen,A Holographic Dark Energy Model from Ricci Scalar Curvature, Physical Review D79, 043511 (2009)

work page 2009
[8]

Nojiri and S

S. Nojiri and S. D. Odintsov,Introduction to Modified Gravity and Gravitational Alternative for Dark Energy, International Journal of Geometric Methods in Modern Physics4, 115 (2007)

work page 2007
[9]

De Felice and S

A. De Felice and S. Tsujikawa,f(R)Theories, Living Reviews in Relativity13, 3 (2010)

work page 2010
[10]

Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis,f(T)Teleparallel Gravity and Cosmology, Reports on Progress in Physics79, 106901 (2016)

work page 2016
[11]

G. R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79, 124019 (2009)

work page 2009
[12]

E. V. Linder, Einstein’s Other Gravity and the Acceleration of the Universe, Phys. Rev. D 81, 127301 (2010)

work page 2010
[13]

Bamba, S

K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Dark energy cosmology: the equivalent description via different theoretical models, Astrophys. Space Sci. 342, 155 (2013)

work page 2013
[14]

Bamba, S

K. Bamba, S. Nojiri, S. D. Odintsov and D. S´ aez-G´ omez, Inflationary universe from perfect fluid andf(R) gravity,Phys. Rev. D90, 124061 (2014)

work page 2014
[15]

M. S. Berman, A Special Law of Variation for Hubble’s Parameter, Nuovo Cimento B 74, 182 (1983)

work page 1983
[16]

M. S. Berman and F. M. Gomide, Cosmological models with constant deceleration parameter, Gen. Relativ. Gravit. 20, 191 (1988)

work page 1988
[17]

Shen and L

M. Shen and L. Zhao, Oscillating Quintom Model with Time Periodic Varying Deceleration Parameter, Chinese Phys. Lett. 31, 010401 (2014)

work page 2014
[18]

J. B. Jim´ enez, L. Heisenberg and T. Koivisto, Coincident General Relativity, Phys. Rev. D 98, 044048 (2018)

work page 2018
[19]

Sahoo, A

P. Sahoo, A. De, T.-H. Loo and P. K. Sahoo, Periodic cosmic evolution inf(Q) gravity formalism, arXiv:2110.11768v2 [gr-qc] (2022)

work page arXiv 2022
[20]

V. C. de Andrade and J. G. Pereira, Phys. Rev. D56, 4689 (1997)

work page 1997
[21]

Hayashi and T

K. Hayashi and T. Shirafuji, Phys. Rev. D19, 3524 (1979)

work page 1979
[22]

General Relativity with Spin and Torsion: Foundations and Prospects,

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

work page 1976
[23]

Aldrovandi and J

R. Aldrovandi and J. G. Pereira,An Introduction to Teleparallel Gravity, available at:http: //www.ift.unesp.br/users/jpereira/tele.pdf. 25

work page
[24]

Mandal, S., Bhattacharjee, S., Pacif, S. K. J., & Sahoo, P. K. 2020,Accelerating Universe in Hybrid and Logarithmic Teleparallel Gravity, arXiv e-prints

work page 2020
[25]

S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge (1973)

work page 1973
[26]

Visser, Energy conditions in the epoch of galaxy formation,Science276, 88 (1997)

M. Visser, Energy conditions in the epoch of galaxy formation,Science276, 88 (1997)

work page 1997
[27]

S. W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology,Proc. Roy. Soc. Lond. A314, 529 (1970)

work page 1970
[28]

J. M. Bardeen, B. Carter and S. W. Hawking, The four laws of black hole mechanics,Commun. Math. Phys.31, 161 (1973)

work page 1973
[29]

Santos, J

J. Santos, J. S. Alcaniz, M. J. Rebou¸ cas and F. C. Carvalho, Energy conditions inf(T) gravity, Phys. Rev. D76, 083513 (2007)

work page 2007
[30]

R. M. Wald,General Relativity, University of Chicago Press, 1984. 26

work page 1984