Cosmological constraints on time-varying cosmological terms: A study of FLRW universe models with $Λ(t)$CDM cosmology
M. Koussour N. Myrzakulov J. Rayimbaev


       




[Email: ]pr.mouhssine@gmail.com
Department of Physics, University of Hassan II Casablanca, Morocco.


[Email: ]nmyrzakulov@gmail.com
L. N. Gumilyov Eurasian National University, Astana 010008,
Kazakhstan.


[Email: ]javlon@astrin.uz
New Uzbekistan University, Mustaqillik Ave. 54, Tashkent 100007, Uzbekistan.
University of Tashkent for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan.
National University of Uzbekistan, Tashkent 100174, Uzbekistan.




This paper explores models of the FLRW universe that incorporate a time-varying cosmological term Λ(t). Specifically, we assume a power-law form for the cosmological term as a function of the scale factor: Λ(t)=Λ_0 a(t)^-α, where Λ_0 represents the present value of the cosmological term. Then, we derive an exact solution to Einstein's field equations within the framework of Λ(t)CDM cosmology and determine the best-fit values of the model parameters using the combined H(z) + SNe Ia dataset and MCMC analysis. Moreover, the deceleration parameter demonstrates the accelerating behavior of the universe, highlighting the transition redshift z_tr, at which the expansion shifts from deceleration to acceleration, with confidence levels of 1-σ and 2-σ. In addition, we analyze the behavior of the Hubble parameter, jerk parameter, and Om(z) diagnostic. Our analysis leads us to the conclusion that the Λ(t)CDM model is consistent with present-day observations.

Keywords: cosmological constraints, Λ(t)CDM, deceleration parameter, dark energy.


Cosmological constraints on time-varying cosmological terms: A study of FLRW universe models with Λ(t)CDM cosmology
    J. Rayimbaev0000-0001-9293-1838
    April 29, 2024
===================================================================================================================









§ INTRODUCTION


Recent research has been dedicated to uncovering the origins of the accelerated expansion observed in the present universe. This accelerated expansion is supported by observational evidence, including astrophysical observations of Type Ia supernovae (SNe Ia) <cit.>, cosmic microwave radiation <cit.>, baryonic acoustic oscillations <cit.>, and measurements of the Hubble constant <cit.>. These observations collectively provide strong evidence for the phenomenon of cosmic acceleration. The accelerating expansion of the universe is attributed to a new form of energy with negative pressure, commonly referred to as dark energy (DE) <cit.>. DE constitutes a significant component of the universe's energy field <cit.>. The role of DE in driving the accelerated expansion of the universe has been the subject of extensive research in astrophysics. However, the nature of DE remains a challenging problem in theoretical physics, presenting a significant puzzle for cosmologists and physicists alike. The cosmological constant, characterized by a time-independent equation of state ω=-1, stands as the earliest and most straightforward candidate for DE. However, from a theoretical perspective, it encounters two significant challenges: the fine-tuning problem and the cosmic coincidence problem <cit.>. These issues have spurred further exploration into alternative models of DE that can offer more satisfactory explanations for the observed accelerated expansion of the universe. In addition to the cosmological constant, several other dynamical DE models have been proposed with time-dependent equations of state to explain the accelerated expansion of the universe. These models include quintessence <cit.>, phantom energy <cit.>, K-essence <cit.>, Chaplygin gas <cit.>, and tachyon  fields <cit.>. In addition, there are interacting DE models, such as holographic models and agegraphic models <cit.>, which offer alternative approaches to understanding the nature of DE and its role in the evolution of the universe. On the other hand, there are modified gravity models that have attracted many researchers recently <cit.>. 

In the same context, a variety of models have been proposed in cosmology, incorporating time-varying Λ(t) or vacuum decay, often with assumed time dependencies for Λ(t) <cit.>.  These models explore different phenomenological decay laws for Λ(t), obtained through geometric analysis <cit.> or quantum mechanical reasoning <cit.>. Also, researchers have explored interactions between vacuum and matter using different approaches, comparing to recent cosmological data <cit.>. One promising approach to address the limitations of the Λ(t)CDM model is the RVM (Running Vacuum Model. This model is derived from quantum field theory in curved spacetime, expressing the vacuum energy density as a series of powers of the Hubble function H and its derivatives for cosmic time t. The RVM has shown potential in fitting with cosmological observables, outperforming the Λ(t)CDM model in certain scenarios <cit.>. For further details on the RVM and recent theoretical developments, readers can refer to relevant literature <cit.>. Overduin and Cooperstock <cit.> investigated the change in the scale factor in relation to a variable cosmological term, represented by forms like Λ = A t^-l, Λ = B a^-m, Λ = C t^n, and Λ = D q^r (where A, B, C, D, l, m, n, r are constants). Rezaei et al. <cit.> employed phenomenological reasoning to parameterize the time-dependent behavior of Λ(t), expressing it as a power series expansion of the Hubble parameter and its time derivatives: Λ(t) ∝ H, Λ(t) ∝Ḣ, Λ(t) ∝ H^2. Following this, we consider models of the FLRW universe that include a time-varying cosmological term Λ(t). We specifically adopt a power-law form for the cosmological term in terms of the scale factor: Λ(t)=Λ_0 a(t)^-α, where Λ_0 denotes the present value of the cosmological term. Recently, various noteworthy findings within the realm of Λ(t)CDM models using the power law with different values of α have appeared in references <cit.>. 

The present manuscript is organized as follows. 
In Section <ref>, we present the flat FLRW universe in the context of Λ(t)CDM cosmology. In Section <ref>, we obtain an exact solution to Einstein's field equations within the framework of Λ(t)CDM cosmology. This solution is expressed in terms of the Hubble parameter, assuming a power-law form of Λ(t). In Section <ref>, we analyze observational data to determine the best-fit values of the model parameters using a dataset that includes 31 points from the Hubble H(z) dataset and 1048 samples from the SNe Ia dataset. Furthermore, we investigate the behavior of various cosmological parameters, including the Hubble parameter, the deceleration parameter, and the jerk parameter. 
In Section <ref>, we examine the behavior of the Om(z) diagnostic, which helps differentiate between various DE models based on the values constrained by observational data. Finally, we discuss our findings in Section <ref>.



§ OVERVIEW OF Λ(T)CDM COSMOLOGY


Let's start with the assumption that the universe can be described by the homogeneous (meaning the universe is uniform in its distribution of matter), isotropic (meaning it looks the same in all directions), and spatially flat FLRW metric, expressed as <cit.>

    ds^2 = -dt^2 + a(t)^2 [ dr^2 + r^2(dθ^2 + sin^2θ dϕ^2) ],

where t is the cosmic time, r is the comoving radial distance, θ is the polar angle, ϕ is the azimuthal angle, and a(t) is the scale factor of the Universe which depends on time.  This metric is a cornerstone of modern cosmology and is a solution to Einstein's field equations in GR under the assumptions of homogeneity and isotropy on large scales. The scale factor a(t) encodes the expansion of the universe, with a(t_0)=1 representing the present scale of the universe at t=t_0. The dynamics of the Universe, including its expansion history and the evolution of its contents (such as matter, radiation, and DE), are governed by the behavior of the scale factor a(t) as a function of cosmic time.

In addition, the matter content of the Universe is assumed to consist of a perfect fluid, for which the energy-momentum tensor is given by

    𝒯_μν=(ρ +p)u_μu_ν+pg_μν.


Here, ρ represents the energy density of the fluid, p stands for the fluid's pressure, u^μ=(1,0,0,0) denotes the 4-velocity of the fluid elements, and g_μν is the metric tensor. In this study, we concentrate on the late-time universe, where we neglect the impact of radiation, represented by ρ_r=0. Consequently, the total density ρ and total pressure p are described as ρ=ρ_m+ρ_Λ and p=p_m+p_Λ, respectively.  Here, ρ_m represents the density of non-relativistic matter, ρ_Λ represents the density of DE (often associated with the cosmological term, which can vary with time  Λ(t)), p_m=0 is the pressure associated with non-relativistic matter, and p_Λ is the pressure associated with DE. 

Initially, the cosmological term Λ was often considered a constant of nature. This perspective is rooted in the Einstein field equations, which can be written as

    G_μν + Λ  g_μν = κ T_μν ,

where κ=8 π G=1 and G_μν=R_μν- 1/2 R g_μν is the Einstein tensor. By calculating the covariant divergence of equation (<ref>), bearing in mind that the vanishing covariant divergence of the Einstein tensor is ensured by the Bianchi identities, and presuming that the energy-momentum tensor adheres to the conservation law ∇^νT_μν=0, we conclude that the covariant divergence of Λ g_μν must also vanish, implying that Λ is a constant. This reasoning firmly places Λ on the left-hand side of the field equations, providing a geometrical interpretation of the cosmological term.

However, a more recent approach, exemplified by <cit.>, has been to treat Λ as part of the matter content of the universe rather than a constant. This is achieved by moving Λ to the right-hand side of the field equations:

    G_μν = κT̃_μν,

where T̃_μν≡ T_μν - Λ/κ  g_μν is the effective energy-momentum tensor. This reinterpretation allows for the possibility that Λ may vary, as long as the effective energy-momentum tensor T̃_μν satisfies energy conservation <cit.>.

Now, by using the metric and the Einstein field equations, we derive the standard Friedmann equations as <cit.>,

    3H^2   =   ρ_m+ρ_Λ, 
    
    2Ḣ+3H^2   =   -p_Λ,

where H=ȧ/a is the Hubble parameter, which represents the current rate of expansion of the universe.

In the present manuscript, we assume the EoS of the vacuum to be ω_Λ=-1. This choice implies that p_Λ=-ρ_Λ=-Λ(t) <cit.>. However, a recent finding regarding the RVM is that the EoS of vacuum evolves with cosmic history <cit.>. With the given assumptions, we can express the second Friedmann equation (<ref>) as <cit.>

    2Ḣ+3H^2=Λ(t).


Also, we can express the continuity equation for the matter and vacuum energy components of the universe as follows:

    ρ̇_m + 3Hρ_m   = Q,
    ρ̇_Λ   = -Q.

where Q represents the interaction term between pressureless matter and vacuum. Now, the dimensionless density parameters for the cosmological term and matter are defined as

    Ω_Λ 0=Λ_0/3H_0^2,    Ω_m0=(1-Ω_Λ 0).




§ TIME-VARYING COSMOLOGICAL TERM MODEL
 

To solve Eq. (<ref>) for the Hubble parameter H, we require an additional equation. Therefore, we assume a well-motivated power-law form for the cosmological term as a function of the scale factor <cit.>, 

    Λ(t)=Λ_0 a(t)^-α,

where Λ_0 represents the present value of the cosmological term. The case where α has received significant attention in the literature and is motivated by dimensional arguments <cit.>. Another group of researchers has focused on values of m approximately equal to 4, where the cosmological term behaves similarly to ordinary radiation <cit.>.

Furthermore, by using the relation 1+z=1/a(t), we can establish the connection between the cosmic time t and the redshift z as described below,

    d/dt=-H(z)(1+z) d/dz.


Therefore, the time derivative of the Hubble parameter can be expressed in the form of

    Ḣ=-H(z)(1+z) dH(z)/dz.


Using Eqs. (<ref>), (<ref>), and (<ref>), we derive the expression for the Hubble parameter H(z) in terms of the redshift z,

    H(z)=H_0 √((α -3 Ω_m0)(1+z)^3/α -3+3 (Ω_m0-1) (1+z)^α/α -3),

where H_0 represents the present-day Hubble value, corresponding to z=0. In particular, when α=0, the expression for H(z)  simplifies to 

    H(z)=H_0 √(Ω_m0(1+z)^3+(1-Ω_m0)),

which corresponds to a standard ΛCDM model. Hence, the model parameter α=0 serves as a useful indicator of how much the current model deviates from the ΛCDM model.

To describe whether the cosmological expansion is accelerating or decelerating, we introduce the deceleration parameter q, which is defined as

    q=-1-Ḣ/H^2= -1+(1+z)/H(z)dH(z)/dz.


In the present model, the deceleration parameter is formulated as

    q(z)=-1+3 (α -3 Ω_m0)(1+z)^3+3 α  (Ω_m0-1) (1+z)^α/2(α -3 Ω_m0) (1+z)^3+6 (Ω_m0-1) (1+z)^α.


When the condition q<0 is met, the universe will transition into an accelerated phase of expansion. In addition, the jerk parameter is a higher-order cosmological parameter that characterizes the rate of change of the deceleration parameter with respect to time <cit.>. It is defined as <cit.>

    j(z)=(2 q+1) q+(1+z) d q/d z.


The jerk parameter provides information about the rate of change in the acceleration or deceleration of the cosmic expansion, offering insights beyond what is captured by the deceleration parameter alone. In various cosmological models, the transition from the decelerating to the accelerating phase of the universe occurs when the jerk parameter takes on a positive value (j>0). For instance, in ΛCDM model, the jerk parameter j remains constant at j=1 <cit.>.



§ OBSERVATIONAL DATA
 



 §.§ Data and methodology


Now, we evaluate the consistency of the Λ(t)CDM model with recent observational data, focusing on the observational Hubble H(z) data and Type Ia Supernovae (SNe Ia).


  * H(z) dataset: 
In our analysis, we consider 31 Hubble H(z) data points obtained using the differential age (DA) approach, which provides valuable insights into the expansion history of the universe <cit.>.


  * SNe Ia dataset: In addition, our analysis includes 1048 Type Ia supernovae (SNe Ia) luminosity distance estimates derived from various sources, such as the Pan-STARRS 1 (PS1) Medium Deep Survey, the Low-z, SDSS, SNLS, and HST missions, which are part of the Pantheon sample. These data provide a comprehensive view of the universe's expansion history and contribute to our assessment of the Λ(t)CDM model's validity <cit.>.


In our analysis of cosmological observational data, we use the Markov Chain Monte Carlo (MCMC) sampling technique. Our approach builds upon previous studies by incorporating a broad range of data sources and applying more stringent priors on the model parameters. Specifically, we focus on the parameter space θ_s=(H_0,Ω_m0,α) and employ the emcee library <cit.> for parallelized MCMC sampling. Our analysis employs 100 walkers and 1000 steps to ensure robust and reliable results. By combining information from both the Hubble H(z) data and the Pantheon sample of SNe Ia, we extract valuable insights into the model parameters and their associated uncertainties, enhancing our understanding of the universe's expansion history.

We define the χ^2 function for the joint H(z)+SNe Ia dataset as a measure of the goodness-of-fit for our model,

    χ^2_joint=χ^2_H(z)+χ^2_SNe Ia,

where

    χ^2_H(z) = ∑_i=1^31[H(θ_s, z_i)-
    H_obs(z_i)]^2/σ(z_i)^2,

and

    χ^2_SNe Ia =∑_i,j=1 ^1048Δμ_i(
    C_SNe Ia^-1)_ijΔμ_j.


In this context, H(z_i) denotes the theoretical value of the Hubble parameter for a specific model at different redshifts z_i, H_obs(z_i) denotes the observed value of the Hubble parameter, σ(z_i) denotes the observational error, Δμ_i=μ_ th-μ_ obs denotes the difference between the theoretical and observed distance modulus, and C_SNe Ia^-1 is the inverse of the covariance matrix of the Pantheon sample, which characterizes the correlations between different measurements. For further details on these definitions and their application, please refer to the following references: <cit.>.



 §.§ Result analysis and discussions


The model parameters are constrained by minimizing their respective χ^2 values using MCMC with the emcee library. The outcomes are summarized in Tab. <ref>, which showcases the best-fit values and their associated uncertainties. Furthermore, Fig. <ref> displays the 1-σ and 2-σ contour plots for the combined observational data. These visual representations illustrate the relationships between parameters and identify the favored regions of parameter space based on the data. Here, we derive the Hubble constant as H_0 = 68.1_-1.5^+1.5 km/s/Mpc by combining observational data from H(z) and SNeIa. Recent studies have highlighted discrepancies, known as the Hubble tension, between the measurements of the Hubble parameter from the Planck satellite and other independent cosmological probes. The Planck collaboration <cit.> estimated H_0 = 67.4 ± 0.5 km/s/Mpc, while the HST team found H_0 = 74.03 ± 1.42 km/s/Mpc <cit.>. This results in a significant tension of 4.4σ between the two measurements. Further discussions on the Hubble tension and potential solutions are presented in Ref. <cit.>. Comparing our results with those of the Planck collaboration <cit.>, we find a tension of 0.44σ in H_0 from Planck results <cit.> based on our analysis of H(z) and SNeIa. Additional investigations on the Hubble tension can be found in Refs. <cit.>.





The figures (Figs. <ref>, <ref>, and <ref>) depict the variations of the Hubble function H(z), the deceleration parameter q(z), and the jerk parameter j(z) as functions of redshift z. These plots were generated using the values of the model parameters derived from the combined H(z)+SNe Ia dataset, providing information about the evolution of these cosmological quantities across different redshifts (-0.5,3).

As depicted in Fig. <ref>, the Hubble function exhibits a monotonically increasing behavior with respect to redshift (or equivalently, a monotonically decreasing behavior with respect to time). The evolution of H(z) is notably influenced by the model parameters. In particular, when α=0, the model closely resembles the standard ΛCDM model in its behavior and characteristics.



The deceleration parameter q(z), as illustrated in Fig. <ref>, exhibits a significant dependence on the model parameters. The plot in Fig. <ref> demonstrates a smooth transition of q(z) from a decelerated phase (q>0) to an accelerated phase (q<0) of the universe's expansion at the transition redshift z_tr=0.5^+0.02_-0.02 for the best-fit model. However, it is important to note that there is an uninspired phase of the universe as z approaches -1, where q(z) diverges further from the de-Sitter phase for the combined dataset. These results are consistent with findings from several independent studies (see <cit.>), which suggest that the universe underwent a phase transition from decelerating to accelerating expansion at redshifts approximately between z_tr=0.5 and z_tr=1.



The jerk parameter, as depicted in Fig. <ref>, is often utilized to differentiate between various DE models. In this particular case, it exhibits consistently positive behavior throughout the entire history of the universe. The observations suggest that, in this model, the present value of j exceeds 1 for the combined H(z)+SNe Ia dataset, specifically j_0=1.4^+0.02_-0.01 <cit.>. Thus, based on the present model, it is apparent that the considered dynamic DE model is the most likely explanation for the ongoing acceleration and warrants further investigation. However, it is important to note that j(z) is not well constrained in this scenario, as it may be associated with the emergence of abrupt future singularities <cit.>. This aspect introduces additional complexity and highlights the need for more comprehensive studies to fully understand the dynamics of the universe in this context.





§ OM(Z) DIAGNOSTICS
 

In the field of cosmology, a geometric approach is utilized, where the Hubble parameter serves as a critical test for the ΛCDM model. Moreover, the Om(z) diagnostic is employed to effectively distinguish between different DE models and the ΛCDM model by observing the variation in the slope of Om(z) <cit.>. Its simplicity arises from its dependence solely on the first-order derivative of the scale factor, which makes it particularly easy to apply and interpret in cosmological studies. In the case of a spatially flat universe, its expression is given by

    Om( z) =( H( z) /H_0) ^2-1
    /( 1+z) ^3-1,

where H_0 represents the present value of the Hubble parameter. A positive or negative slope of the Om(z) diagnostic identifies a quintessence or phantom model, respectively. Moreover, a constant slope with respect to redshift indicates a DE model corresponding to the cosmological constant. 



Fig. <ref> illustrates that the Om(z) diagnostic exhibits a positive slope for the best-fit values of the model parameters constrained from the combined H(z) + SNe Ia dataset. Consequently, based on the Om(z) test, we can conclude that Λ(t)CDM model demonstrates phantom-type behavior, which refers to a scenario where the equation of state parameter for DE is less than -1.



§ CONCLUSION
 

The confirmed accelerated expansion of the universe is supported by various sets of cosmological observations. Several models proposed in the literature effectively explain the transition from a decelerated phase to the current accelerated phase <cit.>. In this paper, we present models of the FLRW universe that incorporate a time-varying cosmological term, Λ(t), to investigate this transition. Specifically, we have considered a power-law form for the cosmological term as a function of the scale factor: Λ(t)=Λ_0 a(t)^-α, where Λ_0 represents the present value of the cosmological term <cit.>. We have obtained an exact solution to Einstein's field equations within the framework of Λ(t)CDM cosmology. Further, we have determined the best-fit values of the model parameters using the combined H(z) + SNe Ia dataset, which are presented in Tab. <ref> and Fig. <ref>.

In addition to these results, we have conducted a comprehensive analysis of the cosmic expansion of the universe. Our findings reveal that the Hubble function demonstrates a consistently increasing trend with respect to redshift (see Fig. <ref>). Furthermore, our investigation of the deceleration parameter indicates a smooth transition of q(z) from a phase of decelerated expansion to accelerated expansion at the transition redshift z_tr=0.5^+0.02_-0.02 for the best-fit model obtained using the combined H(z) + SNe Ia dataset (see Fig. <ref>). Therefore, our results align with recent observational data that indicate the present-day acceleration of the universe. From Fig. <ref>, it is evident that the jerk parameter for the Λ(t)CDM model remains positive across the entire history of the universe.  Finally, the behaviour of the Om(z) diagnostic, as depicted in Fig. <ref>, suggests that our cosmological Λ(t)CDM model aligns with the phantom scenario. These results indicate that our model is in strong agreement with present-day observations.



§ ACKNOWLEDGMENTS

The authors thank the anonymous referees for their helpful comments that improved the
quality of the manuscript.



§ DATA AVAILABILITY STATEMENT

This article does not introduce any new data.



90

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

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

Komatsu/2011 E. Komatsu et al., Astrophys. J. Supp., 192, 18 (2011).

Planck/2014 Planck Collaboration XVI, Astron. Astrophys., 571, A16 (2014). 

Eisenstein2005 D. J. Eisenstein et al., Astrophys. J., 633, 560-574, (2005).

Percival/2007 W.J. Percival et al., Mon. Not. R. Astron. Soc. Lett., 381, 1053 (2007).

Farooq/2017 O. Farooq et al., Astrophys. J., 835, 26 (2017).

Yu_2018 H. Yu, B. Ratra, and F-Y Wang, Astrophys. J., 856, 3 (2018).

Peebles P.J.E. Peebles and B. Ratra, Rev. Mod. Phys., 75, 559 (2003).

Planck Planck Collaboration, Astron. Astrophys., 641,
A6 (2020).

Weinberg/1989 S. Weinberg, Rev. Mod. Phys., 61, 1 (1989). 

Steinhardt/1999 P.J. Steinhardt et al., Phys. Rev. Lett., 59, 123504 (1999).

RP B. Ratra and P.J.E. Peebles, Phys. Rev. D, 37, 3406 (1998).

M.S. M. Sami and A. Toporensky, Mod. Phys. Lett. A, 19, 1509 (2004).

M.S.-2 M. Sami et al., Phys. Lett. B, 619, 193 (2005).

T.C. T. Chiba et al., Phys. Rev. D, 62, 023511 (2000).

C.A. C. Armendariz-Picon et al., Phys. Rev. Lett., 85, 4438 (2000).

M.C. M. C. Bento et al., Phys. Rev. D, 66, 043507 (2002).

A.Y. A. Y. Kamenshchik et al., Phys. Lett. B, 511, 265 (2001).

T.P. T. Padmanabhan, Phys. Rev. D, 66, 021301 (2002).

F2 V. Fayaz, et al., Eur. Phys. J. Plus, 130, 1-12 (2015).

S10 H. Saadat, Int. J. Theor. Phys., 52, 1027-1032 (2013).

M2 J. Morais et al., Phys. Dark Universe, 15, 7-30 (2017).

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

H.K. H. Kleinert and H.J. Schmidt, Gen. Relativ. Gravit., 34, 1295 (2002).

Odintsov1 S.D. Odintsov, V.K. Oikonomou, Phys.Rev.D, 96,  10, 104049 (2017)

Odintsov2 S. Capozziello, S. Nojiri, S.D. Odintsov, Phys.Lett.B, 781,  99-106 (2018).

T1 S. Capozziello et al., Phys. Rev. D, 84,
043527 (2011).

T2 Di Liu, M. J. Reboucas, Phys. Rev. D, 86,
083515 (2012).

Q0 J. B. Jimenez et al., Phys. Rev. D, 98, 044048 (2018).

Q1 J.B. Jimenez et al., Phys. Rev. D, 101, 103507 (2020).

Oztas A.M. Öztaş, Mon. Not. Roy. Astron. Soc., 481, 2228-2234 (2018).

Vishwakarma R.G. Vishwakarma, Class. Quantum Gravity, 18, 1159-1172 (2001).

Overduin J.M. Overduin and F.I. Cooperstock, Phys. Rev. D, 58, 043506 (1998).

Azri1 H. Azri and A. Bounames, Int. J. Mod. Phys.D, 26, 1750060 (2017).

Azri2 H. Azri and A. Bounames, Gen. Relat. Gravit., 44, 2547-2561 (2012).

Szydlowski M. Szydłowski, Phys. Rev. D, 12, 123538 (2015).

Bruni M. Bruni, R. Maier and D. Wands, Phys. Rev. D, 105, 063532 (2022).

Papagiannopoulos G. Papagiannopoulos et al., Eur. Phys. J. C, 80, 55 (2020).

Benetti1 M. Benetti et al., J. Cosmol. Astropart. Phys., 12, 023 (2019).

Benetti2 M. Benetti et al., J. Cosmol. Astropart. Phys., 08, 014 (2021).

Sola2017 J. Solà, A. Gómez-Valent and J. de Cruz Pérez, Astrophys. J. 836, 43 (2017).

Peracaula2018 J. Solà Peracaula, J. de Cruz Pérez and A. Gomez-Valent, Mon. Not. Roy. Astron. Soc. 478, 4357-4373 (2018).

Tsiapi2019 P. Tsiapi and S. Basilakos, Mon. Not. Roy. Astron. Soc. 485, 2505-2510 (2019).

Mavromatos2021 N. E. Mavromatos and J. Solà Peracaula, Eur. Phys. J. ST 230, 2077-2110 (2021).

Peracaula2023 J. Solà Peracaula, A. Gomez-Valent, J. de Cruz Perez and C. Moreno-Pulido, Universe 9, 262 (2023).

Peracaula2021 J. Solà Peracaula, A. Gómez-Valent, J. de Cruz Perez and C. Moreno-Pulido, EPL 134, 19001 (2021).

Peracaula2018b
J. Solà Peracaula, J. de Cruz Pérez and A. Gómez-Valent, EPL 121, 39001 (2018).

Pulido2020 C. Moreno-Pulido and J. Solà, Eur. Phys. J. C 80, 692 (2020).

Pulido2022a
C. Moreno-Pulido and J. Solà Peracaula, Eur. Phys. J. C 82, 551 (2022).

Pulido2022b
C. Moreno-Pulido and J. Solà Peracaula, Eur. Phys. J. C 82, 1137 (2022).

Pulido2023
C. Moreno-Pulido, J. Solà Peracaula and S. Cheraghchi, Eur. Phys. J. C 83, 637 (2023).

Peracaula2022
J. Solà Peracaula, Phil. Trans. Roy. Soc. Lond. A 380, 20210182 (2022).

Overduin1 J.M. Overduin and F.I. Cooperstock, Phys. Rev. D, 58, 043506 (1998).

Rezaei M. Rezaei, M. Malekjani, and J.S. Peracaula, Phys. Rev. D, 100, 023539 (2019).

Lopez J. L. Lopez and D. V. Nanopoulos, Mod. Phys. Lett. A, 11, 1-7 (1996).

Abdel A.-M. M. Abdel-Rahman, Phys. Rev. D, 45, 3497 (1992).

Chen W. Chen and Y.-S. Wu, Phys. Rev. D, 41, 695 (1990).

Calvao M. O. Calvao et al., Phys. Rev. D, 45, 3869 (1992).

Freese K. Freese et al., Nucl. Phys. B, 287, 797 (1987).

Gasperini M. Gasperini, Phys. Lett. B, 194, 347-349 (1987).

Overduin2 J. M. Overduin, P. S. Wesson and S. Bowyer, Astrophys. J., 404, 1 (1993).

Ryden B. Ryden, Introduction to Cosmology; Addison Wesley: San Francisco CA, USA (2003).

Myrzakulov N. Myrzakulov, M. Koussour and D.J. Gogoi, Eur. Phys. J. C, 83,  594 (2023).

Macedo H.A.P. Macedo et al., Eur. Phys. J. C, 83, 1144 (2023).

Moreno C. Moreno-Pulido and J. Solà Peracaula, Eur. Phys. J. C, 82, 1137 (2022).

Socorro J. Socorro, M. D'oleire, and Luis O. Pimentel, Astrophys. Space Sci., 360, 20 (2015).

SolaPeracaula2023Arxiv J. Solà Peracaula, arXiv preprint arXiv:2308.13349 (2023).

Sahni:2002fz V. Sahni et al., JETP Lett. 77, 201-206 (2003).

Visser:2004bf  M. Visser, Gen. Relativ. Gravit., 37, 1541-1548 (2004).



Moresco/2015 M. Moresco, Month. Not. R. Astron. Soc., 450, L16-L20 (2015).

Sharov/2018 G.S. Sharov, V.O. Vasilie, Mathematical Modelling and Geometry 6, 1 (2018).

Scolnic/2018 D.M. Scolnic et al., Astrophys. J, 859, 101
(2018).

Chang/2019 Z. Chang et al., Chin. Phys. C, 43, 125102 (2019).

Mackey/2013 D. F. Mackey et al., Publ. Astron. Soc. Pac. 125, 306 (2013).

Koussour1 M. Koussour et al. , Results Phys., 55, 107166 (2023).

Koussour2 M. Koussour et al., Phys. Dark Universe, 42, 101339 (2023).

Koussour3 M. Koussour et al., Eur. Phys. J. C, 83, 1-14 (2023).

Koussour4 N. Myrzakulov et al., Eur. Phys. J. Plus, 138, 852 (2023).

Koussour5 N. Myrzakulov et al., Chinese Phys. C, 47, 115107 (2023).

Riess:2019cxk A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Astrophys. J., 876, 85 (2019).

DiValentino E. D. Valentino et al., Astropart. Phys., 131, 102605 (2021).

Yang:2021eud W. Yang et al., Phys. Rev. D, 104, 063521 (2021).

DiValentino1 E. D. Valentino, S. Pan, W. Yang, and L. A. Anchordoqui, Phys. Rev. D, 103, 123527 (2021).

Magana J. Magana et al., JCAP, 017, 1410 (2014).

Mamon1 A. A. Mamon and S. Das, Eur. Phys. J. C, 75, 244 (2015).

Mamon2 A. A. Mamon and S. Das, Int. J. Mod. Phys. D, 25, 1650032 (2016).

Mamon3 A. Al Mamon and K. Bamba, Eur. Phys. J. C, 78, 862 (2018).

Dabrowski M. P. Dabrowski, Phys. Lett. B, 625, 184-188 (2005).

Pan S. Pan et al., Mon. Not. R. Astron. Soc., 477, 1189-1205 (2018).

Sahni V. Sahni, A. Shafieloo and A. A. Starobinsky, Phys. Rev. D, 78, 103502 (2008).


