Modified power law cosmology: theoretical scenarios and observational constraints

Figures(19) / Tables(1)

Get Citation
L.K. Sharma, Suresh Parekh, Saibal Ray, Anil Kumar Yadav, Maxim Khlopov and Kalyani C.K. Mehta. Modified power law cosmology: theoretical scenarios and observational constraints[J]. Chinese Physics C.
L.K. Sharma, Suresh Parekh, Saibal Ray, Anil Kumar Yadav, Maxim Khlopov and Kalyani C.K. Mehta. Modified power law cosmology: theoretical scenarios and observational constraints[J]. Chinese Physics C. shu
Milestone
Received: 2024-05-07
Article Metric

Article Views(272)
PDF Downloads(10)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Modified power law cosmology: theoretical scenarios and observational constraints

    Corresponding author: L.K. Sharma, lokesh.sharma@gla.ac.in
    Corresponding author: Suresh Parekh, thesureshparekh@gmail.com
    Corresponding author: Saibal Ray, saibal.ray@gla.ac.in
    Corresponding author: Anil Kumar Yadav, abanilyadav@yahoo.co.in
    Corresponding author: Maxim Khlopov, khlopov@apc.in2p3.fr
    Corresponding author: Kalyani C.K. Mehta, kalyani.c.k.mehta@gmail.com
  • 1. Department of Physics, GLA University, Mathura 281406, Uttar Pradesh, India
  • 2. Department of Physics, SP Pune University, Pune 411007, Maharastra, India
  • 3. Centre of Cosmology, Astrophysics and Space Science (CCASS), GLA University, Mathura 281406, Uttar Pradesh, India
  • 4. Department of Physics, United College of Engineering and Research, Greater Noida 201 306, India
  • 5. Institute of Physics, Southern Federal University, 194 Stachki, Rostov-on-Don 344090, Russian Federation & National Research Nuclear University, MEPHI, Moscow, Russian Federation & Virtual Institute of Astroparticle Physics 10, rue Garreau, 75018 Paris, France
  • 6. Department of Physics, Eberhard Karls University of Tubingen, Germany

Abstract: This research paper examines a cosmological model in flat space-time via $ f(R,G) $ gravity where R and G are respectively the Ricci scalar and Gauss-Bonnet invariant. Our model assumes that $ f(R,G) $ is an exponential function of G combined with a linear combination of R. We scrutinize the observational limitations under a power law cosmology that relies on two parameters - $ H_0 $, the Hubble constant, and q, the deceleration parameter, utilizing the 57-point $ H(z) $ data, 8-point BAO data, 1701-point Pantheon plus data, joint data of $ H(z) $ + Pantheon, and joint data of $ H(z) $ + BAO + Pantheon plus. The outcomes for $ H_0 $ and q are realistic within observational ranges. We have also addressed Energy Conditions, $ Om(z) $ analysis and cosmographical parameters like Jerk, Lerk and Snap. Our estimate of $ H_0 $ is remarkably consistent with various recent Planck Collaboration studies that utilize the ΛCDM model. According to our study, power law cosmology within the context of $ f(R,G) $ gravity provides the most comprehensive explanation of the important aspects of cosmic evolution.

    HTML

    I.   INTRODUCTION
    • Strong evidence for the universe's accelerated expansion is provided by several current standard observations, e.g. type Ia Supernovae (SNIa) [1, 2], the cosmic microwave background (CMB) [3] radiation, the Planck satellite [4]. It is observed that modified gravity could provide a more accurate description of the universe's accelerating expansion. As far as we are aware, modified gravity offers a straightforward gravitational substitute for the dark energy paradigm. These theories of dark energy are based on expanding the Einstein-Hilbert action with gravitational components. This has the effect of altering the universe's evolution, either early or late. In modified gravity, there are numerous examples of these models in the literature [57]. The Universe expanded at an incredibly fast rate during the inflationary phase [8]. During the inflationary era, the size of the Universe increases with an exponential rate, and hence the Universe expanded quite quickly and gained a larger size in a relatively small time. In Ref. [6], the various forms of inflationary theories are investigated in context of modified gravity. In general the bouncing cosmological models are used to describe the phenomenon of early universe, alternative to the inflationary scenario [9, 10]. Recently Mukherjee et al [11] have mapped Einstein and Jordan frames where the Einstein frame Universe describes the late-time evolution and investigated that a perturbed stable bounce is also possible for late-time evolution of the Universe. A uniform description of this could be provided by modified gravity. A phantom fluid or field is needed to explain the accelerated expansion in standard general relativity.

      Modified gravity is another explanation for the universe's late-time acceleration. In the initial phase $ f(R) $ gravity has been exploited with R in the Einstein-Hilbert action which is the scalar curvature. This notion is easy to understand, workable and very effective. However, at present General Relativity (GR) has numerous variations. As a result we obtain $ f(R,T) $ theory if the Lagrangian is a function of both R and the energy-momentum tensor's trace T [1233]. In this gravity theory, some of the basic aspects are as follows: (i) here the trace of the energy-momentum tensor T and (ii) the Ricci scalar R which have considerable intrinsic features to the matter Lagrangian. Moreover, the quantum field effect as well as the particle creation potentiality are some other attributes to $ f(R,T) $ gravity. All these aspects of modified gravity theories are available in the following review work [34]. In order to account for heat conduction, viscosity and quantum effects, the T term is introduced.

      There is another explanation for the late time cosmic acceleration. Observational restrictions have been applied to $ f(R,T) $ gravity. However, $ f(R,G) $ gravity presents an intriguing substitute for $ f(R) $ gravity and G refers to the Gauss-Bonnet invariant. Numerous studies in the literature demonstrate that inflation and late-time acceleration can be explained by $ f(R,G) $ gravity [3549]. In particular, the authors of Ref. [49] formulated the relaxed Universe within the context of modified gravity and investigated an unconventional approach for addressing the old cosmological constant problem viz fine tuning in a class of F(R,G) models. Here, we confine our-self for analyzing the fate of the Universe and its dynamics at present epoch in the framework of $ f(R,G) $ gravity. Also, the authors of Ref. [50] have investigated a vacuum structure for scalar cosmological perturbations in $ f(R,G) $ gravity and found a new instability which can arise with in the structure if the background is not de-Sitter. The scalar type cosmological perturbations for $ f(R,G) $ gravity with a single scalar field are given in Ref. [51].

      Understanding of the late-time acceleration of the Universe is largely dependent on the mainstream cosmological model. Furthermore, it is important to note that finding the correct cosmological model of late time acceleration is still a tough task at all. The late-time acceleration era is known as the dark energy era [5255], and up to date, there are many proposals that try to model the late time phenomenon with using a scalar field [56, 57], while other models use alternative gravity in its numerous forms [58]. The age, horizon, and fuzziness issues in the standard model are successfully resolved by models based on a power-law of the scale factor [5962]. In general the expansion rate of the Universe is described by the Hubble constant $ H_{0} $. In the recent past, we have observed the statistically significant tensions in $ H_{0} $ which refers to the difference between its direct local distance ladder measurements and consideration of the standard ΛCDM model. For example there is approximately 4.4 σ tension in value of $ H_{0} $ determined by SH0ES measurement $ H_0 = 73.04\pm 1.04{\rm \,km\, s^{-1}\, Mpc^{-1}} $ (68% CL) [63] and $ H_0 = 67.27\pm 0.60{\rm \,km\, s^{-1}\, Mpc^{-1}} $ (68% CL) [4]. This discrepancy in the value of $ H_{0} $ is referred as $ H_{0} $ tension. Some important researches on $ H_{0} $ tension are given in Refs. [6478].

      In view of the above mentioned motivation, the plan of the present study is outlined as: in Section 2, a brief mathematical overview of the metric and $ f(R,G) $ gravity theory along with the solution to the field equations have been provided. In Section 3, observational analysis has been executed within the observational constraints of the model parameters. The physical parameters involved in the model are presented by the help of plots in Section 4. At last, Section 5 is designed for relevant comments on the entire investigation.

    II.   THE ACTION AND COSMOLOGICAL SOLUTIONS

      A.   Field Equations

    • In four-dimensional space-time, the modified Gauss-Bonnet gravity operates as

      $ \begin{aligned} S = \int\left[\frac{f(R,G)}{2\kappa}\right]\sqrt{-g}d^4x + S_m, \end{aligned} $

      (1)

      where $ \kappa = 8 \pi G $ and $ S_m $ is the matter Lagrangian which depends upon $ g_{\mu\nu} $ and matter fields. The Gauss-Bonnet invariant G is defined as $ G = R^2 + R_{\mu\nu\alpha\zeta}R^{\mu\nu\alpha\zeta} - 4R_{\mu\nu}R^{\mu\nu} $. The Gauss-Bonnet invariant is obtained from $ R_{\mu\nu\alpha\zeta} $, $ R_{\mu\nu} = R^{\zeta}_{\mu\zeta\nu} $ and $ R = g^{\alpha\zeta}R_{\alpha\zeta} $.

      From Eq. (1), the gravitational field equations are derived as

      $ \begin{aligned}[b] R_{\mu\nu} & - \frac{1}{2} F(G) + (2RR_{\mu\nu} - 4R_{\mu\alpha}R^{\alpha}_{\nu} + 2R^{\alpha\zeta\tau}_{\mu}R_{\nu\alpha\zeta\tau} \\&- 4g^{i\alpha}G^{i\zeta}R_{\mu i\nu j}R_{\alpha\zeta})F'(G) \\ & + 4[\nabla_{\alpha}\nabla_{\nu}F'(G)]R^{\alpha}_{\mu} - 4g_{\mu\nu}[\nabla_{\alpha} \nabla_{\zeta} F'(G)] R^{\alpha\zeta} \\&+ 4 [\nabla_{\alpha} \nabla_{\zeta} F'(G)] g^{i\alpha} g^{j\zeta} R^{\mu}_{i\nu j} \\ & + 2g_{\mu\nu} [\square F'(G)] R - 2 [\nabla_{\mu} \nabla_{\nu} F'(G)] R \\&- 4[\square F'(G)]R_{\mu\nu} + 4[\nabla_{\mu}\nabla_{\alpha}F'(G)]R^{\alpha}_{\nu} = \kappa T^{m}_{\mu\nu}, \end{aligned} $

      (2)

      where $ T^{m}_{ij} $ is the energy momentum tensor arising from $ S_m $.

      The flat FLRW space-time metric is

      $ \begin{aligned} ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2), \end{aligned} $

      (3)

      where the symbols have their usual meanings.

      Now, we calculate the Einstien field equations using Eqs. 2 and 1 as

      $ \begin{aligned} F(G) + 6H^2 - GF'(G) + 24H^3 \dot{G}F''(G) = 2\kappa\rho, \end{aligned} $

      (4)

      $ \begin{aligned}[b] &6H^2 + 4\dot{H} + F(G) + 16H\dot{G}(\dot{H} + H^2)F''(G) - GF'(G) \\&+ 8H^2\ddot{G}F''(G) + 8H^2\dot{G}^2 F'''(G) = -2\kappa p, \end{aligned} $

      (5)

      where $ H = \dfrac{\dot{a}(t)}{a(t)} $ is the Hubble parameter and $ \dot{a}(t) \equiv \dfrac{da}{dt} $.

      Also, we have

      $ \begin{aligned} R = 6(2H^2 + \dot{H}), \end{aligned} $

      (6)

      $ \begin{aligned} G = 24H^2(H^2 + \dot{H}). \end{aligned} $

      (7)

      In the present model, we are taking $ F(R,G) = R + f(G) $. The function $ f(G) $ has the possibility to describe the inflationary era and to yield a transition from early deceleration phase to late time acceleration, as well as a natural crossing of the phantom divide. In Literature, various choices of $ f(G) $ are found there of, $ f(G) = f_{0}G^{\beta} $ [79] with constant $ f_{0} $ and β. Here, we assume $ f(G) = \alpha e^{-G} $ [80] with constant $ \alpha > 0 $ and G is Gauss-Bonnet invariant. Therefore

      $ \begin{aligned} F(R,G) = R + \alpha e^{-G}. \end{aligned} $

      (8)

      The second term of Eq. (8) dominates over the Einstein's term R for $ \alpha > 0 $.

    • B.   Power law cosmology

    • We have a system of four equations (4) – (8) with five unknown variables, namely H, G, $ F(G) $, ρ and p. Thus, one can not solve these equation in general therefore in order to get explicit solution, we need at least one physical assumption among unknown parameters. In the literature, it is common to use the law o variation of Hubble's parameter which yield the power law form of scale factor [81] as follows:

      $ \begin{aligned} a(t) = a_0 \left(\frac{t}{t_0}\right)^{\zeta}, \end{aligned} $

      (9)

      where $ a_0 $ represents the current value of the scale factor and ζ is a dimensionless constant.

      This form of $ a(t) $ describes the power law cosmology and consistent with the late time acceleration of the Universe. Some useful applications of power law cosmology are read in Refs. [82, 83]. Moreover, using the definitions $ a = \dfrac{a_{0}}{1+z} $ and $ H = \dfrac{\dot{a}}{a} = -\dfrac{1}{1+z}\dfrac{dz}{dt} $ in Eq. (9), we obtain the following form of $ H(z) $ in term of z which is

      $ \begin{aligned} H = H_0 (1+z) ^{\frac{1}{\zeta}}. \end{aligned} $

      (10)

      where $ H_{0} $ denotes the present value of Hubble parameter.

      It is worthwhile to note that by bounding Eq. (10) with OHD, BAO and Pantheon plus compilation of SN Ia data sets, we have constrained the values of $ H_{0} $ and ζ in subsequent section.

      We take into account cosmological characteristics like the pressure, the energy density, the EOS parameter, the Hubble parameter, the deceleration parameter, etc. to comprehend the history of the universe. A dimensionless variable known as the deceleration parameter may be used to calculate the universe's acceleration or deceleration phase. The definition of the deceleration parameter q is

      $ \begin{aligned} q = - \frac{\ddot{a}}{aH^{2}}. \end{aligned} $

      (11)

      Now, the following three cases may arise: (i) if $ q > 0 $, then the phase of the universe is decelerating, (ii) if $ q < 0 $, it is accelerating and (iii) if $ q = 0 $, it is expanding continuously. As a result, Eqs. (9) – (11) provide

      $ \begin{aligned} q = \frac{1}{\zeta} - 1. \end{aligned} $

      (12)

      So, using the deceleration parameter q and redshift, we can describe the Hubble parameter as follows

      $ \begin{aligned} H(z) = H_0 (1+z)^{(1+q)}. \end{aligned} $

      (13)

      The motivation of assuming H in form of Eq. (10) or Eq. (13) for the following two reasons: i) This form is comparable to the standard ΛCDM model $ H = H_{0}\left[ \Omega_{m}(1+Z)^{3} + \Omega_{\Lambda}\right]^{\frac{1}{2}} $ because for $ z = 0 $ and $ \Omega_{m} + \Omega_{\Lambda} = 1 $, $ H = H_{0} $ while parameterizing (13), one can obtain $ H = H_{0} $ for $ z = 0 $ and ii) this parameterization also leads to the acceleration behavior of the universe as $ q = \dfrac{1}{\zeta} - 1 $ for $ \zeta > 1 $.

      Let us now obtain the expressions for the energy density and the pressure by solving Eqs. (4) and (5), which are given as

      $ \begin{aligned} \rho = \frac{\alpha e^{24H_0^{4}q(z+1)^{4q+4}}(24H_0^{4}q(z+1)^{4q+4}(96H_0^{4}(q+1)(z+1)^{4q+4}-1)+1)}{2\kappa} + \frac{6H_0^{2}(z+1)^{2q+2}}{2\kappa}, \end{aligned} $

      (14)

      $ { \begin{array}{l} p = \dfrac{\alpha e^{24H_0 ^{4}q(z+1)^{4q+4}}(24H_0 ^{4}q(z+1)^{4q+4}(3072H_0 ^{8}q(q+1)^{2}(z+1)^{8q+8}+ 16H_0 ^{4} (q+1)(9q+5)(z+1)^{4q+4}+1)-1}{2\kappa} + \dfrac{2H_0^2 (2q-1)(z+1)^{2q+2}}{2\kappa},\\ \omega = \dfrac{p}{\rho} \end{array}} $

      (15)

      It should be noted that α is a positive constant and we have chosen $ \alpha = 1 $ for the graphical analysis of the physical parameters of the proposed model and $ \kappa = 8\pi G_{N} = 1 $. The values of $ H_0 $ and q are obtained by bounding Eq. (13) with OHD, BAO and Pantheon compilation of SN Ia data sets using the Markov Chain Monte Charlo (MCMC) method along with minimizing the technique $ \chi^{2} $.

    III.   OBSERVATIONAL CONSTRAINTS ON MODEL PARAMETERS
    • In this section, observational data sets are utilized to restrict the values of $ H_0 $ and q that occur in the tilted Hubble parametrization. In this model, we employ the $ H(z) $, BAO, and Pantheon data sets, as well as their combined data collections. The $ H(z) $ data points are given in [84]. The information on BAO and Pantheon compilation of SN Ia data are sourced from [89] and [9093] respectively.

    • A.   Observed Hubble Data (OHD) set

    • We utilized the 57-point OHD data from [84]. To ensure robust parameter estimation and account for systematic effects in the Cosmic Chronometer (CC) data, we include the full covariance matrix as provided by [85]. More details of cosmic chronometer covariance estimate are available at https://gitlab.com/mmoresco/CCcovariance. This covariance matrix accounts for systematic correlations between redshift bins due to common calibrations, assumptions in stellar population synthesis models, and other potential sources of uncertainty. To achieve reliable parameter constraints, we modify the chi-squared function to incorporate the covariance matrix as follows:

      $ \chi^2_{\mathrm{CCh}} = (\mathbf{H}_{\text{obs}} - \mathbf{H}_{\text{model}})^T \cdot \mathbf{C}^{-1} \cdot (\mathbf{H}_{\text{obs}} - \mathbf{H}_{\text{model}}), $

      where $ \mathbf{H}_{\text{obs}} $ and $ \mathbf{H}_{\text{model}} $ represent the observed and theoretical Hubble parameter values, respectively, and $ \mathbf{C} $ denotes the covariance matrix. This framework ensures a statistically rigorous comparison of observed data with theoretical predictions, accounting for systematic correlations and enhancing the robustness of parameter estimation.

      The CC method offers a model-independent approach to measuring the Hubble parameter, $ H(z) $, as a function of redshift by utilizing the differential ages of passively evolving galaxies ($ dt $). Redshift measurements, derived from spectroscopy of extragalactic objects, achieve high precision ($ \delta z / z \leq 0.001 $). However, the primary challenge lies in accurately estimating $ dt $, which relies heavily on well-constrained stellar population synthesis models [86]. This technique directly probes the Universe’s expansion history without assuming any prior cosmological model, making it a valuable tool for observational cosmology.

      In this work, we adopt the methodology outlined in [85], incorporating statistical and systematic corrections to ensure reliable results. The CC data used in our analysis span redshifts from 0.07 to 1.26, capturing the Universe's expansion dynamics across a crucial epoch. By combining these observations with the covariance matrix, we address systematic correlations, thereby providing robust constraints on cosmological parameters such as $ H_0 $ and q.

      This integration of CC data with a covariance matrix not only enhances the statistical rigor of our analysis but also aligns with best practices in the field [85, 87]. Our approach ensures that both statistical and systematic uncertainties are rigorously addressed, paving the way for reliable insights into the Universe's expansion history.

    • B.   BAO Data set

    • Let us utilize the BAO data to evaluate and verify the probable predictions of our cosmological models at various redshift values. This will obviously offer a unique method to examine the expansion parameters of the presently accelerating universe at low redshift values. Here the BAO dataset has been obtained from the current surveys, e.g. 6dFGS, SDSS and WiggleZ, spanning in the specific redshift range $ 0.106 < z < 0.73 $. The basic idea behind this is as follows: the dimensionless amount serves to obtain a clear-cut indication of the primordial baryon-photon acoustic oscillations in the matter power spectrum. Hence

      $ \begin{aligned} A(z) = \sqrt{\Omega_m}[H(z_i)/H_0]^{-1/3} \left[ \frac{1}{z_i} \int_{0}^{z_i} \frac{H_0}{H(z)}dz \right]^{2/3}. \end{aligned} $

      (16)
    • C.   Pantheon plus (PP)

    • For the redshift range of $ 0.001 < z < 2.26 $, we utilize the Pantheon plus data compilation [88]. The Pantheon+ analysis of 1701 light curves of 1550 distinct Type Ia supernovae (SNe Ia) ranging in redshift from z = 0.001 to 2.26. The investigation of the expansion rate heavily relies on SNe Ia.

      To assess the theoretically expected apparent magnitude (m) and absolute magnitude ($ M_b $) with respect to colour and stretch, we thus compute the distance modulus $ mu_Th(z_i) $ as follows:

      $ \begin{aligned} \mu(z) = -M_{b} + m = \mu_0 + 5logD_L(z), \end{aligned} $

      (17)

      where $ D_L(z) $ and $ \mu_0 $ are respectively the luminosity distance and the nuisance parameter. Furthermore, the absolute magnitude $ M_{b} $ is fixed or treated as a free parameter because of its strong correlation $ H_0 $.

      Therefore, $ \mu_{0} $ in Eq. (17) is read as

      $ \begin{aligned} \mu_0 = 5log\left(\frac{H_0^{-1}}{1Mpc}\right) + 25, \end{aligned} $

      (18)

      Thus, $ D_{L} $, in present case, for a geometrically flat universe reads as

      $ \begin{aligned} D_{L} = (1+z)\int^{z}_{0}\frac{H_{0}}{H(z^{\prime})}dz^{\prime} \end{aligned} $

      (19)

      Now, the minimum $ \chi^2 $ function is given as

      $ \begin{aligned} \chi^2_{PP}(H_0,q) = \sum_{i=1}^{1701}\left[\frac{\mu_{th}(H_0,q,z_i) - \mu_{obs}(z_i)}{\sigma_\mu(z_i)}\right]^2. \end{aligned} $

      (20)
    • D.   Joint OHD + Pantheon plus Data set

    • By performing a joint statistical analysis using $ H(z) $ and Pantheon data sets, we obtain stronger constraints. Therefore, the chi-sq function for joint data sets can be written as

      $ \begin{aligned} \chi^2_{Joint} = \chi^2_{OHD} + \chi^2_{PP}. \end{aligned} $

      (21)
    • E.   Joint OHD + BAO + PP Data set

    • By performing a joint statistical analysis using $ H(z) $, BAO and Pantheon data sets, we obtain even stronger and more reliable constraints. Therefore, the chi-sq function for joint data sets can be written as

      $ \begin{aligned} \chi^2_{Joint} = \chi^2_{OHD} + \chi^2_{BAO} + \chi^2_{PP}. \end{aligned} $

      (22)
    IV.   RESULTS UNDER THE $ f(R,G) $ GRAVITY MODEL

      A.   Parameter Estimation

    • The two-dimensional contour plots for $ H_0 $ and q parameter using the data sets OHD, BAO, Pantheon and their combinations OHD + Pantheon and OHD + BAO + Pantheon are shown in the Figs. 1, 2, 3, 4 and 5 respectively. Their combined nature can be seen in Fig. 6. The obtained values of $ H_0 $ and q by implementing the observational data sets are presented in Table 1. We observe that the obtained values of q at present epoch are $ \sim -0.1 $ but its value is quite small in other investigations [94]. It is worthwhile to note that we obtain $ H_{0} \sim 68 $ by restricting the proposed model with various cosmological data sets via MCMC method along with minimizing $ \chi^{2} $ technique. In the two decades since, $ H_{0} $ measurements with smaller error bars are available i) $ H_{0} = 67.9 \pm 1.5 $ km/s/Mpc from Planck collaboration [90] and ii) $ H_{0} = 73.04 \pm 1.04 $ km/s/Mpc from the supernovae and $ H_{0} $ for equation of state dark energy (SH0ES) project [95]. It is worthwhile to mention that Mehrabi and Rezaei [96] have constrained $ H_{0} \sim 72 $ by utilizing SNIa data and shown it consistency with ΛCDM model while in this paper, our $ H_{0} $ is closure to the Planck result [90] and a little far from Ref. [96] due to inconsistency in expression of $ H(z) $. Therefore, in spite of describing the late time acceleration of the Universe, the power-law cosmology is not a complete package to study the whole dynamics and eventual fate of the Universe.

      Figure 1.  (color online) In this figure we have exhibited one-dimensional marginalized distribution and two-dimensional contours by using the $ H(z) $ dataset.

      Figure 2.  (color online) In this figure we have exhibited one-dimensional marginalized distribution and two-dimensional contours by using the BAO dataset.

      Figure 3.  (color online) In this figure we have exhibited one-dimensional marginalized distribution and two-dimensional contours by using the Pantheon dataset.

      Figure 4.  (color online) In this figure we have exhibited one-dimensional marginalized distribution and two-dimensional contours by using the combination of $ H(z) $ and Pantheon+ dataset.

      Figure 5.  (color online) In this figure we have exhibited one-dimensional marginalized distribution and two-dimensional contours by using the combination of $ H(z) $, BAO and Pantheon dataset.

      Figure 6.  (color online) In this figure we have exhibited one-dimensional marginalized distribution and two-dimensional contours by using the combined variability across all dataset combinations.

      Parameter H(z) BAO Pantheon plus $ H(z)_1 $ $ H(z)_2 $
      $ H_0 $ $ 66.998^{+0.094}_{-0.094} $ $ 67.003^{+0.100}_{-0.105} $ $ 69.990^{+0.096}_{-0.102} $ $ 68.989^{+0.102}_{-0.100} $ $ 67.995^{+0.103}_{-0.099} $
      q $ -0.614^{+0.113}_{-0.097} $ $ -0.606^{+0.085}_{-0.087} $ $ -0.611^{+0.107}_{-0.099} $ $ -0.617^{+0.096}_{-0.092} $ $ -0.104^{+0.098}_{-0.100} $

      Table 1.  The parameter values exstructed from different datasets (i.e. $ H(z) $, BAO, Pantheon, $ H(z)_1=H(z) $ + Pantheon, $ H(z)_2=H(z) $ + BAO + Pantheon). We have executed here MCMC and Bayesian analysis.

    • B.   Energy Conditions

    • Energy conditions (ECs) or similarly construct cosmic principles that explain the distribution of matter and energy across the universe. They are based on Einstein's gravitational equations and replicate the rules of the cosmos. These circumstances indicate the distribution of matter and energy in space. Hence the energy conditions can be expressed as follows:

      (i) Weak Energy Condition (WEC): $ \rho \geq 0 $, $ \rho+p \geq 0 $

      (ii) Null Energy Condition (NEC): $ \rho + p \geq 0 $,

      (iii) Strong Energy Condition (SEC): $ \rho + 3p \geq 0 $,

      (iv) Dominant Energy Condition (DEC): $ \rho - p \geq 0 $.

      All the energy conditions separately as well as jointly are shown in Figs. 15 - 19 by using the Bayesian analysis of the parameters. Except for SEC, our results show that NEC, WEC and DEC are all satisfied. The SEC violation is justified by the Universe's fastest growth. Therefore, the $ f(R,G) $ theory of gravity has potential to explain the current scenario of the late-time acceleration without any need for the cosmological constant as well as dark energy component in the energy budget of the Universe. It is to note that the distribution of the energy density ρ with respect to time t is shown in Fig. 7, whereas the distribution of the pressure is shown in Fig. 8.

      Figure 15.  (color online) In this figure we have exhibited the Weak Energy Conditions (WEC) vs the redshift (z) for all the combined datasets.

      Figure 19.  (color online) In this figure we have exhibited all the energy conditions vs time.

      Figure 7.  (color online) In this figure we have exhibited the dynamic variation of the energy density (ρ) over the redshift (z) under various parameter conditions derived from distinct combinations of the $ H(z) $, BAO and Pantheon datasets.

      Figure 8.  (color online) In this figure we have exhibited the dynamic variation of the pressure (p) over the redshift (z) under various parameter conditions derived from distinct combinations of the $ H(z) $, BAO and Pantheon datasets.

    • C.   Om(z) parameter

    • When assessing various dark energy hypotheses in academic works, researchers commonly use the state finder parameters $ r - s $ and the Om diagnosis. The important $ Om(z) $ parameter is formed when the Hubble parameter H and the cosmic redshift z combine which can be defined as

      $ Om(z) = \frac{[\dfrac{H(z)}{H_0}]^2 - 1}{(1+z)^3 - 1}, $

      (23)

      where $ H_0 $ corresponds to the current value of the Hubble parameter. According to Shahalam et al. [97], the negative, zero, and positive values of $ Om(z) $ indicate the quintessence ($ \omega \ge -1 $), ΛCDM, and phantom ($ \omega \le-1 $) dark energy (DE) hypotheses, respectively.

      The $ Om(z) $ parameter can be provided for our model as follows:

      $ Om(z) = \frac{(1+z)^{2/b}-1}{(1+z)^3-1}. $

      (24)
    • D.   Cosmographic Parameters

    • Many cosmological parameters, given as higher-order derivatives of the scalar component, are examined to comprehend the universe's expansion history better. As a result, these characteristics are extremely useful for investigating the dynamics of the cosmos. For example, the Hubble parameter H depicts the universe's expansion rate, the deceleration parameter q depicts the universe's phase transition whereas the jerk parameter j, snap parameter s and lerk parameter l are required to study dark energy theories and their dynamics. These are as follows:

      $ \begin{aligned} H = \frac{\dot{a}}{a} \end{aligned} $

      (25)

      $ \begin{aligned} q = \frac{\ddot{a}}{aH^2} \end{aligned} $

      (26)

      $ \begin{aligned} j = \frac{\dddot{a}}{aH^3} \end{aligned} $

      (27)

      $ \begin{aligned} s = \frac{\dddot{a}}{aH^4} \end{aligned} $

      (28)

      $ \begin{aligned} l = \frac{\ddddot{a}}{aH^5} \end{aligned} $

      (29)
    • E.   State Finder Diagnotics

    • Basically, state finder diagnostics help us to obtain hidden features of the status of dark energy and thus mysteries attached to the history of the universe. As we employ a cosmic compass, these diagnostics lead us through the complexities of cosmic evolution. The r and s parameters are used in state finder diagnostics. Using these characteristics, we can gain a better understanding of the evolution of the universe. Consider them cosmic metres that offer data on the expansion of the universe and its constituent components. These are basically dimensionless parameters which encapsulate the essence of the cosmic development and thus serve as a filter to aid in our understanding of the underlying dynamics of the universe.

      Now, the general mathematical expression for the required parameter, expressed in terms of H, is as follows:

      $ \begin{aligned} r = \frac{\ddot{\dot{a}}}{aH^3}, \end{aligned} $

      (30)

      whereas the equations for r and s in our model, when expressed in terms of q, become

      $ \begin{aligned} r = 2q^2 + q, \end{aligned} $

      (31)

      $ \begin{aligned} s = \frac{-1 + r}{3(-\dfrac{1}{2}+q)}. \end{aligned} $

      (32)

      The scale factor trajectories in the resulting model may be shown in Fig. 10 to follow a specific set of paths. Our strategy is consistent with the results for the cosmic diagnostic pair from power law cosmology. The pioneer investigation on state finders are given in Refs. [98109]. The evolutionary trajectory in $ r - s $ plane of state finders pairs helps us in enhancing the accuracy of the model and its classification among various type of dark energy models as discussed in the Refs. [98109]. The behaviours of proposed model $ r - q $, $ s - q $ and $ r - s $ planes have been depicted in Fig. 10. From Fig. 10, we observe that the proposed model behaves like SCDM model $ (r = 1, s = 1) $ at initial epoch and approaches towards ΛCDM model $ (r = 1, s = 0) $ at late time.

      Figure 10.  (color online) In this figure we have exhibited the features of the State Finder plots of $ r-q $, $ s-q $ and $ r-q $.

      Figure 9.  (color online) In this figure we have exhibited variation of the equation of state parameter (ω) vs the redshift (z) which demonstrates that dark energy contributes to the accelerated expansion of the universe, however with a bit variations with the redshift and thus potentially leading to interesting cosmological consequences.

    V.   DISCUSSION AND CONCLUSION
    • The motivation behind the research paper was to examine the Ricci scalar R and the Gauss-Bonnet invariant G to characterize a cosmological model in flat space-time via $ f(R,G) $ gravity. Here we wanted to investigate the observational limitations under a power law cosmology that relies on two parameters - $ H_0 $ (the Hubble constant) and q (the deceleration parameter) utilizing the 57-point $ H(z) $ data, 8-point BAO data, 1048-point Pantheon data, joint data of $ H(z) $ + Pantheon and joint data of $ H(z) $ + BAO + Pantheon. The outcomes for $ H_0 $ and q are realistic within observational ranges. As can be noted that our estimate of $ H_0 $ is remarkably consistent with various recent Planck Collaboration studies under the ΛCDM model.

      We have shown via several graphical demonstrations (Figs. 1 - 19 and Table 1) that the obtained values for $ H_0 $ by bounding the proposed model with OHD, BAO and Pantheon compilation of SN Ia data satisfactorily favor its corresponding value observed in the Plank collaboration [90]. Along with this graphical presentations, we have also analyzed the model by studying the energy conditions, the jerk parameter, the lerk parameter and the Om diagnostics as well as the state finder diagnostic tools. According to our study, the power law cosmology within the context of $ f(R,G) $ gravity provides the most comprehensive explanation for the important aspects of cosmic evolution. Furthermore, at the final stage of this paper, we have seen the work of Singh et al. [80] that prescribes for a cosmological model with power law under the framework of modified theory with higher order curvature term. They [80] have obtained $ H_{0} = 68.119^{+0.028}_{-0.12} $ $ km\; s^{-1}Mpc^{-1} $, $ q = -0.109^{+0.014}_{-0.014} $; $ H_{0} = 70.5^{+1.3}_{-0.98} $ $ km\; s^{-1}Mpc^{-1} $, $ q = -0.25^{+0.15}_{-0.15} $ and $ H_{0} = 69.103^{+0.019}_{-0.10} $ $ km\; s^{-1} Mpc^{-1} $, $ q = -0.132^{+0.014}_{-0.014} $ by using $ H(z) $ data, Pantheon plus compilation of SN Ia data and joint data of H(z) + Pantheon plus respectively. In this paper, the constrained values from the proposed model are as follows: $ H_0 = 68.001^{+0.093}_{-0.087} $ km s$ ^{-1} $ Mpc$ ^{-1} $, $ q = -0.106^{+0.009}_{-0.009} $; $ H_0 = 67.973^{+0.101}_{-0.105} $ km s$ ^{-1} $ Mpc$ ^{-1} $, $ q = -0.099^{+0.010}_{-0.010} $; $ H_0 = 67.995^{+0.086}_{-0.111} $ km s$ ^{-1} $ Mpc$ ^{-1} $, $ q = -0.100^{+0.010}_{-0.010} $; $ H_0 = 67.980^{+0.012}_{-0.097} $ km s$ ^{-1} $ Mpc$ ^{-1} $, $ q = -0.110^{+0.010}_{-0.011} $; $ H_0 = 68.018^{+0.093}_{-0.104} $ km s$ ^{-1} $ Mpc$ ^{-1} $, $ q = -0.104^{+0.010}_{-0.011} $ by using $ H(z) $ data, 8-point BAO data, 1048-point Pantheon data, joint data of $ H(z) $ + Pantheon and joint data of $ H(z) $ + BAO + Pantheon respectively. As a suitable methodology the welknown and effective Markov Chain Monte Carlo (MCMC) has been uniquely employed in the present investigation.

      Figure 11.  (color online) In this figure we have exhibited variations of $ Om(z) $ with z across for different combined datasets by considering the β values obtained from each dataset.

      Figure 12.  (color online) In this figure we have exhibited the feature of the Jerk parameter j vs z. Here values of j at z = 0 are as follows: for $ H(z) $ = 7.127 $ s^{-3} $, for BAO = 6.663 $ s^{-3} $, for Pantheon = 6.731 $ s^{-3} $, for $ H(z) $ + Pantheon = 7.387 $ s^{-3} $ and for $ H(z) $ + BAO + Pantheon = 6.997 $ s^{-3} $

      Figure 13.  (color online) In this figure we have exhibited the feature of the Lerk parameter l vs z.

      Figure 14.  (color online) In this figure we have exhibited the feature of the Snap parameter s vs z.

      Figure 16.  (color online) In this figure we have exhibited the Null Energy Conditions (NEC) vs the redshift (z) for all the combined datasets.

      Figure 17.  (color online) In this figure we have exhibited the Strong Energy Conditions (SEC) vs the redshift (z) for all the combined datasets.

      Figure 18.  (color online) In this figure we have exhibited the Dominant Energy Conditions (DEC) vs the redshift (z) for all the combined datasets.

      At this point it is worthwhile to mention that the proposed model has been minimized $ H_{0} $ tensions and it is calibrated only $ 0.68 \sigma $, $ 1.19 \sigma $, $ 1.191 \sigma $, $ 1.23 \sigma $ and $ 1.17 \sigma $ for $ H(z) $ data, 8-point BAO data, 1048-point Pantheon data, joint data of $ H(z) $ + Pantheon and joint data of $ H(z) $ + BAO + Pantheon respectively when we analyzed the estimated values of $ H_{0} $ in this paper with the value of $ H_{0} $ obtained by Plank Collaboration [4]. Moreover, because of the constant value of deceleration parameter q in power-law cosmology, it could not able to describe the red-shift transition and the proposed model fails to explain the early deceleration phase of the universe. However, one can describes the early deceleration phase of the Universe by choosing appropriate value of ζ in $ q = \dfrac{1}{\zeta} - 1 $. But at the same time one can note that the model fails to explain the late-time acceleration of the Universe.

      Finally, even though we have obtained so many useful features, the power-law cosmology seems not a completely packagable technique to study to entire dynamics and eventual fate of the Universe. In this connection, however there may have some scopes to explore thermodynamical aspects, especially entropy, of the late-time acceleration of the Universe and following the works in the Refs. [80, 110, 111] may be tackled in future projects.

    Data Availability
    • In the present study, we have used available observational data only where no data has been produced in any new form.

    Conflict of Interest
    • The authors declare no conflict of interest.

    ACKNOWLEDGEMENT
    • The authors are thankful to the reviewer for his valuable comments and constructive suggestions. SR gratefully acknowledges support from the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India under its Visiting Research Associateship Programme as well as the facilities under ICARD, Pune at CCASS, GLA University, Mathura, India. LKS is also thankful to IUCAA for approving a short visit there when the idea of the present work has been conceived.

Reference (111)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return