Showing papers on "Deceleration parameter published in 2021"
TL;DR: In this paper, it was shown that the supernova absolute magnitude $M_B$ that is used to derive the local $H_0$ constraint is not compatible with the absolute magnitude of supernova, BAO and CMB data, and this disagreement is not solved by a sudden phantom transition at very low redshift.
Abstract: A dark-energy which behaves as the cosmological constant until a sudden phantom transition at very-low redshift ($z 4$\sigma$ disagreement between the local and high-redshift determinations of the Hubble constant, while maintaining the phenomenological success of the $\Lambda$CDM model with respect to the other observables Here, we show that such a hockey-stick dark energy cannot solve the $H_0$ crisis The basic reason is that the supernova absolute magnitude $M_B$ that is used to derive the local $H_0$ constraint is not compatible with the $M_B$ that is necessary to fit supernova, BAO and CMB data, and this disagreement is not solved by a sudden phantom transition at very-low redshift We make use of this example to show why it is preferable to adopt in the statistical analyses the prior on $M_B$ as an alternative to the prior on $H_0$ The three reasons are: i) one avoids potential double counting of low-redshift supernovae, ii) one avoids assuming the validity of cosmography, in particular fixing the deceleration parameter to the standard model value $q_0=-055$, iii) one includes in the analysis the fact that $M_B$ is constrained by local calibration, an information which would otherwise be neglected in the analysis, biasing both model selection and parameter constraints We provide the priors on $M_B$ relative to the recent Pantheon and DES-SN3YR supernova catalogs We also provide a Gaussian joint prior on $H_0$ and $q_0$ that generalizes the prior on $H_0$ by SH0ES
131 citations
TL;DR: In this paper, an interacting model of Barrow holographic dark energy (BHDE) was proposed and the evolution of a spatially flat FLRW universe composed of pressureless dark matter and BHDE that interact with each other through a well-motivated interaction term.
Abstract: In this paper, using Barrow entropy, we propose an interacting model of Barrow holographic dark energy (BHDE). In particular, we study the evolution of a spatially flat FLRW universe composed of pressureless dark matter and BHDE that interact with each other through a well-motivated interaction term. Considering the Hubble horizon as the IR cutoff, we then study the evolutionary history of important cosmological parameters, particularly, the density parameter, the equation of state parameter and the deceleration parameter in the BHDE model and find satisfactory behaviors in the model. We perform a detailed study on the dynamics of the field equations by studying the asymptotic behavior of the field equations, while we write the analytic expression for the scale factor with the use of Laurent series. Finally, we study the implications of gravitational thermodynamics in the interacting BHDE model with the dynamical apparent horizon as the cosmological boundary. In particular, we study the viability of the generalized second law by assuming that the apparent horizon is endowed with Hawking temperature and Barrow entropy.
60 citations
TL;DR: In this paper, the authors analyzed the Barrow holographic dark energy (BHDE) in the framework of the flat FLRW Universe by considering the various estimations of Barrow exponent.
Abstract: We have analyzed the Barrow holographic dark energy (BHDE) in the framework of the flat FLRW Universe by considering the various estimations of Barrow exponent $\triangle$. Here we define BHDE, by applying the usual holographic principle at a cosmological system, for utilizing the Barrow entropy rather than the standard Bekenstein-Hawking. To understand the recent accelerated expansion of the universe, considering the Hubble horizon as the IR cut-off. The cosmological parameters, especially the density parameter ($\Omega_{_D}$), the equation of the state parameter ($\omega_{_D}$), energy density ($\rho_{_{D}}$) and the deceleration parameter($q$) are studied in this manuscript and found the satisfactory behaviors. Moreover, we additionally focus on the two geometric diagnostics, the statefinder $(r,s)$ and $O_{m}(z)$ to discriminant BHDE model from the $\Lambda CDM$ model. Here we determined and plotted the trajectories of evolution for statefinder $(r, s)$, $(r,q)$ and $O_{m}(z)$ diagnostic plane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018 observational information. Finally, we have explored the new Barrow exponent $\triangle$, which strongly affects the dark energy equation of state that can lead it to lie in the quintessence regime, phantom regime, and exhibits the phantom-divide line during the cosmological evolution.
37 citations
TL;DR: In this article, the authors used the newly proposed f(Q,T) gravity, where Q is the non-metricity, and T is the trace of the energymomentum tensor.
Abstract: New high-precision observations are now possible to constrain different gravity theories. To examine the accelerated expansion of the Universe, we used the newly proposed f(Q, T) gravity, where Q is the non-metricity, and T is the trace of the energy–momentum tensor. The investigation is carried out using a parameterized effective equation of state with two parameters, m and n. We have also considered the linear form of $$f(Q,T)= Q+bT$$
, where b is constant. By constraining the model with the recently published 1048 Pantheon sample, we were able to find the best fitting values for the parameters b, m, and n. The model appears to be in good agreement with the observations. Finally, we analyzed the behavior of the deceleration parameter and equation of state parameter. The results support the feasibility of f(Q, T) as a promising theory of gravity, illuminating a new direction towards explaining the Universe dark sector.
29 citations
TL;DR: In this paper, the dynamical aspects of some accelerating models are investigated in the framework of an extension of symmetric teleparallel gravity dubbed as f(Q, T) gravity.
Abstract: The dynamical aspects of some accelerating models are investigated in the framework of an extension of symmetric teleparallel gravity dubbed as f(Q, T) gravity. In this gravity theory, the usual Ricci tensor in the geometrical action is replaced by a functional f(Q, T) where Q is the non-metricity and T is the trace of the energy-momentum tensor. Two different functional forms are considered in the present work. In order to model the Universe, we have considered a signature flipping deceleration parameter simulated by a hybrid scale factor (HSF). The dynamical parameters of the model are derived and analysed. We discuss the role of the parameter space in getting viable cosmological models. It is found that, the models may be useful as suitable geometrical alternatives to the usual dark energy approach.
27 citations
TL;DR: In this article, the cosmological jerk parameter was reconstructed from observational data independent of a fiducial cosmology model using the Cosmic Chronometer data as well as the Supernovae data (the Pantheon compilation).
Abstract: The cosmological jerk parameter j is reconstructed in a non-parametric way from observational data independent of a fiducial cosmological model. The Cosmic Chronometer data as well as the Supernovae data (the Pantheon compilation) are used for the purpose. The reconstructed values are found to be consistent with the standard $$\Lambda $$
CDM model within the $$2\sigma $$
confidence level. The model dependent sets like Baryon Acoustic Oscillation and the CMB Shift data are also included thereafter, which does not significantly help in improving or de-proving the confidence level in favour of $$\Lambda $$
CDM. The deceleration parameter q is also reconstructed from the same data sets. This is used to find the effective equation of state parameter for the model independent datasets only. $$\Lambda $$
CDM model is excluded for some part of the evolution in $$1\sigma $$
, but is definitely included in $$2\sigma $$
in the domain (
$$0 \le z \le 2.36$$
) of all the reconstructions.
21 citations
TL;DR: In this article, the THDE model using an IR cut off as Ricci horizon has been investigated in the context of a flat Friedmann-Lema i ^ tre-Robertson-Walker Universe by describing an accelerated expansion stage which Universe is experiencing.
18 citations
TL;DR: Xu et al. as discussed by the authors considered the Friedmann-Robertson-Walker (FRW) model in the framework of $f(Q,T)$ gravity, an extension of symmetric teleparallel gravity, and determined the numerical solutions of cosmological implications such as Hubble parameter, deceleration parameter, apparent magnitude and luminosity distance.
Abstract: In this paper, we considered the study of Friedmann-Robertson-Walker (FRW) model in the framework of $f(Q,T)$ gravity, an extension of symmetric teleparallel gravity, recently defined by Y. Xu et al. \cite{Xu}. The non-linear model $f(Q,T)=-\alpha Q-\beta T^2$, where $\alpha>0$ and $\beta>0$ are constants, is taken into account. The equation of state of perfect fluid is assumed and 31 points of Hubble data are used to constrain the value of model parameter. To explore the evolution of the universe, the numerical solutions of cosmological implications such as Hubble parameter, deceleration parameter, apparent magnitude and luminosity distance are determined and the energy conditions are examined. The theoretical results of Hubble parameter are compared with $\Lambda$CDM model. Further, 57 Supernova data (42 from Supernova cosmology project and 15 from Calan/ Tolono supernova survey) are also used to have consistent results of apparent magnitude and luminosity distance.
17 citations
TL;DR: In this article, a Tsallis holographic dark energy model in a flat Friedmann-Lamatire-Rbertson-Walker space-time geometry in the context of higher derivative theory of gravity is presented.
16 citations
TL;DR: In this paper, a non-flat Friedmann-Robertson-Walker universe with a pressureless dark matter (DM) and Barrow holographic dark energy (BHDE) whose IR cutoff is the apparent horizon was studied.
15 citations
TL;DR: In this paper, the authors investigated a bulk viscous anisotropic universe and constrained its model parameters with recent $H(z)$ and Pantheon compilation data and estimated the present value of Hubble's constant using cosmic chronometric technique.
Abstract: In this paper, we have investigated a bulk viscous anisotropic Universe and constrained its model parameters with recent $H(z)$ and Pantheon compilation data. Using cosmic chronometric technique, we estimate the present value of Hubble's constant as $H_{0} = 69.39 \pm 1.54~km~s^{-1}Mpc^{-1}$, $70.016 \pm 1.65~km~s^{-1}Mpc^{-1}$ and $69.36 \pm 1.42~km~s^{-1}Mpc^{-1}$ by bounding our derived model with recent $H(z)$ data, Pantheon and joint $H(z)$ and Pantheon data respectively. The present age of the Universe is specified as $t_0= 0.9796H_0^{-1}\sim 13.79$ Gyrs. The model favours a transitioning Universe with the transition red-shift as $z_{t} = 0.73$. We have reconstructed the jerk parameter using the observational data sets. From the analysis of the jerk parameter, it is observed that, our derived model shows a marginal departure from the concordance $\Lambda$CDM model.
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TL;DR: In this article, a new Tsallis holographic dark energy model was constructed using the concepts of Tsalis entropy and future event horizon, and the parameters $c$ and $\delta$ were used to characterize various aspects of the model.
Abstract: Tsallis entropy is a generalization of the Boltzmann-Gibbs entropy in statistical theory which uses a parameter $\delta$ to measure the deviation from the standard scenario quantitatively. Using concepts of Tsallis entropy and future event horizon, we construct a new Tsallis holographic dark energy model. The parameters $c$ and $\delta$ will be used to characterize various aspects of the model. Analytical expressions for various cosmological parameters such as the differential equation describing the evolution of the effective dark energy density parameter, the equation of state parameter and the deceleration parameter are obtained. The equation of state parameter for the current model exhibits the pure quintessence behaviour for $c>1$, quintom behaviour for $c<1$ whereas the $\Lambda$CDM model is recovered for $c=1$. To analyze the thermal history of the universe, we obtained the expression for the deceleration parameter and found that for $z \approx 0.6$, the phase transits from deceleration to acceleration.
TL;DR: In this article, the authors explore the recently proposed new Tsallis agegraphic dark energy model in a flat FLRW Universe by taking the conformal time as IR cutoff with interaction.
Abstract: In this work, we explore the recently proposed new Tsallis agegraphic dark energy model in a flat FLRW Universe by taking the conformal time as IR cutoff with interaction. The deceleration parameter of the interacting new Tsallis agegraphic dark energy model provides the phase transition of the Universe from decelerated to accelerated phase. The EoS parameter of the model shows a rich behaviour as it can be quintessence-like or phantom-like depending on the interaction (
$$b^2$$
) and parameter B. The evolutionary trajectories of the statefinder parameters and $$(\omega _D, \omega _D^{'})$$
planes are plotted by considering the initial condition $$\Omega _{D}^{0} =0.73$$
, $$H_{0}= 67$$
according to $$\Lambda $$
CDM observational Planck 2018 data for different $$b^2$$
and B. The model shows both quintessence and Chaplygin gas behaviour in the statefinder (r, s) and (r, q) pair planes for different $$b^2$$
and B.
TL;DR: In this paper, the authors presented and analyzed cosmological models with an anisotropic variable parameter, and constructed two different models with respect to the scale factors, such as power law scale factor and hybrid scale factor.
Abstract: In this article, we present and analyze cosmological models with an anisotropic variable parameter. We have set up the field equations with the space time in the form of Bianchi I metric with an f(R, T) gravity. The functional form for the f(R, T) gravity has been assumed to be $$f(R,T)=R+2f(T)$$
, where R and T are, respectively, the Ricci scalar and trace of the energy–momentum tensor. Two different models are constructed with respect to the scale factors, such as power law scale factor and hybrid scale factor. Moreover, the anisotropic parameter taken here in the form of hyperbolic function further gives clarity on the behavior of equation of state parameter. It is to note that when the values of the coefficient constant vanish, the model yields the isotropic universe. For both the cases, the deceleration parameter, state finder diagnostic pair and energy conditions have been obtained and analyzed which provide physical plausibility of the models.
TL;DR: In this article, an alternative route to explain the late-time acceleration considering a bulk viscous fluid with viscosity coefficient ζ = ζ 0 + ζ 1 H+ ζ 2 H 2, where ζ0, ζ1, and ζ2 are constants in the framework of modified gravity.
TL;DR: In this paper, the evolution of the cosmic parameters and planes are investigated in the framework of the DGP braneworld model, where the interaction between the Barrow holographic dark energy model (whose infrared cutoff scale is set by Hubble and event horizons) and pressureless dark matter are considered.
Abstract: In the present manuscript, the evolution of the cosmic parameters and planes are being investigated in the framework of the DGP braneworld model. In this scenario, the interaction Γ between the Barrow holographic dark energy model (whose infrared cutoff scale is set by Hubble and event horizons) and pressureless dark matter are considered. We check the behavior of different cosmological parameters such as Hubble, equation of state, deceleration and squared speed of sound from the early matter-dominated era until the late-time acceleration. It is found that the range of Hubble parameter lies in the interval 95−35+35 (for Hubble horizon) and 97−23+23 (for event horizon). For both horizons, the equation of state parameter favors the phantom dominant era as well as the ΛCDM model while the deceleration parameter illustrates the accelerated expansion of the universe. Furthermore, stability of the underlying model is found through squared speed of sound. Furthermore, it is observed that ω−ωϑ′ plane corresponds to freezing and thawing region for Hubble and event horizons, respectively. Furthermore, statefinder plane shows the ΛCDM and Chaplygin gas behavior for both models. Finally, we investigate the thermodynamical nature of the underlying model through Barrow entropy as horizon entropy and found validity for both horizons.
TL;DR: In this article, the authors analyzed holographic principle inspired dark energy models, namely, holographic dark energy (HDE) model, Tsallis HDE model, and the R e ´ nyi HEDE model by using the statefinder parameter diagnostic.
TL;DR: In this article, the authors presented the luminosity distance series expansion to third order in redshift for a general space-time with no assumption on the metric tensor or the field equations prescribing it.
Abstract: We present the luminosity distance series expansion to third order in redshift for a general space-time with no assumption on the metric tensor or the field equations prescribing it. It turns out that the coefficients of this general 'Hubble law' can be expressed in terms of a finite number of physically interpretable multipole coefficients. The multipole terms can be combined into effective direction dependent parameters replacing the Hubble constant, deceleration parameter, curvature parameter, and 'jerk' parameter of the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) class of metrics. Due to the finite number of multipole coefficients, the exact anisotropic Hubble law is given by 9, 25, 61 degrees of freedom in the $\mathcal{O}(z)$, $\mathcal{O}(z^2)$, $\mathcal{O}(z^3)$ vicinity of the observer respectively, where $z\!:=\,$redshift. This makes possible model independent determination of dynamical degrees of freedom of the cosmic neighbourhood of the observer and direct testing of the FLRW ansatz. We argue that the derived multipole representation of the general Hubble law provides a new framework with broad applications in observational cosmology.
TL;DR: In this paper, an interacting model of the Renyi holographic dark energy in the Brans-Dicke theory of gravity using Renyi entropy in a spatially flat Friedmann-Lemaitre-Robertson-Walker Universe considering the infrared cut-off as the Hubble horizon was constructed.
Abstract: In this work, we construct an interacting model of the Renyi holographic dark energy in the Brans-Dicke theory of gravity using Renyi entropy in a spatially flat Friedmann-Lemaitre-Robertson-Walker Universe considering the infrared cut-off as the Hubble horizon. In this setup, we then study the evolutionary history of some important cosmological parameters, in particular, deceleration parameter, Hubble parameter, equation of state parameter, and Renyi holographic dark energy density parameter in both nonflat Universe and flat Universe scenarios and also observe satisfactory behaviors of these parameters in the model. We find that during the evolution, the present model can give rise to a late-time accelerated expansion phase for the Universe preceded by a decelerated expansion phase for both flat and nonflat cases. Moreover, we obtain as , which indicates that this model behaves like the cosmological constant at the future. The stability analysis for the distinct estimations of the Renyi parameter and coupling coefficient has been analyzed. The results indicate that the model is stable at the late time.
TL;DR: In this paper, Tsallis holographic dark energy in a flat Friedmann-Robertson-Walker model under the framework of f (R, T ) gravity is analyzed. But the effects of this model in a non interactive universe are studied by taking different IR cut-offs that include particle horizon, event horizon, conformal age of the universe and GO (Granda-Oliveros) horizon.
TL;DR: In this paper, a bulk viscous universe with dominance of dark energy in Bianchi type I space-time and constrained its parameter with observational $H(z)$ data (OHD) and joint OHD and Pantheon compilation of SN Ia data were investigated.
Abstract: In this paper, we have investigated a bulk viscous universe with dominance of dark energy in Bianchi type I space-time and constrained its parameter with observational $H(z)$ data (OHD) and joint OHD and Pantheon compilation of SN Ia data. We constrain Hubble's constant ${H}_{0}=69.3{8}_{\ensuremath{-}1.54}^{+1.57}$ and ${H}_{0}=70.01{7}_{\ensuremath{-}1.60}^{+1.62}\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ by bounding this model with OHD and joint OHD and Pantheon compilation of SN Ia data respectively. These estimated values of ${H}_{0}$ have $1.27\ensuremath{\sigma}$ and $1.69\ensuremath{\sigma}$ tension with its corresponding Planck collaboration value. We have constrained the best fit value of model parameter $\ensuremath{\zeta}$ as $\ensuremath{\zeta}=0.00002{7}_{\ensuremath{-}0.00052}^{+0.00052}$ and $\ensuremath{\zeta}=0.00003{3}_{\ensuremath{-}0.00296}^{+0.00296}$ respectively for the above observational datasets. We also have obtained a kinematic expression for deceleration parameter and constrained its present value ${q}_{0}=\ensuremath{-}0.58{2}_{\ensuremath{-}0.023}^{+0.021}$ and transition redshift ${z}_{t}=0.72{3}_{\ensuremath{-}0.16}^{+0.34}$. Furthermore, we observe that the derived model exhibits the properties of transitioning the universe and its present age is ${t}_{0}=14.0{4}_{\ensuremath{-}0.32}^{+0.33}\text{ }\text{ }\mathrm{Gyrs}$. The kinematics of the Om(z) parameter and the jerk parameter are also discussed and the analysis of these parameters shows a marginal departure of the derived model from the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model of the universe.
TL;DR: In this paper, the authors construct interacting and non-interacting Tsallis holographic dark energy models in an anisotropic and homogeneous Bianchi type-V I 0 space time within the framework of a scalar tensor theory.
TL;DR: In this article, the Bianchi Type-I cosmological model with varying cosmologically constant Λ in f(R, T) theory of gravity was investigated, which leads to a time dependent deceleration parameter.
TL;DR: In this article, the authors investigated the dynamical behavior of Kaluza-Klein FRW type dark energy cosmological models in the framework of a scalar-tensor theory of gravitation formulated by Saez and Ballester.
TL;DR: In this paper, after reconstructing the redshift evolution of the Hubble function by adopting Gaussian process techniques, they estimate the best-fit parameters for some flat Friedmann cosmological models based on a modified Chaplygin gas interacting with dark matter.
Abstract: In this paper, after reconstructing the redshift evolution of the Hubble function by adopting Gaussian process techniques, we estimate the best-fit parameters for some flat Friedmann cosmological models based on a modified Chaplygin gas interacting with dark matter. In fact, the expansion history of the Universe will be investigated because passively evolving galaxies constitute cosmic chronometers. An estimate for the present-day values of the deceleration parameter, adiabatic speed of sound within the dark energy fluid, effective dark energy, and dark matter equation of state parameters is provided. By this, we mean that the interaction term between the two dark fluids, which breaks the Bianchi symmetries, will be interpreted as an effective contribution to the dark matter pressure similarly to the framework of the “Generalized Dark Matter”. We investigate whether the estimates of the Hubble constant and of the present-day abundance of dark matter are sensitive to the dark matter–dark energy coupling. We will also show that the cosmic chronometers data favor a cold dark matter, and that our findings are in agreement with the Le Châtelier–Braun principle according to which dark energy should decay into dark matter.
TL;DR: In this paper, the authors discussed the cosmic evolution of the anisotropic and spatially homogeneous Kantowski-Sachs universe with Renyi holographic dark energy in the background of general relativity.
TL;DR: In this article, the authors examined the Kantowski-Sachs space-time in the presence of strange quark matter with the appearance and nonappearance of strings in f(R) gravity.
Abstract: In this paper, we have examined the Kantowski–Sachs space-time in the presence of strange quark matter with the appearance and non-appearance of strings in f(R) gravity. Here, R is the Ricci scalar of the space-time. Exact solutions of the field equations are obtained by considering the following conditions (i) hybrid scale factor, (ii) the proportionality of shear scalar (
$$\sigma$$
) with expansion scalar (
$$\theta$$
) and (iii) power-law between the scalar-on function F(R) and average scale factor a(t). We have investigated some physical and geometrical properties of the models, and their behavior is thoroughly studied with the help of graphical representation.
TL;DR: In this paper, an early dark energy (EDEDE) model has been proposed to explain the late accelerated expansion of the universe, where the EDE model has a non-negligible contribution at high redshift (as opposed to the cosmological constant that only plays a role once the structure is formed).
TL;DR: In this paper, the LRS Bianchi-I spatially homogeneous and anisotropic perfect fluid cosmological model is studied in f ( R, T ) theory by specific choice of f( R, T ) = R + α R n + λ T ǫ + β R n+ β β n + β β β N + β Nǫ T, and the scale factor is considered as an increasing function of time by using scale factor and the proportionality of shear scalar of the space-time with expansion scalar.
Abstract: The LRS Bianchi-I spatially homogeneous and anisotropic perfect fluid cosmological model is studied in f ( R , T ) theory by specific choice of f ( R , T ) = R + α R n + λ T In this paper, to get the deceleration parameter as a function of time (t), the scale factor is considered as an increasing function of time By using scale factor and the proportionality of shear scalar ( σ ) of the space–time with expansion scalar ( θ ), the solution of the field equations is obtained, also to explain the accelerating expansion of the universe, the constant n plays a key role It is observed that, for n = 2 , 25 , 3 , 35 , 4 , the pressure and energy density are negative and positive respectively For the remaining values of n, the pressure is positive, and the energy density is negative Also, we have discussed all energy conditions for n = 2 , 25 , 3 , 35 , 4 , and it is interesting to see that NEC, DEC’s are satisfied, while infringement of SEC accelerates the expansion of the universe Moreover, we have observed that n = 2 is the best fit for the present model, and for various values of ν , the r - q and r - s planes have been studied Finally, the LRS Bianchi type-I cosmological model obtained, and presented here is compatible with the recent cosmological observational data
TL;DR: In this paper, the authors have constructed non-interacting and interacting Tsallis holographic dark energy models by solving the field equations using the relationship between the metric potentials.
Abstract: In this work devoted to the investigation of the Tsallis holographic dark energy (IR cut-off is Hubble radius) in homogeneous and anisotropic Kantowski–Sachs Universe within the frame-work of Saez–Ballester scalar tensor theory of gravitation. We have constructed non-interacting and interacting Tsallis holographic dark energy models by solving the field equations using the relationship between the metric potentials. This relation leads to a viable deceleration parameter model which exhibits a transition of the Universe from deceleration to acceleration. In interacting case, we focus on sign-changeable interaction between Tsallis holographic dark energy and dark matter. The dynamical parameters like equation of state parameter, energy densities of Tsallis holographic dark energy and dark matter, deceleration parameter, and statefinder parameters of the models are explained through graphical representation. And also, we discussed the stability analysis of the our models.