Hubble Parameter Research Articles

Modelling the accelerating universe with f(Q) gravity: observational consistency

In this paper, we present a cosmological model within the framework of symmetric teleparallel gravity, focusing on f(Q) gravity, where Q represents the non-metricity scalar. Utilizing cosmological datasets, we derive an accelerating cosmological model by constraining its free parameters. To achieve this, we determine the parametric form of the Hubble parameter using a well-motivated f(Q) function. Remarkably, all obtained values fall within the range suggested by cosmological observations. By employing the best-fit parameters, we calculate the present geometrical parameters and demonstrate the accelerating behaviour of the Universe. Furthermore, we thoroughly examine the evolutionary behaviours of the Universe, noting that our model converges to the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model at late times. Finally, we investigate the energy conditions and find a violation of the strong energy condition, which could provide a valuable understanding of the nature of dark energy.

Can Stiff Matter Solve the Hubble Tension?

A new form of the mathematical expression for the co-moving volume element of a flat universe with cosmological constant, cold matter, and stiff matter is presented. It is used to determine the constraints from the Planck measurements of the Hubble parameter on the amount of stiff matter in the universe. These constraints are used to investigate whether the presence of stiff matter can solve the Hubble tension. It is found that the Planck measurements lead to an upper bound on the present value of the density parameter of stiff matter ΩS0<5⋅10−23, and that this is too small to solve the Hubble tension. Report. The main objective of this article is to introduce a novel mathematical expression for the co-moving volume element in a flat universe that includes a cosmological constant, cold matter, and stiff matter. This expression is utilized to derive constraints on the amount of stiff matter in the universe based on the Planck measurements of the Hubble parameter. These constraints are then examined to assess whether stiff matter could potentially resolve the Hubble tension. The findings indicate that the Planck measurements impose an upper limit on the current value of the density parameter of stiff matter, ΩS0<5⋅10−23, which is insufficient to resolve the Hubble tension.

A method for obtaining cosmological models consistency relations and Gaussian Processes testing

In the present work, we apply consistency relation tests to several cosmological models, including the flat and non-flat ΛCDM models, as well as the flat XCDM model. The analysis uses a non-parametric Gaussian Processes method to reconstruct various cosmological quantities of interest, such as the Hubble parameter H(z) and its derivatives from H(z) data, as well as the comoving distance and its derivatives from SNe Ia data. We construct consistency relations from these quantities which should be valid only in the context of each model and test them with the current data. We were able to find a general method of constructing such consistency relations in the context of H(z) reconstruction. In the case of comoving distance reconstruction, there were not a general method of constructing such relations and this work had to write an specific consistency relation for each model. From H(z) data, we have analyzed consistency relations for all the three above mentioned models, while for SNe Ia data we have analyzed consistency relations only for flat and non-flat ΛCDM models. Concerning the flat ΛCDM model, some inconsistency was found, at more than 2σ c.l., with the H(z) data in the interval 1.8 ≲ z ≲ 2.4, while the other models were all consistent at this c.l. Concerning the SNe Ia data, the flat ΛCDM model was consistent in the 0 < z < 2.5 interval, at 1σ c.l., while the nonflat ΛCDM model was consistent in the same interval, at 2σ c.l.

Axions and primordial magnetogenesis: the role of initial axion inhomogeneities

The relic density of dark matter in the ΛCDM model restricts the parameter space for a cosmological axion field, constraining the axion decay constant, the initial amplitude of the axion field and the axion mass. It is shown via lattice simulations how the relic density of axion-like particles with masses close to the one of the QCD axion is affected by axion-gauge field interactions and by initial axion inhomogeneities. For pre-inflationary axions, once the Hubble parameter becomes smaller than the axion mass, the latter starts to oscillate, and part of its energy density is spent producing gauge fields via parametric resonance. If the gauge fields are dark photons and Standard Model photons, the energy density of dark photons becomes higher than the one of the axion, while the high conductivity of the primordial plasma damps the oscillations of the photon field. Such a scenario allows for the production of small-scale, primordial magnetic fields, and it is found that the relic density of axions with a low decay constant are within the bounds set by the ΛCDM model, while GUT-scale axions are far too abundant. It is also shown that initial inhomogeneities of the axion field can change substantially the gauge field production, boosting or suppressing (depending on the axion parameters and couplings) the magnetogenesis mechanism with respect to an homogeneous axion field. It is found that when the axion mass is far lighter than the QCD axion model and the initial axion field is inhomogeneous, weak but cosmologically relevant magnetic field seeds can be generated on scales of the order of 0.1 kpc.

Addressing the rd tension using late-time observational measurements in a novel deceleration parametrization

This paper introduces a novel cosmological model aimed at probing the accelerated expansion of the late Universe through a unique parametrization of the deceleration parameter. We aim to constrain key cosmic parameters by integrating recent measurements of the Hubble parameter obtained from various observational methods, including cosmic chronometers, Type Ia Supernovae, Gamma-Ray Bursts (GRB), Quasars, and Baryon Acoustic Oscillations (BAO) from recent galaxy surveys. With a redshift range spanning 0.106<z<2.33 and incorporating the latest Hubble constant measurement from Riess in 2022, our analysis yields optimal fit values for the Hubble parameter (H0) and sound horizon (rd). Notably, we uncover an inconsistency in H0 values derived from late-time observational measurements, reflecting the well-known H0 tension. In terms of rd, while there is close agreement between Joint analysis and Joint analysis with R22, discrepancies arise upon gradual inclusion of BAO and BAO with R22 datasets. Our model demonstrates excellent fit to observed data and aligns well with the standard ΛCDM paradigm at higher redshifts. However, its most intriguing aspect lies in predicting a super-accelerated expansion in the distant future, in contrast to the de Sitter phase predicted by ΛCDM. Additionally, unique behaviors in the jerk parameter hint at novel dynamics beyond traditional cosmological models. Statefinder and Om Diagnostics tests were conducted, and comparison using the Akaike information criterion indicates neither model can be ruled out based on the latest observational measurements. These findings propose our cosmological model as a compelling alternative to ΛCDM, offering fresh insights into dark energy's nature and the cosmos' future.

Gravitational lensing of dark energy models and ΛCDM using observational data in loop quantum cosmology

This paper investigates the accelerated cosmic expansion in the late Universe by examining two dark energy models, viscous modified Chaplygin gas (VsMCG) and variable modified Chaplygin gas (VMCG), within loop quantum cosmology alongside the ΛCDM model. The objective is to constrain cosmic parameters using the ΛCDM model and 30 of the latest H(z) measurements from cosmic chronometers (CC), including Type Ia Supernovae, Gamma-Ray Bursts (GRB), Quasars, and 24 uncorrelated baryon acoustic oscillations (BAO) measurements across a redshift range from 0.106 to 2.33. The latest Hubble constant measurement from Riess in 2022 is included to enhance constraints. In the ΛCDM, VsMCG, and VMCG frameworks, best-fit parameters for the Hubble parameter (H0) and sound horizon (rd) are obtained. The results highlight significant disparities between H0 and rd values from late-time observational measurements, reflecting the known H0 and rd tensions. The gravitational lensing optical depth of the two dark energy models is studied by plotting log⁡(τ(zl)/H0−3τN) vs zl. The probability of finding gravitational lenses (optical depth) in both models increases with lens redshift zl. The change in optical depth behavior for different parameter constraints is graphically analyzed. A joint analysis of VsMCG and VMCG with ΛCDM is conducted. While the models diverge in the early Universe, they are indistinguishable at low redshift. Using the Akaike information criteria, the analysis indicates that neither dark energy model can be dismissed based on the latest observations.

Late-time constraints on interacting dark energy: Analysis independent of H0, rd, and MB

We investigated a possible interaction between cold dark matter and dark energy, corresponding to a well-known interacting dark energy model discussed in the literature within the context of resolving the Hubble tension. We put constraints on it in a novel way, by creating new likelihoods with an analytical marginalization over the Hubble parameter H0, the sound horizon rd, and the supernova absolute magnitude MB. Our aim is to investigate the impacts on the coupling parameter of the interacting model, ξ, and the equation of state of dark energy w and the matter density parameter Ωm, 0. The late-time cosmological probes used in our analysis include the PantheonPlus (calibrated and uncalibrated), cosmic chronometers, and baryon acoustic oscillation samples and the Pantheon for comparison. Through various combinations of these datasets, we demonstrate hints of an up to 2σ deviation from the standard Λ cold dark matter model.

Phase space analysis of torsion cosmology

In this paper, we study the dynamical stability analysis by considering the torsion field [Formula: see text] which is proportional to the Hubble parameter and barotropic equation-of-state parameter in the framework of the homogenous and isotropic FLRW metric with non-zero torsion in the scenario of dark matter and dark energy interacting terms. We discuss the stability of the critical points corresponding to the eigenvalues in the presence of dust and radiation at quintessence, [Formula: see text]CDM and phantom regimes by utilizing the different linear and nonlinear interaction models which depend on the energy density. As a result, we observe that the first linear model shows the stable behavior throughout the universe for dust and radiation phase. The other linear and nonlinear models show the stable critical points, computing with the eigenvalues at phantom and [Formula: see text]CDM era with the best fit value of interaction strength and coupling constant for dust and radiation.

Galaxy dynamics tracing quantum cosmology beyond [formula omitted]CDM below the de Sitter scale of acceleration

It is proposed that the baryonic Tully–Fisher relation (bTFR) and JWST ’Impossible galaxies’ at cosmic dawn are unified in weak gravity by the trace J of the Schouten tensor below the Sitter scale of acceleration adS=cH, where c is the velocity of light and H is the Hubble parameter. Across adS, J parametrizes short-period galaxy rotation curves and fast gravitational collapse beyond the predictions of ΛCDM. The sensitivity of weak gravitation to J=16R, where R is the Ricci scalar tensor, is derived in infrared gravitation from a consistent limit to quantum gravity, reducing to general relativity in the limit of a small Planck constant. For the first time, it identifies the exact relation a0=c2J/2π of the Milgrom parameter across all redshifts, accounting for the bTFR and early galaxy formation accelerated by a factor ∼J1/8. It predicts a0′(0)<0 at the present epoch.

Cosmographic analysis of holographic nonzero torsion framework

In this paper, we use the cosmographic approach to discuss Friedmann space-time in the presence of torsion. For this, we explore equations of motion that explain creation in an isotropic and homogeneous cosmic backdrop with nonzero torsion. Here, we consider the energy density of holographic dark energy model [Formula: see text] where [Formula: see text] is a dimensionless parameter depending on redshift parameter [Formula: see text] is the reduced Planck mass and [Formula: see text] represents the holographic length scale. We examine this dark energy model with both constant and variable holographic length scales in terms of Hubble parameter to determine the best-fit scale. The interaction between dark sector components is taken to evaluate cosmographic parameters, like Hubble, equation of state, deceleration, jerk, snap, lerk and statefinder parameters. We consider four [Formula: see text] parametrizations, which are Chevallier–Polarski–Linder, Jassal–Bagla–Padmanabhan, Wetterich and Ma–Zhang for both cases. We obtain consistent results for specific choices of constant parameters in the underlying scenario.

Trapped early dark energy

As a prospective resolution of the Hubble tension, early dark energy (EDE) suffers from the coincidence problem, why EDE is active just at matter-radiation equality (equivalently why the slope of EDE potential is required to approximately equal to the Hubble parameter at that time). In this paper we present a dark-matter-trapped EDE mechanism, by which the bound on the slope of EDE potential can be relieved. We show how this mechanism can work, and discuss the possibility that after inflation ended EDE can settle down at the initial conditions required by observations.

Constraints on the parameters of modified Chaplygin–Jacobi and modified Chaplygin–Abel gases in f(T) gravity

In this work, we explore the parameter constraints of two dark energy models, namely the Modified Chaplygin–Jacobi gas (MCJG) and Modified Chaplygin–Abel gas (MCAG), within the context of [Formula: see text] gravity model in a non-flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe. Our investigation involves comparing the equation of state for the MCJG and MCAG dark energy models with the equation of state derived from the [Formula: see text] gravity model. To derive constraints for the dark energy and [Formula: see text] gravity models, we use recent astronomical datasets, including [Formula: see text] data, type Ia supernovae observations, Gamma Ray Bursts data, quasar data, and Baryon Acoustic Oscillation (BAO) measurements. We present the reduced Hubble parameter in terms of observable parameters such as [Formula: see text] (density parameter of radiation), [Formula: see text] (density parameter of dark matter), [Formula: see text] (density parameter associated with spatial curvature), [Formula: see text] (density parameter of Modified Chaplygin–Jacobi gas), [Formula: see text] (density parameter of Modified Chaplygin–Abel gas), and [Formula: see text] (present value of the Hubble parameter). We explore the cosmological evolution through various cosmic diagnostic parameters, including the deceleration parameter, [Formula: see text] diagnostic, and statefinder diagnostic pair [Formula: see text]. These diagnostic parameters offer valuable insights into the expansion dynamics and the nature of dark energy in the universe. We have also assessed the viability of the models using the information criteria. Our aim is to shed light on the nature of dark energy and its connection to the [Formula: see text] gravity model, and ultimately gain a deeper understanding of the underlying mechanisms driving the accelerated expansion of our universe.

Probing cosmic background dynamics with a cosmological-model-independent method

ABSTRACT The Hubble constant $H_0$ tension has emerged as the most serious crisis in modern cosmology, potentially indicating that the $\Lambda$CDM model may not describe our Universe accurately. In this paper, we establish a new, cosmological-model-independent method to study the cosmic background dynamics. Using the latest Pantheon+ Type Ia supernova (SN Ia) sample and the model-independent SN Ia sample (P+1690), we derive values for the luminosity distance, the Hubble parameter, and the deceleration parameter at five different redshift points ranging from 0.12 to 0.52. Our analysis shows that results obtained from the Pantheon+ sample align with the predictions of the $\Lambda$CDM model within 2$\sigma$ confidence level (CL), while those obtained from the P+1690 sample exhibit deviations of about $2\sim 3\sigma$ CL. Furthermore, we explore the equation of state (EoS) of dark energy and find that while the EoS values from the Pantheon+ sample remain consistent with $-1$ within 2$\sigma$ CL, the P+1690 sample does not conform to this standard. These findings remain unchanged after the inclusion of the Hubble parameter measurements in our analysis. Our results indicate that the $\Lambda$CDM model remains compatible with the Pantheon+ SN Ia and the Hubble parameter measurements at 2$\sigma$ CL.

Dissipative quintessential cosmic inflation

In this paper we construct a dissipative quintessential cosmic inflation. For this purpose, we add a multiplicative dissipative term in the standard quintessence field Lagrangian. We consider the specific form of dissipation as the time integral including the Hubble parameter and an arbitrary function that describes the dissipative properties of the quintessential scalar field. Inflation parameters and observables are calculated under slow-roll approximations and a detailed calculation of the cosmological perturbations is performed in this setup. We consider different forms of potentials and calculate the scalar spectral index and tensor-to-scalar ratio for a constant as well as variable dissipation function. To check the reliability of this model, a numerical analysis on the model parameters space is done in confrontation with recent observational data. By comparing the results with observational joint datasets at 68% and 95% confidence levels, we obtain some constraints on the model parameters space, specially the dissipation factor with e-folds numbers N=55 and N=60. As some specific results, we show that the power-law potential with a constant dissipation factor and N=60 is mildly consistent with observational data in some restricted domains of the model parameter space with very small and negative dissipation factor and a negligible tensor-to-scalar ratio. But this case with N=55 is consistent with observation considerably. For power-law potential and variable dissipation factor as Q=αϕn, the consistency with observation is also considerable with a reliable tensor-to-scalar ratio. The quadratic and quartic potentials with variable dissipation function as Q=αϕn are consistent with Planck2018 TT, TE, EE+lowE+lensing data at the 68% and 95% levels of confidence for some intervals of the parameter n.

Observational constraints on FLRW, Bianchi type I and V brane models

This study explores the compatibility of Covariant Extrinsic Gravity (CEG), a braneworld scenario with an arbitrary number of non-compact extra dimensions, with current cosmological observations. We employ the chi-square statistic and Markov Chain Monte Carlo (MCMC) methods to fit the Friedmann–Lemaître–Robertson–Walker (FLRW) and Bianchi type-I and V brane models to the latest datasets, including Hubble, Pantheon+ Supernova samples, Big Bang Nucleosynthesis (BBN), Baryon Acoustic Oscillations (BAO), and the structure growth rate, fσ8(z). Parameters for FLRW universe consist Ω0(b),Ω0(cd),Ω0(k),H0,γ,σ8, while for the Bianchi model are Ω0(b),Ω0(cd),Ω0(β),H0,γ,Ω0(θ),σ8. By comparing our models to observational data, we determine the best values for cosmological parameters. For the FLRW model, these values depend on the sign of γ (which gives the time variation of gravitational constant in Hubble time unit): γ>0 yields γ=0.00008−0.00011+0.00015, and Ω0(k)=0.014−0.022+0.024 and γ<0 leads to γ=−0.0226−0.0062+0.0054, and Ω0(k)=0.023−0.041+0.039. It should be noted that in both cases Ω0(k)>0, which represents a closed universe. Similarly, for the Bianchi type-V brane model, the parameter values vary with the sign of γ, resulting in γ=0.00084−0.00021+0.00019, Ω0(β)=0.0258−0.0063+0.0052, and Ω0θ(×10−5)=4.19−0.75+0.67 (as with the density parameter of stiff matter) for γ>0, and γ=−0.00107−0.00020+0.00019, Ω0(β)=0.0259−0.0062+0.0050, and Ω0θ(×10−5)=4.17−0.98+0.91 for γ<0. In both cases Ω0(β)>0, which represents the Bianchi type-V, because in the Bianchi type-I, β=0. Subsequently, utilizing these obtained best values, we analyze the behavior of key cosmological parameters such as the Hubble parameter, deceleration parameter, distance modulus, equation of state, and density parameters that characterize both matter and the geometric component of dark energy, as functions of redshift. Our results notably show that the FLRW model with γ<0 is more compatible with observational data than the Bianchi model, based on various statistical criteria.

Constant-roll inflation with tachyon field in the holographic braneworld

Abstract We study inflation driven by the tachyon field in the holographic braneworld by assuming the second slow-roll parameter η is constant. The parameter η can be either defined by the tachyon scalar field and the Hubble parameter or by the Hubble parameter only. By assuming a constant η, we derive and numerically solve a differential equation for the Hubble expansion rate. We calculate numerically the scalar spectral index and the tensor-to-scalar ratio. We confront the results with the observational data and find some constraints on the free model parameters. The swampland conjectures are discussed in the context of the constant-roll inflation, with some accent on the holographic model.

Quintessential f(G) gravity with statistically fitting of H(z)

Inspired by high-energy physics and by considering standard gravity theory as the low-energy limit of quantum theory, the modification of general relativity as [Formula: see text] has been proposed. This paper is dedicated to investigating the cosmological model of modified Gauss–Bonnet gravity in four dimensions. General field equations for the four-dimensional Friedmann– Robertson–Walker metric are given in detail. Also, it is well known that Hubble parameter [Formula: see text] measurements are helpful for the inference of cosmological model parameters. We try to explain the MCMC statistical approach used to calculate Hubble’s parameter. The resulting form of [Formula: see text] was limited using Cosmic Chronometric data, Standard Candles SN Ia from Pantheon, and Baryon Acoustic Oscillations to obtain the best match values by using the regression equation. Finally, to examine the late-time dynamics of our Universe, the behavior of physical and kinematic variables like the deceleration parameter [Formula: see text], State-finder parameters [Formula: see text], diagnostic parameter ([Formula: see text]), equation of state parameter as well as the behavior of energy conditions versus redshift with the parameter values are presented.

On direct estimation of density parameters and Hubble constant for ΛCDM universe using Hubble measurements

The set of cosmological density parameters ( Ω0mh02 , Ω0kh02 , Ω0Λh02 ) and Hubble constant ( hˆ0 ) are useful for fundamental understanding of the Universe from many perspectives. In this article, we propose a new procedure to estimate these parameters for Λ cold dark matter (ΛCDM) universe in the Friedmann-Robertson-Walker (FRW) background. We generalize the two-point statistics first proposed by Sahni et al (2008) to the three point case and estimate the parameters using currently available Hubble parameter (H(z)) values in the redshift range 0.07 ≤ z ≤ 2.36 measured by differential age (DA) and baryon acoustic oscillation (BAO) techniques. All the parameters are estimated assuming the general case of non-flat universe. Using both DA and BAO data we obtain Ω0mh02=0.1485±0.0065 , Ω0kh02=−0.0137±0.017 , Ω0Λh02=0.3126±0.0145 and hˆ0=0.6689±0.0021 . These results are in satisfactory agreement with the Planck results. An important advantage of our method is that to estimate the value of any one of the independent cosmological parameters one does not need to use the values for the rest of them. Each parameter is obtained solely from the measured values of Hubble parameters at different redshifts without any need to use values of other parameters. Such a method is expected to be less susceptible to the undesired effects of degeneracy issues between cosmological parameters during their estimations. Moreover, there is no requirement of assuming spatial flatness in our method.

Observational constrained Weyl type [formula omitted] gravity cosmological model and the dynamical system analysis

Using the cosmological date sets, the cosmological parameters are constrained in this paper, with well known phantom behavioural Hubble parameter. To understand the dynamics of the Weyl type f(Q,T) gravity, functional form of f(Q,T) has been introduced, where Q and T respectively represents the nonmetricity scalar that depends on the Weyl vector (Q=−6w2) and trace of energy–momentum tensor. Using the constrained values of the parameters, the other geometrical parameters are analysed and the accelerating behaviour has been shown. The dynamical parameters and energy conditions are also analysed. Further to get the complete evolutionary behaviour of the Universe, the dynamical system analysis has been performed.

A comprehensive data-driven odyssey to explore the equation of state of dark energy

For the first time, we reconstruct the dark energy equation of the state parameter w from the combination of background and perturbation observations, specifically combining the Hubble parameter data from cosmic chronometer observations and the logarithmic growth rate data from the growth rate observations. We do this analysis using posterior Gaussian process regression without considering any specific cosmological model or parametrization. However there are three main assumptions: (I) a flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric is considered for the cosmological background, (II) there is no interaction between dark energy and matter sectors, and (III) for the growth of inhomogeneity, sub-Hubble approximation and linear perturbations are considered. This study is unique in the sense that the reconstruction of w is independent of any derived parameters such as the present values of the matter-energy density parameter and Hubble parameter. From the reconstruction, we look at how the dark energy equation of state evolves between redshifts 0 and 1.5, finding a slight hint of dynamical behavior in dark energy. However, the evidence is not significant. We also find a leaning towards non-phantom behavior over phantom behavior. We observe that the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document}CDM model (w=-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(w=-1)$$\\end{document} nearly touches the lower boundary of the 1σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} confidence region in the redshift range 0.6≲z≲0.85.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.6 \\lesssim z \\lesssim 0.85.$$\\end{document} However, it comfortably resides within the 2σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} confidence region in the whole redshift range under investigation, 0≤z≤1.5.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le z \\le 1.5.$$\\end{document} Consequently, the non-parametric, model-independent reconstruction of dark energy provides no compelling evidence to deviate from the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document}CDM model when considering cosmic chronometer and growth rate observations.