Scalar Field Cosmology with Logarithmic Deceleration Parameter ^†

Abstract

In this study, we propose a novel framework using a logarithmic parametrization of the deceleration parameter $q\left(z\right)$. This specific form facilitates the reconstruction of the expansion history in a closed analytical form, providing crucial theoretical tractability. We aim to constrain the free parameters of this logarithmic model using a robust combination of datasets (CC, Pantheon+ SNe Ia, BAO, and the R19 prior) to reconstruct fundamental dynamic quantities and determine the transition redshift ($z_{tr}$) with high precision.

1. Introduction

The late-time acceleration of the universe, first evidenced by observations of Type Ia Supernovae at the end of the 20th century [1,2], remains one of the most profound puzzles in modern cosmology. The standard cosmological model, known as the $\Lambda$CDM model, provides a remarkably consistent fit to a wide range of observations, e.g., [3]. However, it is plagued by deep theoretical issues, most notably the cosmological constant problem [4], and the “cosmic coincidence” problem [5].

To address these shortcomings, various dynamical dark energy models have been proposed. Among these, scalar field theories such as quintessence [6,7], k-essence [8], and phantom fields [9] offer a flexible framework where the energy density of the vacuum can evolve over cosmic time. In these models, the acceleration is typically driven by a minimally coupled scalar field $\varphi$ rolling down a potential $V\left(\varphi \right)$, providing a more natural explanation for the observed expansion dynamics than a static constant [10,11].

Parallel to these theoretical developments, model-independent reconstruction methods have gained significant attention. These methods aim to extract information about the expansion of the universe directly from observational data without assuming a specific dark energy fluid [12]. A common strategy is kinematic reconstruction via the parametrization of the deceleration parameter $q\left(z\right)$, which describes the rate of change of the Hubble expansion [13,14].

In this paper, we use a logarithmic parametrization of the deceleration parameter: $q\left(z\right)=q_0+q_{1}ln(1+z)$. This specific functional form is motivated by its ability to provide high theoretical tractability, allowing fundamental quantities such as the Hubble parameter $H\left(z\right)$ and the scalar potential $V\left(\varphi \right)$ to be expressed in closed analytical forms. Such mathematical convenience is essential for performing robust statistical analyses against recent datasets, including the Pantheon+ Supernovae sample [15], Cosmic Chronometers, and Baryon Acoustic Oscillations (BAO) [16]. Furthermore, this approach allows us to investigate the transition redshift ($z_{tr}$) and address the persistent “Hubble tension”—the discrepancy between local measurements and CMB-based predictions of $H_0$ [17,18].

2. Theoretical Framework

We consider a spatially flat, homogeneous, and isotropic universe described by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric:

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left[d{r}^2+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2)\right]$$

(1)

where $a\left(t\right)$ is the cosmic scale factor. In the framework of General Relativity, the dynamics of such a universe are governed by the Einstein Field Equation $G_{\mu \nu }={\kappa }^2{T}_{\mu \nu }$, where ${\kappa }^2=8\pi G$. In our model, the total energy-momentum tensor $T_{\mu \nu }$ accounts for both non-relativistic matter (baryonic and dark matter) and a minimally coupled canonical scalar field $\varphi$, representing the dark energy component [6].

The action for the scalar field is given by

$$S_{\varphi }=\int d^{4}x\sqrt{-g}\left[-{\displaystyle \frac{1}{2}}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V\left(\varphi \right)\right]$$

(2)

where $V\left(\varphi \right)$ is the scalar potential. For a homogeneous field $\varphi \left(t\right)$, the energy density ${\rho }_{\varphi }$ and pressure $p_{\varphi }$ are defined as

$${\rho }_{\varphi }={\displaystyle \frac{1}{2}}{\dot{\varphi }}^2+V\left(\varphi \right),p_{\varphi }={\displaystyle \frac{1}{2}}{\dot{\varphi }}^2-V\left(\varphi \right)$$

(3)

The evolution of the universe is then determined by the Friedmann equations:

$$3H^2={\kappa }^2({\rho }_m+{\rho }_{\varphi })$$

(4)

$$-2\dot{H}-3H^2={\kappa }^2{p}_{\varphi }$$

(5)

where $H=\dot{a}/a$ is the Hubble parameter and ${\rho }_m$ is the matter density following the conservation law ${\rho }_m={\rho }_{m0}{(1+z)}^3$.

The scalar field’s dynamics are further governed by the Klein–Gordon equation, $\ddot{\varphi }+3H\dot{\varphi }+dV/d\varphi =0$, which ensures the conservation of the dark energy sector. By relating the kinematic deceleration parameter $q\left(z\right)=-1-\dot{H}/H^2$ to these field equations, we can reconstruct the scalar potential $V\left(\varphi \right)$ and the field’s kinetic energy as functions of redshift [10]. This mapping is essential for understanding how the proposed logarithmic expansion history translates into a physical potential responsible for the observed cosmic acceleration.

3. Logarithmic Parametrization

The core of our model-independent approach lies in the functional form of the deceleration parameter $q\left(z\right)$. While many studies utilize linear or Taylor expansions of $q\left(z\right)$ in terms of the scale factor or redshift [13,14], such forms can sometimes lead to divergences at high redshifts or lack analytical solutions for the Hubble parameter. To overcome these limitations, we propose a logarithmic ansatz:

$$q\left(z\right)=q_0+q_{1}ln(1+z)$$

(6)

where $q_0$ represents the present value of the deceleration parameter ($z=0$), and $q_1$ characterizes its evolution over cosmic time. This specific parametrization is physically motivated by its slow-varying nature at higher redshifts compared to power-law models, providing a more stable reconstruction of the expansion history [12].

The primary advantage of this logarithmic form is that it allows for an exact integration of the relation between the Hubble parameter and the deceleration parameter, defined as $H\left(z\right)=H_{0}exp\left[{\int }_0^z{\displaystyle \frac{1+q\left(x\right)}{1+x}}dx\right]$. Substituting our ansatz into this integral yields a closed-form analytical expression for the Hubble parameter:

$$H\left(z\right)=H_0{(1+z)}^{1+q_0}exp\left[{\displaystyle \frac{q_1}{2}}{\left(ln(1+z)\right)}^2\right]$$

(7)

This explicit dependency is crucial for deriving the scalar field potential $V\left(\varphi \right)$ and the kinetic energy ${\dot{\varphi }}^2$ without numerical approximations. Furthermore, the transition from the past decelerated phase ($q>0$) to the current accelerated epoch ($q<0$) is easily identified by setting $q\left(z_{tr}\right)=0$, which leads to the simple relation

$$z_{tr}=exp\left(-{\displaystyle \frac{q_0}{q_1}}\right)-1$$

(8)

This analytical clarity not only simplifies the MCMC likelihood calculations but also provides a transparent mapping between the kinematic expansion and the underlying scalar field dynamics.

4. Observational Data and Statistical Analysis

To determine the optimal values for the free parameters of our logarithmic model ($H_0,q_0,q_1$), we employ a robust Bayesian inference framework. We perform a combined ${\chi }^2$ minimization using Markov Chain Monte Carlo (MCMC) simulations, which allow for a full exploration of the parameter space and the identification of degeneracies between variables. The total likelihood function is defined as ${\mathcal{L}}_{tot}\propto exp(-{\chi }_{tot}^2/2)$, where the total ${\chi }^2$ is the sum of the contributions from each independent dataset:

$${\chi }_{tot}^2={\chi }_{CC}^2+{\chi }_{SC}^2+{\chi }_{BAO}^2+{\chi }_{R19}^2$$

(9)

The individual components of the observational suite are described as follows:

• Cosmic Chronometers (CCs): We utilize a sample of 31 measurements of the Hubble parameter $H\left(z\right)$ obtained through the differential aging method of passively evolving galaxies [19]. Unlike distance-based probes, CC data provide a direct measurement of $H\left(z\right)$ without assuming a fiducial cosmological model, making them ideal for model-independent reconstructions.
• Standard Candles (SCs): We incorporate the Pantheon+ compilation, which consists of 1701 light curves from Type Ia Supernovae (SNe Ia) spanning the redshift range $0.001<z<2.26$ [15].
• Baryon Acoustic Oscillations (BAOs): To capture the large-scale structure signals, we include BAO distance measurements from various surveys, including the 6dFGS, SDSS, and extended Baryon Oscillation Spectroscopic Survey (eBOSS) [16].
• R19 Prior: Given the ongoing tension in the measurement of the Hubble constant, we include a Gaussian prior on $H_0$ based on the SH0ES collaboration’s local distance ladder results ($H_0=74.03\pm 1.42$ km/s/Mpc) [17].

By combining these complementary probes (TotR: CC + SC + BAO + R19), we significantly reduce the statistical uncertainties and break the parameter degeneracies inherent in single-dataset analyses, leading to the high-precision constraints presented in the subsequent section.

5. Results and Discussion

The joint MCMC analysis, utilizing the integrated observational suite (CC + SC + BAO + R19), provides tight constraints on the model’s free parameters. The marginalized posterior distributions and the corresponding $1\sigma$ and $2\sigma$ confidence contours demonstrate a well-converged parameter space. From this statistical analysis, the best-fit value for the Hubble constant is found to be $H_0=71.41\pm 0.98$ km/s/Mpc, while the present-day deceleration parameter is constrained to $q_0=-0.55\pm 0.02$. Additionally, the evolution parameter is determined as $q_1=0.95\pm 0.04$. These results signify a robust reconstruction of the cosmic expansion history. Specifically, the negative value of $q_0$ confirms that the universe is currently undergoing an accelerated expansion phase, which is consistent with the latest SNe Ia observations [15].

One of the most significant outcomes of our logarithmic parametrization is the precise determination of the transition redshift. By solving the condition $q\left(z_{tr}\right)=0$, we find $z_{tr}\approx 0.785$. This value indicates that the universe transitioned from a decelerating, matter-dominated phase to its current accelerating epoch approximately 6–7 billion years ago. This result is in excellent agreement with standard $\Lambda$CDM predictions and recent model-independent studies, which typically place $z_{tr}$ in the range $0.5<z<0.9$ [14,20].

The obtained value of $H_0=71.41\pm 0.98$ km/s/Mpc is of particular interest in the context of the “Hubble tension”. This value sits comfortably between the low $H_0$ values derived from CMB data ($H_0\approx 67.4$ km/s/Mpc) [3] and the higher local measurements from the SH0ES collaboration ($H_0\approx 74.03$ km/s/Mpc) [17]. By yielding an intermediate value, our logarithmic scalar field model provides a viable theoretical pathway to alleviate the persistent discrepancy between early- and late-time observations such as the persistent $H_0$ tension [18].

The reconstructed physical quantities, such as the dimensionless energy density ${\rho }_{\varphi }$ and the pressure $p_{\varphi }$, confirm the physical viability of the model. The scalar field potential $V\left(\varphi \right)$ and the kinetic energy term ${\dot{\varphi }}^2$ are consistently positive, ensuring that the model does not violate basic energy conditions. Furthermore, the equation of state parameter ${\omega }_{\varphi }\left(z\right)$ exhibits dynamic behavior.

6. Conclusions

In this study, we have presented a comprehensive reconstruction of the late-time cosmic expansion history using a novel logarithmic parametrization of the deceleration parameter, $q\left(z\right)=q_0+q_{1}ln(1+z)$. By working within the framework of Scalar Field Cosmology under General Relativity, we have demonstrated that this specific functional form serves as a robust and analytically tractable tool for exploring dark energy dynamics. Our conclusions are summarized as follows:

Looking ahead, we plan to extend this research by incorporating data from future high-precision surveys such as Euclid and DESI. Furthermore, analyzing the evolution of cosmic perturbations within this logarithmic framework will be essential to test its performance at the level of structure formation and to further validate its viability against the standard cosmological paradigm.