[Paper Review] Observational constraints on the jerk parameter with the data of the Hubble parameter

Abdulla Al Mamon, Kazuharu Bamba|arXiv (Cornell University)|May 8, 2018

Cosmology and Gravitation Theories9 citations

TL;DR

This study uses a model-independent kinematic approach to constrain the jerk parameter $j$ and deceleration parameter $q$ via a generalized parametrization of $q(z)$, employing the latest 41-point Hubble parameter dataset ($0.07 \leq z \leq 2.36$). It finds consistent evidence for a transition from decelerated to accelerated expansion, with the $\Lambda$CDM model marginally disfavored at $1\sigma$ for most $\alpha$ values, while the model remains well-constrained by $H(z)$ data at low redshifts.

ABSTRACT

We study the accelerated expansion phase of the universe by using the { extit{kinematic approach}}. In particular, the deceleration parameter $q$ is parametrized in a model-independent way. Considering a generalized parametrization for $q$, we first obtain the jerk parameter $j$ (a dimensionless third time derivative of the scale factor) and then confront it with cosmic observations. We use the latest observational dataset of the Hubble parameter $H(z)$ consisting of 41 data points in the redshift range of $0.07 \leq z \leq 2.36$, larger than the redshift range that covered by the Type Ia supernova. We also acquire the current values of the deceleration parameter $q_0$, jerk parameter $j_0$ and transition redshift $z_t$ (at which the expansion of the universe switches from being decelerated to accelerated) with $1σ$ errors ($68.3\%$ confidence level). As a result, it is demonstrate that the universe is indeed undergoing an accelerated expansion phase following the decelerated one. This is consistent with the present observations. Moreover, we find the departure for the present model from the standard $Λ$CDM model according to the evolution of $j$. Furthermore, the evolution of the normalized Hubble parameter is shown for the present model and it is compared with the dataset of $H(z)$.

Motivation & Objective

• To investigate the cosmic acceleration phase using a kinematic approach without assuming a specific cosmological model.
• To constrain the jerk parameter $j$ and deceleration parameter $q$ using a generalized, model-independent parametrization of $q(z)$.
• To test the viability of the $\Lambda$CDM model against observational $H(z)$ data at the current epoch.
• To determine the transition redshift $z_t$ where the universe shifts from deceleration to acceleration.

Proposed method

• A generalized parametrization of the deceleration parameter $q(z)$ is adopted, which reduces to $q \propto z$, $q \propto \ln(1+z)$, and $q \propto z/(1+z)$ for specific $\alpha$ values.
• The jerk parameter $j(z)$ is analytically derived from the parametrized $q(z)$ using kinematic relations involving time derivatives of the scale factor.
• The model is constrained using $\chi^2$ minimization with the latest 41-point $H(z)$ dataset spanning $0.07 \leq z \leq 2.36$, covering higher redshifts than Type Ia supernovae.
• Best-fit values of $q_0$, $j_0$, $z_t$, and model parameters are obtained at $1\sigma$ confidence level for different $\alpha$ values.
• The evolution of the normalized Hubble parameter $h(z) = H(z)/H_0$ is computed and compared with the $H(z)$ data to validate the model.
• The $\Lambda$CDM model is used as a benchmark, but the model is not forced to match it a priori, allowing $j_0$ to be data-driven.

Experimental results

Research questions

• RQ1 Does the kinematic model with a generalized $q(z)$ parametrization support a transition from decelerated to accelerated expansion?
• RQ2 How well does the derived jerk parameter $j(z)$ match observational $H(z)$ data across the redshift range $0.07 \leq z \leq 2.36$?
• RQ3 Is the $\Lambda$CDM model consistent with the $H(z)$ data at the $1\sigma$ confidence level for the current model?
• RQ4 How do the best-fit values of $q_0$, $j_0$, and $z_t$ vary with the parametrization parameter $\alpha$?
• RQ5 To what extent does the model deviate from $\Lambda$CDM in the current epoch, as indicated by $j_0$?

Key findings

• The model shows a smooth transition from deceleration to acceleration for all values of $\alpha$, with $z_t$ ranging from $0.522$ to $0.884$ at $1\sigma$.
• The current deceleration parameter $q_0$ varies with $\alpha$, taking values between $-0.410$ and $0.803$ at $1\sigma$, indicating a consistent acceleration phase.
• The current jerk parameter $j_0$ is found to deviate from the $\Lambda$CDM value of 1, with only $\alpha = 0.5$ and $\alpha = 0.3$ marginally consistent with $\Lambda$CDM at $1\sigma$.
• For $\alpha = 0.5$, $j_0 = 0.803^{+0.061}_{-0.01}$, and for $\alpha = 0.3$, $j_0 = 0.938^{+0.036}_{-0.05}$, both within $1\sigma$ of $\Lambda$CDM.
• The model's predicted $h(z)$ evolution is in good agreement with the $H(z)$ dataset across all $\alpha$ values, confirming consistency with low-redshift observations.
• The $\bar{\chi}^2$ values indicate comparable goodness-of-fit for all $\alpha$ values, suggesting no significant preference for any specific parametrization.