[Paper Review] A dark energy with higher order derivatives of $H$ in the modified gravity $f(R,T)$

Antonio Pasqua, Surajit Chattopadhyay+1 more|arXiv (Cornell University)|Jun 5, 2013

Cosmology and Gravitation Theories3 citations

TL;DR

This paper proposes a dark energy (DE) model in the $f(R,T) = \mu R + \nu T$ modified gravity framework, where DE density depends on $H^2$, $\dot{H}$, and $\ddot{H}$, generalizing Ricci and Granda-Oliveros DE models. The model exhibits quintom-like behavior with $\omega < -1$ transitioning to $\omega > -1$, supports late-time acceleration, and is classically stable in early universe but unstable today.

ABSTRACT

In this paper, we consider a recently proposed model of Dark Energy (DE) which contains three terms (one proportional to the squared Hubble parameter, one to the first derivative with respect to the cosmic time of the Hubble parameter and one proportional to the second derivative with respect to the cosmic time of the Hubble parameter) in the light of the $f\left(R,T ight)$ model of modified gravity, considering the particular model $f\left(R,T ight) = μR + νT$, with $μ$ and $ν$ two free positive constant parameters. Here $R$ and $T$ are the curvature and torsion scalars, respectively. In this work, we have found that the Hubble parameter $H$ exhibits a decaying behavior until redshifts of the order of $z\approx-0.5$ (when it starts to increase) and the time derivative of the Hubble parameter goes from negative to positive values for different redshifts. The equation of state (EoS) parameter of DE and the effective EoS parameter exhibit a transition from $ω-1$ (then the EoS parameters have a quintom-like behavior). We have also found that the said model can attain the late time accelerated phase of the universe. Using the statefinder parameters $r$ and $s$, we derived that the considered model can attaining the $Λ$CDM phase of the universe and can interpolate between dust and $Λ$CDM phase of the universe. Finally, studying the squared speed of sound $v_s^2$, we have seen that the model under consideration is classically stable in the earlier stage of the universe, but classically unstable in the current stage.

Motivation & Objective

• To investigate a dark energy model with energy density dependent on $H^2$, $\dot{H}$, and $\ddot{H}$ within the $f(R,T)$ modified gravity framework.
• To examine whether this model can reproduce the observed late-time accelerated expansion of the universe.
• To analyze the classical stability of the model via the squared speed of sound.
• To assess the model's ability to interpolate between dust and $\Lambda$CDM phases using statefinder parameters.
• To determine the behavior of the equation of state (EoS) parameter and its transition from quintessence to phantom-like regimes.

Proposed method

• Formulate a dark energy density as $\rho_{\text{DE}} = \varepsilon \frac{\ddot{H}}{H} + \lambda \dot{H} + \theta H^2$, with $\varepsilon, \lambda, \theta > 0$, ensuring dimensional consistency.
• Adopt the $f(R,T) = \mu R + \nu T$ gravity model, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy tensor.
• Derive the modified Friedmann equations from the $f(R,T)$ action, incorporating interaction between dark energy and pressureless dark matter.
• Compute the Hubble parameter $H(z)$ and its time derivatives $\dot{H}$, $\ddot{H}$ as functions of redshift $z$.
• Calculate the equation of state (EoS) parameter $\omega = p/\rho$ and effective EoS $\omega_{\text{eff}}$ to analyze dynamical behavior.
• Use statefinder parameters $r$ and $s$ to characterize the cosmological evolution and compare with $\Lambda$CDM and dust phases.

Experimental results

Research questions

• RQ1 Can a dark energy model with $H^2$, $\dot{H}$, and $\ddot{H}$ terms in $f(R,T)$ gravity reproduce the observed late-time acceleration of the universe?
• RQ2 Does the equation of state parameter $\omega$ exhibit a quintom-like transition from $\omega < -1$ to $\omega > -1$?
• RQ3 What is the behavior of the squared speed of sound $v_s^2 = \dot{p}/\dot{\rho}$, and does it indicate classical stability or instability?
• RQ4 Can the model's statefinder trajectories $r(z), s(z)$ reach or interpolate between the $\Lambda$CDM and dust phases?
• RQ5 How do the parameters $\varepsilon$, $\lambda$, and $\theta$ influence the transition redshift and stability of the model?

Key findings

• The Hubble parameter $H(z)$ decreases from high redshifts until $z \approx -0.5$, after which it increases, indicating a transition from deceleration to acceleration.
• The time derivative $\dot{H}$ transitions from negative to positive values at different redshifts depending on $\varepsilon$, signaling a change in the acceleration rate.
• For $\varepsilon = 2$ and $\varepsilon = 3$, the EoS parameter $\omega$ transitions from $\omega < -1$ to $\omega > -1$, exhibiting quintom-like behavior; for $\varepsilon = 4$, $\omega$ remains negative but does not cross $-1$.
• The effective EoS parameter $\omega_{\text{eff}}$ always shows a transition from quintessence to phantom-like behavior, confirming dynamic dark energy.
• The model successfully reaches the late-time accelerated phase of the universe, transitioning from a dark matter-dominated decelerated phase.
• The squared speed of sound $v_s^2$ is positive (indicating classical stability) for $z \lesssim 0.5$, but becomes negative in the current universe ($z \approx 0$), implying classical instability at present.