Unified Dark Matter and Dark Energy in a model of Non-Canonical Scalar-Tensor Theory

TL;DR

The paper analyzes a non-canonical scalar-tensor theory that generalizes Brans-Dicke gravity by substituting the standard kinetic term with $L(X, \phi) = \lambda X^{\alpha} \phi^{\beta} - V(\phi)$. In a spatially flat FRW setting, it demonstrates that the kinetic term alone yields a power-law expansion with a maximum exponent $n_{\max} = (1+\sqrt{3})/4$, and that for $\alpha \ge 18$ the model can reproduce the GR-like dust-dominated expansion $a(t) \propto t^{2/3}$; introducing a linear potential $V(\phi) = V_0 \phi$ then yields ΛCDM-like evolution, with larger $\alpha$ bringing the predictions closer to GR. The work also shows that with $V(\phi)=V_0\phi$, the scalar field can mimic both cold dark matter and a cosmological constant, effectively unifying the dark sector within this scalar-tensor framework. The authors further discuss the potential observational implications and emphasize the need to study cosmological perturbations to constrain the model parameters.

Abstract

We consider a model of non-canonical scalar-tensor theory in which the kinetic term in the Brans-Dicke action is replaced by a non-canonical scalar field Lagrangian $\mathcal{L}(X, φ)= λX^αφ^β- V(φ)$ where $X = (1/2) \partial_μ φ\partial^μ φ$ and $α$, $β$ and $λ$ are parameters of the model. This can be considered as a simple non-canonical generalization of the Brans-Dicke theory with a potential term which corresponds to a special case of this model with the values of the parameter $α= 1$, $β= -1$ and $λ= 2w_{_{BD}}$ where $w_{_{BD}}$ is the Brans-Dicke parameter. Considering a spatially flat Friedmann-Robertson-Walker Universe with scale factor $a(t)$, it is shown that, in the matter free Universe, the kinetic term $λX^αφ^β$ can lead to a power law solution $a(t)\propto t^{n}$ but the maximum possible value of $n$ turns out to be $(1+\sqrt{3})/4 \approx 0.683$. When $α\geq 18$, this model can lead to a solution $a(t)\propto t^{2/3}$, thereby mimicking the evolution of scale factor in a cold dark matter dominated epoch with Einstein's General Relativity (GR). With the addition of a linear potential term $V(φ) = V_{0}φ$, it is shown that this model mimics the standard $Λ$CDM model type evolution of the Universe. The larger the value of $α$, the closer the evolution of $a(t)$ in this model to that in the $Λ$CDM model based on Einstein's GR. The purpose of this paper is to demonstrate that this model with a linear potential can mimic the GR based $Λ$CDM model. However, with an appropriate choice of the potential $V(φ)$, this model can provide a unified description of both dark matter and dynamical dark energy, as if it were based on Einstein's GR.

Figures (8)

• Figure 1: Plot of $N'$ for different integer values of $\alpha$ from $\alpha = 2$ to $50$. Note that $N'$ as determined by Eq. (\ref{["Eqn: N'"]}) gives the upper bound on $n$ in the solution $a(t) \propto t^n$ for each value of $\alpha$. The blue dots represent the function $N'(\alpha)$ while the red dashed line indicates the constant value $N' = 2/3$. It is evident from this figure that only when $\alpha \geq18$ it is possible to have a solution $a(t) \propto t^{2/3}$.
• Figure 2: Plot of $f(\tau)$ as a function of $\tau$ for $\alpha = 18$ and $\alpha = 100$ where $f(\tau)$ is defined in Eq. (\ref{['Eqn: f(tau)']}). Evidently $f(\tau) = 1$ which confirms that the solutions $\bar{a}(\tau)$ and $\bar{\phi}(\tau)$ obtained from Eqs. (\ref{['Eqn: Friedman Eqn-2 dimensionless']}) and (\ref{['Eqn: phi prime prime dimensionless']}) are consistent with Eq. (\ref{['Eqn: Friedman Eqn-1 dimensionless']}).
• Figure 3: Plot of the ratio $a(\tau)/a_{_\Lambda}(\tau)$ as a function of $\tau$ for $\alpha = 18$ and $\alpha = 100$. These plots clearly illustrates that $a(\tau) \approx a_{_\Lambda}(\tau)$ for $\alpha \geq 18$. The larger the value of $\alpha$ closer is $a(\tau)$ to $a_{_\Lambda}(\tau)$. Both the solutions are observationally indistinguishable for $\alpha = 100$.
• Figure 4: Plot of the ratio $H(z)/H_{_\Lambda}(z)$ as a function of redshift $z$ for $\alpha = 18$ and $\alpha = 100$. Here $H(z)$ is the Hubble parameter as a function of redshift $z$ in the model considered in this paper whereas $H_{_\Lambda}(z)$ is corresponding function in the GR- based $\Lambda$CDM model.
• Figure 5: Plot of the deceleration parameter $q$ as a function of scale factor $a$ for $\alpha = 18$ and $\alpha = 100$. The blue line shows the $q$ as a function of $a$ in the non-canonical scalar-tensor model considered in this paper, while the red dashed line gives $q$ for the GR based $\Lambda$CDM model. The two curves nearly overlaps for $\alpha = 18$ while for $\alpha = 100$ the two curves are virtually indistinguishable.