Non-minimal derivative coupling gravity in cosmology

Abstract

We give a brief review of the non-minimal derivative coupling (NMDC) scalar field theory in which there is non-minimal coupling between the scalar field derivative term and the Einstein tensor. We assume that the expansion is of power-law type or super-acceleration type for small redshift. The Lagrangian includes the NMDC term, a free kinetic term, a cosmological constant term and a barotropic matter term. For a value of the coupling constant that is compatible with inflation, we use the combined WMAP9 (WMAP9 + eCMB + BAO + $H_0$) dataset, the PLANCK + WP dataset, and the PLANCK TT, TE, EE + lowP + Lensing + ext datasets to find the value of the cosmological constant in the model. Modeling the expansion with power-law gives a negative cosmological constants while the phantom power-law (super-acceleration) expansion gives positive cosmological constant with large error bar. The value obtained is of the same order as in the $\Lambda$CDM model, since at late times the NMDC effect is tiny due to small curvature.

Appendices

Appendix 1: Equation of state parameter for power-law case

In this part, we apply the power-law expansion $a = a_0 \left({t}/{t_0}\right)^\alpha$ to the NMDC cosmology. The equation of state parameter in Eq. (11) takes the form

$$\begin{aligned} w_\phi =\frac{\dot{\phi }^2(t^2 - 9\kappa \alpha ^2)\left[1 - \frac{2\kappa \alpha (t^2 + 9\kappa \alpha ^2)}{(t^2 - 3\kappa \alpha ^2)(t^2 - 9\kappa \alpha ^2)}\right] - \frac{4\kappa \alpha \dot{\phi }V_{,\phi }t^3}{t^2 - 3\kappa \alpha ^2} - 2V(\phi )t^2}{\dot{\phi }^2(t^2 - 3\kappa \alpha ^2) + 2V(\phi )t^2}. \end{aligned}$$

(24)

Eq. (18) takes the form,

$$\begin{aligned} \dot{\phi }^2 = \frac{F_1(t, \phi , \dot{\phi })}{(t^2 - 9\kappa \alpha ^2)}. \end{aligned}$$

(25)

Substituting Eq. (25) into the equation of state parameter, Eq. (24), we obtain

$$\begin{aligned} w_\phi = \frac{ F_1(t, \phi , \dot{\phi }) \left[1 - \frac{2\kappa \alpha (t^2 + 9\kappa \alpha ^2)}{(t^2 - 3\kappa \alpha ^2)(t^2 - 9\kappa \alpha ^2)}\right] - \frac{4\kappa \alpha V_{,\phi }\dot{\phi }t^3}{(t^2 - 3\kappa \alpha ^2)} - 2V(\phi )t^2}{ F_1(t, \phi , \dot{\phi }) + 2V(\phi )t^2} \end{aligned}$$

(26)

where

$$\begin{aligned} F_1(t, \phi , \dot{\phi }) = \frac{\frac{2\kappa \alpha V_{,\phi }\dot{\phi }t^3}{(t^2 - 9\kappa \alpha ^2)} \,-\, \rho _\mathrm{m,0}\frac{t_0^{3\alpha }}{(t^{3\alpha - 2})} \,+\, \frac{\alpha }{(4\pi G)}}{1 - \frac{\kappa \alpha (t^2+9\kappa \alpha ^2)}{(t^2 - 3\kappa \alpha ^2)(t^2 - 9\kappa \alpha ^2)}} \end{aligned}$$

(27)

Appendix 2: Equation of state parameter for phantom power-law case

Apply the phantom power-law expansion (super-acceleration), $a = a_0\big [(t_\mathrm{s} - t)/$ $(t_\mathrm{s} - t_0)\big ]^\beta ,$ The kinetic term can be written as

$$\begin{aligned} \dot{\phi }^2 = - \frac{F_2(t, \phi , \dot{\phi })}{\left[(t_\mathrm{s} - t)^2 + 9\kappa \beta ^2\right]} \end{aligned}$$

(28)

The equation of state parameter of a phantom power-law expansion is

$$\begin{aligned} w_\phi = \frac{ F_2(t, \phi , \dot{\phi }) \left[1+ \frac{2\kappa \beta \left[(t_\mathrm{s} - t)^2 - 9\kappa \beta ^2\right]}{\left[(t_\mathrm{s} - t)^2 + 3\kappa \beta ^2\right]\left[(t_\mathrm{s} - t)^2 + 9\kappa \beta ^2\right]}\right] - \frac{4\kappa \beta V_{,\phi }\dot{\phi }(t_\mathrm{s} - t)^3}{\left[(t_\mathrm{s} - t)^2 + 3\kappa \beta ^2\right]} - 2V(\phi )(t_\mathrm{s} - t)^2}{ F_2(t, \phi , \dot{\phi }) + 2V(\phi )(t_\mathrm{s} - t)^2}\nonumber \\ \end{aligned}$$

(29)

where

$$\begin{aligned} F_2(t, \phi , \dot{\phi }) = \frac{\frac{2\kappa \beta V_{,\phi }\dot{\phi }(t_\mathrm{s} - t)^3}{(t_\mathrm{s} - t)^2 + 3\kappa \beta ^2} - \rho _\mathrm{m,0}\frac{(t_\mathrm{s} - t_0)^{3\beta }}{(t_\mathrm{s} - t)^{3\beta - 2}} + \frac{\beta }{4\pi G}}{1 + \frac{\kappa \beta \left[(t_\mathrm{s} - t)^2 - 9\kappa \beta ^2\right]}{ \left( \left[(t_\mathrm{s} - t)^2 + 3\kappa \beta ^2\right]\left[(t_\mathrm{s} - t)^2 + 9\kappa \beta ^2\right] \right) } } \end{aligned}$$

(30)

With constant potential in form of $V(\phi ) = {\Lambda }/{8\pi G}$ hence $V_{, \phi } = 0$ for both cases.