Anisotropic bouncing scenario in $F(X)-V(\phi)$ model

Abstract

We investigate the cosmology of a class of model with noncanonical scalar field and matter in an anisotropic time dependent background. Writing the Einstein Equations in terms of dimensionless dynamical variables appropriately defined for bouncing solutions, we find all the fixed points. While evolving the dynamical variables to their stable fixed points numerically, solutions satisfying non singular bounce are found.

Appendix

1.1 A.1 Stability analysis of fixed points

The stability of a fixed point is determined from the behaviour of a small perturbation around that fixed point. We get the set of fixed points $\tilde{x}_{c}$, $\tilde{y}_{c}$, $\tilde{z}_{c}$ and ${\sigma}_{c}$ by solving the set of Eqs. (20) simultaneously (where the subscript $c$ denotes fixed points). Now, if we define the slopes of the dynamical variables $\tilde{x}$, $\tilde{y}$, $\tilde{z}$ and ${\sigma }$ as $f(\tilde{x},\tilde{y},\tilde{z},\sigma)$, $g(\tilde{x},\tilde {y},\tilde{z},\sigma)$, $h(\tilde{x}, \tilde{y},\tilde{z},\sigma)$ and $i(\tilde{x},\tilde{y},\tilde{z},\sigma)$. The set of equations we need to solve to obtain the fixed point is

$$ \begin{aligned} &f(\tilde{x},\tilde{y},\tilde{z},{\sigma}) \equiv \frac{\mathrm{d} \tilde{x}}{\mathrm{d} \tilde{N}}=0 , \\ &g(\tilde{x},\tilde{y},\tilde{z},{\sigma}) \equiv \frac{\mathrm{d} \tilde{y}}{\mathrm{d} \tilde{N}}=0 , \\ &h(\tilde{x},\tilde{y},\tilde{z},{\sigma}) \equiv \frac{\mathrm{d} \tilde{z}}{\mathrm{d} \tilde{N}}=0 , \\ &i(\tilde{x},\tilde{y},\tilde{z},{\sigma}) \equiv \frac{\mathrm{d} {\sigma}}{\mathrm{d} \tilde{N}}=0 , \end{aligned} $$

(20)

where,

$$\begin{aligned} &f(\tilde{x},\tilde{y},\tilde{z},\sigma) \\ &\quad \equiv -\frac{3}{2}\bigl[(w_{k}-w_{m}) (\operatorname{\mathrm{sign}} \rho_{k})+(1+w_{m}) \bigl(\tilde{x}^{2}-\tilde{y}| \tilde{y}|\bigr) \\ &\qquad {}+(1-w_{m})\tilde{z}^{2}\bigr]+\frac{3}{2} \tilde{x}\bigl[(w_{k}+1)\tilde{x}-\sigma \tilde{y}|\tilde{y}| \operatorname{\mathrm{sign}}(\rho_{k})\bigr], \\ &g(\tilde{x},\tilde{y},\tilde{z},\sigma) \equiv \frac{3}{2}\tilde{y}\bigl[- \sigma+(w_{k}+1) \tilde{x} - \sigma\tilde{y} |\tilde{y}| (\operatorname{\mathrm{sign}} \rho_{k})\bigr], \\ &h(\tilde{x},\tilde{y},\tilde{z},\sigma) \equiv -3\tilde{z}\tilde{x}+3\tilde{z} \tilde{x}(1+w_{k})-3\tilde{z}\tilde{y}|\tilde{y}|\operatorname{ \mathrm{sign}}(\rho_{k}), \\ &i(\tilde{x},\tilde{y},\tilde{z},\sigma) \\ &\quad \equiv \frac{3}{2}\frac{[2 \varXi(w_{k}+1)+(w_{k}-1)]}{2(2\sigma+1)(w_{k}+1)}\bigl[(w_{k}+1) \tilde{x}-\sigma\tilde{y}^{2}\bigr]. \end{aligned}$$

(21)

The corresponding fixed point for $\tilde{\varOmega}_{m}$ can be found using the constraint Eq. (12).

The stability of the fixed points can be examined from the evolution of small perturbations around fixed points. Now, if $(\tilde{x}_{c}, \tilde{y}_{c}, \tilde{z}_{c}, {\sigma}_{c})$ is a fixed point and $\delta\tilde{x}= \tilde{x}-\tilde{x}_{c}$, $\delta\tilde {y}=\tilde{y}-\tilde{y}_{c}$, $\delta\tilde{z}= \tilde{z}-\tilde{z}_{c}$ and $\delta\sigma= \sigma -\sigma_{c}$ be the respective perturbation around it, then the evolution of the perturbation is determined by

$$ \begin{aligned} &\delta\dot{\tilde{x}}= \dot{\tilde{x}}= f(\tilde{x}_{c}+ \delta\tilde{x},\tilde{y}_{c}+\delta\tilde{y},\tilde{z}_{c}+ \delta\tilde{z}, {\sigma} +\delta{\sigma} ), \\ &\delta\dot{\tilde{y}}= \dot{\tilde{y}}= g(\tilde{x}_{c}+\delta \tilde{x},\tilde{y}_{c}+\delta\tilde{y},\tilde{z}_{c}+\delta \tilde{z},{\sigma}+\delta{\sigma}), \\ &\delta\dot{\tilde{z}}= \dot{\tilde{z}}= h(\tilde{x}_{c}+\delta \tilde{x},\tilde{y}_{c}+\delta\tilde{y},\tilde{z}_{c}+\delta \tilde{z},{\sigma}+\delta{\sigma}), \\ &\delta\dot{ {\sigma}}= \dot{\sigma}= h(\tilde{x}_{c}+\delta \tilde{x},\tilde{y}_{c}+\delta\tilde{y},\tilde{z}_{c}+\delta \tilde{z},{\sigma}+\delta{\sigma}). \end{aligned} $$

(22)

The evolution equations, up to first order, for these perturbations are

$$\begin{aligned} {\left ( \textstyle\begin{array}{cccc} \delta\dot{\tilde{x}} \\ \delta\dot{\tilde{y}} \\ \delta\dot{\tilde{z}} \\ \delta\dot{\sigma}\end{array}\displaystyle \right ) }= \mathbf{A} {\left ( \textstyle\begin{array}{cccc} \delta{\tilde{x}} \\ \delta{\tilde{y}} \\ \delta {\tilde{z}} \\ \delta\sigma \end{array}\displaystyle \right )} \end{aligned}$$

(23)

where the matrix is

$$\begin{aligned} {\mathbf{A}}= {\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \frac{\partial f}{\partial\tilde{x}} & \frac{\partial f}{\partial \tilde{y}} & \frac{\partial f}{\partial\tilde{z}} & \frac{\partial f}{\partial\sigma} \\ \frac{\partial g}{\partial\tilde{x}} & \frac{\partial g}{\partial \tilde{y}} & \frac{\partial g}{\partial\tilde{z}} & \frac{\partial g}{\partial\sigma} \\ \frac{\partial h}{\partial\tilde{x}} & \frac{\partial h}{\partial \tilde{y}} & \frac{\partial h}{\partial\tilde{z}} & \frac{\partial h}{\partial\sigma}\\ \frac{\partial i}{\partial\tilde{x}} & \frac{\partial i}{\partial \tilde{y}} & \frac{\partial i}{\partial\tilde{z}} & \frac{\partial i}{\partial\sigma} \end{array}\displaystyle \right )} \end{aligned}$$

(24)

is the Jacobian matrix and is evaluated at the fixed point $(\tilde {x}_{c},\tilde{y}_{c},\tilde{z}_{c},\sigma_{c})$ and hence each entry of $\mathbf{A}$ is a number. The solution of the system of equations can be found by diagonalizing the matrix $\mathbf{A}$. A non trivial solution exists only when the determinant $| \mathbf{A}- \lambda\mathbf{I}|$ is zero. Thus, solving this equation in $\lambda$ we would get all the eigen values of the system corresponding to each fixed points.