Role of adiabatic index on the evolution of spherical gravitational collapse in Palatini f(R) gravity

Abstract

The purpose of this paper is to study role of adiabatic index (stiffness parameter) on the evolution of an anisotropic collapsing spherical self-gravitating system in Palatini f(R) gravity. In this scenario, we construct dynamical equation with the help of contracted Bianchi identities of the effective dark sources as well as usual matter. The perturbation approach is applied on the physical variables which consequently perturbes the Ricci scalar. We discuss instabilities both for zero and non-zero expansion. It is shown that instability range of the given system depends upon energy density profile, anisotropic pressure and chosen f(R) model in both scenarios. However, the adiabatic index is not helpful in expansion-free instability analysis at N and pN regimes.

Appendix

$$\begin{aligned} D_0 =& \biggl[A \biggl\{ \frac{-1}{(HA)^2} \biggl( \dot{f'_R} -\frac{A'}{A}\dot{f_R}- \frac{\dot{H}}{H}f'_R-\frac{5}{2} \frac{\dot{f_R}f'_R}{f_R} \biggr) \biggr\} _{,1} - \frac{1}{A} \biggl\{ \frac{f-Rf_R}{2}- \frac{f''_R}{H^2} +\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{H}}{H}+ \frac{9}{4} \frac{\dot{f_R}}{f_R} \biggr) \\ &{}+\frac{f'_R}{H^2} \biggl( \frac{H'}{H} +\frac{1}{4}\frac{f'_R}{f_R}-\frac{2}{r} \biggr) \biggr\} _{,0} + \frac{\dot{f_R}}{f_RA} \biggl\{ \frac{3\ddot{f_R}}{2A^2} - \frac{(f-Rf_R)}{2}+\frac{3f''_R}{2H^2} -\frac{\dot{f_R}}{A^2} \biggl( \frac{3 \dot{H}}{2H}+\frac{3\dot{A}}{2A}+6\frac{\dot{f_R}}{f_R} \biggr) \\ &{} - \frac{f'_R}{H^2} \biggl(\frac{3H'}{2H} + \frac{3f'_R}{2f_R}+\frac{3A'}{2A}-\frac{3}{r} \biggr) \biggr\} + \frac{\dot{H}}{A^2H} \biggl\{ \frac{f''_R}{H^2}+\frac{\ddot{f_R}}{A^2}- \frac{\dot{f_R}}{A^2} \biggl(\frac{5\dot{f_R}}{2f_R}+\frac{\dot{A}}{A}+ \frac{\dot{H}}{H} \biggr) - \frac{f'_R}{H^2} \biggl(\frac{A'}{A}+ \frac{5}{2}\frac{f'_R}{f_R} +\frac{H'}{H} \biggr) \biggr\} \\ &{} - \frac{1}{AH^2} \biggl(\dot{f'_R} - \frac{5}{2}\frac{\dot{f_R}f'_R}{f_R}-\frac{A'}{A}\dot{f_R} - \frac{\dot{H}}{H}f'_R \biggr) \biggl(\frac{3A'}{A} +\frac{3f'_R}{f_R}+ \frac{H'}{H} +\frac{2}{r} \biggr) \biggr], \end{aligned}$$

(A.1)

$$\begin{aligned} D_1 =& \biggl[H \biggl\{ \frac{-1}{(HA)^2} \biggl( \dot{f'_R} -5\frac{\dot{f_R}f'_R}{f_R}-\frac{A'}{A} \dot{f_R}-\frac{\dot{H}}{H} f'_R \biggr) \biggr\} _{,0} + \frac{1}{H} \biggl\{ \frac{f-Rf_R}{2}+ \frac{\ddot{f_R}}{A^2} -\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{A}}{A}+ \frac{1}{4}\frac{ \dot{f_R}}{f_R} \biggr) \\ &{}-\frac{f'_R}{H^2} \biggl( \frac{2}{r}+\frac{A'}{A} +\frac{9}{4}\frac{f'_R}{f_R} \biggr) \biggr\} _{,1} + \frac{A'}{HA} \biggl\{ \frac{f''_R}{H^2}+ \frac{\ddot{f_R}}{ A^2}-\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{5f_R}}{2f_R}+ \frac{\dot{A}}{A} +\frac{\dot{H}}{H} \biggr) -\frac{f'_R}{H^2} \biggl( \frac{5}{2}\frac{f'_R}{f_R}+\frac{A'}{A} +\frac{H'}{H} \biggr) \biggr\} \\ &{}+ \frac{f'_R}{Hf_R} \biggl\{ \frac{f-Rf_R}{2}+ \frac{3\ddot{f_R}}{2A^2} -\frac{3\dot{f_R}}{2A^2} \biggl(\frac{\dot{f_R}}{f_R}+ \frac{\dot{A}}{A} +\frac{\dot{H}}{H} \biggr) -\frac{f'_R}{H^2} \biggl( \frac{3}{r}+\frac{3A'}{ 2A} - \frac{3H'}{2H}+\frac{6f'_R}{f_R} \biggr)+ \frac{3f''_R}{2H^2} \biggr\} \\ &{} +\frac{2}{Hr} \biggl\{ \frac{f''_R}{H^2}- \frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{H}}{H} \biggr)-\frac{f'_R}{H^2} \biggl( \frac{1}{r}+\frac{5}{2}\frac{f'_R}{f_R} + \frac{H'}{H} \biggr) \biggr\} - \frac{1}{HA^2} \biggl(\dot{f'_R}- \frac{5}{2}\frac{ \dot{f_R}f'_R}{f_R}-\frac{A'}{A}\dot{f_R}- \frac{\dot{H}}{H}f'_R \biggr) \\ &{}\times \biggl( \frac{\dot{A}}{A}+\frac{3\dot{H}}{H}+\frac{3\dot{f_R}}{f_R} \biggr) \biggr], \end{aligned}$$

(A.2)

$$\begin{aligned} D_2 =&\frac{-2{\epsilon}}{H_0^2} \biggl[\frac{-eR_0H_0^2}{2}-e''+ \frac{2hR''_0}{H_0} +R'_0 \biggl\{ \frac{2{\epsilon}}{1+2{\epsilon}R_0} \biggl(e' -\frac{2e{\epsilon}R'_0}{1+2{\epsilon}R_0} \biggr) + \biggl(\frac{h}{H_0} \biggr)' \biggr\} \\ &{}+ \biggl(e'-\frac{2hR'_0}{H_0} \biggr) \biggl( \frac{2{\epsilon}R'_0}{1+2{\epsilon}R_0}+\frac{H'_0}{H_0}-\frac{2}{r} \biggr) \biggr]+A_0^2 \biggl[\frac{-2{\epsilon}}{A_0^2H_0^2} \biggl\{ e'-\frac{A'_0}{A_0}e - \frac{hR'_0}{H_0}-\frac{5e{\epsilon}R'_0}{1+2{\epsilon}R_0} \biggr\} \biggr]_{,1} \\ &{}+ \frac{2e'{\epsilon}^2}{(1+2{\epsilon}R_0)} \biggl[ \frac{3R''_0}{H_0^2}+\frac{R_0^2}{2} - \frac{2R'_0}{H_0^2} \biggl(\frac{3A'_0}{2A_0}+ \frac{3H'_0}{2H_0}+\frac{3{\epsilon}R'_0}{ 1+2{\epsilon}R_0}-\frac{3}{r} \biggr) \biggr]+ \frac{2{\epsilon}h}{H_0^3} \biggl[R''_0-R'_0 \biggl(\frac{A'_0}{A_0} + \frac{5{\epsilon}R'_0}{1+2{\epsilon}R_0} +\frac{H'_0}{H_0} \biggr) \biggr] \\ &{}-\frac{2{\epsilon}}{H_0^2} \biggl(e' -\frac{A'_0}{A_0}e- \frac{hR'_0}{H_0}-\frac{5e{\epsilon}R'_0}{1+2{\epsilon}R_0} \biggr) \biggl(\frac{3A'_0}{A_0}+\frac{6{\epsilon}R'_0}{1+2{\epsilon}R_0} + \frac{H'_0}{H_0}+\frac{2}{r} \biggr) \end{aligned}$$

(A.3)

$$\begin{aligned} D_3 =&\frac{\epsilon\ddot{T}}{A_0^2} \biggl(-e'+ \frac{eA'_{0}}{A_{0}} +\frac{hR'_0}{H_0}+\frac{5e{\epsilon}R'_0}{1+2{\epsilon}R_0} \biggr)+ T \biggl( \frac{a}{A_0} \biggr)' \frac{2{\epsilon}}{H_0^2} \biggl[ R''_0+R'_0 \biggl(\frac{H'_0}{H_0}+\frac{A'_0}{A_0} + \frac{5R'_0\epsilon}{(1+2R_0\epsilon)} \biggr) \biggr] \\ &{}+\frac{2e{\epsilon}A'_0\ddot{T}}{A_0^3} +\frac{2{\epsilon}TA'_0}{H_0^2A_0} \biggl[e''-\frac{2hR''_0}{H_0} + R'_0 \biggl\{ \biggl( \frac{a}{A_0} \biggr)'+ \biggl(\frac{h}{H_0} \biggr)' +\frac{5{\epsilon}}{1+2{\epsilon}R_0} \biggl(e'- \frac{2eR'_0\epsilon}{(1+2R_0\epsilon)} \biggr) \biggr\} \\ &{}+ \biggl(e' - \frac{2hR'_0}{H_0} \biggr) \biggl(\frac{A'_0}{A_0} + \frac{H'_0}{H_0}+\frac{5{\epsilon}R'_0}{(1+2R_0\epsilon)} \biggr) \biggr]+\frac{4{\epsilon}T}{H_0^2} \biggl[e''-\frac{2hR''_0}{H_0} + \biggl(e' - \frac{2hR'_0}{H_0} \biggr) \biggl(\frac{1}{r} + \frac{H'_0}{H_0}+\frac{5{\epsilon}R'_0}{(1+2R_0\epsilon)} \biggr) \\ &{}+R'_0 \biggl\{ \biggl(\frac{h}{H_0} \biggr)'+\frac{2{\epsilon}}{ (1+2R_0\epsilon)} \biggl(e'-\frac{2eR'_0\epsilon}{(1+2R_0\epsilon)} \biggr) \biggr\} \biggr]+\frac{2T{\epsilon}}{(1+2R_0\epsilon)} \biggl(e'- \frac{2eR'_0\epsilon}{(1+2R_0\epsilon)} \biggr) \biggl[ \frac{3{\epsilon}R''_0}{H_0^2} - \frac{{\epsilon}R_0^2}{2} \\ &{} -\frac{2{\epsilon}R'_0}{H_0^2} \biggl( \frac{A'_0}{A_0} +\frac{3H'_0}{2H_0}+\frac{2}{r}+\frac{12{\epsilon}R'_0}{(1+2R_0\epsilon)} \biggr) \biggr] +\frac{6{\epsilon}^2eR'_0\ddot{T}}{A_0^2(1+2R_0\epsilon)} +\frac{2{\epsilon}eR'_0T}{(1+2R_0\epsilon)} \biggl[\frac{3{\epsilon}}{H_0^2} \biggl(e''- \frac{2hR''_0}{H_0} \biggr) -{\epsilon}eR_0 \\ &{}-\frac{2{\epsilon}R'_0}{H_0^2} \biggl\{ \biggl(\frac{a}{A_0} \biggr)'+\frac{3}{2} \biggl(\frac{h}{H_0} \biggr)' +\frac{12{\epsilon}}{ (1+2R_0\epsilon)} \biggl(e'-\frac{2eR'_0\epsilon}{(1+2R_0\epsilon)} \biggr) \biggr\} -\frac{2{\epsilon}}{H_0^2} \biggl(e'-\frac{2hR'_0}{H_0} \biggr) \biggl(\frac{A'_0}{A_0} +\frac{3H'_0}{2H_0}+ \frac{2}{r} \\ &{}+\frac{12{\epsilon}R'_0}{(1+2R_0\epsilon)} \biggr) \biggr]+2{\epsilon}\ddot{T} \biggl( \frac{e}{A_0^2} \biggr)'+T \biggl[ -{\epsilon}R_0 -\frac{2{\epsilon}R'_0}{H_0^2} \biggl\{ \biggl(\frac{a}{A_0} \biggr)'+\frac{9{\epsilon}}{ 2(1+2R_0\epsilon)} \biggl(e'- \frac{2eR'_0\epsilon}{(1+2R_0\epsilon)} \biggr) \biggr\} \\ &{}-\frac{2{\epsilon}}{H_0^2} \biggl(e' - \frac{2hR'_0}{H_0} \biggr) \biggl( \frac{A'_0}{A_0}+\frac{2}{r}+\frac{9{\epsilon}R'_0}{2(1+2R_0\epsilon)} \biggr) \biggr], \end{aligned}$$

(A.4)

$$\begin{aligned} D_4 =&-{\epsilon}eR_0+\frac{2{\epsilon}e''}{H_0^2}+4h{ \epsilon}\frac{R''_0}{H^3_0} -\frac{{\epsilon}^2R'_0}{H_0^2(1+2{\epsilon}R_0)} \biggl(e'- \frac{2eR'_0\epsilon}{1+2{\epsilon}R_0} \biggr)-\frac{2R'_0{\epsilon}}{ {H_0}^2} \biggl(\frac{4{\epsilon}e}{1+2R_0\epsilon} \biggr)' + \biggl( \frac{{\epsilon}R'_0}{2(1+2{\epsilon}R_0)} -\frac{2}{r}+\frac{H'_0}{H_0} \biggr) \\ &{}\times \biggl( \frac{2{\epsilon}e'}{H_0^2} +\frac{16eR'_0\epsilon^2}{H^2_0(1+2R_0\epsilon)} \biggr) -A_0^2 \biggl[\frac{2}{A_0^2H_0^2}(1-2R_0 \epsilon) \biggl(-e\epsilon\frac{A'_0}{A_0} +e'\epsilon + \frac{4eR'_0\epsilon^2}{(1+2R_0\epsilon)} -\frac{5eR'_0\epsilon^2}{H_0^2(1+2{\epsilon}R_0)} \biggr) \biggr]' \\ &{} -\frac{3e\epsilon}{A_0(1+2{\epsilon}R_0)} \biggl\{ \frac{R_0^2\epsilon}{3} -\frac{2R''_0\epsilon}{H_0^2}+ \frac{5R'_0\epsilon}{H_0^2r} -2\frac{R'_0H'_0\epsilon}{H_0^3}-\frac{4{\epsilon}^2R'^2_0}{ H_0^2(1+2{\epsilon}R_0)} \biggr\} + \frac{8e\epsilon}{A_0} \biggl(-\frac{R''_0\epsilon}{H_0^2} +\epsilon\frac{R'_0A'_0}{A_0H^2_0} \\ &{} - \frac{5R'^2_0\epsilon^2}{H_0^2(1+2R_0\epsilon)} +\frac{{\epsilon}R'_0H'_0}{H_0^3} \biggr)\frac{1}{(1+2R_0\epsilon)} + \frac{\epsilon}{A_0H_0^2} \biggl(2e' - 2e\frac{A'_0}{A_0}+\frac{R'_0{\epsilon}e}{1+2{\epsilon}R_0} -10 \frac{eR'_0\epsilon^2}{(1+2R_0\epsilon} \biggr) \\ &{} \times\biggl(3\frac{A'_0}{A_0}+\frac{H'_0}{H_0}+ \frac{3}{r} -\frac{6R'_0\epsilon}{1+2R_0\epsilon} \biggr). \end{aligned}$$

(A.5)

The coefficient ω ² in the differential equation (39) is given by

$$\begin{aligned} \omega^2(r) =& \biggl(\frac{A_0^2(1+2{\epsilon}R_0)}{e\epsilon} \biggr) \biggl[ \frac{1}{rH_0^2} \biggl\{ \biggl(\frac{a}{A_0} \biggr)'- \frac{hA'_0}{ A_0H_0} \biggr\} -\frac{h}{r^2H_0^3} \biggr]-\frac{A_0^2}{2e} \biggl[\frac{9\epsilon^2R'_0}{H_0^2(1+2R_0\epsilon)} \biggl(\frac{2e {\epsilon}R'_0}{(1-2R_0\epsilon)}-e' \biggr)+\frac{2R_0'}{H_0^2} \biggl( \frac{a}{A_0} \biggr)' \\ &{} + \frac{2}{H_0^2} \biggl(e' -\frac{2hR_0'}{H_0} \biggr) \biggl(\frac{9{\epsilon}R_0'}{2 (1+2R_0\epsilon)}+ \frac{A_0'}{A_0}+\frac{2}{r} \biggr) -{\epsilon}eR_0^2 +2e \biggl\{ \frac{R_0^2}{2}+\frac{2R_0'}{H_0^2} \biggl(\frac{A_0'}{A_0}+ \frac{9{\epsilon}R_0'}{2(1+2R_0 \epsilon)} +\frac{2}{r} \biggr) \biggr\} \biggr] \end{aligned}$$

(A.6)

$$\begin{aligned} P'_{r0} =&\frac{2}{\kappa}{ \epsilon}R''_0 \biggl(\frac{2}{r}+ \frac{9}{2}R'_0 \epsilon(1-2R_0 \epsilon) \biggr) +\frac{2{\epsilon}R'_0}{r\kappa} \biggl\{ \frac{9}{2}{\epsilon}R''_0(1-2{ \epsilon}R_0)-\frac{2}{r} \biggr\} +\frac{{\epsilon}^2R'^2_0r}{2\kappa}(1-2{ \epsilon}R_0) \biggl(\frac{3}{r} +\frac{29}{2}{ \epsilon}R'_0(1-2{\epsilon}R_0) \biggr) \\ &{} -\frac{3\mathcal{G}{\epsilon}^2}{r^3\kappa}R_0R'_0(1-2{ \epsilon}R_0) m_0\bigl(m_0-rm'_0 \bigr)+\frac{2{\epsilon}R''_0}{r\kappa} -\frac{2{\epsilon}R'_0}{r\kappa} \biggl(\frac{1}{r}+5{ \epsilon}R'_0 (1-2{\epsilon}R_0) \biggr) - \frac{{\epsilon}^2R_0^2R'_0}{2\kappa}(1-2{\epsilon}R_0) \\ &{}+ \biggl(\frac{10{\epsilon}^2R'^2_0}{{\kappa}H^2_0}(1-2{ \epsilon}R_0) -\frac{2{\epsilon}R'_0}{r{\kappa}H_0^2} - \frac{4{\epsilon}R''_0}{r^2\kappa} \biggr) \bigl(1-2{\epsilon}R_0-2r{ \epsilon}R'_0\bigr) \biggl(\frac{1}{4r}\bigl(1-4{ \epsilon}^2R_0^2\bigr) \bigl(2{ \kappa}P_{r0}-{\epsilon}R_0^2\bigr) \\ &{}- \frac{2R_0m_0^2}{r^5} \bigl(4+8{\epsilon}R_0+9r{ \epsilon}R'_0\bigr) (1-2{\epsilon}R_0)+ \frac{3}{2} \frac{\mathcal{G}m_0^2R_0}{r^5}\bigl(4+{\epsilon}R_0+9r{ \epsilon}R'_0\bigr) (1-2{\epsilon}R_0) \biggr) \\ &{}- \frac{R_0}{2r^3} \bigl(4+8{\epsilon}R_0+9r{\epsilon}R'_0 \bigr) \bigl(1-2{\epsilon}R_0-2r{\epsilon}R'_0 \bigr) (1-2{\epsilon}R_0) \biggl\{ P_{r0}+\frac{4{\epsilon}R'_0m_0}{{\kappa}H_0^2r^3} \bigl(m_0-rm'_0\bigr) \biggr\} \\ &{}- \frac{R_0m_0}{2r^4}\bigl(1-2{\epsilon}R_0-2r{\epsilon}R'_0 \bigr) \bigl(4+8{\epsilon}R_0 +9r{\epsilon}R'_0\bigr) (1-2{ \epsilon}R_0) \biggl\{ \mu_0+\frac{{\epsilon}^2m_0R_0^3}{r^3\kappa} - \frac{2{\epsilon}R'_0}{H_0^2\kappa}\bigl(m_0-rm'_0\bigr) \biggr\} \end{aligned}$$

(A.7)

$$\begin{aligned} &{}-\frac{\mathcal{G}}{\mathcal{C}^2} \biggl[-9{\epsilon}R'_0(1-2R_0 \epsilon)\mu_0 -\frac{4{\epsilon}R''_0m_0}{r\kappa} \biggl(\frac{2}{r}+ \frac{9}{2} {\epsilon}R'_0(1-2{ \epsilon}R_0) \biggr) + \frac{4{\epsilon}R'_0}{r^2\kappa}\bigl(m_0-rm'_0 \bigr) \biggl(\frac{3}{r} +\frac{9}{2}{\epsilon}R'_0(1-2{ \epsilon}R_0) \biggr) \\ &{}-\frac{4m_0{\epsilon}R'_0\mathcal{G}}{r^2{\kappa}} \biggl(\frac{9}{2}{\epsilon}R''_0(1-2{ \epsilon}R_0) -\frac{2}{r} \biggr) -\frac{1}{r} \bigl(m_0-rm'_0\bigr)\frac{{\epsilon}^2R'^2_0}{\kappa} (1-2{\epsilon}R_0) - \frac{m_0{\epsilon}^2R'^2_0}{\kappa}(1-2{\epsilon}R_0) \\ &{}\times \biggl\{ \frac{3}{r}+\frac{29}{2}{\epsilon} R'_0(1-2{ \epsilon}R_0) \biggr\} +\frac{4\mathcal{G}}{{\kappa}r^4} {\epsilon}^2R_0m_0^2R'_0 \bigl(m_0-rm'_0 \bigr) (1-2{\epsilon}R_0)+ \biggl\{ \mu_0 + \frac{{\epsilon}^2m_0R''_0}{r^3\kappa} -\frac{2{\epsilon}R'_0}{r^2H_0^2\kappa}\bigl(m_0-rm'_0 \bigr) \biggr\} \\ &{}\times \biggl\{ \frac{1}{4r}\bigl(1-2{ \epsilon}R_0-2r{\epsilon}R'_0\bigr) \bigl(1-4{\epsilon}^2R^2_0\bigr) \bigl(2{ \kappa}P_{r0}-{\epsilon}R^2_0\bigr)- \frac{2m_0^2R_0}{r^5} \bigl(1-2{\epsilon}R_0-2r{ \epsilon}R'_0\bigr) (1-2{\epsilon}R_0) \\ &{}\times \bigl(4+8{\epsilon}R_0-9r{\epsilon}R'_0 \bigr)+\frac{3}{2}\frac{\mathcal{G}m_0^2R_0}{r^5} \bigl(1 - 2{\epsilon}R_0-2r{\epsilon}R'_0 \bigr) (1-2{\epsilon}R_0) \bigl(4+8{\epsilon}R_0 -9r{ \epsilon}R'_0\bigr) \biggr\} \\ &{}+ \biggl\{ \frac{10{\epsilon}^2R'^2_0}{H_0^2\kappa} (1 - 2{\epsilon}R_0)-\frac{2{\epsilon}R'_0}{r{\kappa}H_0^2} - \frac{4{\epsilon}R''_0}{r^2\kappa} \biggr\} \biggl\{ \frac{m_0}{2}\bigl(1 -4{ \epsilon}^2R^2_0\bigr) \bigl(2{ \kappa}P_{r0}-{\epsilon}R_0^2\bigr) + \frac{m_0}{r^2\mathcal{G}} \bigl(1-2{\epsilon}R_0-2r{ \epsilon}R'_0\bigr) \\ &{}\times\bigl(1+2{\epsilon}R_0-4{ \epsilon}^2R^2_0 -8{\epsilon}^3R_0^3 \bigr) + \frac{3m_0^3}{r^6}\mathcal{G}R_0\bigl(1-2{ \epsilon}R_0-2r{\epsilon}R'_0\bigr) \bigl(2+8{\epsilon}R_0-9r{\epsilon}R'_0 \bigr) (1-2{\epsilon}R_0) \biggr\} \\ &{}+ \biggl\{ P_{r0} +\frac{4{\epsilon}R'_0m_0}{{\kappa}H_0^2r^3} \bigl(m_0-rm'_0\bigr) \biggr\} \biggl\{ \frac{R_0m_0}{2r^4}\bigl(1-2{\epsilon}R_0-2r{\epsilon}R'_0 \bigr) \bigl(4 + 8{\epsilon}R_0-9r{\epsilon}R'_0 \bigr) (1-2{\epsilon}R_0) \\ &{}-\frac{m_0R_0}{r^4} \bigl(1-2{ \epsilon}R_0-2r{\epsilon}R'_0\bigr) \bigl(4+8{\epsilon}R_0 - 9r{\epsilon}R'_0\bigr) (1-2{ \epsilon}R_0) \biggr\} -2\mathcal{G}\frac{m_0^2}{r^4}R_0 \bigl(1-2{\epsilon}R_0-2r{\epsilon}R'_0 \bigr) \\ &{}\times \bigl(4+8{\epsilon}R_0 - 9r{\epsilon}R'_0\bigr) (1-2{ \epsilon}R_0)-\frac{4{\epsilon}R''_0m_0}{{\kappa}r^2} +\frac{2{\epsilon}R'_0}{r^3\kappa} \bigl(m_0-rm'_0\bigr)+\frac{4m_0{\epsilon}R'_0}{ r^2\kappa} \biggl(\frac{1}{r}+5{\epsilon}R'_0(1-2{ \epsilon}R_0) \biggr) \biggr] \end{aligned}$$

(A.8)

$$\begin{aligned} &{}+\frac{\mathcal{G}}{\mathcal{C}^4} \biggl[9{\epsilon}R'_0(1-2R_0 \epsilon) (P_{{\bot}0}-P_{r0})+\frac{{\epsilon}R_0R'_0}{\kappa\mathcal{G}} + \frac{{\epsilon}^2R_0^2R'_0}{2\kappa}(1-2R_0\epsilon) - \mathcal{G}\frac{2{\epsilon}R'_0}{r^3\kappa}\bigl(m_0-rm'_0 \bigr) \\ &{}+\mathcal{G}\frac{4m_0{\epsilon}R'_0}{r^3\kappa}\bigl(m_0-rm'_0 \bigr)-\mathcal{G}^2 \frac{2\epsilon^2R_0R'_0m_0^3}{r^5\kappa} \bigl(m_0-rm'_0 \bigr) (1-2{\epsilon}R_0)+ \biggl\{ P_{r0} + \frac{4{\epsilon}R'_0m_0}{{\kappa}H_0^2r^3}\bigl(m_0-rm'_0\bigr) \biggr\} \\ &{}\times \biggl\{ \frac{1}{4r}\bigl(1-2{\epsilon}R_0 - 2r{\epsilon}R'_0\bigr) \bigl(1-4rR^2_0\epsilon^2\bigr) \bigl(2{ \kappa}P_{r0}-{\epsilon}R^2_0\bigr)- \frac{7}{2}\frac{R_0m^2_0}{r^5} \bigl(1-2{\epsilon}R_0-2r{ \epsilon}R'_0\bigr) \bigl(4+8R_0\epsilon+9R'_0r \epsilon\bigr) \biggr\} \\ &{} +\frac{R_0m_0}{2r^4}(1-2R_0\epsilon) \bigl(1-2{\epsilon}R_0-2r{\epsilon}R'_0 \bigr) \bigl(4 + 8{\epsilon}R_0-9r{\epsilon}R'_0 \bigr)-\frac{R_0m_0}{2r^4}(1-2R_0\epsilon) \bigl(1-2{ \epsilon}R_0-2r{\epsilon}R'_0\bigr) \\ &{}\times \bigl(4+8{\epsilon}R_0 - 9r{\epsilon}R'_0\bigr) \biggl\{ \mu_0+\frac{{\epsilon}^2m_0R''_0}{r^3\kappa} -\frac{2{\epsilon}R'_0}{r^2H_0^2\kappa}\bigl(m_0-rm'_0 \bigr) \biggr\} + \biggl(\frac{10{\epsilon}^2R'^2_0}{H^2_0\kappa}(1 - 2{\epsilon}R_0) -\frac{2{\epsilon}R'_0}{H_0^2\kappa}- \frac{4{\epsilon}R''_0}{r^2\kappa} \biggr) \\ &{}\times \biggl\{ \frac{\mathcal{G}m_0^2}{r}\bigl(1-4{ \epsilon}^2R_0^2\bigr) \bigl(1-2{ \epsilon}R_0 -2r{\epsilon}R'_0\bigr) \bigl(2{\kappa}P_{r0}-{ \epsilon}R^2_0\bigr)+\frac{2m_0}{r} \bigl(1+2{ \epsilon}R_0-4{\epsilon}^2R^2_0-8{ \epsilon}^3R_0^3\bigr) \\ &{} +\mathcal{G}^2 \frac{2m_0^4R_0}{r^7} \bigl(1-2{\epsilon}R_0-2r{ \epsilon}R'_0\bigr) \bigl(4-8{\epsilon}R_0-9r{ \epsilon}R'_0\bigr) (1-2{\epsilon}R_0) \biggr\} -\frac{2m_0^2}{r} \biggl(\frac{R_0m_0}{2r^4} \bigl(1-2{\epsilon}R_0-2r{ \epsilon}R'_0\bigr) \\ &{}\times\bigl(4-8{\epsilon}R_0-9r{ \epsilon}R'_0\bigr) (1-2{\epsilon}R_0) \biggr) \biggr]. \end{aligned}$$

(A.9)

The portion Φ which is common in both Eqs. (55) and (58) is given as

$$\begin{aligned} \varPhi =&2\epsilon \biggl[\frac{{\psi}re'}{r+2m_0} \biggr]' - \biggl\{ a_0 \biggl(\frac{r^2}{r-m_0} \biggr)'+ \frac{\epsilon}{1+2\epsilon R_0} \biggl(e'-\frac{2eR_0'\epsilon}{1+2{\epsilon}R_0} \biggr) \biggr\} \biggl(\frac{eR_0^2}{2\kappa} +\mu_0+\biggr)-\frac{{\epsilon}^2R_0^2}{\kappa(1+2\epsilon R_0)} \biggl(e'- \frac{2e\epsilon R_0'}{1+2\epsilon R_0} \biggr) \\ &{}+\frac{e'\omega^2r\epsilon}{r-2m_0}-a_0 \biggl( \frac{r^2}{r-m_0} \biggr)' \biggl(\frac{2r\epsilon}{r+2m_0} \biggr) \biggl[R_0''+R_0' \biggl\{ \frac{5R_0'\epsilon}{1+2\epsilon R_0}-\frac{m_0-rm_0'}{r(r-2m_0)} \biggr\} \biggr] \\ &{} - \frac{4r\epsilon}{r+2m_0} \biggl[e''+e' \biggl\{ \frac{1}{r}-\frac{m_0-rm_0'}{r(r-2m_0)} +\frac{5R_0'\epsilon}{1+2R_0\epsilon} \biggr\} +R_0' \biggl\{ \frac{-2\epsilon}{1+2R_0\epsilon} \biggl( \frac{2eR_0'\epsilon}{1+2R_0\epsilon} - e' \biggr) \biggr\} \biggr] \\ &{}- \frac{2\epsilon}{1+2R_0\epsilon} \biggl( e'-\frac{2eR_0'\epsilon}{1+2R_0\epsilon} \biggr) \biggl[ \frac{3rR_0''\epsilon}{r+2m_0}-\frac{\epsilon R_0^2}{2}-\frac{2rR_0'\epsilon}{r+2m_0} \biggl( \frac{-3}{2} \frac{m_0-rm_0'}{r(r-2m_0)}+ \frac{2}{r} + \frac{12R_0'\epsilon}{1+2R_0\epsilon} \biggr) \biggr] \\ &{} -\frac{6\epsilon^2\omega^2R_0'er}{(r-2m_0)(1+2\epsilon R_0)} -\frac{2er\epsilon R_0'}{1+2\epsilon R_0} \biggl[\frac{3e''r\epsilon}{r+2m_0}-\epsilon e R_0- \frac{2\epsilon R_0'r}{r+2m_0} \biggl\{ a_0 \biggl(\frac{r^2}{r+m_0} \biggr)'-\frac{12\epsilon}{1+2\epsilon R_0} \biggl(\frac{2eR_0'\epsilon}{1+2\epsilon R_0} - e' \biggr) \biggr\} \\ &{}- \frac{2e'r\epsilon}{r+2m_0} \biggl( \frac{2}{r}-\frac{3}{2} \frac{m_0-rm_0'}{r(r-2m_0)} +\frac{12R_0'\epsilon}{1+2\epsilon R_0} \biggr) \biggr] -\frac{2e\omega^2\epsilon}{r} \biggl(\frac{r^2}{r-m_0} \biggr)' + \biggl[eR_0{\epsilon}+\frac{2rR_0'\epsilon}{r+2m_0} \biggl\{ a_0 \biggl(\frac{r}{r-m_0} \biggr)' \\ &{}+ \frac{9\epsilon}{1+2\epsilon R_0} \biggl(e'-\frac{2eR_0'\epsilon}{1+2\epsilon R_0} \biggr) \biggr\} + \frac{2re'\epsilon}{r+2m_0} \biggl( \frac{2}{r}+ \frac{9R_0'\epsilon}{2(1+2\epsilon R_0)} \biggr) \biggr]'-\psi \biggl[a_0 \biggl( \frac{r^2}{r-m_0} \biggr)'\frac{2rR_0'\epsilon}{r+2m_0} + \frac{2er\omega^2\epsilon}{r-2m_0} \\ &{}+\frac{2r\epsilon}{ r+2m_0} \biggl\{ e''+R_0'' \biggl\{ a_0 \biggl(\frac{r}{r-m_0} \biggr)' + \frac{5\epsilon}{1+2\epsilon R_0} \biggl(e'-\frac{2eR_0'\epsilon}{1+2\epsilon R_0} \biggr) \biggr\} + e' \biggl(\frac{5R_0'\epsilon}{1+2\epsilon R_0}- \frac{m_0-rm_0'}{r(r-2m_0)} \biggr) \biggr\} \\ &{}+\frac{4R_0'\epsilon^2}{H_0^2(1+2\epsilon R_0)} \biggl(e'- \frac{2eR_0'\epsilon}{1+2\epsilon R_0} \biggr) - \frac{4ee'rR_0'\epsilon^2}{(r+2m_0)(1+2\epsilon R_0)} \biggr] -\frac{2e'r\epsilon\psi^2}{r+2m_0}. \end{aligned}$$

The perturbation of Eq. (12) gives

$$\begin{aligned} \bar{P}_{\bot} =& \biggl[\frac{1}{\kappa}(1+2{ \epsilon}R_0) \biggl[\frac{a_0}{ H_0r} \biggl(\frac{r}{A_0} \biggr)'+\frac{4{\epsilon}}{r} \biggl(\frac{e_0r}{ (1+2{\epsilon}R_0)} \biggr)'+\frac{a_0H'_0}{H_0^3} \biggl(\frac{r}{A_0} \biggr)' + \frac{a_0}{H_0^2} \biggl(\frac{r}{A_0} \biggr)' \biggr] \\ &{}-\frac{1}{ \kappa} \biggl[-{\epsilon}eR_0+ \frac{{\epsilon}^2R'^2_0}{2H_0^2(1+2{\epsilon} R_0)} \biggl(e'+\frac{2{\epsilon}eR'_0}{(1+2{\epsilon}R_0)} \biggr) - \frac{2{\epsilon}a_0R'_0}{H_0^2} \biggl(\frac{r}{A_0} \biggr)' -\frac{8{\epsilon}^2R'_0}{H_0^2} \biggl(\frac{e}{(1+2{\epsilon}R_0)} \biggr)' \\ &{}+\frac{2{\epsilon}}{H_0^2} \biggl(e'+ \frac{8{\epsilon}eR'_0}{(1+2{\epsilon} R_0)} \biggr) \biggl(\frac{H'_0}{H_0}-\frac{1}{r}+ \frac{{\epsilon}R'_0}{ 2(1+2{\epsilon}R_0)} \biggr) \biggr]+\frac{2{\epsilon}e}{\kappa}(1+2{\epsilon} R_0) \biggl(\frac{A''_0}{A_0}-\frac{H'_0}{rH_0} \biggr) +\frac{(1+2{\epsilon}R_0)}{\kappa} \biggl[\frac{8{\epsilon}e}{(1+2 {\epsilon}R_0)} \\ &{}-8\epsilon \biggl(\frac{e}{(1+2{\epsilon}R_0)} \biggr)' \biggr]\psi +\psi \biggl[ \frac{8e{\epsilon}H'_0}{\kappa} + \frac{8{\epsilon}e}{H_0^2\kappa}-\frac{2{\epsilon}}{H_0^2 \kappa} \biggl(e'+\frac{8{\epsilon}eR'_0}{(1+2{\epsilon}R_0)} \biggr) \biggr] -\frac{2e\omega^2\epsilon}{A_0^2\kappa} \biggr]T=A_1T\quad\mbox{(say)}. \end{aligned}$$

(A.10)

The terms Ω _1e and Ω ₁ mentioned in Eqs. (55) and (59) are

$$\begin{aligned} \varOmega_{1e} =&\frac{-\omega^2{\epsilon}r}{r-2m_0} \biggl( \frac{e\epsilon R_0'}{1+2R_0\epsilon} \biggr)-\frac{4r\epsilon}{r+2m_0} \biggl[\frac{8\epsilon eR_0''}{1+2R_0\epsilon} + \frac{8eR_0'\epsilon}{1+2R_0\epsilon} \biggl\{ -\frac{m_0-rm_0'}{ r(r-2m_0)} +\frac{5R_0'\epsilon}{ 1+2R_0\epsilon} \biggr\} \\ &{}-\frac{4{\epsilon}e(r+m_0)}{r^2(1+2{\epsilon}R_0)} \biggl(\frac{r^2}{r+m_0} \biggr)' \biggr]-\frac{2e\epsilon R_0'}{1+2R_0\epsilon} \biggl[\frac{24r\epsilon^2eR''_0}{(r+2m_0)(1+2R_0\epsilon)} +\frac{2rR_0'\epsilon}{r+2m_0} \biggl\{ \frac{6{\epsilon}e(r+m_0)}{ r^2(1+2R_0\epsilon)} \biggr\} ' \\ &{}-\frac{16erR_0'\epsilon^2}{(r+2m_0)(1+2R_0\epsilon)} \biggl\{ \frac{-3(m_0-rm_0')}{2r(r-2m_0)}+ \frac{2}{r} +\frac{12R_0'\epsilon}{1+2R_0\epsilon} \biggr\} \biggr] - \biggl[ \frac{16erR_0'\epsilon^2}{(r+2m_0) (1+2R_0\epsilon)} \biggl\{ \frac{9R_0'\epsilon}{2(1+2R_0\epsilon)} +\frac{2}{r} \biggr\} \biggr]' \\ &{}-\psi \biggl[\frac{16re\epsilon^2R''_0}{(r+2m_0)(1+2R_0\epsilon)} +\frac{8eR_0'r\epsilon}{(1+2R_0\epsilon)} \biggl\{ \frac{-(m_0-rm_0')}{r(r-2m_0)}+\frac{5R_0'\epsilon}{ 1+2R_0\epsilon} \biggr\} -\frac{32e^2R_0'^2r\epsilon^3}{ (r+2m_0)(1+2R_0\epsilon)^2} \biggr] \\ &{}+16{\epsilon}^2 \biggl[ \frac{eR_0' \psi}{(1+2R_0\epsilon)} \biggr]' +\frac{16\epsilon^2R'_0}{r+2m_0}\psi^2, \end{aligned}$$

$$\begin{aligned} \varOmega_1 =&\frac{\omega^2 \epsilon r}{r-2m_0} \biggl( \frac{h_0rR_0'}{H_0}-\frac{5e\epsilon R_0'}{ 1+2R_0\epsilon} \biggr)+\frac{4r\epsilon}{r+2m_0} \biggl[ \frac{2h_0rR_0''}{H_0}+\frac{2h_0rR_0'}{H_0} \biggl\{ \frac{-(m_0-rm_0')}{ r(r-2m_0)}+\frac{5R_0'\epsilon}{1+2R_0\epsilon} \biggr\} \\ &{}-R_0'h_0 \biggl(\frac{r^2}{r+2m_0} \biggr)' \biggr]-\frac{2e \epsilon R_0'}{1+2\epsilon R_0} \biggl[\frac{3r\epsilon}{r+2m_0} \biggl(\frac{-2hrR_0''}{ r+m_0} \biggr)- \frac{2\epsilon R_0'r}{r+2m_0} \biggl\{ \frac{3}{2} h_0 \biggl( \frac{r^2}{r+2m_0} \biggr)' \biggr\} ' \\ &{}+ \frac{4\epsilon h_0r^2R_0'}{r+3m_0} \biggl(\frac{-3(m_0-rm_0')}{2r(r-2m_0)} +\frac{2}{r}+ \frac{12\epsilon R_0'}{1+2R_0\epsilon} \biggr) \biggr] + \biggl[\frac{4\epsilon h_0r^2R_0'}{r+3m_0} \biggl( \frac{2}{r} +\frac{9\epsilon R_0'}{2(1+2\epsilon R_0)} \biggr) \biggr]' +\psi \biggl[\frac{2\epsilon Tr}{r+2m_0} \biggl(\frac{2h_0rR_0''}{H_0} \biggr) \\ &{}+ \frac{2h_0r^2R_0'}{ r+m_0} \biggl\{ \frac{-(m_0-rm_0')}{r(r-m_0)}+\frac{5\epsilon R_0'}{1+2\epsilon R_0} \biggr\} -\frac{4\epsilon^2eR_0'r}{(r+2m_0)(1+2\epsilon R_0)} \biggl(\frac{2h_0r^2R_0'}{r+m_0} \biggr) \biggr] -4\epsilon \biggl(\frac{{\psi}h_0R_0'r^2}{ (r+m_0)} \biggr)' \\ &{}+\psi^2 \frac{4\epsilon h_0r^2R_0'}{r+3m_0}. \end{aligned}$$