Note on gauge invariance of second order cosmological perturbations

$ \tilde{x}^{\mu} = x^{\mu} + \xi^{(1) \mu} + \frac{1}{2} \xi^{(2) \mu} +{\cal{O}} \left(\xi^{(3)}\right) ,$

$ \tilde{{\cal{Q}}}^{(1)} = {\cal{Q}}^{(1)} +{\cal{L}}_{\xi_1} {\cal{Q}}^{(0)}, \tag{2a}$

$ \tilde{{\cal{Q}}}^{(2)} = {\cal{Q}}^{(2)} + 2{\cal{L}}_{\xi_1} {\cal{Q}}^{(1)} + ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) {\cal{Q}}^{(0)}, \tag{2b}$

$ \tilde{g}_{\mu \nu}^{(1)} = g_{\mu \nu}^{(1)} +{\cal{L}}_{\xi_1} g_{\mu \nu}^{(0)},\tag{3a}$

$ \tilde{g}^{(2)}_{\mu \nu} = g_{\mu \nu}^{(2)} + 2{\cal{L}}_{\xi_1} g_{\mu \nu}^{(1)} + ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) g_{\mu \nu}^{(0)} . \tag{3b}$

$ g_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} = g_{\mu \nu}^{(1)} -{\cal{L}}_X g_{\mu \nu}^{(0)}, \tag{4a}$

$ g^{(\mathrm{\rm{GI}, 2)}}_{\mu \nu} = g_{\mu \nu}^{(2)} - 2{\cal{L}}_X g_{\mu \nu}^{(1)} - ({\cal{L}}_Y -{\cal{L}}_X^2) g_{\mu \nu}^{(0)},\tag{4b} $

$ \tilde{X}^{\mu} = X^{\mu}+ \xi_1^{\mu}, \tag{5a}$

$ \tilde{Y}^{\mu} = Y^{\mu}+ \xi_2^{\mu} + [\xi_1, X]^{\mu} . \tag{5b}$

$\tilde{T}_{\mu \nu}^{(1)} = T_{\mu \nu}^{(1)} +{\cal{L}}_{\xi_1} T_{\mu \nu}^{(0)}, \tag{6a}$

$ \tilde{T}^{(2)}_{\mu \nu} = T_{\mu \nu}^{(2)} + 2{\cal{L}}_{\xi_1} T_{\mu \nu}^{(1)} + ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) T_{\mu \nu}^{(0)} .\tag{6b} $

$ T_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} = T_{\mu \nu}^{(1)} -{\cal{L}}_X T_{\mu \nu}^{(0)}, \tag{7a}$

$ T^{(\mathrm{\rm{GI}, 2)}}_{\mu \nu} = T_{\mu \nu}^{(2)} - 2{\cal{L}}_X T_{\mu \nu}^{(1)} - ({\cal{L}}_Y -{\cal{L}}_X^2) T_{\mu \nu}^{(0)}, \tag{7b}$

$ {\cal{Q}}^{(\mathrm{\rm{GI}, 1)}} = {\cal{Q}}^{(1)} -{\cal{L}}_X {\cal{Q}}^{(0)}, \tag{8a}$

$ {\cal{Q}}^{(\mathrm{\rm{GI}, 2)}} = {\cal{Q}}^{(2)} - 2{\cal{L}}_X {\cal{Q}}^{(1)} - ({\cal{L}}_Y -{\cal{L}}_X^2) {\cal{Q}}^{(0)} . \tag{8b}$

$ \begin{aligned}[b] 0 =& G_{\mu \nu} - \kappa T_{\mu \nu} \\ \approx &G^{(0)}_{\mu \nu} - \kappa T_{\mu \nu}^{(0)} + G^{(1)}_{\mu \nu} - \kappa T_{\mu \nu}^{(1)} + \frac{1}{2} \left(G_{\mu \nu}^{(2)} - \kappa T_{\mu \nu}^{(2)}\right) \\ = &G_{\mu \nu}^{(0)} - \kappa T_{\mu \nu}^{(0)} + \left(G^{(\mathrm{\rm{GI}, 1)}} - \kappa T_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} +{\cal{L}}_X \left(G_{\mu \nu}^{(0)} - \kappa T_{\mu \nu}^{(0)}\right)\right) \\ &+ \frac{1}{2} \left(G_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} - \kappa T_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} + 2{\cal{L}}_X \left(G_{\mu \nu}^{(1)} - \kappa T_{\mu \nu}^{(1)}\right)\right. \\ &\left.+ \left({\cal{L}}_Y -{\cal{L}}_X^2\right) \left(G_{\mu \nu}^{(0)} - \kappa T^{(0)}_{\mu \nu}\right)\right), \end{aligned} $

$ G_{\mu \nu}^{(0)} = \kappa T_{\mu \nu}^{(0)}, \tag{10a}$

$ G_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} = \kappa T_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}}, \tag{10b}$

$ G_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} = \kappa T_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} . \tag{10c}$

$ g_{\mu \nu}^{(0)} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} = a^2 (\eta) (- \mathrm{d} \eta^2 + \delta_{ij} \mathrm{d} x^i \mathrm{d} x^j),$

$ \begin{array}{l} g_{\mu \nu}^{(n)} = a^2 \left( \begin{array}{ccc} - 2 \phi^{(n)} & & \partial_i b^{(n)} + \nu_i^{(n)}\\ \partial_i b^{(n)} + \nu_i^{(n)} & & - 2 \psi^{(n)} \delta_{ij} + 2 \partial_i \partial_j {e}^{(n)} + \partial_i c_j^{(n)} + \partial_j c_i^{(n)} + h_{ij}^{(n)} \end{array} \right) , \end{array} $

$ \partial_i \nu^{i (n)} = 0,\tag{13a}$

$ \partial_i c^{i (n)} = 0, \tag{13b}$

$ \delta^{jk} \partial_k h_{ij}^{(n)} = 0, \tag{13c}$

$ \delta^{ij} h_{ij}^{(n)} = 0. \tag{13d}$

$ \begin{aligned}[b] \tilde{g}^{(1)}_{00} &= - 2 a^2 \tilde{\phi}^{(1)} \\ &= - 2 a^2 \phi^{(1)} - 2 a^2 \left( \partial_0 + \frac{\dot{a}}{a} \right) \xi_1^0,\end{aligned} \tag{14a}$

$ \begin{aligned}[b] \tilde{g}^{(1)}_{0i} & = a^2 (\partial_i \tilde{b}^{(1)} + \tilde{\nu}_i^{(1)}) \\ & = a^2 (\partial_i b^{(1)} + \nu_i^{(1)}) + a^2 (\delta_{ij} \partial_0 \xi_1^j - \partial_i \xi^0_1), \end{aligned}\tag{14b} $

$ \begin{aligned}[b] \tilde{g}^{(1)}_{ij} =& a^2 (- 2 \tilde{\psi}^{(1)} \delta_{ij} + 2 \partial_i \partial_j \tilde{e}^{(1)} + \partial_i \tilde{c}_j^{(1)} + \partial_j \tilde{c}_i^{(1)} + \tilde{h}_{ij}^{(1)}) \\ =& a^2 (- 2 \psi^{(1)} \delta_{ij} + 2 \partial_i \partial_j {e}^{(1)} + \partial_i c_j^{(1)} + \partial_j c_i^{(1)} + h_{ij}^{(1)}) \\ & + a^2 \left( (\delta_{ik} \partial_j + \delta_{jk} \partial_i) (\xi^k_{1, T} + \delta^{kl} \partial_l \xi_{1,S}) + \frac{2 \dot{a}}{a} \delta_{ij} \xi^0_1 \right) . \end{aligned}\tag{14c}$

$ \tilde{\phi}^{(1)} = \phi^{(1)} + \left( \partial_0 + \frac{\dot{a}}{a} \right) \xi_1^0, \tag{15a}$

$ \tilde{b}^{(1)} = b^{(1)} + \partial_0 \xi_{1,S} - \xi_1^0,\tag{15b} $

$ \tilde{\nu}_i^{(1)} = \nu_i^{(1)} + \delta_{ij} \partial_0 \xi^j_{1, T}, \tag{15c}$

$ \tilde{\psi}^{(1)} = \psi^{(1)} - \frac{\dot{a}}{a} \xi_1^0, \tag{15d}$

$ \tilde{e}^{(1)} = {e}^{(1)} + \xi_{1,S}, \tag{15e}$

$ \tilde{c}_i^{(1)} = c_i^{(1)} + \delta_{ik} \xi^k_{1, T}, \tag{15f}$

$ \tilde{h}_{ij}^{(1)} = h_{ij}^{(1)} .\tag{15g} $

$ g_{00}^{(\mathrm{\rm{GI}, 1)}} = - 2 a^2 \Phi^{(1)}, \tag{16a}$

$ g_{0 i}^{(\mathrm{\rm{GI}, 1)}} = a^2 V_i^{(1)}, \tag{16b}$

$ g_{ij}^{(\mathrm{\rm{GI}, 1)}} = a^2 \left(- 2 \Psi^{(1)} \delta_{ij} + H_{ij}^{(1)}\right), \tag{16c}$

$ \Phi^{(1)} = \phi^{(1)} - \left( \partial_0 + \frac{\dot{a}}{a} \right) X^0, \tag{17a}$

$ \Psi^{(1)} = \psi^{(1)} + \frac{\dot{a}}{a} X^0, \tag{17b}$

$ V^{(1)}_i = \nu_i^{(1)} - \delta_{ij} \partial_0 X^j_T, \tag{17c}$

$ H_{ij}^{(1)} = h_{ij}^{(1)}, \tag{17d}$

$ \begin{aligned}[b] a^2 V_i^{(1)} =& g_{0 i}^{(1)} -{\cal{L}}_X g_{0 i}^{(0)} \\ =& a^2 \left(\partial_i \left(b^{(1)} - \partial_0 X_S + X^0\right)\right. \\ &\left.+ \left(\nu_i^{(1)} - \delta_{ji} \partial_0 X^j_T\right)\right) . \end{aligned} $

$ \begin{array}{l} b^{(1)} - \partial_0 X_S + X^0 = 0. \end{array} $

$ \begin{aligned}[b] a^2 (- 2 \Psi^{(1)} \delta_{ij} \!+\! H_{ij}^{(1)}) \!=& g_{ij}^{(1)} -{\cal{L}}_X g_{ij}^{(0)} \\ = &a^2 \Biggr(\! -\! 2 \delta_{ij} \left( \psi^{(1)} \!+\! \frac{\dot{a}}{a} X^0 \right) + 2 \partial_i \partial_j \left({e}^{(1)} \!-\! X_S\right) \\ &+ \left(\delta^k_j \partial_i + \delta^k_i \partial_j\right) \left(c_k^{(1)} - \delta_{lk} X^l_T\right) + h_{ij}^{(1)}\Biggr), \end{aligned} $

$ \begin{array}{l} {e}^{(1)} - X_S = 0, \end{array} \tag{21a} $

$ c_k^{(1)} - \delta_{lk} X_T^l = 0. \tag{21b}$

$ X^0 = \partial_0 {e}^{(1)} - b^{(1)}, \tag{22a} $

$ X^i = \delta^{ik} \left(c_k^{(1)} + \partial_k {e}^{(1)}\right) . \tag{22b} $

$ \Phi^{(1)} = \phi^{(1)} - \frac{1}{a} \partial_0 \left(a \left(\partial_0 {e}^{(1)} - b^{(1)}\right)\right), \tag{23a} $

$ \Psi^{(1)} = \psi^{(1)} - \frac{\dot{a}}{a} \left(\partial_0 {e}^{(1)} - b^{(1)}\right), \tag{23b} $

$ V_i^{(1)} = \nu_i^{(1)} - \partial_0 c_i^{(1)}, \tag{23c} $

$ H_{ij}^{(1)} = h_{ij}^{(1)} . \tag{23d} $

$ g_{00}^{(\mathrm{\rm{GI}, 1)}} = 0,\tag{24a} $

$ g_{0 i}^{(\mathrm{\rm{GI}, 1)}} = 0, \tag{24b}$

$ \begin{aligned}[b] g_{ij}^{(\mathrm{\rm{GI}, 1)}} =& a^2 \left(- 2 \Psi^{(1)} \delta_{ij} + 2 \partial_i \partial_j E^{(1)}\right.\\ &\left.+ \partial_i C_j^{(1)} + \partial_j C_i^{(1)} + H_{ij}^{(1)}\right), \end{aligned}\tag{24c} $

$ \Psi^{(1)} = \psi^{(1)} + \frac{\dot{a}}{a} X^0, \tag{25a} $

$ E^{(1)} = {e}^{(1)} - X_S, \tag{25b}$

$ C_i^{(1)} = c_i^{(1)} - \delta_{ik} X_T^k, \tag{25c}$

$ H_{ij}^{(1)} = h_{ij}^{(1)} . \tag{25d} $

$X^0 = \frac{1}{a} \int \mathrm{d} \eta \{a \phi^{(1)} \}, \tag{26a} $

$X^j \!=\! \delta^{ji} \int \mathrm{d} \eta \left\{ \nu_i^{(1)} \!+\! \partial_i b^{(1)} \!+\! \frac{1}{a}\! \int \mathrm{d} \eta' \left\{a \partial_i \phi^{(1)} \right\} \right\} . \tag{26b}$

$ \Psi^{(1)} = \psi^{(1)} + \frac{\dot{a}}{a^2} \int \mathrm{d} \eta \{a \phi^{(1)} \}, \tag{27a} $

$ E^{(1)} = {e}^{(1)} + \int \mathrm{d} \eta \left\{ b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta' \left\{a \phi^{(1)} \right\} \right\}, \tag{27b} $

$ C_i^{(1)} = c_i^{(1)} - \int \nu_i^{(1)} \mathrm{d} \eta, \tag{27c} $

$ H_{ij}^{(1)} = h_{ij}^{(1)} . \tag{27d}$

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, A, 1)}} - g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 1)}} = &(g_{\mu \nu}^{(1)} -{\cal{L}}_{X^A} g_{\mu \nu}^{(0)}) - (g_{\mu \nu}^{(1)} -{\cal{L}}_{X^B} g_{\mu \nu}^{(0)}) \\ =& {\cal{L}}_{(X^B - X^A)} g_{\mu \nu}^{(0)} . \end{aligned} $

$ \begin{array}{l} g_{\mu \nu}^{(\mathrm{\rm{GI}, A, 1)}} = g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 1)}} -{\cal{L}}_{Z_1^{\mathrm{AB}}} g_{\mu \nu}^{(0)} . \end{array} $

$ \begin{array}{l} Z^{\mu, \mathrm{AB}}_1 = \hat{N}_1^{\mu} {\cal{A}}^{(1)} + \hat{N}_2^{\mu} {\cal{B}}^{(1)} + \hat{N}_3^{\mu i} {\cal{C}}_i^{(1)}, \end{array} $

$ {\cal{A}}^{(1)} \equiv \partial_0 \left( \frac{a^2}{\dot{a}} \psi^{(1)} \right) + a \phi^{(1)}, \tag{31a} $

$ {\cal{B}}^{(1)} \equiv \partial_0 {e}^{(1)} - b^{(1)} + \frac{a}{\dot{a}} \psi^{(1)}, \tag{31b}$

$ {\cal{C}}_i^{(1)} \equiv \nu_i^{(1)} - \partial_0 c_i^{(1)} . \tag{31c} $

$ X^0 = \partial_0 {e}^{(1)} - b^{(1)} + Z^0_1, \tag{32a}$

$ X^j = \delta^{jk} (c_k^{(1)} + \partial_k {e}^{(1)}) + Z^j_1 . \tag{32b} $

$ g_{00}^{(\mathrm{\rm{GI}, 1)}} = - 2 a^2 \Phi^{(1)}, \tag{33a}$

$ g_{0 i}^{(\mathrm{\rm{GI}, 1)}} = a^2 (\partial_i B^{(1)} + V_i^{(1)}), \tag{33b} $

$ \begin{aligned}[b] g_{ij}^{(\mathrm{\rm{GI}, 1)}} =& a^2 \left(- 2 \Psi^{(1)} \delta_{ij} + 2 \partial_i \partial_j E^{(1)}\right.\\ &\left.+ \partial_i C_j^{(1)} + \partial_j C_i^{(1)} + H_{ij}^{(1)}\right),\end{aligned} \tag{33c} $

$ \Phi^{(1)} = \phi^{(1)} - \frac{1}{a} \partial_0 (a (\partial_0 {e}^{(1)} - b^{(1)} + Z^0_1)), \tag{34a}$

$ B^{(1)} = Z^0_1 - \partial_0 \Delta^{- 1} \partial_j Z^j_1, \tag{34b}$

$ \Psi^{(1)} = \psi^{(1)} + \frac{\dot{a}}{a} (\partial_0 {e}^{(1)} - b^{(1)} + Z^0), \tag{34c} $

$ E^{(1)} = \Delta^{- 1} \partial_j Z^j_1, \tag{34d} $

$ V_i^{(1)} = \nu_i^{(1)} - \partial_0 c_i^{(1)} - \left(\delta_{ik} - \partial_i \Delta^{- 1} \partial_k\right) \partial_0 Z^k_1, \tag{34e}$

$ C_i^{(1)} = - \left(\delta_{ik} - \partial_i \Delta^{- 1} \partial_k\right) Z^k_1, \tag{34f} $

$ H_{ij}^{(1)} = h_{ij},\tag{34g} $

$ Z^{0, \mathrm{AB}}_1 = \frac{1}{a} \int {\cal{A}}^{(1)} \mathrm{d} \eta -{\cal{B}}^{(1)}, \tag{35a} $

$ \begin{aligned}[b] Z^{j, \mathrm{AB}}_1 =& \delta^{jk} \partial_k \int \mathrm{d} \eta \left\{ \frac{1}{a} \int {\cal{A}}^{(1)} \mathrm{d} \eta' \right\} \\ &- \delta^{jk} \partial_k \int {\cal{B}}^{(1)} \mathrm{d} \eta + \delta^{ji} \int {\cal{C}}_i^{(1)} \mathrm{d} \eta . \end{aligned} \tag{35b}$

$ \begin{aligned}[b] \tilde{g}^{(2)}_{00} = &- 2 a^2 \tilde{\phi}^{(2)} \\ =& - 2 a^2 \phi^{(2)} + 2{\cal{L}}_{\xi_1} g_{00}^{(1)} + \left({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2\right) g_{00}^{(0)}, \end{aligned}\tag{36a} $

$ \begin{aligned}[b] \tilde{g}_{0 i}^{(2)} = &a^2 \left(\partial_i \tilde{B}^{(2)} + \tilde{\nu}_i^{(2)}\right) \\ =& g_{00}^{(2)} + 2{\cal{L}}_{\xi_1} g_{0 i}^{(1)} + \left({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2\right) g_{0 i}^{(0)} \\ =& a^2 \left(\partial_i \left(b^{(2)} + \Delta^{- 1} \partial^j \Xi_{0 j}\right) + \nu_i^{(2)} + \left(\delta_i^j - \partial_i \Delta^{- 1} \partial^j\right) \Xi_{0 j}\right) \\ \equiv & a^2 \left(\partial_i \left(b^{(2)} + \Delta^{- 1} \partial^j \Xi_{0 j}\right) + \nu_i^{(2)} +{\cal{T}}^j_i \Xi_{0 j}\right), \end{aligned} \tag{36b}$

$ \begin{aligned}[b] \tilde{g}^{(2)}_{ij} =& a^2 (- 2 \delta_{ij} \tilde{\psi}^{(2)} + 2 \partial_i \partial_j \tilde{e}^{(2)} + \partial_i \tilde{c}^{(2)}_j + \partial_j \tilde{c}_i^{(2)} + \tilde{h}_{ij}^{(2)}) \\ =& g_{ij}^{(2)} + 2{\cal{L}}_{\xi_1} g_{ij}^{(1)} + ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) g_{ij}^{(0)} \\ =& a^2 \Biggr(- 2 \delta_{ij} \psi^{(2)} + 2 \partial_i \partial_j {e}^{(2)} + \partial_i {c}^{(2)}_j + \partial_j c_i^{(2)} + h_{ij}^{(2)} \\ & + \frac{1}{2} \delta_{ij} \left(\delta^{kl} - \partial^k \Delta^{- 1} \partial^l\right) \Xi_{kl} + \frac{1}{2} \partial_i \partial_j \Delta^{- 1} \left(\left(3 \Delta^{- 1} \partial^k \partial^l\right.\right. \\ &\left.\left.- \delta^{kl}\right) \Xi_{kl}\right) + \partial_j \Delta^{- 1} \partial^l \left(\left(\delta^k_i - \partial_i \Delta^{- 1} \partial^k\right) \Xi_{kl}\right) \\ & + \!\partial_i \Delta^{- 1} \partial^k \left( \left(\delta^l_j \!-\! \partial_j \Delta^{- 1} \partial^l\right) \Xi_{kl}\right)\! +\! \Biggr( \left(\delta^k_i \!-\! \partial_i \Delta^{- 1} \partial^k\right) \left(\delta^l_j \right. \\ &\left. - \partial_j \Delta^{- 1} \partial^l\right) - \frac{1}{2} \left(\delta_{ij} - \partial_i \Delta^{- 1} \partial_j\right) \left(\delta^{kl} - \partial^k \Delta^{- 1} \partial^l\right) \Biggr) \Xi_{kl} \Biggr) \\ =& a^2 \Biggr( - 2 \delta_{ij} \psi^{(2)} + 2 \partial_i \partial_j {e}^{(2)} + \partial_i {c}^{(2)}_j + \partial_j c_i^{(2)} + h_{ij}^{(2)} \\ &+ \frac{1}{2} \delta_{ij} {\cal{T}}^{kl} \Xi_{kl} + \partial_i \partial_j \Delta^{- 1} \left(\left( \partial^k \Delta^{- 1} \partial^l - \frac{1}{2} {\cal{T}}^{kl} \right) \Xi_{kl}\right) \\ &+ \partial_j \Delta^{- 1} \partial^l {\cal{T}}^k_i \Xi_{kl} + \partial_i \Delta^{- 1} \partial^k {\cal{T}}^l_j \Xi_{kl} \\ &+ \left( {\cal{T}}^k_i {\cal{T}}^l_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{kl} \right) \Xi_{kl} \Biggr), \end{aligned} \tag{36c}$

$ \Xi_{\mu \nu} \equiv \frac{2{\cal{L}}_{\xi_1} g_{\mu \nu}^{(1)} + ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) g_{\mu \nu}^{(0)}}{a^2} . \tag{37} $

$ \tilde{\phi}^{(2)} = \phi^{(2)} - \frac{1}{2} \Xi_{00}, \tag{38a} $

$ \tilde{b}^{(2)} = b^{(2)} + \Delta^{- 1} \partial^j \Xi_{0 j}, \tag{38b} $

$ \tilde{\nu}^{(2)}_i = \nu_i^{(2)} +{\cal{T}}^j_i \Xi_{0 j}, \tag{38c}$

$ \tilde{\psi}^{(2)} = \psi^{(2)} - \frac{1}{4} {\cal{T}}^{kl} \Xi_{kl}, \tag{38d} $

$ \tilde{e}^{(2)} = {e}^{(2)} + \frac{1}{2} \Delta^{- 1} \left(\left( \partial^k \Delta^{- 1} \partial^l - \frac{1}{2} {\cal{T}}^{kl} \right) \Xi_{kl}\right), \tag{38e} $

$ \tilde{c}_j^{(2)} = c_j^{(2)} + \Delta^{- 1} \partial^k {\cal{T}}^l_j \Xi_{kl},\tag{38f} $

$ \tilde{h}_{ij}^{(2)} = h^{(2)}_{ij} + \left( {\cal{T}}^k_i {\cal{T}}^l_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{kl} \right) \Xi_{kl} . \tag{38g} $

$ \Phi^{(2)} = \phi^{(2)} - \left( \partial_0 + \frac{\dot{a}}{a} \right) Y^0 +{\cal{X}}_{00}, \tag{39a} $

$ B^{(2)} = b^{(2)} - \partial_0 Y_S + Y^0 - \Delta^{- 1} \partial^j {\cal{X}}_{0 j}, \tag{39b} $

$ V_i^{(2)} = \nu_i^{(2)} - \delta_{ij} \partial_0 Y^j_T -{\cal{T}}^j_i {\cal{X}}_{0 j},\tag{39c}$

$ \Psi^{(2)} = \psi^{(2)} + \frac{\dot{a}}{a} Y^0 + \frac{1}{4} {\cal{T}}^{kl} {\cal{X}}_{kl} \tag{39d} $

$ E^{(2)} \!=\! {e}^{(2)} \!-\! Y_S \!-\! \frac{1}{2} \Delta^{- 1} \left(\left( \partial^k \Delta^{- 1} \partial^l \!-\! \frac{1}{2} {\cal{T}}^{kl} \right) {\cal{X}}_{kl}\right), \tag{39e} $

$ C_j^{(2)} = c_j^{(2)} - \delta_{jk} Y^k_T - \Delta^{- 1} \partial^k {\cal{T}}^l_j {\cal{X}}_{kl}, \tag{39f} $

$ H_{ij}^{(2)} = h_{ij}^{(2)} - \left( {\cal{T}}^k_i {\cal{T}}^l_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{kl} \right) {\cal{X}}_{kl},\tag{39g} $

$ {\cal{X}}_{\mu \nu} \equiv \frac{2{\cal{L}}_X g_{\mu \nu}^{(1)} -{\cal{L}}_X^2 g_{\mu \nu}^{(0)}}{a^2} , \tag{40}$

$ \begin{aligned}[b] Y^0 =& \partial_0 {e}^{(2)} - b^{(2)} + \Delta^{- 1} \partial^j {\cal{X}}_{0 j} - \frac{1}{2} \Biggr( \partial^k \Delta^{- 2} \partial^l \partial_0 \\ &- \frac{1}{2} \Delta^{- 1}{\cal{T}}^{kl}\partial_0 \Biggr) {\cal{X}}_{kl}, \end{aligned}\tag{41a} $

$ \begin{aligned}[b] Y^i =& \delta^{ij} (\partial_j {e}^{(2)} + c_j^{(2)}) + \Biggr( \frac{1}{4} \partial^i\partial^k \Delta^{- 2} \partial^l \\ &+ \frac{1}{4}\delta^{kl} \partial^i \Delta^{- 1} - \delta^{li} \Delta^{- 1} \partial^k \Biggr) {\cal{X}}_{kl} . \end{aligned}\tag{41b} $

$ g^{(\mathrm{\rm{GI}, 2)}}_{00} = - 2 a^2 \Phi^{(2)}, \tag{42a} $

$ g_{0 i}^{(\mathrm{\rm{GI}, 2)}} = a^2 V_i^{(2)}, \tag{42b} $

$ g_{ij}^{(\mathrm{\rm{GI}, 2)}} = - 2 a^2 \delta_{ij} \Psi^{(2)} + a^2 H_{ij}^{(2)}, \tag{42c} $

$ \begin{aligned}[b] \Phi^{(2)} = &\phi^{(2)} - \left( \frac{\dot{a}}{a} + \partial_0 \right) \left(\partial_0 {e}^{(2)} - b^{(2)}\right) +{\cal{X}}_{00} \\ &- \left(\frac{\dot{a}}{a} + \partial_0 \right) \left(\Delta^{- 1} \partial^j {\cal{X}}_{0 j} - 3 \Delta^{- 2} \partial^k \partial^l {\cal{X}}_{kl} \right.\\ & \left.+ \delta^{kl} \Delta^{- 1}{\cal{X}}_{kl}\right), \end{aligned} \tag{43a}$

$ \begin{aligned}[b] \Psi^{(2)} =& \psi^{(2)} + \frac{\dot{a}}{a} (\partial_0 {e}^{(2)} - b^{(2)})\\ &+ \frac{\dot{a}}{a} (\Delta^{- 1} \partial^j {\cal{X}}_{0 j} - 3 \Delta^{- 2} \partial^k \partial^l {\cal{X}}_{kl} \\ & + \delta^{kl} \Delta^{- 1}{\cal{X}}_{kl}) +{\cal{T}}^{kl} {\cal{X}}_{kl}, \end{aligned} \tag{43b}$

$ V_i^{(2)} = \nu_i^{(2)} - \partial_0 c_i^{(2)} + \Delta^{- 1} \partial^k {\cal{T}}^l_i \partial_0 {\cal{X}}_{kl} -{\cal{T}}^j_i {\cal{X}}_{0 j},\tag{43c} $

$ H_{ij}^{(2)} = h^{(2)}_{ij} - \left( {\cal{T}}^k_i {\cal{T}}^l_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{kl} \right) {\cal{X}}_{kl} . \tag{43d}$

$ Y^0 = \frac{1}{a} \int \mathrm{d} \eta \{ a \phi^{(2)} + \frac{1}{2} a{\cal{X}}_{00} \}, \tag{44a} $

$ \begin{aligned}[b] Y^k =& \delta^{ki} \int \mathrm{d} \eta \left\{ \nu_i^{(2)} + \partial_i b^{(2)} + \frac{1}{a} \int \mathrm{d} \eta \left\{a \partial_i \phi^{(2)} \right\} \right\} \\ &- \int \mathrm{d} \eta \left\{ \delta^{kj} {\cal{X}}_{0 j} - \frac{1}{2 a} \int \mathrm{d} \eta \left\{a \partial^k {\cal{X}}_{00} \right\} \right\} . \end{aligned}\tag{44b} $

$ g_{00}^{(\mathrm{\rm{GI}, 2)}} = 0, \tag{45a} $

$ g_{0 i}^{(\mathrm{\rm{GI}, 2)}} = 0, \tag{45b} $

$ \begin{aligned}[b] g_{ij}^{(\mathrm{\rm{GI}, 2)}} = & a^2 \left(- 2 \Psi^{(2)} \delta_{ij} + 2 \partial_i \partial_j E^{(2)} \right.\\ & \left.+ \partial_i C_j^{(2)} + \partial_j C_i^{(2)} + H_{ij}^{(2)}\right),\end{aligned} \tag{45c} $

$ \Psi^{(2)} \!=\! \psi^{(2)} \!+\! \frac{\dot{a}}{a^2} \int \mathrm{d} \eta \left\{ a \phi^{(2)} \!+\! \frac{1}{2} a{\cal{X}}_{00} \right\} \!+\!{\cal{T}}^{kl} {\cal{X}}_{kl}, \tag{46a} $

$ \begin{aligned}[b] E^{(2)} =& {e}^{(2)} - \int \mathrm{d} \eta \left\{ b^{(2)} + \frac{1}{a} \int \mathrm{d} \eta' \left\{a \phi^{(2)} \right\} \right\} \\ & + \int \mathrm{d} \eta \left\{ \Delta^{- 1} \partial^j {\cal{X}}_{0 j} - \frac{1}{2 a} \int \mathrm{d} \eta' \left\{a{\cal{X}}_{00} \right\} \right\} \\ & - \frac{1}{4} (3 \Delta^{- 2} \partial^k \partial^l - \delta^{kl}\Delta^{- 1}) {\cal{X}}_{kl}, \end{aligned} \tag{46b}$

$ C_j^{(2)}\! \!=\! c_j^{(2)} \!-\!\! \!\int \nu_j^{(2)} \mathrm{d} \eta \!+\!{\cal{T}}^k_j \!\!\!\int\!\! \!{\cal{X}}_{0 k} \mathrm{d} \eta \!-\! \Delta^{- 1} \partial^k {\cal{T}}^l_j {\cal{X}}_{kl}, \tag{46c}$

$ H_{ij}^{(2)} = h^{(2)}_{ij} - \left( {\cal{T}}^k_i {\cal{T}}^l_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{kl} \right) {\cal{X}}_{kl} . \tag{46d} $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, A, 2)}} - g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 2)}} =& g_{\mu \nu}^{(2)} - 2{\cal{L}}_{X^A} g_{\mu \nu}^{(1)} - \left({\cal{L}}_{Y^A} -{\cal{L}}_{X^A}^2\right) g_{\mu \nu}^{(0)} \\ &- \left(g_{\mu \nu}^{(2)} - 2{\cal{L}}_{X^B} g_{\mu \nu}^{(1)} - \left({\cal{L}}_{Y^B} -{\cal{L}}_{X^B}^2\right) g_{\mu \nu}^{(0)}\right) \\ = &2{\cal{L}}_{(X^B - X^A)} \left(g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 1)}} +{\cal{L}}_{X^B} g_{\mu \nu}^{(0)}\right) \\ &+ \left({\cal{L}}_{(Y^B - Y^A)} -{\cal{L}}_{X^B}^2 -{\cal{L}}_{X^A}^2\right) g_{\mu \nu}^{(0)} \\ =& - 2{\cal{L}}_{(X^A - X^B)} g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 1)}} \\ &- \left({\cal{L}}_{(Y^A - Y^B + [X^A, X^B])}\right.\\ &\left.-{\cal{L}}_{(X^A - X^B)}^2\right) g_{\mu \nu}^{(0)} . \end{aligned} $

$ \begin{array}{l} g_{\mu \nu}^{(\mathrm{\rm{GI}, A, 2)}} \!=\! g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 2)}} \!-\! 2{\cal{L}}_{Z^{\mathrm{AB}}_1} g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 1)}} \!-\! ({\cal{L}}_{Z_2^{\mathrm{AB}}} \!-\!{\cal{L}}_{Z^{\mathrm{AB}}_1}^2) g_{\mu \nu}^{(0)} . \end{array} $

$ \begin{aligned}[b]Z_2^{0, \mathrm{AB}} =& \hat{M}_1^0 {\cal{A}}^{(2)} + \hat{M}_2^0 {\cal{B}}^{(2)} + \hat{M}_3^{0 i} {\cal{C}}_i^{(2)} \\& + \hat{M}^{0, \sigma \kappa}_4 {\cal{D}}_{\sigma \kappa}^{(2)}, \end{aligned} \tag{49a}$

$\begin{aligned}[b] Z_2^{k, \mathrm{AB}} =& \hat{M}_1^k {\cal{A}}^{(2)} + \hat{M}_2^k {\cal{B}}^{(2)} + \hat{M}_3^{ki} {\cal{C}}_i^{(2)} \\&+ \hat{M}^{k, \sigma \kappa}_4 {\cal{D}}_{\sigma \kappa}^{(2)},\end{aligned} \tag{49b} $

$ \begin{aligned}[b] {\cal{A}}^{(2)} \equiv &\partial_0 \left( \frac{a^2}{\dot{a}} \psi^{(2)} \right) + a \phi^{(2)} + \frac{1}{4} \partial_0 \left( \frac{a^2}{\dot{a}} {\cal{T}}^{kl} {\cal{X}}_{kl} \right) \\ &+ \frac{1}{2} a{\cal{X}}_{00}, \end{aligned}\tag{50a} $

$ \begin{aligned}[b] {\cal{B}}^{(2)} \equiv &\partial_0 {e}^{(2)} - b^{(2)} + \frac{a}{\dot{a}} \psi^{(2)} + \Delta^{- 1} \partial^j {\cal{X}}_{0 j} \\ &+ \! \frac{1}{4} \left( \frac{a}{\dot{a}} {\cal{T}}^{kl} \!-\! 3 \Delta^{- 2} \partial^k \partial^l \partial_0 \!+\! \Delta^{- 1}\delta^{kl}\partial_0 \right) {\cal{X}}_{kl}, \end{aligned} \tag{50b}$

$ {\cal{C}}_k^{(2)} \equiv \nu_k^{(2)} - \partial_0 c_k^{(2)} + \Delta^{- 1} \partial^i {\cal{T}}^l_k \partial_0 {\cal{X}}_{il} -{\cal{T}}^l_k {\cal{X}}_{0 l}, \tag{50c} $

$ {\cal{D}}_{\mu \nu}^{(2)} \equiv \frac{1}{a^2}{\left(2{\cal{L}}_{Z_1^{\mathrm{AB}}} g_{\mu \nu}^{(\mathrm{\rm{GI}, B, 1)}} -{\cal{L}}_{Z_1^{\mathrm{AB}}}^2 g_{\mu \nu}^{(0)}\right)} . \tag{50d} $

$ Z^{0, \mathrm{AB}}_2 = \frac{1}{a} \int {\cal{A}}^{(2)} \mathrm{d} \eta -{\cal{B}}^{(2)} + \frac{1}{2 a} \int \mathrm{d} \eta \left\{a{\cal{D}}_{00}^{(2)} \right\}, \tag{51a}$

$ \begin{aligned}[b] Z_2^{j, \mathrm{AB}} = &\delta^{jk} \partial_k \int \mathrm{d} \eta \left\{ \frac{1}{a} \int {\cal{A}}^{(2)} \mathrm{d} \eta' \right\} \\ &- \delta^{jk} \partial_k \int {\cal{B}}^{(2)} \mathrm{d} \eta + \delta^{j k} \int {\cal{C}}_k^{(2)} \mathrm{d} \eta \\ &- \!\int \mathrm{d} \eta \left\{ \delta^{j k} {\cal{D}}_{0 k}^{(2)} \!-\! \frac{1}{2 a} \!\int \!\mathrm{d} \eta \left\{a \partial^j {\cal{D}}_{00}^{(2)} \right\} \right\} . \end{aligned} \tag{51b}$

$ Z^{0, \mathrm{AB}}_2 = \frac{1}{a} \int {\cal{A}}^{(2)} \mathrm{d} \eta -{\cal{B}}^{(2)} , \tag{52a}$

$ \begin{aligned}[b] Z_2^{j, \mathrm{AB}} = &\delta^{jk} \partial_k \int \mathrm{d} \eta \left\{ \frac{1}{a} \int {\cal{A}}^{(2)} \mathrm{d} \eta' \right\} \\ &- \delta^{jk} \partial_k \int {\cal{B}}^{(2)} \mathrm{d} \eta + \delta^{jk} \int {\cal{C}}_k^{(2)} \mathrm{d} \eta . \end{aligned}\tag{52b} $

$ \begin{aligned}[b] Z^{0,\rm{AB}}_2 = & \Delta^{- 1} \partial^j {\cal{D}}_{0 j}^{(2)} \\ &- \frac{1}{2} \left( \partial^k \Delta^{- 2} \partial^l\partial_0 - \frac{1}{2} \Delta^{- 1}{\cal{T}}^{kl}\partial_0 \right) {\cal{D}}_{kl}^{(2)}, \end{aligned} \tag{53a} $

$ Z^{i, \rm{AB}}_2\! \!=\! \left(\! \frac{1}{4} \partial^i\partial^k \Delta^{- 2} \partial^l \!\!+\!\! \frac{1}{4}\delta^{kl} \!\Delta^{- 1}\partial^i \!-\! \delta^{li}\! \Delta^{- 1} \partial^k \right)\! {\cal{D}}_{kl}^{(2)} , \tag{53b} $

$ T_{\mu \nu}^{(\mathrm{\rm{GI})}} \equiv T_{\mu \nu}^{(0)} + T_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} + \frac{1}{2} T_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} +{\cal{O}} \left(T_{\mu \nu}^{(\mathrm{\rm{GI}, 3)}}\right), \tag{54} $

$ T^{(0)}_{\mu \nu} = u_{\mu}^{(0)} u_{\nu}^{(0)} (\rho^{(0)} + P^{(0)}) + g_{\mu \nu}^{(0)} P^{(0)}, \tag{55a}$

$ \begin{aligned}[b] T^{(\mathrm{\rm{GI}, 1)}}_{\mu \nu} =& u_{\mu}^{(\mathrm{\rm{GI}, 1)}} u_{\nu}^{(0)} \Bigr(\rho^{(0)} + P^{(0)}\Bigr) + u^{(0)}_{\mu} u^{(\mathrm{\rm{GI}, 1)}}_{\nu} \Bigr(\rho^{(0)} + P^{(0)}\Bigr) \\ &+ u_{\mu}^{(0)} u_{\nu}^{(0)} \Bigr(\rho^{(\mathrm{\rm{GI}, 1)}} + P^{(\mathrm{\rm{GI}, 1)}}\Bigr) \\ &+ g_{\mu \nu}^{(0)} P^{(\mathrm{\rm{GI}, 1)}} + g_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} P^{(0)}, \end{aligned}\tag{55b} $

$ \begin{aligned}[b] T^{(\rm{GI}, 2)}_{\mu \nu} =& u_{\mu}^{(\rm{GI}, 1)} u_{\nu}^{(\rm{GI}, 1)} \Bigr(\rho^{(0)} + P^{(0)}\Bigr) + u_{\mu}^{(\rm{GI}, 1)} u_{\nu}^{(0)} \Bigr(\rho^{(\rm{GI}, 1)} \\ &+ P^{(\rm{GI}, 1)}\Bigr) + u_{\mu}^{(0)} u_{\nu}^{(\rm{GI}, 1)} \Bigr(\rho^{(\rm{GI}, 1)} + P^{(\rm{GI}, 1)}\Bigr)\\ &\times u_{\mu}^{(\mathrm{\rm{GI}, 2)}} u_{\nu}^{(0)} \Bigr(\rho^{(0)} + P^{(0)}\Bigr) + u^{(0)}_{\mu} u^{(\mathrm{\rm{GI}, 2)}}_{\nu} \Bigr(\rho^{(0)} + P^{(0)}\Bigr) \\ &+ u_{\mu}^{(0)} u_{\nu}^{(0)} \Bigr(\rho^{(\mathrm{\rm{GI}, 2)}} + P^{(\mathrm{\rm{GI}, 2)}}\Bigr) + g_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} P^{(\mathrm{\rm{GI}, 1)}}\\ &+ g_{\mu \nu}^{(0)} P^{(\mathrm{\rm{GI}, 2)}} + g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} P^{(0)} . \end{aligned} \tag{55c}$

$\rho^{(\mathrm{\rm{GI}, 1)}} = \rho^{(1)} -{\cal{L}}_X \rho^{(0)}, \tag{56a} $

$ P^{(\mathrm{\rm{GI}, 1)}} = P^{(1)} -{\cal{L}}_X P^{(0)}, \tag{56b} $

$ u_{\mu}^{(\mathrm{\rm{GI}, 1)}} = u_{\mu}^{(1)} -{\cal{L}}_X u_{\mu}^{(0)}, \tag{56c} $

$ \rho^{(\mathrm{\rm{GI}, 2)}} = \rho^{(2)} - 2{\cal{L}}_X \rho^{(1)} - \left({\cal{L}}_Y -{\cal{L}}_X^2\right) P^{(0)}, \tag{56d} $

$ P^{(\mathrm{\rm{GI}, 2)}} = P^{(2)} - 2{\cal{L}}_X P^{(1)} - \left({\cal{L}}_Y -{\cal{L}}_X^2\right) P^{(0)}, \tag{56e}$

$ u_{\mu}^{(\mathrm{\rm{GI}, 2)}} = u^{(2)}_{\mu} - 2{\cal{L}}_X u_{\mu}^{(1)} - \left({\cal{L}}_Y -{\cal{L}}_X^2\right) u^{(0)}_{\mu}. \tag{56f}$

$ P^{(0)} = w \rho^{(0)},\tag{57a} $

$ P^{(\mathrm{\rm{GI}},1)} = c_s^2 \rho^{(\mathrm{\rm{GI}},1)} , \tag{57b}$

$ P^{(\mathrm{\rm{GI}},2)} = (c_s^{(2)})^2 \rho^{(\mathrm{\rm{GI}},2)}.\tag{57c} $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI})}} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} =& - a^2 \left(1 + 2 \Phi^{(1)}\right) \mathrm{d} \eta^2 + 2 a^2 V_i^{(1)} \mathrm{d} \eta \mathrm{d} x^i \\ &+ a^2 \delta_{ij} \left(1 - 2 \Psi^{(1)}\right) \mathrm{d} x^i \mathrm{d} x^j, \end{aligned} \tag{58}$

$ \dot{{\cal{H}}} = - \frac{1}{2} (1 + 3 w) {\cal{H}}^2, \tag{59a}$

$ \rho^{(0)} = \frac{3{\cal{H}}^2}{\kappa a^2}, \tag{59b} $

$ \rho^{(\mathrm{\rm{GI}, 1)}} = \frac{- 6{\cal{H}}^2 \Phi^{(1)} - 6{\cal{H}} \partial_0 \Psi^{(1)} + 2 \Delta \Psi^{(1)}}{\kappa a^2}, \tag{59c} $

$ \upsilon_i^{(\mathrm{\rm{GI}, 1)}} \!=\! - V_i^{(1)} \!+\! \frac{\Delta V_i^{(1)} \!-\! 4 \partial_i ({\cal{H}} \Phi^{(1)} \!+\! 4 \partial_0 \Psi^{(1)})}{6 (1 \!+\! w) {\cal{H}}^2}, \tag{59d} $

$ \Phi^{(1)} = \Psi^{(1)}, \tag{60a} $

$ V^{(1)}_i = 0, \tag{60b} $

$ \partial_0^2 \Phi^{(1)} + 3 (1 + c_s^2) {\cal{H}} \partial_0 \Phi^{(1)} + 3 (c^2_s - w) {\cal{H}}^2 \Phi^{(1)} - c_s^2 \Delta \Phi^{(1)} = 0. $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} = &2 a^2 \Phi^{(2)} \mathrm{d} \eta^2 + 2 a^2 V_i^{(2)} \mathrm{d} \eta \mathrm{d} x^i \\ &+ a^2 \left(- 2 \delta_{ij} \Psi^{(2)} + H_{ij}^{(2)}\right) \mathrm{d} x^i \mathrm{d} x^j. \end{aligned} $

$ \begin{aligned}[b] \rho^{(\rm{GI}, 2)} = &\frac{2}{3 (1 + w) \kappa a^2 {\cal{H}}^2} \Bigr(9 (1 + w) {\cal{H}}^4 \Bigr(4 \Bigr(\Phi^{(1)}\Bigr)^2 - \Phi^{(2)}\Bigr) \\ &- 9 (1 + w) {\cal{H}}^3 \partial_0 \Psi^{(2)} +{\cal{H}}^2 \Bigr(9 (1 + w) \Bigr(\partial_0 \Phi^{(1)}\Bigr)^2 \\ &+ 24 (1 + w) \Phi^{(1)} \Delta \Phi^{(1)} + 3 (1 + w) \Delta \Psi^{(2)}\\ &+ (5 + 9 w) \partial_i \Phi^{(1)} \partial^i \Phi^{(1)}\Bigr) - 8{\cal{H}} \partial_i \partial_0 \Phi^{(1)} \partial^i \Phi^{(1)} \\ &- 4 \partial_i \partial_0\Phi^{(1)} \partial^i \partial_0\Phi^{(1)}\Bigr) . \end{aligned} $

$ \begin{array}{l} {\cal{G}}_{i j} +{\cal{S}}_{i j} = 0 , \end{array} $

$ \begin{aligned}[b] {\cal{S}}_{i j} = &\delta_{i j} \left( \frac{4 (c_s^{(2)})^2}{3 (1 + w)} \partial_k \Phi^{(1)} \partial^k \Phi^{(1)} - 12 \left(c_s^2 + \left(c_s^{(2)}\right)^2 - 2w\right) {\cal{H}}^2 \left(\Phi^{(1)}\right)^2 - 4 \left(5 + 3 c_s^2\right) {\cal{H}} \Phi^{(1)} \partial_0 \Phi^{(1)} \right. \\ &+ \frac{8 (c_s^{(2)})^2}{3 (1 + w) {\cal{H}}} \partial_k \partial_0 \Phi^{(1)} \partial^k \Phi^{(1)} - \left(1 + 3 \left(c_s^{(2)}\right)^2\right) (\partial_0 \Phi^{(1)})^2 - 4 \Phi^{(1)} \partial_0^2 \Phi^{(1)} - 4 \Phi^{(1)} \Delta \Phi^{(1)} - 3 \partial_k \Phi^{(1)} \partial^k \Phi^{(1)} \\ &\left.- 8 \left(c_s^{(2)}\right)^2 \Phi^{(1)}\Delta \Phi^{(1)} - 3 \left(c_s^{(2)}\right)^2 \partial_k \Phi^{(1)} \partial^k \Phi^{(1)} + \frac{4 (c_s^{(2)})^2}{3 (1 + w) {\cal{H}}^2} \partial_k \partial_0 \Phi^{(1)} \partial^k \partial_0 \Phi^{(1)} + 4 c_s^2 \Phi^{(1)} \Delta \Phi^{(1)} \right) \\ &+ 4 \Phi^{(1)} \partial_i \partial_j \Phi^{(1)} - \frac{4}{3 (1 + w) {\cal{H}}} \left(\partial_i \partial_0 \Phi^{(1)} \partial_j \Phi^{(1)} + \partial_i \Phi^{(1)} \partial_0 \partial_j \Phi^{(1)}\right) \\ &+ \left( 2 - \frac{4}{3 (1 + w)} \right) \partial_i \Phi^{(1)} \partial_j \Phi^{(1)} - \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Phi^{(1)} \partial_i \partial_0 \Phi^{(1)}, \end{aligned} $

$ \begin{aligned}[b] {\cal{G}}_{i j} = & \frac{1}{4} \partial_0^2 H_{i j}^{(2)} + \frac{1}{2} {\cal{H}} \partial_0 H_{i j}^{(2)} - \frac{1}{4} \Delta H_{i j}^{(2)} + \delta_{i j} \left( 3 \left(\left(c_s^{(2)}\right)^2 - w\right) {\cal{H}}^2 \Phi^{(2)} +{\cal{H}} \partial_0 \Phi^{(2)} + \frac{1}{2} \Delta \Phi^{(2)} \right.\\ & + \partial_0^2 \Psi^{(2)} + \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)}\left.- \frac{1}{2} \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \Delta \Psi^{(2)} \right) - \frac{1}{2} {\cal{H}} \partial_i V_j^{(2)} \\ & - \frac{1}{4} \partial_i \partial_0 V_j^{(2)} - \frac{1}{2} {\cal{H}} \partial_j V_i^{(2)}- \frac{1}{4} \partial_j \partial_0 V_i^{(2)} - \frac{1}{2} \partial_i \partial_j \Phi^{(2)} + \frac{1}{2} \partial_i \partial_j \Psi^{(2)} . \end{aligned} $

$ \Lambda_{k l}^{i j} {\cal{G}}_{i j} = \frac{1}{4} \partial_0^2 H_{k l}^{(2)} + \frac{1}{2} {\cal{H}} \partial_0 H_{k l}^{(2)} - \frac{1}{4} \Delta H_{k l}^{(2)} ,\tag{67a} $

$ \Delta^{- 1} \partial^l {\cal{T}}^k_i {\cal{G}}_{k l} = \frac{1}{4} (\partial_0 + 2{\cal{H}}) V_i^{(2)} , \tag{67b} $

$ \frac{1}{2} \Delta^{- 1} \left( \partial^k \Delta^{- 1} \partial^l - \frac{1}{2} {\cal{T}}^{k l} \right) {\cal{G}}_{k l} = \frac{1}{4} \left(\Psi^{(2)} - \Phi^{(2)}\right), \tag{67c} $

$ \begin{aligned}[b] \frac{1}{4} {\cal{T}}^{i j} {\cal{G}}_{i j} = & \frac{1}{2} \left( 3 \left(\left(c_s^{(2)}\right)^2 - w\right) {\cal{H}}^2 \Phi^{(2)} +{\cal{H}} \partial_0 \Phi^{(2)} + \frac{1}{2} \Delta \Phi^{(2)}\right. \\ &+ \partial_0^2 \Psi^{(2)} + \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)} \\ &\left.- \frac{1}{2} \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \Delta \Psi^{(2)} \right) , \end{aligned} \tag{67d}$

$\partial_0^2 H_{ij}^{(2)} + 2{\cal{H}} \partial_0 H_{ij}^{(2)} - \Delta H_{ij}^{(2)} = - 4 \Lambda^{kl}_{ij} {\cal{S}}_{kl}, \tag{68a}$

$ (\partial_0 + 2{\cal{H}}) V_i^{(2)} = - 4 \Delta^{- 1} {\cal{T}}^k_i \partial^l {\cal{S}}_{k l} , \tag{68b}$

$ \Psi^{(2)} - \Phi^{(2)} = - 2 \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) {\cal{S}}_{i j} , \tag{68c} $

$ \begin{aligned}[b] \partial_0^2 \Psi^{(2)} +& \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)} - \frac{1}{2} \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \Delta \Psi^{(2)} \\ +& 3 \left(\left(c_s^{(2)}\right)^2 - w\right) {\cal{H}}^2 \Phi^{(2)} +{\cal{H}} \partial_0 \Phi^{(2)} \\ +& \frac{1}{2} \Delta \Phi^{(2)}= - \frac{1}{2} {\cal{T}}^{i j} {\cal{S}}_{i j} , \end{aligned} \tag{68d}$

$ \begin{aligned}[b] \Lambda^{i j}_{k l} {\cal{S}}_{i j} =& \Lambda^{i j}_{k l} \left( \left( 2 + \frac{4}{3 (1 + w)} \right) \Phi^{(1)} \partial_i \partial_j \Phi^{(1)} \right.\\ &+ \frac{8}{3 (1 + w) {\cal{H}}} \Phi^{(1)} \partial_0 \partial_i \partial_j \Phi^{(1)} \\ &\left.- \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Phi^{(1)} \partial_j \partial_0 \Phi^{(1)} \right) , \end{aligned} $

$ \begin{aligned}[b] \Delta^{- 1} {\cal{T}}^k_i \partial^l {\cal{S}}_{k l} =& \Delta^{- 1} {\cal{T}}^k_i \partial^l \left( \left( 2 - \frac{4}{3 (1 + w)} \!\right) \partial_k \Phi^{(1)} \partial_l \Phi^{(1)} \right.- \frac{4}{3 (1 + w) {\cal{H}}} (\partial_k \partial_0 \Phi^{(1)} \partial_l \Phi^{(1)} + \partial_k \Phi^{(1)} \partial_0 \partial_l \Phi^{(1)}) \\ &+ 4 \Phi^{(1)} \partial_k \partial_l \Phi^{(1)} \left.- \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_k \partial_0 \Phi^{(1)} \partial_l \partial_0 \Phi^{(1)} \right) , \end{aligned} $

$ \begin{aligned}[b] \frac{1}{4} {\cal{T}}^{i j} {\cal{S}}_{i j} =& \frac{1}{2} \left( \frac{4 \left(c_s^{(2)}\right)^2}{3 (1 \!+\! w)} \partial_k \Phi^{(1)} \partial^k \Phi^{(1)} \!-\! 12 \left(c_s^2\!+\! \left(c_s^{(2)}\right)^2 \!-\! 2w\right) {\cal{H}}^2 \left(\Phi^{(1)}\right)^2 \!-\! 4 \left(5 \!+\! 3 c_s^2\right) {\cal{H}} \Phi^{(1)} \partial_0 \Phi^{(1)} \!-\! 8 \left(c_s^{(2)}\right)^2 \Phi^{(1)}\Delta \Phi^{(1)}\right. \\ &+ \frac{8 (c_s^{(2)})^2}{3 (1 + w) {\cal{H}}} \partial_k \partial_0 \Phi^{(1)} \partial^k \Phi^{(1)} - \left(1 + 3 \left(c_s^{(2)}\right)^2\right) \left(\partial_0 \Phi^{(1)}\right)^2 - 4 \Phi^{(1)} \partial_0 ^2\Phi^{(1)} - 4 \Phi^{(1)} \Delta \Phi^{(1)} - 3 \partial_k \Phi^{(1)} \partial^k \Phi^{(1)} \\ &\left.- 3 \left(c_s^{(2)}\right)^2 \partial_k \Phi^{(1)} \partial^k \Phi^{(1)} + \frac{4 \left(c_s^{(2)}\right)^2}{3 (1 + w) {\cal{H}}^2} \partial_k \partial_0 \Phi^{(1)} \partial^k \partial_0 \Phi^{(1)} + 4 c_s^2 \Phi^{(1)} \Delta \Phi^{(1)} \right)\\ & + \frac{1}{4} {\cal{T}}^{i j} \left( \left( 2 + \frac{4}{3 (1 + w)} \right) \Phi^{(1)} \partial_i \partial_j \Phi^{(1)} + \frac{8}{3 (1 + w) {\cal{H}}} \Phi^{(1)} \partial_0 \partial_i \partial_j \Phi^{(1)} - \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Phi^{(1)} \partial_j \partial_0 \Phi^{(1)} \right) , \end{aligned} $

$ \begin{aligned}[b] \frac{1}{2} \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) {\cal{S}}_{i j} =& \frac{1}{2} \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) \left( \left( 2 - \frac{4}{3 (1 + w)} \right) \partial_i \Phi^{(1)} \partial_j \Phi^{(1)} + 4 \Phi^{(1)} \partial_i \partial_j \Phi^{(1)} \right. \\ &- \frac{4}{3 (1 + w) {\cal{H}}} \left(\partial_i \partial_0 \Phi^{(1)} \partial_j \Phi^{(1)} + \partial_i \Phi^{(1)} \partial_0 \partial_j \Phi^{(1)}\right) \left. - \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Phi^{(1)} \partial_j \partial_0 \Phi^{(1)} \right) . \end{aligned} $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} =& a^2 \left(- 2 \delta_{ij} \Psi^{(2)} + 2 \partial_i \partial_j E^{(2)}\right. \\ &\left.+ \partial_i C_j^{(2)} + \partial_j C_i^{(2)} + H_{ij}^{(2)}\right) \mathrm{d} x^i \mathrm{d} x^j . \end{aligned} $

$ \begin{aligned}[b] \rho^{(\rm{GI}, 2)} = &\frac{2}{3 (1 + w) \kappa a^2 {\cal{H}}^2} (36 (1 + w) {\cal{H}}^4 (\Phi^{(1)})^2 \\ &- 3 (1 + w) {\cal{H}}^3 (3 \partial_0 \Psi^{(2)} - \Delta \partial_0 E^{(2)}) \\ &+{\cal{H}}^2 (9 (1 + w) (\partial_0 \Phi^{(1)})^2 + 24 (1 + w) \Phi^{(1)} \Delta \Phi^{(1)} \\ &+ 3 (1 + w) \Delta \Psi^{(2)} + (5 + 9 w) \partial_i \Phi^{(1)} \partial^i \Phi^{(1)}) \\ &- 8{\cal{H}} \partial_i \partial_0 \Phi^{(1)} \partial^i \Phi^{(1)} - 4 \partial_i \partial_0\Phi^{(1)} \partial^i \partial_0\Phi^{(1)}) . \end{aligned} $

$ \begin{array}{l} {\cal{G}}_{i j} +{\cal{S}}_{i j} = 0, \end{array} $

$ \begin{aligned}[b] {\cal{G}}_{i j} =& \frac{1}{4} \partial_0^2 H_{i j}^{(2)} + \frac{1}{2} {\cal{H}} \partial_0 H_{i j}^{(2)} - \frac{1}{4} \Delta H_{i j}^{(2)} + \delta_{i j} \left( \partial_0^2 \Psi^{(2)} + \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)} - \frac{1}{2} \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \Delta \Psi^{(2)} - \Delta \left( \frac{1}{2} \left(\partial_0 + 2{\cal{H}}\right) \partial_{\eta} E^{(2)}\right.\right.\\ & + \left(c_s^{(2)}\right)^2 {\cal{H}} \partial_{\eta} E^{(2)} \Biggr) \Biggr) + \frac{1}{2} {\cal{H}} \partial_i \partial_0 C_j^{(2)} + \frac{1}{4} \partial_i \partial_0^2 C_j^{(2)} + \frac{1}{2} {\cal{H}} \partial_j \partial_0 C_i^{(2)}+ \frac{1}{4} \partial_j \partial_0^2 C_i^{(2)} +{\cal{H}} \partial_i \partial_j \partial_0 E^{(2)} + \frac{1}{2} \partial_j \partial_j \partial_0^2 E^{(2)} + \frac{1}{2} \partial_i \partial_j \Psi^{(2)} . \end{aligned} $

$\partial_0^2 H_{i j}^{(2)} + 2{\cal{H}} \partial_0 H_{ i j}^{(2)} - \Delta H_{i j}^{(2)} = - 4 \Lambda^{kl}_{ij} {\cal{S}}_{kl}, \tag{77a} $

$ (\partial_0 + 2{\cal{H}}) \partial_0 C_i^{(2)} = - 4 \Delta^{- 1} {\cal{T}}^k_i \partial^l {\cal{S}}_{k l}, \tag{77b} $

$ (\partial_0 + 2{\cal{H}}) \partial_0 E^{(2)} + \Psi^{(2)} = - 2 \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) {\cal{S}}_{i j}, \tag{77c}$

$ \begin{aligned}[b] \partial_0^2 \Psi^{(2)} &+ \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)} - \frac{1}{2} (1 + 2 (c_s^{(2)})^2) \Delta \Psi^{(2)} \\ &- \Delta \left( \frac{1}{2} (\partial_0 + 2{\cal{H}}) \partial_0 E^{(2)} + \left(c_s^{(2)}\right)^2 {\cal{H}} \partial_0 E^{(2)} \right) \\ =& - \frac{1}{2} {\cal{T}}^{i j} {\cal{S}}_{i j} . \end{aligned}\tag{77d} $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI})}} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} = &- a^2 \mathrm{d} \eta^2 + a^2 (\delta_{ij} (1 - 2 \Psi^{(1)}) \\ &+ 2 \partial_i \partial_j E^{(1)} + \partial_i C_j^{(1)} + \partial_j C_i^{(1)}) \mathrm{d} x^i \mathrm{d} x^j , \end{aligned} \tag{78}$

$ \rho^{(\mathrm{\rm{GI}, 1)}} = \frac{- 6{\cal{H}} \partial_0 \Psi^{(1)} + 2{\cal{H}} \Delta \partial_0 E^{(1)} + 2 \Delta \Psi^{(1)}}{\kappa a^2}, \tag{79a}$

$ \upsilon_i^{(\mathrm{\rm{GI}, 1)}} = - \frac{\Delta \partial_0 C_i^{(1)} + 4 \partial_i \partial_0 \Psi^{(1)}}{6 (1 + w) {\cal{H}}^2} . \tag{79b} $

$ C_i^{(1)} = 0, \tag{80}$

$ \Psi^{(1)} + 2{\cal{H}} \partial_0 E^{(1)} + \partial_0^2 E^{(1)} = 0, \tag{81a} $

$ \partial_0^2 \Psi^{(1)} +{\cal{H}} (2 + 3 c_s^2) \partial_0 \Psi^{(1)} - c_s^2 \Delta \Psi^{(1)} -{\cal{H}}c_s^2 \Delta \partial_0 E^{(1)} = 0. \tag{81b}$

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} =& 2 a^2 \Phi^{(2)} \mathrm{d} \eta^2 + 2 a^2 V_i^{(2)} \mathrm{d} \eta \mathrm{d} x^i \\ & + a^2 (- 2 \delta_{ij} \Psi^{(2)} + H_{ij}^{(2)}) \mathrm{d} x^i \mathrm{d} x^j, \end{aligned} $

$ \begin{aligned}[b] \rho^{(\rm{GI}, 2)} =& - \frac{1}{3 (1 + w) \kappa a^2 {\cal{H}}^2} \Big(18 (1 + w) {\cal{H}}^4 \Phi^{(2)} + 8 \partial_i \partial_0 \Psi^{(1)} \partial^i \partial_0 \Psi^{(1)} + 6 (1 + w) {\cal{H}}^3 (3 \partial_0 \Psi^{(2)} - 4 \partial_0 \Psi^{(1)} \Delta E^{(1)} \\ &+ 4 \Psi^{(1)} (3 \partial_0 \Psi^{(1)} - \Delta \partial_0 {e}^{(1)}) + 4 \partial_i \partial_j \partial_0 E^{(1)} \partial^i \partial^j E^{(1)}) - 3 (1 + w) {\cal{H}}^2 (6 (\partial_0 \Psi^{(1)})^2-4 \partial_0 \Psi^{(1)} \Delta \partial_0 E^{(1)} \\ & + 16 \Psi^{(1)} \Delta \Psi^{(1)} + 2 \Delta \Psi^{(2)} + 6 \partial_i \Psi^{(1)} \partial^i \Psi^{(1)} - 4 \partial_i \Delta E^{(1)} \partial^i \Psi^{(1)} + \Delta \partial_0 E^{(1)} \Delta \partial_0 E^{(1)} - 4 \Delta E^{(1)} \Delta \Psi^{(1)} \\ &- \partial_i \Delta E^{(1)} \partial^i \Delta E^{(1)} - 4 \partial_i \partial_j \Psi^{(1)} \partial^i \partial^j E^{(1)} - \partial_i \partial_j \partial_0 E^{(1)} \partial^i \partial^j \partial_0 E^{(1)} + \partial_k \partial_i \partial_j E^{(1)} \partial^i \partial^j \partial^k E^{(1)})\Big). \end{aligned} $

$ \begin{array}{l} {\cal{G}}_{i j} +{\cal{S}}_{i j} = 0, \end{array} $

$ \begin{aligned}[b] {\cal{S}}_{i j} = &\delta_{i j} \Biggr( - 12 \left(c_s^2 - \left(c_s^{(2)}\right)^2\right) {\cal{H}} \Psi^{(1)} \partial_0 \Psi^{(1)} + \left( 1 - 3 \left( c_s^{(2)} \right)^2 \right) \left(\partial_0 \Psi^{(1)}\right)^2 - 2 \left(1 - 2 c_s^2 + 4 \left(c_s^{(2)}\right)^2\right) \Psi^{(1)} \Delta \Psi^{(1)} \\ &- \left(2 + 3 \left(c_s^{(2)}\right)^2\right) \partial_k \Psi^{(1)} \partial^k \Psi^{(1)} + \frac{4 \left(c_s^{(2)}\right)^2}{3 (1 + w) {\cal{H}}^2} \partial_k \partial_0 \Psi^{(1)} \partial^k \partial_0 \Psi^{(1)} - \left(1 + \left(c_s^{(2)}\right)^2\right) \Delta \partial_0 E^{(1)} \Delta \partial_0 E^{(1)}\\ &+ \left(1 + \left(c_s^{(2)}\right)^2\right) \partial_k \Delta E^{(1)} \partial^k \Delta E^{(1)} + 8 \left(1 + \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_k \partial_l \partial_0 E^{(1)} \partial^k \partial^l E^{(1)} + 4 \partial_k \partial_l \partial_0^2 E^{(1)} \partial^k \partial^l E^{(1)}\\ &+ \left(3 + \left(c_s^{(2)}\right)^2\right) \partial_k \partial_l \partial_0 E^{(1)} \partial^k \partial^l \partial_0 E^{(1)} - \left(1 + \left(c_s^{(2)}\right)^2\right) \partial_m \partial_l \partial_k E^{(1)} \partial^m \partial^l \partial^k E^{(1)} \\ &- 2 \partial_0^2 \Psi^{(1)} \Delta E^{(1)} - \left(1 - 2 \left(c_s^{(2)}\right)^2\right) \partial_0 \Psi^{(1)} \Delta \partial_0 E^{(1)} + \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \partial_k \Delta E^{(1)} \partial^k \Psi^{(1)} \\ &+ 2 \left(1 + \left(c_s^{(2)}\right)^2\right) \Delta E^{(1)} \Delta \Psi^{(1)} + 2 \left(c_s^{(2)}\right)^2 \partial_k \partial_l \Psi^{(1)} \partial^k \partial^l E^{(1)} \\ &- 2 - 4{\cal{H}} \Biggr( \partial_0 \Psi^{(1)} \left( 1 + \left(c_s^{(2)}\right)^2 \right) \Delta E^{(1)} + \Psi^{(1)} \left(\left(c_s^{(2)}\right)^2 - c_s^2\right) \Delta \partial_0 E^{(1)} \Biggr) \Biggr)\\ &+ 3 \partial_i \Psi^{(1)} \partial_j \Psi^{(1)} - \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Psi^{(1)} \partial_j \partial_0 \Psi^{(1)} + 2 \Psi^{(1)} \partial_i \partial_j \Psi^{(1)}\\ &- 2 \partial_k \partial_j \partial_0 E^{(1)} \partial^k \partial_i \partial_0 E^{(1)} - \partial_k \Delta E^{(1)} \partial^k \partial_i \partial_j E^{(1)} + \partial_k \partial_l \partial_j E^{(1)} \partial^k \partial^l \partial_i E^{(1)}\\ &- 2 \Delta \partial_0^2 E^{(1)} \partial_i \partial_j E^{(1)} - 4{\cal{H}} \left(1 + c_s^2\right) \Delta \partial_0 E^{(1)} \partial_i \partial_j E^{(1)} + \Delta \partial_0 E^{(1)} \partial_i \partial_j \partial_0 E^{(1)}\\ &- \partial_k \partial_i \partial_j E^{(1)} \partial^k \Psi^{(1)} + 2 \partial_k \partial_j \Psi^{(1)} \partial^k \partial_i E^{(1)} + 2 \partial_k \partial_i \Psi^{(1)} \partial^k \partial_j E^{(1)} + 12 (1 + c_s^2) {\cal{H}} \partial_0 \Psi^{(1)} \partial_i \partial_j E^{(1)}\nonumber\\ &+ 6 \partial_0^2 \Psi^{(1)} \partial_i \partial_j E^{(1)} - 4 \left(1 + c_s^2\right) \Delta \Psi^{(1)} \partial_i \partial_j E^{(1)} + \partial_0 \Psi^{(1)} \partial_i \partial_j \partial_0 E^{(1)} - 2 \Delta E^{(1)} \partial_i \partial_j \Psi^{(1)} . \end{aligned} \tag{85}$

$ \partial_0^2 H_{ij}^{(2)} + 2{\cal{H}} \partial_0 H_{ij}^{(2)} - \Delta H_{ij}^{(2)} = - 4 \Lambda^{kl}_{ij} {\cal{S}}_{kl}, \tag{86a}$

$ (\partial_0 + 2{\cal{H}}) V_i^{(2)} = - 4 \Delta^{- 1} {\cal{T}}^k_i \partial^l {\cal{S}}_{k l} , \tag{86b}$

$ \Psi^{(2)} - \Phi^{(2)} = - 2 \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) {\cal{S}}_{i j} , \tag{86c}$

$ \begin{aligned}[b] \partial_0^2 \Psi^{(2)}& + \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)} \\&- \frac{1}{2} \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \Delta \Psi^{(2)} \\ &+ 3 \left(\left(c_s^{(2)}\right)^2 - w\right) {\cal{H}}^2 \Phi^{(2)} \\&+{\cal{H}} \partial_0 \Phi^{(2)} + \frac{1}{2} \Delta \Phi^{(2)} \\ =& - \frac{1}{2} {\cal{T}}^{i j} {\cal{S}}_{i j} , \end{aligned}\tag{86d} $

$ \begin{aligned}[b] \Lambda^{i j}_{k l} {\cal{S}}_{i j} = &\Lambda^{i j}_{k l} \Biggr( -\Psi^{(1)} \partial_i \partial_j \Psi^{(1)} + \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_0 \Psi^{(1)} \partial_i \partial_j \partial_0 \Psi^{(1)}+ \Bigr(- 5 \partial^m \Psi^{(1)} \partial_m - 6 c_s^2 {\cal{H}} \partial_0 \Psi^{(1)} \\ &- 2 \left(1 - c_s^2\right) \Delta \Psi^{(1)} + \partial_0 \Psi^{(1)} \partial_0- 2 \Psi^{(1)} \Delta\Bigr) \partial_i \partial_j E^{(1)} + 2 \partial_m \partial_0 E^{(1)} \partial^m \partial_i \partial_j \partial_0 E^{(1)}+ \Delta \partial_0 E^{(1)} \partial_i \partial_j \partial_0 E^{(1)} \\ &- \partial_m \Delta E^{(1)} \partial^m \partial_i \partial_j E^{(1)}- \partial_n \partial_m E^{(1)} \partial^n \partial^m \partial_i \partial_j E^{(1)} + 2 c_s^2 {\cal{H}} \Delta \partial_0 E^{(1)} \partial_i \partial_j E^{(1)} \Biggr) , \end{aligned} \qquad \qquad \quad $

$ \begin{aligned}[b] \Delta^{- 1} {\cal{T}}^i_m \partial^j {\cal{S}}_{i j} = & \Delta^{- 1} {\cal{T}}^i_m \partial^j \Biggr( 3 \partial_i \Psi^{(1)} \partial_j \Psi^{(1)} - \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Psi^{(1)} \partial_j \partial_0 \Psi^{(1)} + 2 \Psi^{(1)} \partial_i \partial_j \Psi^{(1)} - 2 \partial_k \partial_j \partial_0 E^{(1)} \partial^k \partial_i \partial_0 E^{(1)} \\ &- \partial_k \Delta E^{(1)} \partial^k \partial_i \partial_j E^{(1)} + \partial_k \partial_l \partial_j E^{(1)} \partial^k \partial^l \partial_i E^{(1)}- 2 \Delta \partial_0^2 E^{(1)} \partial_i \partial_j E^{(1)} - 4{\cal{H}} \left(1 + c_s^2\right) \Delta \partial_0 E^{(1)} \partial_i \partial_j E^{(1)} \\ & + \Delta \partial_0 E^{(1)} \partial_i \partial_j \partial_0 E^{(1)} - \partial_k \partial_i \partial_j E^{(1)} \partial^k \Psi^{(1)}+ 2 \partial_k \partial_j \Psi^{(1)} \partial^k \partial_i E^{(1)} + 2 \partial_k \partial_i \Psi^{(1)} \partial^k \partial_j E^{(1)} + 12 (1 \\ &+ c_s^2) {\cal{H}} \partial_0 \Psi^{(1)} \partial_i \partial_j E^{(1)} + 6 \partial_0^2 \Psi^{(1)} \partial_i \partial_j E^{(1)} - 4 \left(1 + c_s^2\right) \Delta \Psi^{(1)} \partial_i \partial_j E^{(1)} + \partial_0 \Psi^{(1)} \partial_i \partial_j \partial_0 E^{(1)} - 2 \Delta E^{(1)} \partial_i \partial_j \Psi^{(1)} \Biggr) , \end{aligned} $

$ \begin{aligned}[b] \frac{1}{4} {\cal{T}}^{i j} {\cal{S}}_{i j} = & \frac{1}{2} \Biggr( - 12 \left(c_s^2 - \left(c_s^{(2)}\right)^2\right) {\cal{H}} \Psi^{(1)} \partial_0 \Psi^{(1)} + \left( 1 - 3 \left( c_s^{(2)} \right)^2 \right) \left(\partial_0 \Psi^{(1)}\right)^2 - 2 \left(1 - 2 c_s^2 + 4 \left(c_s^{(2)}\right)^2\right) \Psi^{(1)} \Delta \Psi^{(1)} \\ &- \left(2 + 3 \left(c_s^{(2)}\right)^2\right) \partial_k \Psi^{(1)} \partial^k \Psi^{(1)} + \frac{4 (c_s^{(2)})^2}{3 (1 + w) {\cal{H}}^2} \partial_k \partial_0 \Psi^{(1)} \partial^k \partial_0 \Psi^{(1)} - \left(1 + \left(c_s^{(2)}\right)^2\right) \Delta \partial_0 E^{(1)} \Delta \partial_0 E^{(1)} \\ &+ \left(1 + \left(c_s^{(2)}\right)^2\right) \partial_k \Delta E^{(1)} \partial^k \Delta E^{(1)} + 8 \left(1 + \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_k \partial_l \partial_0 E^{(1)} \partial^k \partial^l E^{(1)} + 4 \partial_k \partial_l \partial_0^2 E^{(1)} \partial^k \partial^l E^{(1)} \\ &+ \left(3 + \left(c_s^{(2)}\right)^2\right) \partial_k \partial_l \partial_0 E^{(1)} \partial^k \partial^l \partial_0 E^{(1)} - \left(1 + \left(c_s^{(2)}\right)^2\right) \partial_m \partial_l \partial_k E^{(1)} \partial^m \partial^l \partial^k E^{(1)} - 2 \partial_0^2 \Psi^{(1)} \Delta E^{(1)}\\ &- \left(1 - 2 \left(c_s^{(2)}\right)^2\right) \partial_0 \Psi^{(1)} \Delta \partial_0 E^{(1)} + \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \partial_k \Delta E^{(1)} \partial^k \Psi^{(1)} + 2 \left(1 + \left(c_s^{(2)}\right)^2\right) \Delta E^{(1)} \Delta \Psi^{(1)}\\ &+ 2 \left(c_s^{(2)}\right)^2 \partial_k \partial_l \Psi^{(1)} \partial^k \partial^l E^{(1)} - 4{\cal{H}} \big( \partial_0 \Psi^{(1)} \left( 1 + \left(c_s^{(2)}\right)^2 \right) \Delta E^{(1)} \\ &+ \Psi^{(1)} \left(\left(c_s^{(2)}\right)^2 - c_s^2\right) \Delta \partial_0 E^{(1)} \big) \Biggr) \frac{1}{4} {\cal{T}}^{i j} \Biggr( -\Psi^{(1)} \partial_i \partial_j \Psi^{(1)} + \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_0 \Psi^{(1)} \partial_i \partial_j \partial_0 \Psi^{(1)}\\ &+\left(- 5 \partial^m \Psi^{(1)} \partial_m - 6 c_s^2 {\cal{H}} \partial_0 \Psi^{(1)} - 2 \left(1 - c_s^2\right) \Delta \Psi^{(1)} + \partial_0 \Psi^{(1)} \partial_0 - 2 \Psi^{(1)} \Delta \right) \partial_i \partial_j E^{(1)}+ 2 \partial_m \partial_0 E^{(1)} \partial^m \partial_i \partial_j \partial_0 E^{(1)}\\ & + \Delta \partial_0 E^{(1)} \partial_i \partial_j \partial_0 E^{(1)} - \partial_m \Delta E^{(1)} \partial^m \partial_i \partial_j E^{(1)}- \partial_n \partial_m E^{(1)} \partial^n \partial^m \partial_i \partial_j E^{(1)} + 2 c_s^2 {\cal{H}} \Delta \partial_0 E^{(1)} \partial_i \partial_j E^{(1)} \Biggr) , \end{aligned} $

$ \begin{aligned}[b] \frac{1}{2} \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) {\cal{S}}_{i j} =& \frac{1}{2} \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) \Biggr( 3 \partial_i \Psi^{(1)} \partial_j \Psi^{(1)} - \frac{4}{3 (1 + w) {\cal{H}}^2} \partial_i \partial_0 \Psi^{(1)} \partial_j \partial_0 \Psi^{(1)} \\ &+ 2 \Psi^{(1)} \partial_i \partial_j \Psi^{(1)}- 2 \partial_k \partial_j \partial_0 E^{(1)} \partial^k \partial_i \partial_0 E^{(1)} - \partial_k \Delta E^{(1)} \partial^k \partial_i \partial_j E^{(1)} \\ &+ \partial_k \partial_l \partial_j E^{(1)} \partial^k \partial^l \partial_i E^{(1)} - 2 \Delta \partial_0^2 E^{(1)} \partial_i \partial_j E^{(1)} - 4{\cal{H}} \left(1 + c_s^2\right) \Delta \partial_0 E^{(1)} \partial_i \partial_j E^{(1)} \\ & + \Delta \partial_0 E^{(1)} \partial_i \partial_j \partial_0 E^{(1)} - \partial_k \partial_i \partial_j E^{(1)} \partial^k \Psi^{(1)}+ 2 \partial_k \partial_j \Psi^{(1)} \partial^k \partial_i E^{(1)} + 2 \partial_k \partial_i \Psi^{(1)} \partial^k \partial_j E^{(1)} \\ &+ 12 \left(1 + c_s^2\right) {\cal{H}} \partial_0 \Psi^{(1)} \partial_i \partial_j E^{(1)} + 6 \partial_0^2 \Psi^{(1)} \partial_i \partial_j E^{(1)} \\ &- 4 (1 + c_s^2) \Delta \Psi^{(1)} \partial_i \partial_j E^{(1)} + \partial_0 \Psi^{(1)} \partial_i \partial_j \partial_0 E^{(1)} - 2 \Delta E^{(1)} \partial_i \partial_j \Psi^{(1)}\Biggr) . \end{aligned} $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} = & a^2 \left(- 2 \delta_{ij} \Psi^{(2)} + 2 \partial_i \partial_j E^{(2)} \right.\\ &\left.+ \partial_i C_j^{(2)} + \partial_j C_i^{(2)} + H_{ij}^{(2)}\right) \mathrm{d} x^i \mathrm{d} x^j. \end{aligned} $

$ \begin{aligned}[b] \rho^{(\rm{GI}, 2)} = &- \frac{1}{3 (1 + w) \kappa a^2 {\cal{H}}^2} \Bigr(8 \partial_i \partial_0 \Psi^{(1)} \partial^i \partial_0 \Psi^{(1)} \\ &+ 6 (1 + w) {\cal{H}}^3 \Bigr(3 \partial_0 \Psi^{(2)} - \Delta \partial_0 E^{(2)} - 4 \partial_0 \Psi^{(1)} \Delta E^{(1)} \\ &+ 4 \Psi^{(1)} \Bigr(3 \partial_0 \Psi^{(1)} - \Delta \partial_0 E^{(1)}\Bigr) + 4 \partial_i \partial_j \partial_0 E^{(1)} \partial^i \partial^j E^{(1)}\Bigr) \\ &- 3 (1 + w) {\cal{H}}^2 \Bigr(6 \Bigr(\partial_0 \Psi^{(1)}\Bigr)^2 - 4 \partial_0 \Psi^{(1)} \Delta \partial_0 E^{(1)} \\ &+ 16 \Psi^{(1)} \Delta \Psi^{(1)} + 2 \Delta \Psi^{(2)} + 6 \partial_i \Psi^{(1)} \partial^i \Psi^{(1)} \\ &- 4 \partial_i \Delta E^{(1)} \partial^i \Psi^{(1)} + \Delta \partial_0 E^{(1)} \Delta \partial_0 E^{(1)} \\ &- 4 \Delta E^{(1)} \Delta \Psi^{(1)} - \partial_i \Delta E^{(1)} \partial^i \Delta E^{(1)} - 4 \partial_i \partial_j \Psi^{(1)} \partial^i \partial^j E^{(1)} \\ &- \partial_i \partial_j \partial_0 E^{(1)} \partial^i \partial^j \partial_0 E^{(1)} + \partial_i \partial_j \partial_k E^{(1)} \partial^i \partial^j \partial^k E^{(1)}\Bigr)\Bigr) . \end{aligned} $

$ \begin{array}{l} {\cal{G}}_{i j} +{\cal{S}}_{i j} = 0, \end{array} $

$ \partial_0^2 H_{i j}^{(2)} + 2{\cal{H}} \partial_0 H_{i j}^{(2)} - \Delta H_{i j}^{(2)} = - 4 \Lambda^{kl}_{ij} {\cal{S}}_{kl}, \tag{94a} $

$ (\partial_0 + 2{\cal{H}}) \partial_0 C_i^{(2)} = - 4 \Delta^{- 1} {\cal{T}}^k_i \partial^l {\cal{S}}_{k l}, \tag{94b} $

$ \left(\partial_0 + 2{\cal{H}}\right) \partial_0 E^{(2)} + \Psi^{(2)} = - 2 \Delta^{- 1} \left( \partial^i \Delta^{- 1} \partial^j - \frac{1}{2} {\cal{T}}^{i j} \right) {\cal{S}}_{i j}, \tag{94c} $

$ \begin{aligned}[b] \partial_0^2 \Psi^{(2)} &+ \left(2 + 3 \left(c_s^{(2)}\right)^2\right) {\cal{H}} \partial_0 \Psi^{(2)} - \frac{1}{2} \left(1 + 2 \left(c_s^{(2)}\right)^2\right) \Delta \Psi^{(2)} \\ &- \Delta \left( \frac{1}{2} \left(\partial_0 + 2{\cal{H}}\right) \partial_0 E^{(2)} + \left(c_s^{(2)}\right)^2 {\cal{H}} \partial_0 E^{(2)} \right) \\ &= - \frac{1}{2} {\cal{T}}^{i j} {\cal{S}}_{i j} . \end{aligned} \tag{94d}$

$ {\cal{L}}_{\zeta} {\cal{Q}} \equiv \lim_{\epsilon \rightarrow 0} \frac{\varphi^{\ast}_{\epsilon} {\cal{Q}} (x) -{\cal{Q}} (x)}{\epsilon}, \tag{A1}$

$ \varphi^{\ast}_{\epsilon} f (x) = f (\tilde{x}), \tag{A2a} $

$ \varphi^{\ast}_{\epsilon} w_{\mu} (x) = \frac{\partial \tilde{x}^{\nu}}{\partial x^{\mu}} w_{\nu} (\tilde{x}), \tag{A2b} $

$ \varphi_{\epsilon, \ast} A^{\mu} (x) = \frac{\partial x^{\mu}}{\partial \tilde{x}^{\nu}} A^{\nu} (\tilde{x}) .\tag{A2c} $

$ {\cal{L}}_{\zeta} f = \zeta^{\nu} \partial_{\nu} f, \tag{A3a} $

$ {\cal{L}}_{\zeta} w_{\mu} = \zeta^{\nu} \partial_{\nu} w_{\mu} + w_{\nu} \partial_{\mu} \zeta^{\mu},\tag{A3b} $

$ {\cal{L}}_{\zeta} A^{\mu} = \zeta^{\nu} \partial_{\nu} A^{\mu} - A^{\nu} \partial_{\nu} \zeta^{\mu} . \tag{A3c} $

$ {\cal{L}}_{\zeta} S_{\mu \nu} = \zeta^{\lambda} \partial_{\lambda} S_{\mu \nu} + S_{\lambda \nu} \partial_{\mu} \zeta^{\lambda} + S_{\mu \lambda} \partial_{\nu} \zeta^{\lambda} . \tag{A4} $

$ \begin{aligned}[b] f (x) \rightarrow &f (\tilde{x}) \approx f \left( x + \xi^{(1)} + \frac{1}{2} \xi^{(2)} \right) \\ = &f + \xi^{(1) \nu} \partial_{\nu} f + \frac{1}{2} \xi^{(2) \nu} \partial_{\nu} f + \frac{1}{2} \xi^{(1) \sigma} \xi^{(1) \rho} \partial_{\sigma} \partial_{\rho} f \\ = &f + \xi^{(1) \nu} \partial_{\nu} f + \frac{1}{2} \left.(\xi^{(2) \nu} \partial_{\nu} f + \xi^{(1) \sigma} \xi^{(1) \rho} \partial_{\sigma} \partial_{\rho} f \right.\\ &\left.+ \xi^{(1) \sigma} \partial_{\sigma} \xi^{(1) \rho} \partial_{\rho} f - \xi^{(1) \sigma} \partial_{\sigma} \xi^{(1) \rho} \partial_{\rho} f\right) \\ = & f +{\cal{L}}_{\xi^{(1)}} f + \frac{1}{2} \left({\cal{L}}_{\xi^{(2)} - \xi^{(1) \nu} \partial_{\nu} \xi^{(1)}} +{\cal{L}}^2_{\xi^{(1)}}\right) f. \end{aligned} \tag{A5} $

$ \xi_1^{\mu} \equiv \xi^{(1) \mu}, \tag{A6a}$

$ \xi_2^{\mu} \equiv \xi^{(2) \mu} - \xi^{(1) \nu} \partial_{\nu} \xi^{(1) \mu} . \tag{A6b} $

$ f (\tilde{x}) = f +{\cal{L}}_{\xi_1} f + \frac{1}{2} \left({\cal{L}}_{\xi_2} +{\cal{L}}^2_{\xi_1}\right) f +{\cal{O}} \left(\xi^{(3)}\right) . \tag{A7}$

$ \begin{aligned}[b] w_{\mu} (x) \rightarrow & \frac{\partial \tilde{x}^{\nu}}{\partial x^{\mu}} w_{\nu} (\tilde{x}) \approx \left( \delta^{\lambda}_{\mu} + \partial_{\mu} \xi^{(1) \lambda} + \frac{1}{2} \partial_{\mu} \xi^{(2) \lambda} \right) \Biggr( w_{\lambda} (x) \\ & + \left( \xi^{(1) \nu} + \frac{1}{2} \xi^{(2) \nu} \right) \partial_{\nu} w_{\lambda} \\ &+ \left( \xi^{(1) \sigma} + \frac{1}{2} \xi^{(2) \sigma} \right) \left( \xi^{(1) \rho} + \frac{1}{2} \xi^{(1) \rho} \right) \partial_{\sigma} \partial_{\rho} w_{\lambda} \Biggr) \\ =& w_{\mu} (x) + \xi^{(1) \nu} \partial_{\nu} w_{\mu} + w_{\lambda} \partial_{\mu} \xi^{(1) \lambda} + \frac{1}{2} w_{\lambda} \partial_{\mu} \xi^{(2) \lambda} \\ &+ \partial_{\mu} \xi^{(1) \lambda} \xi^{(1) \nu} \partial_{\nu} w_{\lambda} \\ &+ \frac{1}{2} \left(\left(\xi^{(2) \nu} \partial_{\nu} w_{\mu} + \xi^{(1) \sigma} \xi^{(1) \rho} \partial_{\sigma} \partial_{\rho} w_{\mu}\right)\right) \\ = & w_{\mu} +{\cal{L}}_{\xi^{(1)}} w_{\mu} + \frac{1}{2} {\cal{L}}_{\xi^{(2)}} w_{\mu} \\ &+ \frac{1}{2} \left({\cal{L}}_{\xi^{(1)}}^2 w_{\mu} - \xi^{(1) \sigma} \partial_{\sigma} \xi^{(1) \rho} \partial_{\rho} w_{\mu} \right)\\ &- w_{\rho} \partial_{\mu} \left(\left(\xi^{(1) \sigma} \partial_{\sigma} \xi^{(1) \rho}\right)\right) \\ =& w_{\mu} +{\cal{L}}_{\xi^{(1)}} w_{\mu} + \frac{1}{2} \left({\cal{L}}_{\xi^{(2)} - \xi^{(1) \sigma} \partial_{\sigma} \xi^{(1)}} +{\cal{L}}_{\xi^{(1)}}^2\right) w_{\mu} \\ =& w_{\mu} +{\cal{L}}_{\xi_1} w_{\mu} + \frac{1}{2} \left({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2\right) w_{\mu} . \end{aligned} \tag{A8}$

$ \begin{aligned}[b] A^{\mu} (x) \rightarrow & \frac{\partial x^{\mu}}{\partial \tilde{x}^{\nu}} A^{\nu} (\tilde{x}) \approx \left( \frac{\partial \tilde{x}^{\mu}}{\partial x^{\nu}} \right)^{- 1} A^{\nu} \left( x + \xi^{(1)} + \frac{1}{2} \xi^{(2)} \right) \\ \approx & A^{\mu} + \xi^{(1) \sigma} \partial_{\sigma} A^{\mu} - A^{\nu} \partial_{\nu} \xi^{(1) \mu} - \partial_{\nu} \xi^{(1) \mu} \xi^{(1) \sigma} \partial_{\sigma} A^{\nu} \\ &- \frac{1}{2} \left(\partial_{\nu} \xi^{(2) \mu} - 2 \partial_{\lambda} \xi^{(1) \mu} \partial_{\nu} \xi^{(1) \lambda}\right) A^{\nu} \\ &+ \frac{1}{2} \Biggr(\left(\xi^{(2) \sigma} \partial_{\sigma} A^{\mu} + \xi^{(1) \sigma} \xi^{(1) \rho} \partial_{\sigma} \partial_{\rho} A^{\mu}\right) \\ =& A^{\mu} +{\cal{L}}_{\xi^{(1)}} A^{\mu} + \frac{1}{2} \left({\cal{L}}_{\xi^{(2)} - \xi^{(1) \nu} \partial_{\nu} \xi^{(1)})} A^{\mu} +{\cal{L}}^2_{\xi^{(1)}} A^{\mu}\right)\Biggr) \\ =& A^{\mu} +{\cal{L}}_{\xi_1} A^{\mu} + \frac{1}{2} \left({\cal{L}}_{\xi_2} A^{\mu} +{\cal{L}}^2_{\xi_1} A^{\mu}\right), \end{aligned} \tag{A9} $

$ \begin{aligned}[b] \left( \frac{\partial \tilde{x}^{\mu}}{\partial x^{\nu}} \right)^{- 1} =& \delta^{\mu}_{\nu} - \partial_{\nu} \xi^{(1) \mu} - \frac{1}{2} \left(\partial_{\nu} \xi^{(2) \mu}\right.\\ &\left.- 2 \partial_{\lambda} \xi^{(1) \mu} \partial_{\nu} \xi^{(1) \lambda}\right) +{\cal{O}} \left(\xi^{(3)}\right) . \end{aligned}\tag{A10} $

$ \begin{aligned}[b] S_{\sigma \rho} (x) \rightarrow & \frac{\partial \tilde{x}^{\mu}}{\partial x^{\sigma}} \frac{\partial \tilde{x}^{\nu}}{\partial x^{\rho}} S_{\mu \nu} (\tilde{x}) \approx S_{\mu \nu} \left( x + \xi^{(1)} + \frac{1}{2} \xi^{(2)} \right) \partial_{\sigma} \Biggr( x^{\mu} \\ &+ \xi^{(1) \mu} + \frac{1}{2} \xi^{(2) \mu} \Biggr) \partial_{\rho} \left( x^{\nu} + \xi^{(1) \nu} + \frac{1}{2} \xi^{(2) \nu} \right) \\ \approx & S_{\sigma \rho} + \xi^{(1) \lambda} \partial_{\lambda} S_{\sigma \rho} + S_{\sigma \nu} \partial_{\rho} \xi^{(1) \nu} + S_{\mu \rho} \partial_{\sigma} \xi^{(1) \mu}\\ &+ \frac{1}{2} \left(\left(\xi^{(2) \lambda} \partial_{\lambda} S_{\sigma \rho} + S_{\mu \rho} \partial_{\sigma} \xi^{(2) \mu} + S_{\sigma \nu} \partial_{\rho} \xi^{(2) \nu} \right.\right.\\ &+ \xi^{(1) \lambda} \xi^{(1) \kappa} \partial_{\lambda} \partial_{\kappa} S_{\sigma \rho} + 2 S_{\mu \nu} \partial_{\sigma} \xi^{(1) \mu} \partial_{\rho} \xi^{(1) \nu} \\ &\left.+ 2 \xi^{(1) \lambda} \partial_{\lambda} S_{\sigma \nu} \partial_{\rho} \xi^{(1) \nu} + 2 \xi^{(1) \lambda} \partial_{\lambda} S_{\mu \rho} \partial_{\sigma} \xi^{(1) \mu}\right) \\ = &S_{\sigma \rho} +{\cal{L}}_{\xi^{(1)}} S_{\sigma \rho} + \frac{1}{2} \left({\cal{L}}_{\xi^{(2)} - \xi^{(1) \nu} \partial_{\nu} \xi^{(1)}} +{\cal{L}}_{\xi^{(1)}}^2\right) S_{\sigma \rho} \\ =& S_{\sigma \rho} +{\cal{L}}_{\xi_1} S_{\sigma \rho} + \frac{1}{2} \left({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2\right) S_{\sigma \rho} . \end{aligned} \tag{A11}$

$ \tilde{x}^\mu \equiv \varphi_\epsilon x^\mu = x^\mu + \frac{{\rm d}\varphi_\epsilon x^\mu}{{\rm d} \epsilon}\left|_{\epsilon = 0}\epsilon + \frac{1}{2}\frac{{\rm d}^2\varphi_\epsilon x^\mu}{{\rm d} \epsilon^2}\right|_{\epsilon = 0}\epsilon^2 + {\cal{O}}\left(\epsilon^3\right)\; , \tag{A12}$

$ \xi^{(1)}\equiv\frac{{\rm d}\varphi_\epsilon x^\mu}{{\rm d} \epsilon}\Biggr|_{\epsilon = 0}\epsilon\; , \tag{A13a}$

$\xi^{(2)}\equiv \frac{{\rm d}^2\varphi_\epsilon x^\mu}{{\rm d} \epsilon^2}\Biggr|_{\epsilon = 0}\epsilon^2\; .\tag{A13b} $

$ \begin{aligned}[b] {\cal{Q}} (x)^{i_1 i_2 \cdots}_{j_1 j_2 \cdots} \rightarrow & \left( \frac{\partial \tilde{x}^{l_1}}{\partial {x}^{j_1}} \frac{\partial \tilde{x}^{l_2}}{\partial {x}^{j_2}} \frac{\partial {x}^{i_1}}{\partial \tilde{x}^{k_1}} \frac{\partial {x}^{i_2}}{\partial \tilde{x}^{k_2}} \cdots \right) {\cal{Q}} (\tilde{x})^{k_1 k_2 \cdots}_{l_1 l_2 \cdots} \approx \left( \delta^{l_1}_{j_1} + \partial_{j_1} \xi^{(1) l_1} + \frac{1}{2} \partial_{j_1} \xi^{(2) l_1} \right) \left(\cdots\right) \Biggr({\cal{Q}} (x)^{k_1 k_2 \cdots}_{l_1 l_2 \cdots} + \xi^{(1) \nu} \partial_{\nu} {\cal{Q}} (x)^{k_1 k_2 \cdots}_{l_1 l_2 \cdots} \\ & + \frac{1}{2} \left((\xi^{(1) \sigma} \xi^{(2) \rho} \partial_{\sigma} \partial_{\rho} {\cal{Q}}(x)^{k_1 k_2 \cdots}_{l_1 l_2 \cdots} + \xi^{(2) \nu} \partial_{\nu} {\cal{Q}}(x)^{k_1 k_2 \cdots}_{l_1 l_2 \cdots}) \right) \Biggr) \approx {\cal{Q}} (x^{\mu})^{i_1 i_2 \cdots}_{j_1 j_2 \cdots} +{\cal{L}}_{\xi^{(1)}} {\cal{Q}} (x^{\mu})^{i_1 i_2 \cdots}_{j_1 j_2 \cdots} + \frac{1}{2} ({\cal{L}}_{\xi^{(2)} - \xi^{(1) \nu} \partial_{\nu} \xi^{(1)}} +{\cal{L}}_{\xi^{(1)}}^2) {\cal{Q}} (x^{\mu})^{i_1 i_2 \cdots}_{j_1 j_2 \cdots} \end{aligned} \tag{B1}$

$ {\cal{Q}} \rightarrow \tilde{{\cal{Q}}} = {\cal{Q}}+{\cal{L}}_{\xi_1} {\cal{Q}}+ \frac{1}{2} ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) {\cal{Q}}+{\cal{O}} (\xi^{(3)}) \tag{B2} $

$ {\cal{Q}} = {\cal{Q}}^{(0)} +{\cal{Q}}^{(1)} + \frac{1}{2} {\cal{Q}}^{(2)} +{\cal{O}} ({\cal{Q}}^{(3)}) . \tag{B3} $

$ \begin{aligned}[b] \tilde{{\cal{Q}}} =& {\cal{Q}}^{(0)} + \left({\cal{Q}}^{(1)} +{\cal{L}}_{\xi_1} {\cal{Q}}^{(0)}\right) \\ &+ \frac{1}{2} \left({\cal{Q}}^{(2)} + 2{\cal{L}}_{\xi_1} {\cal{Q}}^{(1)} + \left({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2\right){\cal{Q}}^{(0)}\right) +{\cal{O}} \left(\xi^{(3)}\right) . \end{aligned} \tag{B4}$

$ \tilde{{\cal{Q}}} = \tilde{{\cal{Q}}}^{(0)} + \tilde{{\cal{Q}}}^{(1)} + \frac{1}{2} \tilde{{\cal{Q}}}^{(2)} +{\cal{O}} (\tilde{{\cal{Q}}}^{(3)} ) . \tag{B5} $

$ g_{\mu \nu}^{(1)} = : g_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} + \mathbb{C}_{\mu \nu}^{(1)} . \tag{C1} $

$ \begin{aligned}[b] g^{(\mathrm{\rm{GI}, 1)}}_{\mu \nu} = \tilde{g}_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}}= \tilde{g}_{\mu \nu}^{(1)} - \tilde{\mathbb{C}}_{\mu \nu}^{(1)} = g_{\mu \nu}^{(1)} +{\cal{L}}_{\xi_1} g_{\mu \nu}^{(0)} - \tilde{\mathbb{C}}_{\mu \nu}^{(1)} . \end{aligned}\tag{C2} $

$ \tilde{\mathbb{C}}^{(1)}_{\mu \nu} = \mathbb{C}^{(1)}_{\mu \nu} +{\cal{L}}_{\xi_1} g_{\mu \nu}^{(0)} . \tag{C3} $

$ \tilde{X}^{\mu} - X^{\mu} = \xi_1^{\mu} . \tag{C4} $

$ g_{\mu \nu}^{(1)} = g_{\mu \nu}^{(\mathrm{\rm{GI}, 1)}} +{\cal{L}}_X g_{\mu \nu}^{(0)} . \tag{C5} $

$ g_{\mu \nu}^{(2)} = : g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}} + \mathbb{C}_{\mu \nu}^{(2)}, \tag{C6} $

$ \begin{aligned}[b] g_{\mu \nu}^{(\mathrm{\rm{GI}, 2)}}& = \tilde{g}_{\mu \nu}^{(\mathrm{\rm{GI},2)}} = \tilde{g}_{\mu \nu}^{(2)} - \tilde{\mathbb{C}}_{\mu \nu}^{(2)} \\ &= 2{\cal{L}}_{\xi_1} g_{\mu \nu}^{(1)} + ({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2) g_{\mu \nu}^{(0)} - \tilde{\mathbb{C}}_{\mu \nu}^{(2)} . \end{aligned} \tag{C7}$

$ \begin{aligned}[b] \tilde{\mathbb{C}}_{\mu \nu}^{(2)} - \mathbb{C}_{\mu \nu}^{(2)} =& 2{\cal{L}}_{\xi_1} g_{\mu \nu}^{(1)} + \left({\cal{L}}_{\xi_2} +{\cal{L}}_{\xi_1}^2\right) g_{\mu \nu}^{(0)} = 2{\cal{L}}_{\tilde{X}} \left(\tilde{g}_{\mu \nu}^{(1)} -{\cal{L}}_{\xi_1} g_{\mu \nu}^{(0)}\right) - 2{\cal{L}}_X g_{\mu \nu}^{(1)} +{\cal{L}}_{\xi_2} g_{\mu \nu}^{(0)} + \left({\cal{L}}_{\tilde{X}}^2 +{\cal{L}}_X^2 -{\cal{L}}_{\tilde{X}} {\cal{L}}_X -{\cal{L}}_X {\cal{L}}_{\tilde{X}}\right) g_{\mu \nu}^{(0)} \\ = &\left(2{\cal{L}}_{\tilde{X}} \tilde{g}_{\mu \nu}^{(1)} - {\cal{L}}^2_{\tilde{X}} g_{\mu \nu}^{(0)}\right) - \left(2{\cal{L}}_X g_{\mu \nu}^{(1)} -{\cal{L}}_X^2 g_{\mu \nu}^{(0)}\right) +{\cal{L}}_{\left(\xi_2 + [\tilde{X}, X]\right)} g_{\mu \nu}^{(0)} \equiv \left(2{\cal{L}}_{\tilde{X}} \tilde{g}_{\mu \nu}^{(1)} + \left({\cal{L}}_{\tilde{Y}} - {\cal{L}}^2_{\tilde{X}}\right) g_{\mu \nu}^{(0)}\right) \\&- \left(2{\cal{L}}_X g_{\mu \nu}^{(1)} + \left({\cal{L}}_Y -{\cal{L}}_X^2\right) g_{\mu \nu}^{(0)}\right), \end{aligned} \tag{C8}$

$ \tilde{Y}^{\mu} - Y^{\mu} = \xi_2^{\mu} + [\xi_1, X]^{\mu} . \tag{C9} $

$ g_{\mu \nu}^{(2)} = g^{(\mathrm{\rm{GI}, 2)}}_{\mu \nu} + 2{\cal{L}}_X g^{(1)}_{\mu \nu} + ({\cal{L}}_Y -{\cal{L}}_X^2) g_{\mu \nu}^{(0)} .\tag{C10} $

$ U_i = U^{(T)}_i + \partial_i U^{(S)}, \tag{D1} $

$ U_i^{(T)} \equiv {\cal{T}}^k_i U_k, \tag{D2a} $

$ U^{(S)} \equiv \Delta^{- 1} \partial^i U_i , \tag{D2b}$

$ \begin{aligned}[b] S_{ij} =& S^{(H)}_{ij} + 2 \delta_{ij} S^{(\Psi)} + 2 \partial_i \partial_j S^{(E)} \\&+ \partial_j S_i^{(C)} + \partial_i S_j^{(C)},\end{aligned} \tag{D3} $

$ S^{(H)}_{ij} \equiv \left( {\cal{T}}^k_i {\cal{T}}^l_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{kl} \right) S_{kl}, \tag{D4a} $

$ S_i^{(C)} \equiv \Delta^{- 1} \partial^l {\cal{T}}^k_i S_{lk} , \tag{D4b}$

$ S^{(\Psi)} \equiv \frac{1}{4} {\cal{T}}^{kl} S_{kl}, \tag{D4c}$

$ S^{(E)} \equiv \frac{1}{2} \Delta^{- 1} \left( \partial^k \Delta^{- 1} \partial^l - \frac{1}{2} {\cal{T}}^{kl} \right) S_{lk}. \tag{D4d}$

$ \begin{aligned}[b] {\cal{X}}_{\mu \nu} =& \frac{4}{a^2}{\cal{H}}X^0 {g}^{(1)}_{\mu \nu} + 2 X^{\sigma} \partial_{\sigma} \left(\frac{1}{a^2}{g}_{\mu \nu}^{(1)}\right) + \frac{2}{a^2} {g}_{\sigma \nu}^{(1)} \partial_{\mu} X^{\sigma} \\ &+ \frac{2}{a^2} {g}_{\mu \sigma}^{(1)} \partial_{\nu} X^{\sigma}- \eta_{\mu \nu} \Big(\Big(4{\cal{H}}^2 + 2 \dot{{\cal{H}}}\Big) \Big(X^0\Big)^2 \\ &+ 2{\cal{H}}X^{\mu} \partial_{\mu} X^0\Big) - 2 \eta_{\sigma \rho} \partial_{\mu} X^{\sigma} \partial_{\nu} X^{\rho} - \eta_{\mu \sigma} \Big(X^{\rho} \partial_{\rho} \partial_{\nu} X^{\sigma} \\ &+ \partial_{\rho} X^{\sigma} \partial_{\nu} X^{\rho} + 4{\cal{H}}X^0 \partial_{\nu} X^{\sigma}\Big) - \eta_{\sigma \nu} \Big(X^{\rho} \partial_{\rho} \partial_{\mu} X^{\sigma} \\ &+ \partial_{\rho} X^{\sigma} \partial_{\mu} X^{\rho} + 4{\cal{H}}X^0 \partial_{\mu} X^{\sigma}\Big) \end{aligned}\tag{E1} $

$ \begin{aligned}[b] \mathcal{X}_{00}^N = & - 2 \left( \frac{2}{a} (\partial_0 (a (\partial_0 {e}^{(1)} - b^{(1)}))) + (\partial_0 {e}^{(1)} - b^{(1)}) \partial_0 + \delta^{ik} (c_k^{(1)} + \partial_k {e}^{(1)}) \partial_i \right) \Biggr( 2 \phi^{(1)} - \frac{1}{a} \partial_0 \left(a (\partial_0 {e}^{(1)} - b^{(1)})\right) \Biggr) \\ & + 2 \delta^{ik} (\partial_0 \left(c_k^{(1)} + \partial_k {e}^{(1)}) \right) \left(\partial_i b^{(1)} + 2 \nu_i^{(1)} - \partial_0 c_i^{(1)}\right), \end{aligned} \tag{E2a}$

$ \begin{aligned}[b] \mathcal{X}_{0 j}^N = & \Biggr( \delta^k_j \Biggr( \frac{2 \dot{a}}{a} \left(\partial_0 {e}^{(1)} - b^{(1)}\right) + \left(\partial_0 {e}^{(1)} - b^{(1)}\right) \partial_0 + \delta^{ik} \left(c_k^{(1)} + \partial_k {e}^{(1)}\right) \partial_i + \left(\partial_0 \left(\partial_0 {e}^{(1)} - b^{(1)}\right)\right) \Biggr) \\ &+ \delta^{ks} \left(\partial_j \left(c_s^{(1)} + \partial_s {e}^{(1)}\right)\right) \Biggr) \left(\partial_k b^{(1)} + 2 \nu_k^{(1)} - \partial_0 c_k^{(1)}\right) - 2 \left( \partial_j \left(\partial_0 {e}^{(1)} - b^{(1)}\right) \right) \Biggr( 2 \phi^{(1)} - \frac{1}{a} \partial_0 \left(a \left(\partial_0 {e}^{(1)} - b^{(1)}\right)\right) \Biggr) \\ & + \delta^{kl} \left(\partial_0 \left(c_l^{(1)} + \partial_l {e}^{(1)}\right)\right) \Biggr( - 2 \delta_{jk} \Biggr( 2 \psi^{(1)} + \frac{\dot{a}}{a} (\partial_0 {e}^{(1)} - b^{(1)}) \Biggr) + 2 \partial_j \partial_k {e}^{(1)} + \partial_j c_k^{(1)} + \partial_k c_j^{(1)} + 2h_{j k}^{(1)} \Biggr), \end{aligned} \tag{E2b}$

$ \begin{aligned}[b] \mathcal{X}_{kl}^N = & \Biggr(\delta^s_k \delta^t_l \Biggr( \frac{2 \dot{a}}{a} \left(\partial_0 {e}^{(1)} - b^{(1)}\right) + \left(\partial_0 {e}^{(1)} - b^{(1)}\right) \partial_0 + \delta^{ik} \left(c_k^{(1)} + \partial_k {e}^{(1)}\right) \partial_i \Biggr) \\ &+ \left(\left(\delta^s_k \delta^{tw} \partial_l + \delta^t_l \delta^{sw} \partial_k\right) \left(c_w^{(1)} + \partial_w {e}^{(1)}\right)\right)\Biggr) \Biggr( - 2 \delta_{st} \Biggr( 2 \psi^{(1)} + \frac{\dot{a}}{a} \left(\partial_0 {e}^{(1)} - b^{(1)}\right) \Biggr) + 2 \partial_s \partial_t {e}^{(1)} \\ &+ \partial_s c_t^{(1)} + \partial_t c_s^{(1)} + 2h_{s t}^{(1)} \Biggr) + \left(\left(\delta^s_k \partial_l + \delta^s_l \partial_k) (\partial_0 {e}^{(1)} - b^{(1)}\right)\right) \left(\partial_s b^{(1)} + 2 \nu_s^{(1)} - \partial_0 c_s^{(1)}\right) . \end{aligned} \tag{E2c}$

$ \begin{aligned}[b] {\cal{X}}_{00}^S =& - 2 \left( 2 \phi^{(1)} + \frac{1}{a} \int \mathrm{d} \eta \left\{a \phi^{(1)} \right\} \partial_0 + \delta^{ij} \int \mathrm{d} \eta \left\{ \nu_i^{(1)} + \partial_i b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta'\left\{a \partial_i \phi^{(1)} \right\} \right\} \partial_j \right) \phi^{(1)} \\ & + 2 \delta^{ij} \left( \nu_j^{(1)} + \partial_j b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta \left\{a \partial_j \phi^{(1)} \right\} \right) \left(\partial_i b^{(1)} + \nu_i^{(1)}\right), \end{aligned}\tag{E3a} $

$ \begin{aligned}[b] {\cal{X}}_{0 j}^S = & \Biggr( \delta^k_j\phi^{(1)} + \frac{\dot{a}}{a^2} \delta^k_j \int \mathrm{d} \eta\left\{a \phi^{(1)} \right\}+ \frac{1}{a} \delta^k_j \int \mathrm{d} \eta \left\{a \phi^{(1)} \right\} \partial_0 + \delta^{il}\delta^k_j \int \mathrm{d} \eta \Biggr\{ \nu_i^{(1)} + \partial_i b^{(1)} \\ & + \frac{1}{a}\int \mathrm{d} \eta' \left\{a \partial_i \phi^{(1)} \right\} \Biggr\} \partial_l + \delta^{lk} \left(\partial_j \int \mathrm{d} \eta \left\{ \nu_l^{(1)} + \partial_l b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta' \left\{a \partial_l \phi^{(1)} \right\} \right\}\right) \Biggr) \left(\partial_k b^{(1)} + \nu_k^{(1)}\right) \\ &+ \frac{2}{a} \phi^{(1)} \int \mathrm{d} \eta \left\{a \partial_j \phi^{(1)} \right\} + \delta^{ik} \left( \nu_i^{(1)} + \partial_i b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta \{a \partial_i \phi^{(1)} \} \right) \Biggr( - 4 \psi^{(1)} \delta_{jk} + 4 \partial_j \partial_k {e}^{(1)} \\ &+ 2 \partial_j c_k^{(1)} + 2 \partial_k c_j^{(1)} + 2 h_{j k}^{(1)} - \frac{2 \dot{a}}{a^2} \delta_{kj} \int \mathrm{d} \eta \left\{a \phi^{(1)} \right\} \\ &+ \left(\delta_{j}^s \partial_k + \delta_{k}^s \partial_j\right) \int \mathrm{d} \eta \left\{ \nu_s^{(1)} + \partial_s b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta' \{a \partial_s \phi^{(1)} \} \right\} \Biggr), \end{aligned}\tag{E3b} $

$ \begin{aligned}[b] {\cal{X}}_{kl}^S =& \Biggr( \delta^s_k \delta^t_l \Biggr( \frac{2 \dot{a}}{a^2} \int \mathrm{d} \eta \{a \phi^{(1)} \}+ \frac{1}{a} \int \mathrm{d} \eta \{a \phi^{(1)} \} \partial_0 + \delta^{ij} \int \mathrm{d} \eta \Biggr\{ \nu_i^{(1)} + \partial_i b^{(1)} \frac{1}{a} \int \mathrm{d} \eta' \{a \partial_i \phi^{(1)} \} \Biggr\} \partial_j \Biggr) \\ &+ \left( \left(\delta^s_k \delta^{ti} \partial_l + \delta^t_l \delta^{s i} \partial_k\right) \int \mathrm{d} \eta \left\{ \nu_i^{(1)} + \partial_i b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta' \{a \partial_i \phi^{(1)} \} \right\} \right) \Bigg) \Biggr( - 4 \psi^{(1)} \delta_{st} + 4 \partial_s \partial_t {e}^{(1)} + 2 \partial_s c_t^{(1)} + 2 \partial_t c_s^{(1)} + 2 h_{s t}^{(1)} \\ &- \left(\delta_{s}^n \partial_t + \delta_{t}^n \partial_s\right) \int \mathrm{d} \eta \left\{ \nu_n^{(1)} + \partial_n b^{(1)} + \frac{1}{a} \int \mathrm{d} \eta' \{a \partial_n \phi^{(1)} \} \right\} - \frac{2 \dot{a}}{a^2} \delta_{st} \int \mathrm{d} \eta \{a \phi^{(1)} \} \Biggr) \\ &+ \frac{1}{a} \left(\partial_s b^{(1)} + \nu_s^{(1)}\right) \int \mathrm{d} \eta \Biggr\{a (\delta^s_k \partial_l + \delta^s_l \partial_k) \phi^{(1)} \Biggr\} . \end{aligned} \tag{E3c}$

$ \begin{aligned}[b] \Lambda^{lm}_{ij} = &{\cal{T}}^l_i {\cal{T}}^m_j - \frac{1}{2} {\cal{T}}_{ij} {\cal{T}}^{lm} \\ = &\left(\delta^l_i - \partial^l \Delta^{- 1} \partial_i\right) \left(\delta^m_j - \partial^m \Delta^{- 1} \partial_j\right)\\ &- \frac{1}{2} \left(\delta_{ij} - \partial_i \Delta^{- 1} \partial_j\right) \left(\delta^{lm} - \partial^l \Delta^{- 1} \partial^m\right) . \end{aligned} \tag{F1}$

$ \begin{aligned}[b] \Lambda^{lm}_{ij} =& \left( \delta^l_i - \frac{k^l k_i}{\left| {\boldsymbol{k}} \right|^2} \right) \left( \delta^m_j - \frac{k^m k_j}{\left| {\boldsymbol{k}} \right|^2} \right) - \frac{1}{2} \left( \delta_{ij} - \frac{k_i k_j}{\left| {\boldsymbol{k}} \right|^2} \right) \left( \delta^{lm} - \frac{k^l k^m}{\left| {\boldsymbol{k}} \right|^2} \right) \\ = &\left(\delta^l_i - n^l n_i\right) \left(\delta^m_j - n^m n_j\right)- \frac{1}{2} \left(\delta_{ij} - n_i n_j\right) \left(\delta^{lm} - n^l n^m\right), \end{aligned} \tag{F2}$

$ {\epsilon}_i {\epsilon}^i = \bar{\epsilon}_i \bar{\epsilon}^i = n_i n^i = 1, \tag{F3a}$

$ {\epsilon}_i \bar{\epsilon}^i = {\epsilon}_i n^i = \bar{\epsilon}_i n^i = 0. \tag{F3b} $

$ \delta^j_i = {\epsilon}_i {\epsilon}^j + \bar{\epsilon}_i \bar{\epsilon}^j + n_i n^j . \tag{F4} $

$ \begin{aligned}[b] \Lambda^{lm}_{ij} =& \left({\epsilon}_i {\epsilon}^l + \bar{\epsilon}_i \bar{\epsilon}^l\right) \left({\epsilon}_j {\epsilon}^m + \bar{\epsilon}_j \bar{\epsilon}^m\right) - \frac{1}{2} \left({\epsilon}_i {\epsilon}_j + \bar{\epsilon}_i \bar{\epsilon}_j\right) \left({\epsilon}^l {\epsilon}^m + \bar{\epsilon}^l \bar{\epsilon}^m\right) \\ =& {\epsilon}_i {\epsilon}_j {\epsilon}^l {\epsilon}^m + {\epsilon}_i \bar{\epsilon}_j {\epsilon}^l \bar{\epsilon}^m + \bar{\epsilon}_i {\epsilon}_j \bar{\epsilon}^l {\epsilon}^m + \bar{\epsilon}_i \bar{\epsilon}_j \bar{\epsilon}^l \bar{\epsilon}^m \\ &- \frac{1}{2} \left({\epsilon}_i {\epsilon}_j {\epsilon}^l {\epsilon}^m + {\epsilon}_i {\epsilon}_j \bar{\epsilon}^l \bar{\epsilon}^m + \bar{\epsilon}_i \bar{\epsilon}_j {\epsilon}^l {\epsilon}^m + \bar{\epsilon}_i \bar{\epsilon}_j \bar{\epsilon}^l \bar{\epsilon}^m\right) \\ =& \frac{1}{2} ({\epsilon}_i \bar{\epsilon}_j {\epsilon}^l \bar{\epsilon}^m + \bar{\epsilon}_i {\epsilon}_j {\epsilon}^l \bar{\epsilon}^m + \bar{\epsilon}_i {\epsilon}_j \bar{\epsilon}^l {\epsilon}^m + {\epsilon}_i \bar{\epsilon}_j \bar{\epsilon}^l {\epsilon}^m) \\ &+ \frac{1}{2} ({\epsilon}_i {\epsilon}_j {\epsilon}^l {\epsilon}^m - {\epsilon}_i {\epsilon}_j \bar{\epsilon}^l \bar{\epsilon}^m - \bar{\epsilon}_i \bar{\epsilon}_j {\epsilon}^l {\epsilon}^m - \bar{\epsilon}_i \bar{\epsilon}_j \bar{\epsilon}^l \bar{\epsilon}^m) \\ =& \frac{1}{\sqrt{2}} ({\epsilon}_i \bar{\epsilon}_j + \bar{\epsilon}_i {\epsilon}_j) \frac{1}{\sqrt{2}} ({\epsilon}^l \bar{\epsilon}^m + \bar{\epsilon}^l {\epsilon}^m) \\ &+ \frac{1}{\sqrt{2}} \left({\epsilon}_i {\epsilon}_j - \bar{\epsilon}_i \bar{\epsilon}_j\right) \frac{1}{\sqrt{2}} \left({\epsilon}^l {\epsilon}^m - \bar{\epsilon}^l \bar{\epsilon}^m\right) =: {\epsilon}^\times_{ij} {\epsilon}^{\times, lm} + {\epsilon}^+_{ij} {\epsilon}^{+, lm}, \end{aligned}\tag{F5} $

$ {\epsilon}^\times_{ij} \equiv \frac{1}{\sqrt{2}} ({\epsilon}_i \bar{\epsilon}_j + \bar{\epsilon}_i {\epsilon}_j), \tag{F6a} $

$ {\epsilon}^+_{ij} \equiv \frac{1}{\sqrt{2}} ({\epsilon}_i {\epsilon}_j - \bar{\epsilon}_i \bar{\epsilon}_j). \tag{F6b} $