Expansion through second order in q-η

Displayed formulas in this passage

• X-E@1#eq:G2 · line 622

  $$\begin{array}{rl}{R^{(1)}}_{\mu \nu }[h]& ={\partial }_{\rho }{\Gamma }_1{}_{\mu \nu }^{\rho }-{\partial }_{\nu }{\Gamma }_1{}_{\rho \mu }^{\rho }=\frac{1}{2}({\partial }_{\rho }{\partial }_{\mu }{h}^{\rho }{}_{\nu }+{\partial }_{\rho }{\partial }_{\nu }{h}^{\rho }{}_{\mu }-\square h_{\mu \nu }-{\partial }_{\mu }{\partial }_{\nu }H),\\ G_{\mathrm{lin}}{}_{\mu \nu }[h]& ={R^{(1)}}_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{\eta }^{\alpha \beta }{R^{(1)}}_{\alpha \beta },\\ {R^{(2)}}_{\mu \nu }[h,h]& ={\partial }_{\rho }{\Gamma }_2{}_{\mu \nu }^{\rho }-{\partial }_{\nu }{\Gamma }_2{}_{\rho \mu }^{\rho }+{\Gamma }_1{}_{\rho \lambda }^{\rho }{\Gamma }_1{}_{\mu \nu }^{\lambda }-{\Gamma }_1{}_{\nu \lambda }^{\rho }{\Gamma }_1{}_{\rho \mu }^{\lambda },\\ G_{2,\mu \nu }[h,h]& ={R^{(2)}}_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{\eta }^{\alpha \beta }{R^{(2)}}_{\alpha \beta }-\frac{1}{2}(h_{\mu \nu }{\eta }^{\alpha \beta }-{\eta }_{\mu \nu }{h}^{\alpha \beta }){R^{(1)}}_{\alpha \beta },\end{array}$$

• X-E@1#eq:G-expansion · line 644

  $$\sqrt{-q}{G}^{\mu \nu }[q]=\kappa {G_{\mathrm{lin}}}^{\mu \nu }[h]+{\kappa }^2({G_{\mathrm{lin}}}^{\mu \nu }[k]+{{\mathcal{G}}_2}^{\mu \nu }[h,h])+O({\kappa }^3),$$

• unlabeled · line 615

  $${\Gamma }_1{}_{\mu \nu }^{\rho }[h]=\frac{1}{2}{\eta }^{\rho \sigma }({\partial }_{\mu }{h}_{\sigma \nu }+{\partial }_{\nu }{h}_{\sigma \mu }-{\partial }_{\sigma }{h}_{\mu \nu }),{\Gamma }_2{}_{\mu \nu }^{\rho }[h,h]=-h^{\rho }{}_{\sigma }{\Gamma }_1{}_{\mu \nu }^{\sigma }[h].$$

• unlabeled · line 651

  $${{\mathcal{G}}_2}^{\mu \nu }[h,h]={\eta }^{\mu \alpha }{\eta }^{\nu \beta }{G}_{2,\alpha \beta }[h,h]+b_1[h]{G_{\mathrm{lin}}}^{\mu \nu }[h]-h^{\mu }{}_{\alpha }{G_{\mathrm{lin}}}^{\alpha \nu }[h]-h^{\nu }{}_{\alpha }{G_{\mathrm{lin}}}^{\mu \alpha }[h].$$