Linearized gravity and perturbation theory

Question

This is a question regarding a calculation in perturbative GR. We have :

$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$

where $h_{\mu\nu}$ is a small perturbation around the flat spacetime metric. In linearized theory, we ignore terms which grow as $h^2$ and higher.

Can you point me to a reference which provides the Curvature tensor, Ricci tensor and Ricci scalar to cubic powers in h, and not just the linear term?

Answer 1

Expansions around an arbitrary background. You can later put background curvatures to zero to get the expansion around flat space.

Ricci scalar up to $\mathcal{O}(h^3)$:

$$R + \epsilon (- h^{\alpha \beta } R_{\alpha \beta } + \nabla_{\beta }\nabla_{\alpha }h^{\alpha \beta } - \nabla_{\beta }\nabla^{\beta }h^{\alpha }{}_{\alpha }) + \epsilon^2 (2 h_{\alpha }{}^{\gamma } h^{\alpha \beta } R_{\beta \gamma } - h^{\alpha \beta } h^{\gamma \lambda } R_{\alpha \gamma \beta \lambda } + h^{\alpha \beta } \nabla_{\beta }\nabla_{\alpha }h^{\gamma }{}_{\gamma } - \tfrac{1}{4} \nabla_{\beta }h^{\gamma }{}_{\gamma } \nabla^{\beta }h^{\alpha }{}_{\alpha } - \nabla_{\alpha }h^{\alpha \beta } \nabla_{\gamma }h_{\beta }{}^{\gamma } + \nabla^{\beta }h^{\alpha }{}_{\alpha } \nabla_{\gamma }h_{\beta }{}^{\gamma } - 2 h^{\alpha \beta } \nabla_{\gamma }\nabla_{\beta }h_{\alpha }{}^{\gamma } + h^{\alpha \beta } \nabla_{\gamma }\nabla^{\gamma }h_{\alpha \beta } - \tfrac{1}{2} \nabla_{\beta }h_{\alpha \gamma } \nabla^{\gamma }h^{\alpha \beta } + \tfrac{3}{4} \nabla_{\gamma }h_{\alpha \beta } \nabla^{\gamma }h^{\alpha \beta }) + \epsilon^3 (-2 h_{\alpha }{}^{\gamma } h^{\alpha \beta } h_{\beta }{}^{\lambda } R_{\gamma \lambda } + h_{\alpha }{}^{\gamma } h^{\alpha \beta } h^{\lambda \sigma} R_{\beta \lambda \gamma \sigma} - \tfrac{3}{4} h^{\alpha \beta } \nabla_{\alpha }h^{\gamma \lambda } \nabla_{\beta }h_{\gamma \lambda } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h^{\gamma }{}_{\gamma } \nabla_{\beta }h^{\lambda }{}_{\lambda } - h^{\alpha \beta } \nabla_{\beta }h^{\lambda }{}_{\lambda } \nabla_{\gamma }h_{\alpha }{}^{\gamma } - h^{\alpha \beta } \nabla_{\beta }h_{\alpha }{}^{\gamma } \nabla_{\gamma }h^{\lambda }{}_{\lambda } - h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\gamma }\nabla_{\beta }h^{\lambda }{}_{\lambda } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h^{\lambda }{}_{\lambda } \nabla^{\gamma }h_{\alpha \beta } + h^{\alpha \beta } \nabla_{\gamma }h_{\alpha }{}^{\gamma } \nabla_{\lambda }h_{\beta }{}^{\lambda } + 2 h^{\alpha \beta } \nabla_{\beta }h_{\alpha }{}^{\gamma } \nabla_{\lambda }h_{\gamma }{}^{\lambda } - h^{\alpha \beta } \nabla^{\gamma }h_{\alpha \beta } \nabla_{\lambda }h_{\gamma }{}^{\lambda } + h^{\alpha \beta } h^{\gamma \lambda } \nabla_{\lambda }\nabla_{\beta }h_{\alpha \gamma } - h^{\alpha \beta } h^{\gamma \lambda } \nabla_{\lambda }\nabla_{\gamma }h_{\alpha \beta } + 2 h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\lambda }\nabla_{\gamma }h_{\beta }{}^{\lambda } - h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\lambda }\nabla^{\lambda }h_{\beta \gamma } + h^{\alpha \beta } \nabla_{\beta }h_{\gamma \lambda } \nabla^{\lambda }h_{\alpha }{}^{\gamma } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\beta \lambda } \nabla^{\lambda }h_{\alpha }{}^{\gamma } - \tfrac{3}{2} h^{\alpha \beta } \nabla_{\lambda }h_{\beta \gamma } \nabla^{\lambda }h_{\alpha }{}^{\gamma })$$

Ricci Tensor up to $\mathcal{O}(h^3)$:

$$ R_{\mu \nu } + \epsilon (\tfrac{1}{2} h_{\nu }{}^{\alpha } R_{\mu \alpha } + \tfrac{1}{2} h_{\mu }{}^{\alpha } R_{\nu \alpha } - h^{\alpha \beta } R_{\mu \alpha \nu \beta } - \tfrac{1}{2} \nabla_{\alpha }\nabla^{\alpha }h_{\mu \nu } + \tfrac{1}{2} \nabla_{\mu }\nabla_{\alpha }h_{\nu }{}^{\alpha } + \tfrac{1}{2} \nabla_{\nu }\nabla_{\alpha }h_{\mu }{}^{\alpha } - \tfrac{1}{2} \nabla_{\nu }\nabla_{\mu }h^{\alpha }{}_{\alpha })+ \epsilon^2 (- \tfrac{1}{2} h^{\beta \gamma } h_{\nu }{}^{\alpha } R_{\mu \beta \alpha \gamma } + h_{\alpha }{}^{\gamma } h^{\alpha \beta } R_{\mu \beta \nu \gamma } - \tfrac{1}{2} h^{\beta \gamma } h_{\mu }{}^{\alpha } R_{\nu \beta \alpha \gamma } - \tfrac{1}{4} \nabla_{\alpha }h^{\beta }{}_{\beta } \nabla^{\alpha }h_{\mu \nu } + \tfrac{1}{2} \nabla^{\alpha }h_{\mu \nu } \nabla_{\beta }h_{\alpha }{}^{\beta } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }\nabla_{\alpha }h_{\mu \nu } - \tfrac{1}{2} \nabla_{\alpha }h_{\nu \beta } \nabla^{\beta }h_{\mu }{}^{\alpha } + \tfrac{1}{2} \nabla_{\beta }h_{\nu \alpha } \nabla^{\beta }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla_{\alpha }h^{\beta }{}_{\beta } \nabla_{\mu }h_{\nu }{}^{\alpha } - \tfrac{1}{2} \nabla_{\beta }h_{\alpha }{}^{\beta } \nabla_{\mu }h_{\nu }{}^{\alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\mu }\nabla_{\beta }h_{\nu \alpha } + \tfrac{1}{4} \nabla_{\mu }h^{\alpha \beta } \nabla_{\nu }h_{\alpha \beta } + \tfrac{1}{4} \nabla_{\alpha }h^{\beta }{}_{\beta } \nabla_{\nu }h_{\mu }{}^{\alpha } - \tfrac{1}{2} \nabla_{\beta }h_{\alpha }{}^{\beta } \nabla_{\nu }h_{\mu }{}^{\alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\nu }\nabla_{\beta }h_{\mu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\nu }\nabla_{\mu }h_{\alpha \beta }) + \epsilon^3 (\tfrac{1}{2} h_{\beta }{}^{\lambda } h^{\beta \gamma } h_{\nu }{}^{\alpha } R_{\mu \gamma \alpha \lambda } - h_{\alpha }{}^{\gamma } h^{\alpha \beta } h_{\beta }{}^{\lambda } R_{\mu \gamma \nu \lambda } + \tfrac{1}{2} h_{\beta }{}^{\lambda } h^{\beta \gamma } h_{\mu }{}^{\alpha } R_{\nu \gamma \alpha \lambda } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \nu } \nabla_{\beta }h^{\gamma }{}_{\gamma } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\alpha }h_{\mu }{}^{\gamma } \nabla_{\beta }h_{\nu \gamma } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \nu } \nabla_{\gamma }h_{\beta }{}^{\gamma } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\alpha }h_{\mu }{}^{\gamma } \nabla_{\gamma }h_{\nu \beta } - \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\gamma }\nabla_{\beta }h_{\mu \nu } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\nu \gamma } \nabla^{\gamma }h_{\mu \alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\nu \beta } \nabla^{\gamma }h_{\mu \alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\alpha \gamma } \nabla^{\gamma }h_{\mu \nu } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\gamma }h_{\alpha \beta } \nabla^{\gamma }h_{\mu \nu } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h^{\gamma }{}_{\gamma } \nabla_{\mu }h_{\nu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\beta }{}^{\gamma } \nabla_{\mu }h_{\nu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\alpha \gamma } \nabla_{\mu }h_{\nu }{}^{\gamma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\gamma }h_{\alpha \beta } \nabla_{\mu }h_{\nu }{}^{\gamma } + \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\mu }\nabla_{\gamma }h_{\nu \beta } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\mu }h_{\alpha }{}^{\gamma } \nabla_{\nu }h_{\beta \gamma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h^{\gamma }{}_{\gamma } \nabla_{\nu }h_{\mu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\beta }{}^{\gamma } \nabla_{\nu }h_{\mu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\alpha \gamma } \nabla_{\nu }h_{\mu }{}^{\gamma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\gamma }h_{\alpha \beta } \nabla_{\nu }h_{\mu }{}^{\gamma } + \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\nu }\nabla_{\gamma }h_{\mu \beta } - \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\nu }\nabla_{\mu }h_{\beta \gamma }) $$

Riemann tensor up to $\mathcal{O}(h^3)$:

$$ R_{\mu \nu \rho \sigma } + \epsilon (\tfrac{1}{2} h_{\sigma }{}^{\alpha } R_{\mu \alpha \nu \rho } - \tfrac{1}{2} h_{\rho }{}^{\alpha } R_{\mu \alpha \nu \sigma } + \tfrac{1}{2} h_{\sigma }{}^{\alpha } R_{\mu \nu \rho \alpha } - \tfrac{1}{2} h_{\rho }{}^{\alpha } R_{\mu \nu \sigma \alpha } - \tfrac{1}{2} h_{\sigma }{}^{\alpha } R_{\mu \rho \nu \alpha } - \tfrac{1}{2} h_{\nu }{}^{\alpha } R_{\mu \rho \sigma \alpha } + \tfrac{1}{2} h_{\rho }{}^{\alpha } R_{\mu \sigma \nu \alpha } + \tfrac{1}{2} h_{\nu }{}^{\alpha } R_{\mu \sigma \rho \alpha } + \tfrac{1}{2} h_{\mu }{}^{\alpha } R_{\nu \rho \sigma \alpha } - \tfrac{1}{2} h_{\mu }{}^{\alpha } R_{\nu \sigma \rho \alpha } - \tfrac{1}{2} \nabla_{\rho }\nabla_{\mu }h_{\nu \sigma } + \tfrac{1}{2} \nabla_{\rho }\nabla_{\nu }h_{\mu \sigma } + \tfrac{1}{2} \nabla_{\sigma }\nabla_{\mu }h_{\nu \rho } - \tfrac{1}{2} \nabla_{\sigma }\nabla_{\nu }h_{\mu \rho }) + \epsilon^2 (- \tfrac{1}{4} \nabla_{\alpha }h_{\nu \sigma } \nabla^{\alpha }h_{\mu \rho } + \tfrac{1}{4} \nabla_{\alpha }h_{\nu \rho } \nabla^{\alpha }h_{\mu \sigma } + \tfrac{1}{4} \nabla^{\alpha }h_{\nu \sigma } \nabla_{\mu }h_{\rho \alpha } - \tfrac{1}{4} \nabla^{\alpha }h_{\nu \rho } \nabla_{\mu }h_{\sigma \alpha } - \tfrac{1}{4} \nabla^{\alpha }h_{\mu \sigma } \nabla_{\nu }h_{\rho \alpha } + \tfrac{1}{4} \nabla_{\mu }h_{\sigma \alpha } \nabla_{\nu }h_{\rho }{}^{\alpha } + \tfrac{1}{4} \nabla^{\alpha }h_{\mu \rho } \nabla_{\nu }h_{\sigma \alpha } - \tfrac{1}{4} \nabla_{\mu }h_{\rho }{}^{\alpha } \nabla_{\nu }h_{\sigma \alpha } + \tfrac{1}{4} \nabla_{\alpha }h_{\nu \sigma } \nabla_{\rho }h_{\mu }{}^{\alpha } - \tfrac{1}{4} \nabla_{\nu }h_{\sigma \alpha } \nabla_{\rho }h_{\mu }{}^{\alpha } - \tfrac{1}{4} \nabla^{\alpha }h_{\mu \sigma } \nabla_{\rho }h_{\nu \alpha } + \tfrac{1}{4} \nabla_{\mu }h_{\sigma \alpha } \nabla_{\rho }h_{\nu }{}^{\alpha } - \tfrac{1}{4} \nabla_{\alpha }h_{\nu \rho } \nabla_{\sigma }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla_{\nu }h_{\rho \alpha } \nabla_{\sigma }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla_{\rho }h_{\nu \alpha } \nabla_{\sigma }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla^{\alpha }h_{\mu \rho } \nabla_{\sigma }h_{\nu \alpha } - \tfrac{1}{4} \nabla_{\rho }h_{\mu }{}^{\alpha } \nabla_{\sigma }h_{\nu \alpha } - \tfrac{1}{4} \nabla_{\mu }h_{\rho \alpha } \nabla_{\sigma }h_{\nu }{}^{\alpha }) + \epsilon^3 (- \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \sigma } \nabla_{\beta }h_{\nu \rho } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \rho } \nabla_{\beta }h_{\nu \sigma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\nu \sigma } \nabla_{\mu }h_{\rho \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\nu \rho } \nabla_{\mu }h_{\sigma \beta } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\sigma \beta } \nabla_{\nu }h_{\rho \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \sigma } \nabla_{\nu }h_{\rho \beta } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \rho } \nabla_{\nu }h_{\sigma \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\rho \alpha } \nabla_{\nu }h_{\sigma \beta } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h_{\nu \sigma } \nabla_{\rho }h_{\mu \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\nu }h_{\sigma \beta } \nabla_{\rho }h_{\mu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\sigma \beta } \nabla_{\rho }h_{\nu \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \sigma } \nabla_{\rho }h_{\nu \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h_{\nu \rho } \nabla_{\sigma }h_{\mu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\nu }h_{\rho \beta } \nabla_{\sigma }h_{\mu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\rho }h_{\nu \beta } \nabla_{\sigma }h_{\mu \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\rho \beta } \nabla_{\sigma }h_{\nu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \rho } \nabla_{\sigma }h_{\nu \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\rho }h_{\mu \alpha } \nabla_{\sigma }h_{\nu \beta }) $$

Computations performed using xAct. You might also find perturbative expansions of curvatures in some PhD theses, though I don't remember coming across one that has them up to the cubic level.