Conventions and equations

Displayed formulas in this passage

• Em02-E@1#eq:action · line 253

  $$S[g,\varphi ]={\int }_M\left[\frac{{M_{\mathrm{Pl}}}^2}{2}R(g)-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V(\varphi )\right]\mathrm{d}{\mathrm{vol}}_g,M_{\mathrm{Pl}}>0.$$

• Em02-E@1#eq:Einstein · line 275

  $$\begin{array}{rl}{M_{\mathrm{Pl}}}^2{G}_{\mu \nu }& ={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -g_{\mu \nu }\left(\frac{1}{2}{g}^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi +V\right),\\ {\square }_g\varphi -V^{\prime }(\varphi )& =0.\end{array}$$

• Em02-E@1#eq:KG · line 275

  $$\begin{array}{rl}{M_{\mathrm{Pl}}}^2{G}_{\mu \nu }& ={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -g_{\mu \nu }\left(\frac{1}{2}{g}^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi +V\right),\\ {\square }_g\varphi -V^{\prime }(\varphi )& =0.\end{array}$$

• unlabeled · line 266

  $$R^{\rho }{}_{\sigma \mu \nu }={\partial }_{\mu }{{\Gamma }^{\rho }}_{\nu \sigma }-{\partial }_{\nu }{{\Gamma }^{\rho }}_{\mu \sigma }+{{\Gamma }^{\rho }}_{\mu \alpha }{{\Gamma }^{\alpha }}_{\nu \sigma }-{{\Gamma }^{\rho }}_{\nu \alpha }{{\Gamma }^{\alpha }}_{\mu \sigma }.$$