Background, action, and clock calibration

Displayed formulas in this passage

• XX-D@1#eq:action · line 239

  $$S[g,\varphi ]=\int \sqrt{-g}\left(\frac{M^2}{2}R[g]-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V(\varphi )\right){\mathrm{d}}^{4}x.$$

• XX-D@1#eq:background · line 249

  $$\begin{array}{rlrlrl}A& =\sqrt{2p}M,& a(t)& =(t/t_{\ast }{)}^p,& H(t)& =p/t,\\ \bar{\varphi }(t)& ={\varphi }_{\ast }+A\log (t/t_{\ast }),& v(t)& =\dot{\bar{\varphi }}(t)=A/t,& & \\ \bar{g}& =-\mathrm{d}{t}^2+a(t{)}^2{\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j,& V(\varphi )& =\frac{M^{2}p(3p-1)}{t_{\ast }^2}{e}^{-2(\varphi -{\varphi }_{\ast })/A}.& & \end{array}$$

• XX-D@1#eq:jet · line 277

  $$g_s=\bar{g}+sh+\frac{s^2}{2}k,{\varphi }_s=\bar{\varphi }+s\phi +\frac{s^2}{2}\psi .$$

• XX-D@1#eq:T · line 283

  $$T_s=t_{\ast }{e}^{({\varphi }_s-{\varphi }_{\ast })/A}.$$

• XX-D@1#eq:rawN · line 288

  $$N_s={\left(-{g_s}^{-1}(\mathrm{d}{\varphi }_s,\mathrm{d}{\varphi }_s)\right)}^{-1/2}.$$

• unlabeled · line 267

  $$\mathcal{M}=I\times {\mathbb{R}}^3,I=(t_{-},t_{+}),0<t_{-}<t_{+}<\infty .$$

• unlabeled · line 295

  $$N_0=t/A.$$

• unlabeled · line 300

  $$\mathrm{d}{V}_{\bar{g}}=a(t{)}^3\mathrm{d}t{\mathrm{d}}^{3}x.$$