TFPT scientific source excerpt
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000029 | \label{thm:gravity-cosmology-closure}
000030 | The same geometric closure sector defines the canonical spectral gravity branch
000031 | \[
000032 | \Gamma_{\mathrm{grav}}
000033 | :=
000034 | -6\chigeo^2\mathcal E_D+\Gamma_D^{(4)}.
000035 | \]
000036 | On admissible geometric fluctuations, let
000037 | \[
000038 | Q_{\mathrm{geo}}
000039 | \]
000040 | denote the geometric BRST/Hodge differential that quotients diffeomorphism and local-frame gauge
000041 | directions, and set
000042 | \[
000043 | P_{\mathrm{phys}}^{\mathrm{grav}}
000044 | :=
000045 | \Pi_{\ker\Delta_{\mathrm{tot}}},
000046 | \qquad
000047 | \Delta_{\mathrm{tot}}
000048 | :=
000049 | \{Q_{\mathrm{adm}}+Q_{\mathrm{geo}},(Q_{\mathrm{adm}}+Q_{\mathrm{geo}})^\dagger\}.
000050 | \]
000051 | On the low-curvature local branch one recovers the effective action
000052 | \[
000053 | \begin{aligned}
000054 | \Gamma_{\mathrm{grav}}
000055 | :={}&
000056 | \int d^4x\sqrt{-g}\Big[
000057 | \frac{\Mpl^2}{2}R
000058 | -\Lambda_{\Seam,\mathrm{ren}}
000059 | +\frac{1}{6M^2}R^2
000060 | +\frac{Z_\chi}{2}(\nabla\chigeo)^2
000061 | \\
000062 | &\qquad
000063 | -U(\chigeo)
000064 | +|D\Phi|^2
000065 | -V(\Phi,\chigeo)
000066 | \\
000067 | &\qquad
000068 | -\sum_a \frac{1}{4g_a^2}\tr F_{a,\mu\nu}F_a^{\mu\nu}
000069 | +\cdots
000070 | \Big].
000071 | \end{aligned}
000072 | \]
000073 | with
000074 | \[
000075 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000076 | \qquad
000077 | Z_H=\frac{1}{16\pi^2},
000078 | \qquad
000079 | \vgeo^2=\frac{Z_H\mu_\Phi^2(\chigeozero)}{\lambda_\Phi}.
000080 | \]
000081 | Thus the low-curvature gravitational coefficient is later read internally as
000082 | \[
000083 | \frac{\Mpl^2}{\lambda_\Sigma^2}=\frac{\rho_\star}{2\pi^2}
000084 | \]
000085 | once the stationary boundary-normalized root has been fixed.
000086 | Its low-curvature Euler--Lagrange equations are
000087 | \[
000088 | \Mpl^2 G_{\mu\nu}
000089 | +\Lambda_{\Seam,\mathrm{ren}} g_{\mu\nu}
000090 | +\frac{1}{3M^2}\mathcal H_{\mu\nu}
000091 | =
000092 | T_{\mu\nu}^{(\chigeo,\Phi,A,\psi)},
000093 | \]
000094 | \[
000095 | Z_\chi\Box\chigeo=U'(\chigeo)+\partial_{\chigeo}V(\Phi,\chigeo),
000096 | \qquad
000097 | D^\mu D_\mu\Phi=\partial_{\Phi^\dagger}V(\Phi,\chigeo),
000098 | \]
000099 | where
000100 | \[
000101 | \mathcal H_{\mu\nu}
000102 | :=
000103 | RR_{\mu\nu}
000104 | -\frac14 g_{\mu\nu}R^2
000105 | -\nabla_\mu\nabla_\nu R
000106 | +g_{\mu\nu}\Box R.
000107 | \]
000108 | On the low-curvature vacuum branch
000109 | \[
000110 | R\ll M^2,