General relativity on the multiverse and nature’s hierarchies

We define “third-derivative” general relativity by promoting the integration measure in Einstein-Hilbert action to be an arbitrary 4-form field strength. We project out its local fluctuations by coupling it to another 4-form field strength. This ensures that the gravitational sector contains only the usual massless helicity-2 propagating modes. Adding the charges to these 4-forms allows for discrete variations of the coupling parameters of conventional general relativity: $G_N,\mathrm{\Lambda },H_0$, and even $⟨\mathsf{Higgs}⟩$ are all variables which can change by jumps. Hence, de Sitter spacetime is unstable to membrane nucleation. Using this instability, we explain how the cosmological constant problem can be solved. The scenario utilizes the idea behind the irrational axion, but instead of an axion it requires one more 4-form field strength and corresponding charged membranes. When the membrane charges satisfy the constraint $\frac{2{\kappa }_{\mathsf{eff}}^2{\kappa }^2|{\mathcal{Q}}_i|}{3{\mathcal{T}}_i^2}<1$, the theory which ensues exponentially favors a huge hierarchy $\mathrm{\Lambda }/M_{\mathrm{Pl}}^4\ll 1$ instead of $\mathrm{\Lambda }/M_{\mathrm{Pl}}^4\simeq 1$. The discharges produce the distribution of the values of $\mathrm{\Lambda }$ described by the saddle point approximation of the Euclidean path integral.

Figure 1

Spherical ($S^4$, top row) and horospherical (also known as hyperbolic; $H^4$, bottom row) sections which are glued together to form instantons. Red ones are the interiors and the blue ones the exterior geometries of the instanton. The $\pm$ are the values of ${\zeta }_{\mathrm{in}/\mathrm{out}}$.

Figure 2

The instanton Baedeker. The instantons fall into four types divided by double lines in the table and counted clockwise from the top corner [42]. The transitions corresponding to empty squares are ruled out kinematically by Eqs. (49) and (50). The top nine are further split by $q=\frac{2{\kappa }_{\mathsf{eff}}^2{\kappa }^2|{\mathcal{Q}}_A|}{3{\mathcal{T}}_A^2}<1$ (pale green) or $q>1$ (pale gold). We keep both since ${\kappa }_{\mathsf{eff}}^2$ might vary independently (we will suppress those variations later on). The “ogrelike” configurations in the right column which are crossed out are allowed kinematically but are suppressed dynamically since their bounce action is huge and positive, $S_{\mathsf{bounce}}\gg 1$, diverging when anti–de Sitter sections are noncompact (see the text).

Figure 3

An illustration of an instanton comprised of two sections of $S^4$. The region around the north pole, shaded red, has a larger curvature radius because ${\mathcal{T}}_A>0$ by Israel junction conditions [56].

Figure 4

The only $q<1$ instanton mediating $dS\to dS$.

Figure 5

The cosmological constant spectral bands for $q=\frac{2{\kappa }_{\mathsf{eff}}^2{\kappa }^2|{\mathcal{Q}}_A|}{3{\mathcal{T}}_A^2}<1$.

Figure 6

Left panel: an illustration of Euclidean space “boiling” in pancosmic general relativity. Each monochromatic pastille is a universe with a locally fixed Planck scale, cosmological constant, and the QFT parameters. Those change from one pastille to another. Right panel: an illustration of a borough of a multiverse of eternal inflation [26] (in Lorentzian signature).

Figure 7

The cosmological constant spectrum with many superselection sectors depicted by differently colored spectral levels. The blue spectrum is tuned to get close to $\mathrm{\Lambda }\simeq 0$, the black and purple spectra are not.