The emergent Big Bang scenario

TL;DR

The paper addresses the cosmological singularity by proposing an emergent Big Bang scenario in a purely Euclidean four-dimensional spacetime endowed with a clock field $\phi$, which generates Lorentzian patches via the effective metric $g_{\mu\nu}=g^{\rm E}_{\mu\nu}-\frac{\partial_\mu\phi\partial_\nu\phi}{M^4}$; time becomes an emergent variable, and the boundary $\Sigma_0$ where $g_{00}$ changes sign replaces the traditional Big Bang with a regular surface. It develops a concrete framework with two free functions ${\cal K}(X_E)$ and $G_4(X_E)$, derives the core field equations in terms of ${\cal G}=G_4-2X_EG'_4$, and identifies necessary conditions for solutions that yield two Lorentzian regions separated by a Euclidean core, including a no-crossing condition for the clock-field gradient. Through a simple explicit model and a general reconstruction procedure, the authors demonstrate realizations of mirror universes and pocket universes with emergent near-de Sitter expansion, and show how ${\cal K}(X_E)$ and $G_4(X_E)$ can be reconstructed from chosen clock-field and geometry profiles. Although speculative, the scenario offers a falsifiable view of the primordial Universe, predicts potential observational signatures from the pre-Lorentzian phase and large-scale Euclidean regions, and motivates further study of perturbations, stability, and phenomenology beyond General Relativity.

Abstract

This letter proposes a new avenue for understanding the cosmological singularity. The standard cosmological model contains a generic initial singularity usually referred to as the {\em big bang}. Herein, we present a novel idea to extend the description of our Universe beyond this limit. The proposal relies on rewriting physics in a purely Riemannian, {\em i.e.} locally Euclidean, 4-dimensional space and the emergence of Lorentzian patches owing to the interaction of all matter fields to a clock field that is responsible for a signature change. If our universe is contained within one of these patches, the initial singularity is replaced by a smooth boundary on which the signature of the physical metric flips. In this letter, we first define the model and draw the necessary conditions on its arbitrary functions for solutions to exist. Next, we prove the existence of solutions that lead to an emergent universe with a primordial (almost) de Sitter phase. To finish, we discuss the consequences of this construction for the universe on scales much larger than our observable universe: a large ``Euclidean sea'' in which Lorentzian islands locally emerge and host an expanding universe potentially similar to ours. While speculative, this scenario has specific features that can be tested, and the present letter sets the basis for further phenomenological investigations.

Figures (7)

• Figure 1: A big bang that never happened. Our universe can be thought as an emergent Lorentzian patch within a 4-dimensional purely Riemannian space. The "big bang" can be understood as the hypersurface $\Sigma_0$ on which the signature of the metric flips. In the simplest version, the geometry and field configurations are assumed to ( i) depend only on one space coordinate, $z$, so that the emergent spacetimes ${\cal M}_{0\pm}$ satisfy the Copernican principle and ( ii) exhibit the $(-,+,+,+)$ signature for the effective metric in the two asymptotic regions.
• Figure 2: Integration of the field equations assuming $G_4$ and ${\cal K}$ of the form (\ref{['e.choice']}) with parameters $(g_0,g_1,k_0,k_2,X_0)=(1, -1, 1, -1.55, 1)$ and the integration constant $J_0=1$. The upper line shows the intermediary functions $(10H_{\rm E}^2,a^3)$ [Left] as a function of $\psi$ obtained from Eq. (\ref{['e.eomaH']}) and $(10 \dot{H}_{\rm E},\dot\psi)$ [Right] as a function of $\psi$ obtained thanks to Eqs. \ref{['e.dpsidH']}. The missing segments in the plot of $\dot\psi$ correspond to regions where $H_{\rm E}^2<0$. The lower plot gives the full solution for the clock field $\psi$ and the geometry $H_{\rm E}$ as functions of $z$.
• Figure 3: The considered solutions to be reconstructed require to specify the profile of the geometry $h$ (Solid line) and $H_{\rm E}$ (Dashed line) [Bottom] as given by Eq. (\ref{['e.sol2']}) and of the clock field $\psi$ [Top] as given by Eq. (\ref{['e.sol1']}) for a model of mirror universes [Left] and pocket universe [Right]. Asymptotically at large $z$, $H_{\rm E}\rightarrow$ const. and $h\sim H_{\rm E} z$ while $\psi\rightarrow const.$, allowing for the emergence of a de Sitter phase at late time in the case of the mirror universe.
• Figure 4: Reconstruction of the free functions ${\cal K}(X_{\rm E})$ [Top] and $G_4(X_{\rm E})$ [Bottom] for a mirror universes [Solid lines] and pocket universe [Dashed lines] assuming $J_0=1$ and $C_0=0$. The mirror universe curve is not defined below the value of $X_E$ corresponding to the minimum value of $\psi$ occurring at $z=0$ in Fig. \ref{['fig:solution']}, and from the form of Eq. \ref{['e.sol1']}, one can see that the values of $\psi$ and consequently $X_E$ are bounded.
• Figure 5: The emergent cosmologies for mirror universes [Solid lines] and pocket universe [Dashed lines]. The scale factor $R$ and the Hubble rate $H$ as a function of the cosmic time $t$ -- assuming $t=0$ on $\Sigma_0$ -- are respectively plotted on the top and bottom frames for the same parameters as in Fig. \ref{['fig:model']}. The red dot-dashed curve represents an exact de Sitter spacetime and the vertical dashed lines mark the limit of the Lorentzian regions ${\cal M}_0$. Contrary to the mirror universes, a pocket universe has a boundary in the future and hence a finite maximum age.