[Paper Review] Crossing the Big Bang singularity

C. Wetterich|arXiv (Cornell University)|Apr 9, 2020

Cosmology and Gravitation Theories68 references4 citations

TL;DR

This paper presents a simple scalar-tensor model of variable gravity with a dynamical Planck mass, showing that cosmological solutions can cross the Big Bang singularity in a regular manner when formulated in a scaling frame with field-dependent Planck mass. The singularity in the Einstein frame is shown to be a field redefinition artifact—non-physical—while the crossing is generic only under fine-tuned parameter conditions, not in generic models with two derivatives.

ABSTRACT

A simple model for a scalar field and gravity admits homogeneous isotropic cosmological solutions which cross the Big Bang singularity. In the scaling frame with field dependent effective Planck mass these solutions are regular. They become singular in the Einstein frame with fixed Planck mass. This field singularity arises since the field transformation of the metric to the Einstein frame is singular at the crossing point of a vanishing scalar field. No physical singularity is present for these solutions. Within general models with no more than two derivatives we find that the possibility of a crossing of the "Big Bang singularity" is not generic. It needs a tuning of model parameters. The present models are not a realistic description of the Universe since they fail to render acceptable inhomogeneities.

Motivation & Objective

• To investigate whether the Big Bang singularity in standard cosmology can be physically crossed, rather than being an artifact of field frame choice.
• To determine if such crossing solutions are generic within models with at most two derivatives in the action.
• To analyze the implications for primordial fluctuations and anisotropies in a model where the singularity is removable via field redefinition.
• To clarify the distinction between physical singularities and field singularities arising from coordinate (frame) choices in gravity theories.
• To assess the viability of such models for describing realistic inflationary cosmology, particularly regarding the primordial fluctuation spectrum.

Proposed method

• The model uses a scalar field χ with a negative kinetic term (B=4) in a variable gravity action, where the Planck mass is proportional to χ.
• The effective action contains only two derivatives, ensuring a well-defined, stable theory without higher-order equations of motion.
• Solutions are derived in the scaling frame, where the metric and scalar field are coupled via χ, and the theory remains regular even when χ→0.
• The Einstein frame is obtained via a Weyl transformation, which becomes singular at χ=0, thereby introducing the apparent Big Bang singularity.
• The field equations are solved explicitly for a homogeneous, isotropic Robertson-Walker metric, showing that χ(t) = μ²t crosses zero at t=0 with finite derivative.
• The behavior of linearized anisotropies and relative fluctuations is analyzed to test the robustness of the crossing solution in the presence of inhomogeneities.

Experimental results

Research questions

• RQ1 Can the Big Bang singularity be crossed in a physically meaningful way within a diffeomorphism-invariant theory with at most two derivatives?
• RQ2 Is the apparent singularity in the Einstein frame a physical feature or an artifact of field redefinition?
• RQ3 Are crossing solutions generic in models with variable gravity and two-derivative actions, or do they require fine-tuning of parameters?
• RQ4 How do primordial fluctuation spectra and anisotropies behave near the crossing point, and can they be consistent with observations?
• RQ5 Can the breakdown of linearized anisotropy equations at χ=0 be resolved by including higher derivative terms or non-linear effects?

Key findings

• The Big Bang singularity in the Einstein frame is not physical but a field singularity arising from a singular Weyl transformation at χ=0.
• The model exhibits a regular solution in the scaling frame where χ crosses zero smoothly, with finite Hubble parameter and derivative, confirming a true crossing of the cosmological singularity.
• Crossing solutions are not generic in two-derivative models; they require fine-tuning of parameters such as B=4, with generic solutions approaching χ=0 only asymptotically in the infinite past.
• The primordial fluctuation spectrum in this model is inconsistent with observations when extrapolated to the crossing point, indicating failure to reproduce Bunch-Davies initial conditions.
• Anisotropic fluctuations diverge near χ=0 in the linear approximation, signaling a breakdown of the linear theory and suggesting the need for non-linear or higher-derivative corrections.
• The results suggest that realistic crossing cosmologies may only emerge if higher derivative terms are included in the effective action, potentially restoring consistency with observations.