[Paper Review] Big Bang in Dipole Cosmology

Alireza Allahyari, Ehsan Ebrahimian+2 more|arXiv (Cornell University)|Jul 28, 2023

Cosmology and Gravitation Theories50 references4 citations

TL;DR

This paper investigates the Big Bang singularity in dipole cosmology, a framework extending FLRW models by introducing a preferred spatial direction (cosmic dipole) via tilt and shear. It shows that the nature of the Big Bang singularity depends on the sign of shear: negative shear leads to a curvature singularity, while positive shear results in a milder 'whimper' singularity where curvature invariants remain finite despite diverging tilt. The analysis extends to a dipole ΛCDM model with radiation, matter, and dark energy, confirming similar singularity types and revealing that relative tilt between radiation and matter remains significant at late times.

ABSTRACT

We continue the study of dipole cosmology framework put forward in \cite{Krishnan:2022qbv}, a beyond FLRW setting that has a preferred direction in the metric which may be associated with a cosmological tilt, a cosmic dipole. In this setup the shear and the tilt can be positive or negative given the dipole direction. We thoroughly analyze evolution of the universe in this setting, particularly focusing on the behaviour near the Big Bang (BB). We first analyze a single fluid model with a generic constant equation of state $w$. While details of the behavior near the BB depends on $w$ and the other initial conditions, we find that when the shear is negative we have a shear dominated BB singularity, whereas for a positive shear we have a much milder singularity, the whimper singularity \cite{Ellis:1974ug}, at which the tilt blows up while curvature invariants remain finite. We then consider dipole $Λ$CDM model which besides the shear has two tilt parameters, one for radiation and one for the pressureless matter. For positive (negative) shear we again find whimper (curvature) singularity near the BB. Moreover, when the tilt parameters have opposite signs, the shear can change sign from negative to positive in the course of evolution of the Universe. We show that the relative tilt of the radiation and the matter generically remains sizable at late times.

Motivation & Objective

• To analyze the behavior of the Big Bang singularity in dipole cosmology, a non-FLRW framework with a preferred spatial direction.
• To determine how the sign of shear and tilt influences the nature of the initial singularity.
• To extend the analysis to the dipole ΛCDM model with multiple fluids (radiation, matter, dark energy) and assess near-Big Bang dynamics.
• To investigate whether the relative tilt between radiation and matter remains non-zero at late times, despite isotropization.

Proposed method

• Formulates dipole cosmology within the tilted cosmology framework of King and Ellis, allowing for a preferred direction via a non-comoving fluid flow.
• Analyzes a single-fluid model with constant equation of state $w$, deriving evolution equations for shear, tilt, and scale factor near the Big Bang.
• Applies the same framework to the dipole ΛCDM model, incorporating three components: radiation (with tilt), pressureless matter (with tilt), and cosmological constant (tilt-inert).
• Uses numerical integration with Planck-constrained parameters to explore all possible initial condition combinations for the singularity structure.
• Classifies initial singularity types as either curvature singularity (shear dominant, finite tilt) or whimper singularity (tilt dominant, finite curvature invariants).
• Performs backward evolution from late-time isotropization to trace the origin of initial conditions and singularity type.

Figure 1 : As an example of SEC-violating case we show $w=-2/3$ . The top-left plot shows evolution of $\beta$ which starts with a large negative value and goes to zero. The top-right plot shows $\sigma/H$ which remains $-3$ for a long time before starting to drop to zero, it has an inflection point

Experimental results

Research questions

• RQ1 How does the sign of shear determine the type of Big Bang singularity in a single-fluid dipole cosmology?
• RQ2 What is the near-Big Bang behavior in the dipole ΛCDM model, particularly regarding the interplay between shear, tilt parameters for radiation and matter, and curvature invariants?
• RQ3 Can the relative tilt between radiation and matter remain sizeable at late times despite overall isotropization?
• RQ4 Does the presence of a cosmic dipole (via tilt) lead to a generic preference for one type of initial singularity over another in the dipole cosmology framework?
• RQ5 How do the initial conditions for shear and tilt evolve to produce the observed late-time isotropization in dipole ΛCDM?

Key findings

• For a single fluid with constant $w$, a negative shear leads to a curvature singularity where matter density diverges while curvature invariants remain finite.
• With positive shear, the model exhibits a 'whimper' singularity: the tilt diverges while curvature invariants stay finite, indicating a milder initial state.
• In the dipole ΛCDM model, the same singularity classification holds: positive shear leads to a whimper singularity, negative shear to a curvature singularity.
• When the tilt parameters for radiation and matter have opposite signs, the shear can change sign during evolution, leading to a transition from curvature to whimper singularity.
• Despite isotropization at late times, the relative tilt between radiation and matter remains finite and non-zero, with $\beta_r$ approaching a finite asymptotic value.
• Backward evolution shows that $K\mathfrak{S} \sim 0^+$ can originate from either a whimper singularity ($\mathfrak{S}=+1$) or a curvature singularity with subsequent shear sign change ($\mathfrak{S}=-1$).

Figure 2 : Plots for $w=-1/3$ case. The top-left plot shows evolution of $\tanh\beta$ which starts with a negative value and $|\beta|$ goes to zero. The top-right plot shows $\sigma/H$ which remains $-3$ for a long time before starting to drop to zero. Both $\beta$ and $\sigma/H$ plot have a $\tanh$