Role of the cosmological constant in the holographic description of the early universe

TL;DR

This paper explores how the cosmological constant influences the holographic description of a radiation-dominated early universe, revealing its roles in entropy, energy, and modular properties of solutions.

Contribution

It introduces a novel perspective on the cosmological constant's role in holography, linking it to entropy bounds and modular properties in early universe models.

Findings

Cosmological constant affects entropy and energy in holographic models.

Solutions involve Weierstrass elliptic functions with modular properties.

The cosmological constant acts as a modular parameter in entropy formulas.

Abstract

We investigate the role of the cosmological constant in the holographic description of a radiation-dominated universe $C_2/R^4$ with a positive cosmological constant $\Lambda$. In order to understand the nature of cosmological term, we first study the newtonian cosmology. Here we find two aspects of the cosmological term: entropy ($\Lambda \to S_{\rm \Lambda}$) and energy ($\Lambda \to E_{\rm \Lambda}$). Also we solve the Friedmann equation parametrically to obtain another role. In the presence of the cosmological constant, the solutions are described by the Weierstrass elliptic functions on torus and have modular properties. In this case one may expect to have a two-dimensional Cardy entropy formula but the cosmological constant plays a role of the modular parameter $\tau(C_2,\Lambda)$ of torus. Consequently the entropy concept of the cosmological constant is very suitable for establishing the holographic entropy bounds in the early universe. This contrasts to the role of the cosmological constant as a dark energy in the present universe.