The Unified First law in "Cosmic Triad" Vector Field Scenario

TL;DR

This paper explores the application of the unified first law to the 'cosmic triad' vector field scenario, analyzing both minimal and non-minimal coupling cases and deriving the corresponding Friedmann equations and energy expressions.

Contribution

It provides a detailed analysis of the unified first law in the 'cosmic triad' vector field scenario, including frame transformations and energy calculations, which was not previously established.

Findings

Correct Friedmann equations derived in both frames

Misner-Sharp energy expressions obtained for the scenario

Entropy in non-minimal coupling case remains undetermined

Abstract

In this letter, we try to apply the unified first law to the "cosmic triad" vector field scenario both in the minimal coupling case and in the non-minimalcoupling case. After transferring the non-minimally coupling action in Jordan frame to Einstein frame, the correct dynamical equation (Friedmann equation) is gotten in a thermal equilibrium process by using the already existing entropy while the entropy in the non-minimal coupled "cosmic triad" scenario has not been derived. And after transferring the variables back to Jordan frame, the corresponding Friedmann equation is demonstrated to be correct. For complete arguments, we also calculate the related Misner-Sharp energy in Jordan and Einstein frames.