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Redundancy of the cosmological evolution equations and its relationship with the initial conditions

Kaushik Bhattacharya, Dipanjan Dey, Priyanka Saha

TL;DR

The paper investigates why FLRW cosmology presents more evolution equations than unknowns, revealing that the Friedmann equation acts as a crucial initial-condition constraint rather than a standalone dynamical equation. By analyzing pairs of equations (Friedmann+continuity, pressure+continuity, acceleration+continuity) and introducing constraint functions $F(t)$ and $\tilde F(t)$, it shows that consistent evolution requires initial data to satisfy the Friedmann constraint, which then persists dynamically. The study extends these ideas to $f(R)$ gravity, where the modified Friedmann equation becomes a higher-order constraint, yet a conformal Einstein frame clarifies the GR-like dynamics and the role of initial conditions. The results establish a coherent logical structure for cosmological dynamics, applicable to both GR and modified gravity, and provide a generalizable operational framework for analyzing the initial-value problem in cosmology.

Abstract

It is known that in Friedmann-Lemaitre-Robertson-Walker cosmology one has more number of dynamical equations, compared to the number of unknown variables. This fact makes some equations redundant. The situation becomes complicated because all the relevant differential equations in cosmology are not of the same order. In this article we study the fate of the redundant equations. We show that this redundancy is inevitable in general relativity and a modern modification of gravity, called $f(R)$ gravity. It is shown that this redundancy predicts a special role to one of the Friedmann equations, which constrains the initial values of the problem. It is shown that the initial constraint condition itself evolves with time. Our method of analyzing the dynamical structure of the theories relies on an operational approach and can be generalized further.

Redundancy of the cosmological evolution equations and its relationship with the initial conditions

TL;DR

The paper investigates why FLRW cosmology presents more evolution equations than unknowns, revealing that the Friedmann equation acts as a crucial initial-condition constraint rather than a standalone dynamical equation. By analyzing pairs of equations (Friedmann+continuity, pressure+continuity, acceleration+continuity) and introducing constraint functions and , it shows that consistent evolution requires initial data to satisfy the Friedmann constraint, which then persists dynamically. The study extends these ideas to gravity, where the modified Friedmann equation becomes a higher-order constraint, yet a conformal Einstein frame clarifies the GR-like dynamics and the role of initial conditions. The results establish a coherent logical structure for cosmological dynamics, applicable to both GR and modified gravity, and provide a generalizable operational framework for analyzing the initial-value problem in cosmology.

Abstract

It is known that in Friedmann-Lemaitre-Robertson-Walker cosmology one has more number of dynamical equations, compared to the number of unknown variables. This fact makes some equations redundant. The situation becomes complicated because all the relevant differential equations in cosmology are not of the same order. In this article we study the fate of the redundant equations. We show that this redundancy is inevitable in general relativity and a modern modification of gravity, called gravity. It is shown that this redundancy predicts a special role to one of the Friedmann equations, which constrains the initial values of the problem. It is shown that the initial constraint condition itself evolves with time. Our method of analyzing the dynamical structure of the theories relies on an operational approach and can be generalized further.
Paper Structure (13 sections, 40 equations)

This paper contains 13 sections, 40 equations.

Table of Contents

  1. Introduction
  2. A single barotropic fluid in FLRW spacetime and the Einstein equation
  3. The various cases in an ever expanding or ever contracting universe
  4. Choosing the Friedmann equation and the continuity equation
  5. Choosing the pressure equation and the continuity equation
  6. Choosing the acceleration equation and the continuity equation
  7. The link between all the choices is in the initial condition
  8. The cases where $\dot{a}(t_0)=0$ and $t_0$ is inside the interval $[t_i\,,\,t_f]$
  9. When $t_0=t_i$
  10. When $t_i < t_0 < t_f$
  11. On the logical consistency of the theory
  12. On the redundancy of dynamical equations and initial conditions in $f(R)$ cosmology
  13. Conclusion