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Challenges for scaling cosmologies

Luca Amendola, Miguel Quartin, Shinji Tsujikawa, Ioav Waga

TL;DR

This work investigates scaling cosmologies with a scalar field coupled to dark matter in a general two-component cosmic setting. By deriving the most general second-order Lagrangian admitting scaling solutions under a field-dependent coupling, the authors show that, up to a field redefinition, the Lagrangian can be written as $p(X,\varphi)=X\,g(Xe^{\lambda\varphi})$, with $\lambda$ determined by background parameters; the constant-coupling form becomes universal after redefinition. Through a phase-space analysis including radiation, they identify four fixed points: A (scalar-field domination), B (accelerated scaling), and C/D (kinetic or φ-MDE solutions), and determine their existence and stability conditions. Their main result is that for a broad class of integer polynomial Lagrangians with variable coupling, a sequence of a matter-dominated era followed by a stable accelerated scaling attractor is not realizable, underscoring the difficulty of solving the coincidence problem within this framework. They note a possible exception in fractional Lagrangians with $0<u<1$, which warrants further investigation.

Abstract

A cosmological model that aims at solving the coincidence problem should show that dark energy and dark matter follow the same scaling solution from some time onward. At the same time, the model should contain a sufficiently long matter-dominated epoch that takes place before acceleration in order to guarantee a decelerated epoch and structure formation. So a successful cosmological model requires the occurrence of a sequence of epochs, namely a radiation era, a matter-dominated era and a final accelerated scaling attractor with $Ω_φ \simeq 0.7$. In this paper we derive the generic form of a scalar-field Lagrangian that possesses scaling solutions in the case where the coupling $Q$ between dark energy and dark matter is a free function of the field $φ$. We then show, rather surprisingly, that the aforementioned sequence of epochs cannot occur for a vast class of generalized coupled scalar field Lagrangians that includes, to our knowledge, all scaling models in the current literature.

Challenges for scaling cosmologies

TL;DR

This work investigates scaling cosmologies with a scalar field coupled to dark matter in a general two-component cosmic setting. By deriving the most general second-order Lagrangian admitting scaling solutions under a field-dependent coupling, the authors show that, up to a field redefinition, the Lagrangian can be written as , with determined by background parameters; the constant-coupling form becomes universal after redefinition. Through a phase-space analysis including radiation, they identify four fixed points: A (scalar-field domination), B (accelerated scaling), and C/D (kinetic or φ-MDE solutions), and determine their existence and stability conditions. Their main result is that for a broad class of integer polynomial Lagrangians with variable coupling, a sequence of a matter-dominated era followed by a stable accelerated scaling attractor is not realizable, underscoring the difficulty of solving the coincidence problem within this framework. They note a possible exception in fractional Lagrangians with , which warrants further investigation.

Abstract

A cosmological model that aims at solving the coincidence problem should show that dark energy and dark matter follow the same scaling solution from some time onward. At the same time, the model should contain a sufficiently long matter-dominated epoch that takes place before acceleration in order to guarantee a decelerated epoch and structure formation. So a successful cosmological model requires the occurrence of a sequence of epochs, namely a radiation era, a matter-dominated era and a final accelerated scaling attractor with . In this paper we derive the generic form of a scalar-field Lagrangian that possesses scaling solutions in the case where the coupling between dark energy and dark matter is a free function of the field . We then show, rather surprisingly, that the aforementioned sequence of epochs cannot occur for a vast class of generalized coupled scalar field Lagrangians that includes, to our knowledge, all scaling models in the current literature.

Paper Structure

This paper contains 15 sections, 74 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The General Lagrangian for Scaling Solutions with Arbitrary Coupling
  3. Phase Space Equations
  4. Critical points
  5. Point A: Scalar-field dominated solutions
  6. Point B: Scaling solutions
  7. Points C and D: Kinetic solutions
  8. Summary of fixed points
  9. Can we have two scaling regimes?
  10. Case of $c_{0}>0$
  11. Positive powers of $Y$
  12. Negative powers of $Y$
  13. Case of $c_{0}<0$
  14. Case of $c_{0}=0$
  15. Conclusions

Figures (3)

  • Figure 1: Phase space for the model (\ref{['model']}) with $u=2$, $c=c_0=1$, $\lambda=4$ and $Q=0.7$ together with the fixed points A, B, C and D. Here and in the following figures the gray area represents the region where $\Omega_{\varphi}>1$. The dotted line corresponds to the singularity given by (\ref{['ysingularity']}) at which the speed of sound diverges.
  • Figure 2: Phase space for the model (\ref{['model']}) with $u=1.1$, $c=c_0=1$, $\lambda=2$ and $Q=2$ together with the fixed points A, B and D (point C lies in the $\Omega_{\varphi}>1$ region). The dotted line corresponds to the singularity given by (\ref{['ysingularity']}) at which the speed of sound diverges.
  • Figure 3: Phase space for the model (\ref{['model']}) with $u=1$, $c=c_0=1$, $\lambda=1.54$ and $Q=1.02$ together with the fixed points A, B, C and D.