Cosmological Tracking Solutions

Cosmological Tracking Solutions

16 Dec 1998 | Paul J. Steinhardt, Limin Wang, and Ivaylo Zlatev
The paper by Steinhardt, Wang, and Zlatev introduces the concept of "tracker fields" to address the "coincidence problem" in quintessence models. Quintessence is a hypothetical form of energy that makes up a significant fraction of the universe's energy density and has a negative pressure. The main issue is that the energy density of quintessence must be set to a very specific initial condition to match the matter energy density today, which seems finely tuned. The authors define tracker fields as solutions to the equation of motion that converge to a common track, regardless of the initial conditions. This property allows a wide range of initial conditions to lead to the same cosmic evolution, avoiding the fine-tuning problem. They derive conditions for the existence of tracker solutions, which depend on the functional form of the potential \( V(Q) \). Specifically, they show that tracking behavior occurs if \( \Gamma \equiv V''V/(V')^2 \) is nearly constant and greater than 1 for \( w_Q < w_B \) or less than 1 for \( (1/2)(1 + w_B) > w_Q > w_B \). The paper also discusses the convergence of solutions from different initial conditions, the stability of tracker solutions, and the constraints on the initial value of \( Q \) and \( \rho_Q \). It highlights that initial conditions where \( Q \) dominates the radiation and matter density are disallowed, but those with equipartition between \( \rho_Q \) and the background energy density are allowed. Finally, the authors explore the \( \Omega_Q - w_Q \) relation, which arises from the tracker solutions. This relation predicts that \( w_Q \) decreases as \( \Omega_Q \) increases, with \( w_Q \) being sufficiently constrained to be cosmologically interesting. They argue that this relation, combined with the constraint \( \Omega_m \geq 0.2 \), creates a significant gap between \( w_Q^{eff} \) and \( -1 \), making it possible to distinguish tracker fields from a cosmological constant in future observations.The paper by Steinhardt, Wang, and Zlatev introduces the concept of "tracker fields" to address the "coincidence problem" in quintessence models. Quintessence is a hypothetical form of energy that makes up a significant fraction of the universe's energy density and has a negative pressure. The main issue is that the energy density of quintessence must be set to a very specific initial condition to match the matter energy density today, which seems finely tuned. The authors define tracker fields as solutions to the equation of motion that converge to a common track, regardless of the initial conditions. This property allows a wide range of initial conditions to lead to the same cosmic evolution, avoiding the fine-tuning problem. They derive conditions for the existence of tracker solutions, which depend on the functional form of the potential \( V(Q) \). Specifically, they show that tracking behavior occurs if \( \Gamma \equiv V''V/(V')^2 \) is nearly constant and greater than 1 for \( w_Q < w_B \) or less than 1 for \( (1/2)(1 + w_B) > w_Q > w_B \). The paper also discusses the convergence of solutions from different initial conditions, the stability of tracker solutions, and the constraints on the initial value of \( Q \) and \( \rho_Q \). It highlights that initial conditions where \( Q \) dominates the radiation and matter density are disallowed, but those with equipartition between \( \rho_Q \) and the background energy density are allowed. Finally, the authors explore the \( \Omega_Q - w_Q \) relation, which arises from the tracker solutions. This relation predicts that \( w_Q \) decreases as \( \Omega_Q \) increases, with \( w_Q \) being sufficiently constrained to be cosmologically interesting. They argue that this relation, combined with the constraint \( \Omega_m \geq 0.2 \), creates a significant gap between \( w_Q^{eff} \) and \( -1 \), making it possible to distinguish tracker fields from a cosmological constant in future observations.
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Understanding Cosmological tracking solutions