Search for dark energy potentials in quintessence theory

Abstract

The time evolution of the equation of state $w$ for quintessence models with a scalar field as dark energy is studied up to the third derivative ($d^{3}w/d{a}^3$) with respect to the scale factor $a$, in order to predict the future observations and specify the scalar potential parameters with the observables. The third derivative of $w$ for general potential $V$ is derived and applied to several types of potentials. They are the inverse power-law ($V=M^{4+\alpha }/Q^{\alpha }$), the exponential ($V=M^4\exp (\beta M/Q)$), the mixed ( $V=M^{4+\gamma }\exp (\beta M/Q)/Q^{\gamma }$), the cosine ($V=M^4(\cos (Q/f)+1)$) and the Gaussian types ($V=M^4\exp (-Q^2/{\sigma }^2)$), which are prototypical potentials for the freezing and thawing models. If the parameter number for a potential form is $n$, it is necessary to find at least for $n+2$ independent observations to identify the potential form and the evolution of the scalar field ($Q$ and $\dot{Q}$). Such observations would be the values of ${\Omega }_Q,w,dw/da.\cdots$, and $d{w}^n/d{a}^n$. From these specific potentials, we can predict the $n+1$ and higher derivative of $w$ ; $d{w}^{n+1}/d{a}^{n+1},\cdots$. Since four of the above mentioned potentials have two parameters, it is necessary to calculate the third derivative of $w$ for them to estimate the predict values. If they are tested observationally, it will be understood whether the dark energy could be described by the scalar field with this potential. At least it will satisfy the necessary conditions. Numerical analysis for $d^{3}w/d{a}^3$ are made under some specified parameters in the investigated potentials, except the mixed one. It becomes possible to distinguish the potentials by the accurate observing $dw/da$ and $d^{2}w/d{a}^2$ in some parameters.

38 pages, 7 figures, Section V is added and improved, PTEP (accepted)

References in corpus (6)

• Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests
• Quintessence: A Review
• Observational Constraints on Quintessence: Thawing, Tracker, and Scaling models
• Hilltop Quintessence
• A one-parameter formula for testing slow-roll dark energy: observational prospects
• Time variation of the Equation of State for Dark Energy

Cited by in corpus (2)

• Mass varying neutrinos, symmetry breaking, and cosmic acceleration
• Constraints for the thawing and freezing potentials