Dynamical analysis in scalar field cosmology

Abstract

We give a general method to find exact cosmological solutions for scalar-field dark energy in the presence of perfect fluids. We use the existence of invariant transformations for the Wheeler De Witt (WdW) equation. We show that the existence of a point transformation under which the WdW equation is invariant is equivalent to the existence of conservation laws for the field equations, which indicates the existence of analytical solutions. We extend previous work by providing exact solutions for the Hubble parameter and the effective dark-energy equation of state parameter for cosmologies containing a combination of perfect fluid and a scalar field whose self-interaction potential is a power of hyperbolic functions. We find solutions explicity when the perfect fluid is radiation or cold dark matter and determine the effects of non-zero spatial curvature. Using the Planck 2015 data, we determine the evolution of the effective equation of state of the dark energy. Finally, we study the global dynamics using dimensionless variables. We find that if the current cosmological model is Liouville integrable (admits conservation laws) then there is a unique stable point which describes the de-Sitter phase of the universe.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

Cosmological evolution and exact solutions in a fourth-order theory of gravity

“…This means that the theory provides us with a cosmological constant term, a dust term and a stiff fluid, equivalently with that of the minimally coupled scalar field [60].…”

f ( R )-gravity from Killing tensors

“…Recently the same analysis has been performed and for a general perfect fluid with nonzero equation of state parameter [57], whereas for some locally rotational spacetimes the Noether point symmetry classification of f (R)-cosmology can be found in [60]. Some other f (R)-models with closed-form solutions can be found in [61], while the application of point transformations in f (R)-gravity in static spherically symmetric spacetimes can be found in [62,63] Another geometric selection rule which is based upon the group invariant transformations of the Wheeler-DeWitt equations was introduced in [64]. It has been shown that the existence of a Lie point symmetry for the WheelerDeWitt equation is equivalent with the existence of a Noetherian conservation law for the classical field equations.…”

Exact solution of the Einstein–Skyrme model in a Kantowski–Sachs spacetime

“…Furthermore, following the method which was established in [19] for scalar field cosmology with a matter source, we determine Noetherian conservation laws for the field equations. The existence of the new conservation laws implies that the field equations are Liouville integrable and with the method of separation of variables we write the analytical solution of the model.…”