Hubble parameter and the potential of the cosmological scalar field

Abstract

We consider a homogeneous isotropic Universe filled with cold matter (with zero pressure) and dynamic dark energy in a form of a scalar field. For known scalar field potential V(φ), the Friedmann equations are reduced to a system of the first order equation for the Hubble parameter H(z) and the second order equation for the scalar field as functions of the redshift z. On the other hand, knowledge of H(z) allows us to get the scalar field potential in a parametric form for a known cold matter content and three dimensional curvature parameter. We analyze when the accepted model mimics the dependence H(z) derived in the framework of the other models, e.g., hydrodynamic ones. Two examples of this mimicry are considered. The first one deals with the case when H2(z)~ Ωm(1+z)3+ΩΛ, but Ωm parameter overestimates the input of the cold matter (dark matter+baryons). The resulting scalar field potential is V(φ)=a+bsinh2(cφ), where the constants a,b,c depend on the Ω – parameters of the problem. In the other example we assume that some part of the dark matter has a non-zero equation of state p=wε, -1<w<1. In this case H2(z)~ Ωdm1(1+z)3(1+w)+ Ωb+Ωdm2)(1+z)3+ΩΛ. The corresponding potentials are defined for positive values of φ. For both signs of w potential V(φ) is a monotonically increasing function with typically an asymptotically exponential behavior; though for some choice of parameters we may have a singularity of V(φ)on a finite interval. Then we consider fitting of the potential for w from the interval [-0.2,0.2] for three different values of Ωdm2 by means of a simple formula Vfit(φ)=p0+p1exp(p2 φ). The dependencies pi(w) are presented and the approximation error is estimated.