Analytical solutions in R + qRn cosmology from singularity analysis

• It is proven that the f(R)=R+qRnf(R)=R+qRn gravity pass the singularity test.
• The solution of the field equation can be expressed in terms of Laurent expansions.
• The scalar-field equivalence potential is calculated and is a new integrable potential.

The integrability of higher-order theories of gravity is of importance in the determining the properties of these models and so their viability as models of reality. An important tool in the establishment of integrability is the singularity analysis. We apply this analysis to the case of fourth-order theory of gravity f(R)=R+qRnf(R)=R+qRn to establish those values of the free parameters q and n   for which integrability in this sense exists. As a preliminary we examine the well-known case of n=4/3n=4/3.