Hořava–Lifshitz F(R¯) Theories and the Swampland

Abstract

The compatibility between the de Sitter Swampland conjecture and Hořava–Lifshitz F(R¯) theories with a flat FLRW metric is studied. We first study the standard f(R) theories and show that the only way in which the dS conjecture can be made independent of R is by considering a power law of the form f(R)∼Rγ. The conjecture and the consistency of the theory puts restrictions on γ to be greater but close to one. For F(R¯) theories described by its two parameters λ and μ, we use the equations of motion to construct the function starting with an ansatz for the scale factor in the Jordan frame of the power law form. By performing a conformal transformation on the three metric to the Einstein frame, we can obtain an action of gravity plus a scalar field by relating the parameters of the theory. The non-projectable and projectable cases are studied and the differences are outlined. The obtained F(R¯) function consists of terms of the form R¯γ with the possibility of having negative power terms. The dS conjecture leads to inequalities for the λ parameter; in both versions, it becomes restricted to be greater but close to 1/3. We can also study the general case in which μ and λ are considered as independent. The obtained F function has the same form as before. The consistency of the theory and the dS conjecture lead to a set of inequalities on both parameters that are studied numerically. In all cases, λ is restricted by μ around 1/3, and we obtain λ→1/3 if μ→0. We consider the f(R) limit μ,λ→1 and we obtain consistent results. Finally, we study the case of a constant Hubble parameter. The dS conjecture can be fulfilled by restricting the parameters of the theory; however, the constraint makes this compatibility exclusive to these kinds of theories.