Hořava–Lifshitz <semantics>F(<mover accent="true">R¯</mover>)</semantics> Theories and the Swampland - Hugo García-Compeán; Daniel Mata-Pacheco; Luis Zapata (Universe, 2023)
The compatibility between the de Sitter Swampland conjecture and Hořava–Lifshitz <semantics>F(<mover accent="true">R¯</mover>)</semantics> theories with a flat FLRW metric is studied. We first study the standard <semantics>f(R)</semantics> 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 <semantics>f(R)∼Rγ</semantics>. The conjecture and the consistency of the theory puts restrictions on <semantics>γ</semantics> to be greater but close to one. For <semantics>F(<mover accent="true">R¯</mover>)</semantics> theories described by its two parameters <semantics>λ</semantics> and <semantics>μ</semantics>, 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 <semantics>F(<mover accent="true">R¯</mover>)</semantics> function consists of terms of the form <semantics><mover accent="true">R¯</mover>γ</semantics> with the possibility of having negative power terms. The dS conjecture leads to inequalities for the <semantics>λ</semantics> parameter; in both versions, it becomes restricted to be greater but close to <semantics>1/3</semantics>. We can also study the general case in which <semantics>μ</semantics> and <semantics>λ</semantics> 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, <semantics>λ</semantics> is restricted by <semantics>μ</semantics> around <semantics>1/3</semantics>, and we obtain <semantics>λ→1/3</semantics> if <semantics>μ→0</semantics>. We consider the <semantics>f(R)</semantics> limit <semantics>μ,λ→1</semantics> 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.