Disforming to Conformal Symmetry

Abstract

We analyse the dynamical properties of disformally transformed theories of gravity.We show that disformal transformation typically introduces novel degrees of freedom, equivalent to the mimetic dark matter, which possesses a Weyl-invariant formulation. We demonstrate that this phenomenon occurs in a wider variety of disformal transformations than previously thought.

The Variational principle, Conformal and Disformal transformations, and the degrees of freedom

“…Note that in their new paper [5] the Authors of the Ref. [4] also give examples which, in my notations here, can be taken as substitutions of the x = z + ẏ kind. In particular, this relation does not change anything for the dynamics of x(t) since its arbitrary variation can be given in terms of δz which has no extra time derivative.…”

“…And then, in their next paper [5], the Authors of the Ref. [4] have used precisely the x 3 -type trick in order to construct yet another mimetic model with an ill-defined number of degrees of freedom.…”

“…Actually, both conditions of non-invertibility (or let's say, singularity conditions, to account for invertible transformations with a zero derivative locus), from the old paper [3] and the one from considering the Jacobian matrix [4], can be traced back to the very same equation ( 4). Indeed, one can easily find…”

“…An important fact recently noticed in the Paper [4] is that, even in absence of a full-fledged local symmetry for the auxiliary metric, the relation between the two metrics depends on the scalar field φ(x) and might fail to be bijective at particular configurations of it. Moreover, some configurations like that seem to be unavoidable for almost any transformation with non-trivial dependence on the gradients.…”

“…It was presented in the Ref. [4] as a blessing, as a benediction to introduce even many more new models of Mimetic Dark Matter and to study the new physics. However, this non-constant rank of the Jacobian matrix of the transformation defining the model seems to me rather as a malediction, with the threats of an ill-defined number of degrees of freedom.…”