Ultra-local models of modified gravity without kinetic term
  We present a class of modified-gravity theories which we call ultra-local
models. We add a scalar field, with negligible kinetic terms, to the
Einstein-Hilbert action. We also introduce a conformal coupling to matter. This
gives rise to a new screening mechanism which is not entirely due to the
non-linearity of the scalar field potential or the coupling function but to the
absence of the kinetic term. As a result this removes any fifth force between
isolated objects in vacuum. The predictions of these models only depend on a
single free function, as the potential and the coupling function are
degenerate, with an amplitude given by a parameter $\alpha \lesssim 10^{-6}$,
whose magnitude springs from requiring a small modification of Newton's
potential astrophysically and cosmologically. This singles out a redshift
$z_{\alpha} \sim \alpha^{-1/3} \gtrsim 100$ where the fifth force is the
greatest. The cosmological background follows the $\Lambda$-CDM history within
a $10^{-6}$ accuracy, while cosmological perturbations are significantly
enhanced (or damped) on small scales, $k \gtrsim 2 h {\rm Mpc}^{-1}$ at $z=0$.
The spherical collapse and the halo mass function are modified in the same
manner. We find that the modifications of gravity are greater for galactic or
sub-galactic structures. We also present a thermodynamic analysis of the
non-linear and inhomogeneous fifth-force regime where we find that the Universe
is not made more inhomogeneous before $z_\alpha$ when the fifth force
dominates, and does not lead to the existence of clumped matter on extra small
scales inside halos for large masses while this possibility exists for masses
$M\lesssim 10^{11} M_\odot$ where the phenomenology of ultra-local models would
be most different from $\Lambda$-CDM.
