  Consistency relations involving the soft limit of the (n + 1)-correlator
functions of dark matter and galaxy overdensities can be obtained, both in real
and redshift space, thanks to the symmetries enjoyed by the Newtonian equations
of motion describing the dark matter and galaxy fluids coupled through gravity.
We study the implications of such symmetries for the theory of galaxy bias and
for the theories of modified gravity. We find that the invariance of the fluid
equations under a coordinate transformation that induces a long-wavelength
velocity constrain the bias to depend only on a set of invariants, while the
symmetry of such equations under Lifshitz scalings in the case of matter
domination allows one to compute the time-dependence of the coefficients in the
bias expansion. We also find that in theories of modified gravity which violate
the equivalence principle induce a violation of the consistency relation which
may be a signature for their observation. Thus, given adiabatic Gaussian
initial conditions, the observation of a deviation from the consistency
relation for galaxies would signal a break-down of the so-called non-local
Eulerian bias model or the violation of the equivalence principle in the
underlying theory of gravity.
