  We analyse the role, on large cosmological scales and laboratory experiments,
of the leading curvature squared contributions to the low energy effective
action of gravity. We argue for a natural relationship $c_0\lambda^2\simeq 1$
at low-energy between the ${\cal R}^2$ coefficients $c_0$ of the Ricci scalar
squared term in this expansion and the dark energy scale $\Lambda=(\lambda
M_{\rm Pl})^4$ in four dimensional Planck mass units. We show how the
compatibility between the acceleration of the expansion rate of the Universe,
local tests of gravity and the quantum stability of the model all converge to
select such a relationship up to a coefficient which should be determined
experimentally. When embedding this low energy theory of gravity into
candidates for its ultraviolet completion, we find that the proposed
relationship is guaranteed in string-inspired supergravity models with modulus
stabilisation and supersymmetry breaking leading to de Sitter
compactifications. In this case, the scalar degree of freedom of ${\cal R}^2$
gravity is associated to a volume modulus. Once written in terms of a
scalar-tensor theory, the effective theory corresponds to a massive scalar
field coupled with the universal strength $\beta=1/\sqrt{6}$ to the matter
stress-energy tensor. When the relationship $c_0\lambda^2\simeq 1$ is realised
we find that on astrophysical scales and in cosmology the scalar field is
ultralocal and therefore no effect arises on such large scales. On the other
hand, the scalar field mass is tightly constrained by the non-observation of
fifth forces in torsion pendulum experiments such as E\"ot-Wash. It turns out
that the observation of the dark energy scale in cosmology implies that the
scalar field could be detectable by fifth force experiments in the near future.
