  It is discussed the possibility of a fine-tuneable contribution to the two
way Doppler acceleration either towards, either outwards the Sun for
heliocentric distances above 20 AU by considering a background described by an
Expanding Locally Anisotropic (ELA) metric. This metric encodes both the
standard local Schwarzschild gravitational effects and the cosmological
Universe expansion effects allowing simultaneously to fine-tune other
gravitational effects at intermediate scales, which may be tentatively
interpreted as a covariant parameterization of either cold dark matter either
gravitational interaction corrections. Are derived bounds for the ELA metric
functional parameter by considering the bounds on the deviation from standard
General Relativity imposed by the current updated limits for the Pioneer
anomaly, taking in consideration both the natural outgassing and on-board
radiation pressure, resulting in an average Doppler acceleration outwards the
Sun of a_p = +0.4^{+2.1}_{-2.0} x 10^{-10} (m/s^2). It is also computed the
mass-energy density for the ELA metric within the bounds obtained and are
discussed the respective contributions to the cosmological mass-energy density
which, for compatibility with the Lambda-CDM model, are included in
Omega_{CDM}.
