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 } \approx + 0.4 ^ { +2.1 } _ { -2.0 } \times 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 } .