Evolution of gravitational orbits in the expanding universe

The gravitational action of the smooth energy-matter components filling in the universe can affect the orbit of a planetary system. Changes are related to the acceleration of the cosmological scale size $R$. In a universe with significant dark matter, a gravitational system expands or contracts according to the amount and equation of state of the dark energy. At present time, the Solar System, according to the $\Lambda \mathrm{CDM}$ scenario emerging from observational cosmology, should be expanding if we consider only the effect of the cosmological background. Its fate is determined by the equation of state of the dark energy alone. The mean motion and periastron precession of a planet are directly sensitive to $\ddot{R}/R$, whereas variations with time in the semimajor axis and eccentricity are related to its time variation. Actual bounds on the cosmological deceleration parameters $q_0$ from accurate astrometric data of perihelion precession and changes in the third Kepler’s law in the Solar System fall short of 10 orders of magnitude with respect to estimates from observational cosmology. Future radio-ranging measurements of outer planets could improve actual bounds by 5 orders of magnitude.

Figure 1

Orbital radius evolution for an Earth-like planet (orbital period $\sim 1\mathrm{year}$). The variation is put to zero at the transition redshift $z_{\mathrm{tr}}$ from a decelerated to an accelerated expansion. We consider flat models with $h=0.7$. The full line is for an Einstein-de Sitter model ${\Omega }_{\mathrm{M}0}=1$ (the acceleration gets asymptotically to zero at $z_{\mathrm{tr}}\to -1$), the dashed line for a $\Lambda \mathrm{CDM}$ model with ${\Omega }_{\mathrm{M}0}=0.3$ and $w_{\mathrm{X}0}=-1$ ($z_{\mathrm{tr}}\simeq 0.67$), the long-dashed line for ${\Omega }_{\mathrm{M}0}=0.3$ and phantom energy with $w_{\mathrm{X}0}=-2$ ($z_{\mathrm{tr}}\simeq 0.51$).

Figure 2

The curves represent the loci of points in the space of dark energy parameters, i.e. the ${\Omega }_{\mathrm{X}0}-w_{\mathrm{X}0}$ plane, for which the radius of a gravitationally bound system does not vary at the present time. Curves from the left to the right are for ${\Omega }_{\mathrm{M}0}=0.2$, 0.3, and 0.4, respectively. The region on the left side of each curve corresponds to positive expansion for the given ${\Omega }_{\mathrm{M}0}$.

Figure 3

Time (in units of ${10}^9/h$ years) needed for the dissociation of an Earth-like system ($P\sim 1\mathrm{year}$) in a phantom cosmology as a function of the equation of state of the dark energy. We have considered a flat model with ${\Omega }_{\mathrm{M}0}=0.3$.

Figure 4

Rate of variation of the orbital radius in the ${\Omega }_{\mathrm{M}0}-w_{\mathrm{X}0}$ plane in units of the rate occurring for a standard flat model with cosmological constant and ${\Omega }_{\mathrm{M}0}=0.3$. We assume flat models.