The theory of statistical thermodynamics tells us the equipartition law of energy does not hold in the limit of very low temperatures. It is found the Debye model is very successful in explaining the experimental results for most of the solid objects. Motivated by this fact, we modify the entropic force formula which is proposed very recently. Since the Unruh temperature is proportional to the strength of the gravitational field, so the modified entropic force formula is an extension of the Newtonian gravity to the weak field. On the contrary, general relativity extends Newtonian gravity to the strong field case. Corresponding to Debye temperature, there exists a Debye acceleration $g_D$. It is found the Debye acceleration is $g_D={10}^{-15}\mathrm{N}{\mathrm{kg}}^{-1}$. This acceleration is very much smaller than the gravitational acceleration ${10}^{-4}\mathrm{N}{\mathrm{kg}}^{-1}$ which is felt by Neptune and the gravitational acceleration ${10}^{-10}\mathrm{N}{\mathrm{kg}}^{-1}$ felt by the Sun. Therefore, the modified entropic force can be very well approximated by the Newtonian gravity in the Solar System and in the Galaxy. With this Debye acceleration, we find the current cosmic speeding up can be explained without invoking any kind of dark energy.

• Received 26 January 2010

DOI:https://doi.org/10.1103/PhysRevD.81.087306