According to hubble's law of universal expansion, the velocity of a galaxy moving away from ours is directly proportional to the distance between the two. Now velocity is increasing in direction away from us. Therefore accelaration and hence a force ( I will assume here gravitational force of repulsion!) acts between the two galaxies. Is this true? If not then why? Please explain.
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With standard Newtonian gravitation a space that is uniformly filled with some gravitating mass-energy acts repulsively. I wrote on this in a Stack Exchange post here. With just Newton's laws and gravity force $\vec F~=~\frac{GMm\vec r}{|\vec r|^3}$ it is not hard to show that gravitation acts repulsively. This constant mass-energy in space is thought to be the quantum vacuum. It acts to cause mass-energy or particles of mass-energy to "fall up," or to repulsively accelerate away from each other.
From a general relativistic perspective is is a bit more subtle. Instead of it being a force, it is that the manifold of space is constantly being stretched apart. Particles of matter are then frame dragged along with this constant stretching of space, where different time slices of space foliate spacetime.
Lawrence B. Crowell
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Hubble's Law does not say that individual galaxies are accelerating away from us.
Rather, it says that in comparing two galaxies, the one further from us is moving away from us faster than the closer one. That's how it got further away from us!
If you were to fire a shotgun vertically upward, all the pellets would be slowing down as they rose, pulled down by earth's gravity. But the pellets that came out of the barrel travelling faster, would always be higher that slower pellets...
DJohnM
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Matter (dark and visible) and radiation have an attractive effect, dark energy has a repulsive effect. The acceleration is proportional to the factor $\rho + 3p$ where $\rho$ is the density and $p$ is the pressure. So, if $p<-\tfrac{1}{3}\rho$ the factor $\rho + 3p$ becomes negative and the gravitational acceleration repulsive. Matter has the equation of state $p \approx 0$, radiation has $p = \tfrac{1}{3}\rho$, dark energy has $p=-\rho$, so only dark energy leads to repulsive gravity.
Photon
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