In the quantum vacuum there are many transient acceleration vectors of mean magnitude a randomly oriented. If the vacuum is viewed from an accelerated frame, the vectors going with the frame appear diminished, and the vectors going against the frame appear enhanced, resulting in a net polarization of the vacuum. If the frame's acceleration g is small, the effect is linear, and if the vacuum is filled with vectors the coefficient of the polarization will be unity. The standard exponential term for suppressing high-energy fluctuations must also be applied. Hence the vacuum polarization is g exp (g/a). The terms of the exponent when multiplied by the dipole moment have the dimensions of energy.

The rest frame of the galaxy, for example, is accelerated with respect to local inertial frames that fall into the center. In this rest frame the vacuum appears polarized and enhances the galaxy's gravitational field g. So we have

g= -GM/r^{2} + g exp (g/a)

where g is understood to be negative. For g much greater than a, the exponential is negligible and Newton's law results. But for g less than a, the exponential can be expanded to 1 + g/a and we get

g^{2 }= aGM/r^{2}

This is precisely the formula found empirically by Milgrom to explain the motion of stars and galaxies in the weak-field region, except the law of gravity is altered, not the law of motion (Scientific American, August 2002). He finds that a is about one Angstrom per second squared, which is near the "surface gravity" of an electron, the field of a one-kilogram mass at one meter, or the field of a galaxy in its outer parts.

The observations can be adequately explained by assuming a plausible amount of ordinary matter M and using the correct quantum law of gravity. There is no need for dark matter.

As space accelerates away from us, the resulting apparent polarization would enhance the acceleration, and indeed might cause the acceleration, once the process has begun, due perhaps to some disturbance long ago. If space is collapsing in some remote region, the same process would enhance the collapse. So the cosmos may consist of interspersed regions of expansion and collapse. When expansion becomes extreme, a big bang would result as virtual particles are ripped out of the vacuum. A collapsing region would produce a big crunch, where matter is crushed back into the vacuum. The whole process is presumably infinite and eternal.

The distance c^{2}/a is about a hundred billion light-years, a plausible estimate of the size of an expanding or collapsing region.

The vacuum polarization would be noticeable nearby, but disappear farther out, due to the exponential factor, leading us to infer that the acceleration of the universe is increasing, eliminating the need for dark energy.

The so-called dark attractor may simply be the nearest collapsing region.

The total gravitational field is equal to the tensor obtained from general relativity plus the vacuum polarization tensor.

These quantities are not really tensors. They just look like tensors.

If one falls freely toward the center of the galaxy, the space in which the center is embedded appears polarized, so one gets the same value of g as before.

One would not expect the amount and distribution of dark matter to be the the same in all galaxies, but one would expect the law of gravity to be the same, as observed.

We have assumed the principle of equivalence, that the acceleration of a test particle is equal to the acceleration of gravity, and that gravity includes vacuum polarization. If a nongravitational force is applied to a body, the acceleration of the body would be reduced by the amount of the vacuum polarization seen by the body. This effect is too small to observe.

A collapsing region of space would appear to have an inward gravitational field, which would have negative energy, which would cancel the energy of the matter being crushed. An expanding region of space would appear to have an outward gravitational field, which would have positive energy,which would be converted into particles in a big bang.

When matter is crushed out of existence in a big crunch, the entropy it carries also vanishes, so the entropy of the entire cosmos does not increase over the ages.

Vacuum polarization would cause a large expanse of empty space to become unstable and develop regions of expansion and collapse, giving rise to the world we know.

A self-sustaining universe might eliminate the need for God, or might suggest that God is a good engineer. Science cannot answer this question.

In this model, the big bang had a finite size in flat space, so if we are somewhat off center, the cosmic background would appear warmer in the direction of the center, as observed.