The a0 Redshift Law

01The evidence

• The radial acceleration relation is uncomfortably tight. Across the SPARC sample of 175 galaxies, the observed acceleration follows a single function of the baryonic acceleration with very little intrinsic scatter, and the fit returns a characteristic scale near 1.2 times ten to the minus ten metres per second squared.SPARC full sample analysis
• Milgrom proposed that the acceleration scale is not a new force but a consequence of the vacuum defining inertia, which connects it to the cosmic horizon rather than to any property of a galaxy.Milgrom, The Modified Dynamics as a Vacuum Effect

02The tension

The scale is treated as a universal constant, fitted once from nearby galaxies and applied everywhere. That is a strange status for a quantity that appears in no fundamental theory. Meanwhile high-redshift rotation curves are only now becoming measurable, and the first samples are small enough that nobody has tested whether the same constant applies.

03The theory

If a0 comes from the vacuum, and the vacuum state in an expanding universe is characterised by the Hubble scale, then a0 must be of order the speed of light times the expansion rate, up to a dimensionless factor. Locally that reproduces the observed value with a factor of about six.

The consequence is that a0 evolves. At redshift two, the expansion rate was roughly three times larger, so the acceleration scale should have been roughly three times larger too. A galaxy of fixed baryonic mass would show a departure from Newtonian dynamics at a larger radius and with a larger amplitude than the local relation predicts.

This is not a free parameter dressed up as a prediction. The shape of the evolution is fixed once the vacuum interpretation is accepted. Only the overall normalisation is adjustable, and that is already pinned by nearby galaxies.

The observable consequence is a redshift-dependent radial acceleration relation. Fit the relation separately in redshift bins and the fitted scale should climb with redshift along a curve fixed by the expansion history. If instead it stays flat, then a0 is a constant of nature with no known origin, and the vacuum interpretation is wrong.

04What this is not

05Predictions

• P1Rotation curves at redshift two should require an acceleration scale about three times the local value, producing a detectable upturn at larger radii than a constant-scale fit allows.
• P2Fitting the radial acceleration relation with a fixed scale should produce scatter that grows with redshift in a predictable way.
• P3The normalisation should satisfy a0 approximately equal to one sixth of the speed of light times the present expansion rate.

06What would refute it

A measured acceleration scale at redshift two within ten per cent of the local value would kill the vacuum interpretation, since the predicted change is a factor of three.

07What it would change

It converts a phenomenological fit into a cosmological prediction, and makes high-redshift rotation curves a test of the vacuum rather than a test of dark matter.