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Method

We took a population of of 20 574 stars for which there are complete and accurate distance and velocity measurements. The spread of velocities in the local population is much greater than the suggested error in radial velocity which would account for the flattening of rotation curves. A test is required which will not be affected by the structure of the velocity distribution, real velocity gradients, bulk streams and moving groups. To test for a signature in such a noisy distribution we binned the population into 20 bins, each containing around 1 000 stars, and tested the velocity components, U, towards the Galactic centre, V, in the direction of Galactic rotation, and W, perpendicular to the Galactic plane. Testing the component on a particular axis avoids correlations arising from the structure of the velocity distribution.

We plotted the component of velocity, vaxis, in the direction of the axis, against the cosine of the angle, θ, subtended by the star with that axis. The four quadrants of the plot represent stars positioned in either direction along the axis (quadrants I & IV opposed to quadrants II & III), and stars approaching (quadrants II & IV) and receding (quadrants I & III). Under the null hypothesis, that there is no systematic error in spectrographic measurement of radial velocity, there should be a 50-50 split of plots with absolute component of velocity increasing or decreasing with abs(cos θ). Trials showing increasing abs(vaxis) with abs(cos θ) were designated passes for the alternate hypothesis, radial velocities are overstated. The plot shows four passes in one bin on the W-axis. Correlations are low but the total number of quadrants with absolute value of the component of velocity increasing with the absolute value of the cosine is significant.

For stars with equal true velocity and different positions, an error in radial velocity would contribute more to vaxis for stars which subtend a narrow angle with the axis (horizontal), and would tend to generate a correlation between vaxis and cos θ in each of the four quadrants. Separate tests are used in each quadrant. Real velocity gradients and bulk streaming motions would bias particular quadrants towards passes or fails, but would still produce a 50% pass rate under the null hypothesis.

Test Results

The overall result from 240 quadrants was 140 passes, leading us to reject the null hypothesis with a confidence of 99.4%. Because outliers in regression have a disproportionate effect on results, it is normal to restrict the population to within 3 or fewer standard deviations of the mean. When we restricted to a velocity ellipsoid containing 14 914 stars representing the bulk of thin disc motions, the the number of passes rose to 154, leading to rejection of the null hypothesis with a confidence of 99.9993%. A velocity ellipsoid has no dependency on space coordinates, so does not introduce a bias.

The result on the W-axis is particularly significant because the Sun is close to the Galactic plane, where abs(W) should be at a maximum. We should therefore expect less than a 50% pass rate under the null hypotheses for this axis. In fact there were 60 passes out of 80 tests on this axis, rising to 63 passes out of 80 tests in the velocity ellipsoid, rejecting the null hypothesis with 99.99999% confidence.

The results from 80 tests on each axis show that the components of radial velocity in the V- and W-directions are overstated, but there is no evidence that the component in the U-direction is overstated. In fact there is some indication that the U-component is not overstated. In consequence we may reject the possibility that the results are due to a systematic understatement of Hipparcos parallax distance which would affect the U- V- and W-directions equally.

For the same reason we may reject the possibility that the result is due to truncation bias arising from the fact that measurement errors in radial velocity are slightly larger than those in transverse velocity; the division of the population into stars approaching and stars receding cuts stars which cross the horizontal axis because of measurement errors, and could potentially produce a bias towards passes. Estimates of the magnitude of this bias show that it is greatly outweighed by random factors in the motions of stars, and the reversed correlation on the U-axis shows that this is not the cause of the correlation on the other axes.

Adjusting radial distances by +10% substantially reduces the pass rate, as would be expected, but even with this increase we found 93 successes out of 160 trials on the V- and W-axes for the population in the velocity ellipsoid, rejecting the null hypothesis with a confidence of 97.6%. This shows that the error in radial velocity is of an order greater than 10% and eliminates the possibility that the result could be caused by systematic measurement errors.

Moving groups could potentially affect results. If a group is localised in space, the stars in a given moving group will appear in the same quadrant in each test. Under the null hypothesis, the chance of the gradient in that quadrant being positive or negative is still 50%, but, because group stars are split among the tests, it could potentially be the case that a particular quadrant repeats the same result in several tests because of group stars. Very high results in a particular quadrant would be evidence that the result is due to a moving group, and reduce the significance of the overall result. In fact no quadrants individually show significantly higher results than others. If the result were caused by moving groups, then it would also be dependent on the choice of axis. We rotated the V- and W-axes through 22.5°, 45° and 67.5°, and were able to reject the null hypothesis in each direction. We therefore reject the possibility that the result is due to moving groups.

Implications

Determination of spectral shift is straightforward, well established, and not in itself open to systematic measurement errors of the type seen in the test. The results cannot be accounted through systematic distance adjustments, because there is no observed correlation in the radial direction of the Galaxy. Velocity components are not expected to vary greatly with position over the distances of stars tested and a simple velocity gradient could not be responsible for the results, because this would produce as many fails as passes. If one rejects the notion that the Sun occupies a preferred position in space such that other stars tend to move radially towards and from the Sun, the principle conclusion one can draw is that there is a systematic overstatement in radial velocities.

The cosmological redshift prediction of general relativity based on classical wave motions is clear, but general relativity does not consider the possibility that photons from astronomical objects should be described using quantum theory. In relational quantum gravity, light from distant stars is treated quantum mechanically. As a result spectral shifts have a cosmological component in addition to the accepted Doppler component. To rigorously test this prediction it is necessary to compare astrometric radial velocities with spectrographic radial velocities for individual stars. This will be possible for near, high velocity, stars with Gaia», but cannot be done at current astrometric precision. We do not know of any other cosmological model which modifies spectral shifts without modifying the laws of classical motion in general relativity. The statistical test described here shows, to very high confidence, that spectroscopic radial velocity is overstated, and can be construed as a success for the prediction of relational quantum gravity.

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