ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O
tokens: with Equation ( 5 ). From Figure 1 it is expected that data points at high enough accelerations do not have any sensitivity to e . Indeed, for data points with −10.3 x 0 −9 the distribution of Δ ⊥ gives a null result. Similarly, for galaxies with , the distribution of e (median value) gives a null result. Figure 4. Distributions of Δ ⊥ and e from fitting Equation ( 5 ) to the SPARC galaxies. The top panels show the distributions of orthogonal residuals Δ ⊥ for three acceleration bins from the MCMC results shown in the middle column of Figure 3 . The mean of the distribution is displaced from zero for lower acceleration bins, indicating declining RCs. The bottom panels show the distributions of the e values fitted to the individual galaxies binned by the median values of x 0 within the galaxies. As expected, for the galaxies in the high acceleration bin (), the data do not have any sensitivity to e and so the distribution has a mean of ∼0. For lower acceleration bins the distributions are shifted to positive e with high statistical significance, indicating a preference for the EFE. The broad distributions are due to the broad individual posteriors on e . Data points at low enough accelerations will have sensitivity to e and distributions with nonzero mean value are expected from Figure 3 . For data points with x 0 −11.3 the distribution of Δ ⊥ has a mean of −0.061 ± 0.008 (a bootstrap error) which is statistically significant at more than 7 σ . For a much larger number of data points with x 0 −10.3, Δ ⊥ has a smaller deviation of −0.035 ± 0.003, but the statistical significance of the deviation is more than 11 σ . Figure 5 shows individual e values and their uncertainties for a subset of 113 galaxies with median . Due to the large uncertainties on e , some galaxies can occasionally return negative values. However, the median value of e is 0.052 ± 0.011 (bootstrap error), which represents ≈5 σ detection of positive e . This value is statistically consistent with the median environmental gravitational field for these galaxies ( (bootstrap error)). Furthermore, based on the robust binomial statistic with equal probabilities for e > 0 and e 0, 78 cases of e > 0 out of 113 is 4 σ away from the expected mean of 56.5 cases. Figure 5. Statistical detection of the EFE. The median values of e from rotation-curve fits are compared with e env values from the large-scale galaxy environment (Desmond et al. 2018 ) for 113 galaxies with . The heat map considers the posterior probabilities of individual e measurements. The distributions of e and e env are shown by the top and right histograms. The median value of e is clearly offset from zero, indicating a 5 σ statistical detection of the EFE. The value of is statistically consistent with (see also Appendix B ). The individual galaxies considered in Figure 2 are indicated: for the golden galaxies at exceptionally high e env values, e is significantly different from zero at 8 σ (NGC 5033) and 11 σ (NGC