These panels illustrate the quantitative effect of Virgo infall. Without any correction to velocity, and plain Hubble diagram (upper left panel) shows a lot of scatter, a smallish Hubble constant, and a tendency for points near Virgo (large, colored points) to lie below points which are far from Virgo (small points). This is in fact just the S-wave, but blurred out because of the variety of projection angle of points on the sky.
When the velocities are progressively corrected for the effect of Virgo infall (measured in terms of its amplitude at the Local Group) we see that (a) the scatter diminishes, (b) the Hubble constant increases, and (c) the points near Virgo come into agreement with the distant points. This effect is maximized at vin = 200 to 300 km/s, and when we go beyond this amplitude to 400 and 500 km/s, we see an inverse S-wave developing, indicating that we have overcorrected the velocities.
I will present four attempts at a formal fit to the large scale flows around Virgo. I will use two samples, the first being all galaxies within 25 Mpc of Virgo (which includes Centaurus, but excludes Fornax), and the second being all the galaxies. I will fit two models, the first being just a Virgo infall model (plus Hubble flow), and the second being Virgo infall plus a quadrupole distortion of the Hubble flow. The motivation is that we certainly don't expect Virgo infall, even modified by an external quadrupole, to perfectly describe the flow field even restricted to within 25 Mpc. On the other hand, there is a lot of covariance between H0 and vin, and unless we include a significant tie to galaxies well beyond the Virgo supercluster we run the risk of incurring substantial, correlated errors in H0 and vin.
The Virgo infall model being used here is the usual spherical, inverse r-squared density profile, infalling as concentric Friedmann universes. The linear models which are often used substantially overestimate infall velocities within the Local Group's position, and therefore will lead to a systematically smaller vin. Virgo infall is significantly non-linear.
The results are summarized in the following table and the next four graphs. The "best guess" from all this fitting arises from an infall plus quadrupole model, and yields an infall velocity of 250 km/s (generated by the mass of Virgo), an net velocity of 390 km/s (the sum of Virgo infall and the quadrupole distortion of the Hubble flow), a distance to Virgo of 1390 km/s, and a residual quadrupole distortion of 0.2 in the Hubble constant. That is, within 25 Mpc or so, the Hubble flow is 1.2 the global flow in the SGX direction, and 0.9 the global flow in the SGY and SGZ directions. This is after the Virgo infall has been fitted and removed.
The first graph shows a Hubble diagram before and after correction for Virgo infall only. The sample fitted consists of galaxies closer to Virgo than 25 Mpc. Open points are galaxies which were not used in the fit.
The second graph shows a Hubble diagram before and after correction for Virgo infall only. The sample fitted consists of all galaxies in the SBF sample.
The third graph shows a Hubble diagrams with H0 * r subtracted before and after correction for Virgo infall and a quadrupole correction to the Hubble flow. The sample fitted consists of galaxies closer to Virgo than 25 Mpc.
The fourth graph shows a Hubble diagrams with H0 * r subtracted before and after correction for Virgo infall and a quadrupole correction to the Hubble flow. The sample fitted consists of all galaxies in the SBF sample.
The confidence contours on H0 and vin illustrated here demonstrate the large covariance between these parameters. It is a bit disappointing that we will never do better than this, given this sample, because the errors are mostly driven by random motions of galaxies, not distance error. A typical random motion might be 200 km/s, which at a distance of 1500 km/s is about 15 percent, much larger than the typical distance error. We can do better, of course, by including more galaxies within the supercluster, such as spirals with TF distances, or by getting more galaxies at large distance. The connoisseur will notice that these results differ little from the Aaronson et al (1982) result. For this application, at least, a fourfold improvement in distance accuracy has bought little improvement in modelling accuracy.
We can also look at the covariance between some of the other model parameters, such as density profile exponent and quadrupole amplitude. Although we fixed the infall exponent at (-)2, as suggested by the distribution of galaxies, we find that the distribution of mass pretty much agrees with this: light traces mass. There is a suggestion that a shallower density profile may fit the data slightly better, but we prefer to defer exploration of that issue until our photometry reconciliation.
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