The linear approximation to this perturbation, which works in the limit of small delta rho/rho is the following:
We can apply this equation to the case of Virgo Infall as schematically shown below:
The Virgo cluster is a nearby cluster or region of mass overdensity. Our galaxy feels the gravitational pull of Virgo and therefore our expansion velocity, with respect to Virgo is retarded. The diagram above shows three components:
For now ,let's just consider our infall into Virgo.
But we can only measure delta rho/rho in light when its really delta rho/rho in mass that is causing our velocity perturbation. Hence bias or the b parameter becomes important.
The measured delta rho/rho in light is 1.9. If we assume there is no bias then we have:
which yields OMEGA = 0.15
Suppose there is bias such that we really area measuring
and hence we only have (OMEGA)/b = 0.15
For OMEGA = 1 this would require b = 6.6 - which is a huge bias; in practice b =2.4 is the maximum which is allowed by observation and this yields OMEGA = 0.36. Recent data, however, strongly suggests that b is not larger than 1.5 which is OMEGA = 0.23.
Hence, small scale infall values are generally too small to be consistent with OMEGA = 1. However, let's be insistent that OMEGA=1 and ask then what the density constrast must be:
if b = 1 then delta rho/rho = 0.6 but delta rho/rho = 1.9 in light and hence there must be bias to recover OMEGA = 1. But the bias would have to be too large.
What about larger scales? Next time we will journey to the Great Attractor to see what we can learn.