23 - Growth of the two-galaxy correlation function

Summary

Two are better than one; because they have a good reward for their labour.

Ecclesiastes

If we look out from any point in an isotropic, homogeneous Universe it must appear to expand in the same way. This expansion, mentioned in Section 21, has the form H(t)x with H = Ṙ/R when the coordinate system is chosen to coincide with a point of zero systematic motion.

To see this pretend to be a ‘fundamental observer’ at point O, moving with the average flow. Looking out at time t to an arbitrary point P along the position vector x = OP, you see the velocity v(x, t) of P relative to you. Another fundamental observer is moving with the flow at O′ and his velocity relative to you is v(s, t) where s = OO′. (The reader may find it helpful to draw a diagram.) He measures the velocity of the same point P, located at x′ = x - s in his coordinate system, and finds it to be v′(x′, t) = v′(x - s, t) = v(x, t) - v(s, t). Now, since the Universe is homogeneous and isotropic, v′ must be the same function of x′ and t that v is of x and t. Therefore, v(x - s, t) = v(x, t) - v(s, t). By inspection the solution of this functional equation is that the velocity is a linear function of position, so it has the form v = f(t)x = Ẉ.