Хабблівське розширення: Як розширюється Всесвіт

In the twentieth century, scientists observing galaxies found that nearly all galaxies had a noticeable Doppler shift toward lower frequencies (usually called redshift). This meant that nearly all observable galaxies were moving away from us. Why was that?

The key observation was this: the farther away galaxies were away from us, the faster they seemed to be receding as well. Moreover, up to several hundred megaparsecs, the distance and velocity of recession seemed to be roughly proportional.

This led to a radical new idea: that space itself was expanding. Rather than treating the Big Bang as a typical explosion that sent matter flying out in all directions, scientists proposed an alternate view in which the Big Bang was the start of the expansion of space itself.

Numerous models have been made to demonstrate what is meant by this expansion of space. The one I find most clarifying is that of raisins embedded in a loaf of bread. The raisins may be thought of as galaxies, and the dough may be thought of as empty space. Rather than the raisins "flying apart," it is actually the expansion of the dough in between them that increases the distance.

Assuming that there is a fair resemblance between the Universe and this raisin bread, a simple geometric argument supports the previously mentioned idea.

Suppose that you have two points on a sphere of radius $R$. The distance between them is $S$. Then, some time later, the sphere has expanded to radius $R+dR$, and the distance is $S+dS$.

The following relation holds, assuming resemblance to the raisin bread model: $$\frac{S+dS}{S}=\frac{R+dR}{R}$$ Cross-multiplying, $$RS+R\cdot dS = RS+S\cdot dR$$ Cancelling on both sides and changing the differentials into the appropriate derivatives with respect to time yields: $$R\frac{dS}{dt}=S\frac{dR}{dt}$$ Note that $\frac{dS}{dt}$ is the rate of recession, which will be denoted $v$, and that $S$ is the distance between the points, which will be denoted $d$.

Calculating the rate of recession, $$v=d\left(\frac{1}{R}\frac{dR}{dt}\right)$$ Note that at a given time, both the radius and the rate of increase of radius are fixed. Thus, we have a clear, direct proportion between the distance and the rate of recession.

The proportionality constant is called Hubble's constant. It is typically denoted $H$, but since the value of $H$ has been measured to change with time, the current value is often specially denoted $H_0$.

The proportion established through observation is known as Hubble's Law. It is stated as follows: $$v=H_0d$$ where $v$ is the speed of the object moving away from us and $d$ is the distance of that object from us. The motion of objects in space solely as a result of this law is called Hubble flow.

Note that the dimension of Hubble's constant is $T^{-1}$. Thus, the SI unit is Hertz (Hz). However, the most commonly used unit in practice is kilometers per second per megaparsec $\left(\frac{km}{s\cdot Mpc}\right)$. This unit is used more due to the astronomically large distances needed in the actual application of Hubble's Law.