# PlanetPhysics/Expanding Universe

### Hubble's Law

A cornerstone of cosmology is the observation of an expanding universe. In 1929, Hubble published the constant of proportionality $H_{0}$  that has become to be known as Hubble's constant . The importance of this constant to the expanding universe is seen through Hubble's law, which states that the velocity of a galaxy is proportional to its distance from any point.

${\vec {v}}=H_{0}{\vec {r}}$

When Hubble discovered the expansion of the Universe, he came up with values of 500 and 530 [km/(s Mpc)]. Although remarkable for the time, astronomers today can measure distance with a lot more accuracy. The Hubble space telescope (HST) key project used the HST to measure the Hubble constant in multiple ways. The final combined result of these different methods, yielded a Hubble constant of 

$H_{0}=72\pm 8\,\,\,\,[kms^{-1}Mpc^{-1}]$

As an example, let us compute the Hubble constant using redshift data from the SIMBAD Astronomical Database and distances from papers involved in the HST key project. The SINBAD data comes in two formats, either it directly gives the velocity or it reports the redshift value, which is related to a nonrelativistic velocity by

$\displaystyle z = \frac{v}$

A table of selected objects and their NGC galaxy, with their receding velocities and distances is given below

 Object (NGC) SINBAD (v) [km/s] Distance [Mpc] type and Source 925 553 9.29 Cepheid variable stars  1326A 1718 18.7 Cepheid variable stars  1365 1662 18.3 Cepheid variable stars  1425 1508.1 22.2 Cepheid variable stars  2090 924.6 12.3 Cepheid variable stars  3031 -42 3.63 Cepheid variable stars  3198 663 14.5 Cepheid variable stars  3319 742.5 14.3 Cepheid variable stars  3351 778.5 10.05 Cepheid variable stars  3621 727 6.3 Cepheid variable stars  4321 1575 16.1 Cepheid variable stars  4414 718.5 19.1 Cepheid variable stars  4535 1964.4 16 Cepheid variable stars  4548 493.2 15.9 Cepheid variable stars  4725 1209.9 12.6 Cepheid variable stars  5457 241 7.4 Cepheid variable stars  7331 819.6 15.1 Cepheid variable stars  \\

Plotting the data and fitting it to a line gives us an estimate of Hubble's Constant. The slope of the line in the below graph is what we want

$H_{0}=78.2\,\,\,\,[kms^{-1}Mpc^{-1}]$

\begin{figure} \includegraphics[scale=.8]{graph.eps} \vspace{20 pt} \end{figure}

As we shall see shortly, Hubble's law does not imply that we are at the center of the universe. On the contrary, it means that there is no center. To demonstrate this concept, we will follow the suggestion of Liddle in . We set up a square grid showing Hubble's law and then transform the origin to another point to see that it also follows Hubble's law. First we set up the grid with equally spaced squares of unit length as shown in the first figure. If you are able to view the attached spreadsheet, note that the constant used was 0.25 with the the length of each side of the squares being 1. This means that at a distance of 1, the velocity vector will be of length 0.25 and at a distance of 2, a length of 0.5, etc.

\begin{figure} \includegraphics[scale=.8]{figure1a.eps} \vspace{20 pt} \end{figure}

The next step is to transform our origin to the red dot at (1,1). This is done by subtracting off the equivalent velocity vector at a distance to (1,1) from all vectors as shown by the pink vectors in the second figure.

\begin{figure} \includegraphics[scale=.8]{figure2a.eps} \vspace{20 pt} \end{figure}

After computing the subtraction (green vectors), we see how it looks like every point is receding away from our new origin. This is true for any other point we choose.

\begin{figure} \includegraphics[scale=.8]{figure3a.eps} \vspace{20 pt} \end{figure}

### Friedmann Equation

• Lots more to come, if you would like to help drop me an email