A distant galaxy has a redshift \(z=5.82\) and a recessional velocity \(v_{r}=287,000 \mathrm{km} / \mathrm{s}\) (about 96 percent of the speed of light). a. If \(H_{\mathrm{o}}=70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\) and if Hubble's law remains valid out to such a large distance, then how far away is this galaxy? b. Assuming a Hubble time of 13.7 billion years, how old was the universe at the look-back time of this galaxy? c. What was the scale factor of the universe at that time?

Redshift is a key concept in cosmology. It refers to the phenomenon where light from distant objects in the universe is shifted towards the red end of the spectrum. This happens because the universe is expanding, stretching the light waves as they travel towards us. The redshift value is often denoted as 'z'. For example, a redshift value of \( z = 5.82 \) tells us that the wavelength of light from the galaxy has been stretched by a factor of 5.82.
To calculate the look-back time, distance, and scale factor, redshift serves as a crucial starting point. It directly connects to how much the universe has expanded since the light left the galaxy.

Hubble's Law is fundamental to understanding the expanding universe. It states that the recessional velocity of a galaxy (\( v_r \)) is proportional to its distance from us (\( d \)). The relationship can be represented by the equation:\[ v_r = H_0 \times d \] where \( H_0 \) is the Hubble constant, representing the rate of expansion of the universe. In our exercise, \( H_0 = 70 \mathrm{km/s/Mpc} \), and the recessional velocity of the galaxy is \( v_r = 287,000 \mathrm{km/s} \). By rearranging the equation:\[ d = \frac{v_r}{H_0} \] and substituting the given values, we find the distance to the galaxy:\[ d = 4100 \mathrm{Mpc} \]

Look-back time is the time it has taken for the light from a distant object, like a galaxy, to reach us. It essentially tells us how far back in time we are observing the universe. Look-back time can be estimated using the redshift value and the Hubble time (the inverse of the Hubble constant, which gives a rough age of the universe).
The formula for look-back time is:
\[ \text{Look-back time} \approx \frac{z}{(1+z)} \times \text{Hubble time} \]\( \text{Hubble time} = 13.7 \text{billion years} \) in this context. Using \( z = 5.82 \), we find:
\[ \text{Look-back time} \approx \frac{5.82}{(1+5.82)} \times 13.7 \text{billion years} \approx 11.8 \text{billion years} \]
This means that we are looking back 11.8 billion years into the past!

The scale factor, denoted as 'a,' describes how the size of the universe has changed over time. A scale factor of 1 corresponds to the current size of the universe. In earlier times, the scale factor was smaller, indicating the universe was more compact. The scale factor can be calculated using redshift with the formula:
\[ a = \frac{1}{1+z} \]
For our example with \( z = 5.82 \):
\[ a = \frac{1}{1+5.82} \approx 0.146 \]
This tells us that the universe was only about 14.6% of its current size when the light from this galaxy was emitted.

Understanding the distance to galaxies is crucial for mapping the universe. Using Hubble's Law, we can calculate the distance through the relationship between recessional velocity and the Hubble constant. Here, we already found the distance via:
\[ d = \frac{v_r}{H_0} \]
Substituting the values of \( v_r = 287,000 \mathrm{km/s} \) and \( H_0 = 70 \mathrm{km/s/Mpc} \), we get:
\[ d = 4100 \mathrm{Mpc} \]
This distance measure helps us understand how far away the galaxy is and provides context for other observations involving redshift and look-back time.