Age of Universe, Redshift & Scale Factor

[kingwinner]

Homework Statement

Assume that the current age of the universe is 13.4 billion years old, and that we live in a matter-dominated, omega_m = 1, critical universe, what is the age of the universe at redshift 0.6? HINT: use the current age of the universe to pin down the proportionality relationship between age and the scale factor

Homework Equations

Some formulas that I have found are:
1+z=R(t_o)/R(t) (redshift and scale factor)
H_o= cz/d (hubble constant)
omega_m=1 means k=0 and matter-dominated together with k=0 critical universe implies that the age of the universe t_o = 2/(3H_o)
For k=0 critical universe R(t) is proportional to t^(2/3)

The Attempt at a Solution

I know that (t_o) proportional to (1/H_o) proportional to (1/z), but I don't know how to connect this with the scale factor...and I have no idea how to "use the current age of the universe to pin down the proportionality relationship between age and the scale factor"


Can someone please help me out? I did all the reading but was still unable to solve this problem, no matter how hard I tried...

Thanks for helping!

 

[anathema]

Thank you for your question. I understand that this problem may seem challenging, but with a few key concepts and equations, you can solve it.

Firstly, let's define some terms. The scale factor, denoted by "a," is a measure of the expansion of the universe at a given time. It is equal to 1 at the current time, and increases as the universe expands. The redshift, denoted by "z," is a measure of how much the wavelength of light has been stretched due to the expansion of the universe.

Now, let's look at the relationship between the scale factor and the age of the universe. As you mentioned, for a matter-dominated universe with k=0, the scale factor is proportional to t^(2/3). This means that as the scale factor increases, the age of the universe also increases. In other words, as the universe expands, it gets older.

Next, let's look at the relationship between the scale factor and the redshift. As you mentioned, 1+z=R(t_o)/R(t). Since we are assuming that we live in a matter-dominated universe with k=0 and omega_m=1, we can simplify this equation to 1+z=1/a. This means that as the scale factor increases, the redshift decreases. In other words, as the universe expands, the light from distant objects becomes less redshifted.

Now, let's use these relationships to solve the problem. We know that the current age of the universe, t_o, is 13.4 billion years. We also know that at redshift z=0.6, the scale factor is a=1/(1+0.6)=0.625. Using the relationship between the scale factor and the age of the universe, we can set up the following equation:

t_o/t=0.625^(2/3)

Solving for t, we get t=13.4/0.625^(2/3)=13.4/0.625=21.44 billion years.

Therefore, the age of the universe at redshift 0.6 is 21.44 billion years. I hope this helps you understand the problem better. If you have any further questions, please don't hesitate to ask.

Scientist

 

[m2003]

I would like to start by acknowledging the complexity of this problem and the various equations and concepts involved. It is clear that you have put in a lot of effort in researching and attempting to solve this problem on your own. I will try my best to provide a clear and concise response.

Firstly, let's define some terms and concepts. The age of the universe refers to the time since the Big Bang, which is currently estimated to be 13.8 billion years old. Redshift is a measure of how much the light from distant objects has been stretched due to the expansion of the universe. It is denoted by the symbol 'z' and is related to the scale factor, denoted by the symbol 'a', which describes how the size of the universe changes over time.

Now, onto the problem at hand. We are given the current age of the universe, t_o = 13.4 billion years, and we are asked to find the age of the universe at a redshift of 0.6. To solve this, we need to use the relationship between redshift and scale factor, which is given by the equation: 1+z = a_o/a, where a_o is the scale factor at the present time and a is the scale factor at a particular time in the past.

We are also given that we live in a matter-dominated, omega_m = 1, critical universe. This means that the matter density of the universe is high enough to counteract the expansion and keep the universe at a critical density (meaning it will neither expand forever nor collapse back on itself). In this type of universe, the age of the universe is given by t_o = 2/(3H_o), where H_o is the Hubble constant, which describes the rate of expansion of the universe.

Now, let's put all of this together. We can use the equation 1+z = a_o/a to find the scale factor at redshift 0.6, which is equal to 0.417. Plugging this value into the equation for the age of the universe in a critical universe, we get t = 2/(3H_o*a^(3/2)). Since we are given that omega_m = 1, we can use the equation H_o = cz/d, where c is the speed of light and d is the distance to the object (in this case, the distance to a redshift of 0