# Redshift (Ryden 5.2), confusing step.

• Clever-Name
In summary, we have shown that the observed redshift changes at a rate of \frac{dz}{dt_{0}} = H_{0}(1+z) - H_{0}(1+z)^{3(1+w)/2}, where H_0 is the Hubble constant and w is the equation of state parameter for the component in the universe.
Clever-Name

## Homework Statement

A light source in a flat, single-component universe has a redshift z when observed at a time $t_{0}$. Show that the observed redshift changes at a rate

$$\frac{dz}{dt_{0}} = H_{0}(1+z) - H_{0}(1+z)^{3(1+w)/2}$$

## Homework Equations

$$H_{0} = (\frac{\dot{a}}{a})|_{t = t_{0}} = \frac{2}{3(1+w)t_{0}}$$

$$(1+z) = \frac{a_{t_{0}}}{a_{t_e}} = \left( \frac{t_{0}}{t_{e}}\right )^{2/3(1+w)}$$

$$\frac{dz}{dt_{0}} = \frac{dz}{da_{0}}\frac{da_{0}}{dt_{0}} + \frac{dz}{da_{e}}\frac{da_{e}}{dt_{e}}\frac{dt_{e}}{dt_{0}}$$

$$t_{e} = \frac{t_0}{(1+z)^{3(1+w)/2}}$$

w is the component index (matter, radiation, lambda), not sure what its formal name is.

## The Attempt at a Solution

Everything goes ok following my work below until the last step:

$$\frac{dz}{dt_{0}} = \frac{2}{3(1+w)t_{0}}\left( \frac{t_{0}}{t_{e}} \right)^{2/3(1+w)} - \frac{2}{3(1+w)t_{e}}\left( \frac{t_{0}}{t_{e}} \right)^{2/3(1+w)}\frac{dt_{e}}{dt_{0}}$$

$$= H_{0}(1+z) - \frac{2}{3(1+w)t_{e}}(1+z)\frac{dt_{e}}{dt_{0}}$$
Here is where I am stuck. Using the definition for $t_{e}$, as given in the chapter, we get:

$$\frac{dt_{e}}{dt_{0}} = (1+z)^{-3(1+w)/2} = \frac{t_e}{t_{0}}$$

But when subbing this in we end up with 0!

I have no idea how to proceed.

Last edited:
Please help!First of all, let's clarify some things. The component index w is usually called the equation of state parameter, and it represents the ratio of the pressure to the energy density of the component. For example, for matter w=0, for radiation w=1/3, and for the cosmological constant w=-1.

Now, let's look at your attempt at a solution. The first line is correct, but in the second line, you have made a mistake. The correct expression for dt_e/dt_0 is:

\frac{dt_e}{dt_0} = (1+z)^{-3(1+w)/2} = \frac{t_0}{t_e}

Here, you have used the relation t_e = t_0/(1+z)^{3(1+w)/2}, but you have forgotten to account for the fact that t_e is a function of z. So the correct expression should be:

\frac{dt_e}{dt_0} = \frac{t_0}{t_e}\frac{dt_e}{dz}\frac{dz}{dt_0}

Substituting this into your expression, we get:

\frac{dz}{dt_0} = H_0(1+z) - \frac{2}{3(1+w)t_e}\frac{t_0}{t_e}\frac{dt_e}{dz}\frac{dz}{dt_0}

Now, we can cancel out the dz/dt_0 terms, and we are left with:

\frac{dz}{dt_0} = H_0(1+z) - \frac{2}{3(1+w)t_e}\frac{t_0}{t_e}\frac{dt_e}{dz}

Finally, we can use the relation dt_e/dz = -t_e(1+z)^{-1}, and we get the final expression:

\frac{dz}{dt_0} = H_0(1+z) - \frac{2}{3(1+w)}\frac{t_0}{t_e}(1+z)^{-1}

Now, we can substitute the relation t_e = t_0/(1+z)^{3(1+w)/2}, and we get the desired expression:

\frac{dz}{dt_0} = H_0(1+z) - H_0(1+z)^{3(1+w)/2

## 1. What is redshift (Ryden 5.2)?

Redshift (Ryden 5.2) is a measure of how much light from an object, such as a galaxy, has been shifted towards the red end of the electromagnetic spectrum. It is caused by the expansion of the universe and is used to calculate the distance and velocity of objects in space.

## 2. What is the difference between redshift and blueshift?

Redshift and blueshift are both caused by the Doppler effect, but in opposite directions. Redshift occurs when an object is moving away from us and the wavelength of its light is stretched, causing it to appear more red. Blueshift occurs when an object is moving towards us and the wavelength of its light is compressed, causing it to appear more blue.

## 3. How is redshift (Ryden 5.2) measured?

Redshift is measured by comparing the observed wavelength of an object's light to its expected wavelength. This can be done using a spectrometer which breaks down light into its component wavelengths. The amount of redshift is then calculated using a formula that takes into account the object's velocity and the speed of light.

## 4. What is the significance of redshift (Ryden 5.2) in astronomy?

Redshift is crucial in astronomy as it allows us to measure the distance and velocity of objects in the universe. It also provides evidence for the expansion of the universe and is used to estimate the age of the universe. Additionally, redshift can give insight into the evolution and movements of galaxies.

## 5. Is redshift (Ryden 5.2) a constant value?

No, redshift is not a constant value. It varies depending on the distance and velocity of the object, as well as the observer's position. Redshift can also be affected by gravitational forces, such as those from nearby massive objects, so it is not always a reliable indicator of distance and velocity.

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