# I Interpreting a scale factor vs. distance graph

#### naushaan

Summary
Hi guys, I'm having trouble with interpreting scale-factor and distance graphs of Type 1a supernovae.
'Imagine that you live in a different universe, which may have a different cosmology to our own. You measure the distances to and redshifts of a large number of Type Ia supernovae, and you use the redshifts to calculate the scale-factor of the universe at the time when the supernova exploded. You get the following graph of distances and scale-factors.'
The question is asking to deduce as much as you can from the plot below:

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#### phinds

Gold Member
So what have you come up with so far?

#### mathman

What is the meaning of scale factor here?

#### Arman777

Gold Member
From the graph you can find the Hubble distance and, if the speed of light is same for that universe, you can extract the Hubble constant for that universe. And by using Hubble constant you can find the Hubble Time. And thats all I guess

#### naushaan

So what have you come up with so far?
i've calculated redshift and velocities so far, but haven't really gotten far with the question otherwise

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#### naushaan

From the graph you can find the Hubble distance and, if the speed of light is same for that universe, you can extract the Hubble constant for that universe. And by using Hubble constant you can find the Hubble Time. And thats all I guess
will do that! thank you

#### phinds

Gold Member
i've calculated redshift and velocities so far, but haven't really gotten far with the question otherwise
And are you going to tell us what you got for those things?

#### Arman777

Gold Member
will do that! thank you
I guess you find the answer. I think that from the graph we can also assume that universe is homogenous and isotropic (?). But I am not sure ..

#### naushaan

And are you going to tell us what you got for those things?
so for redshift i got 0.53, and from then i calculated the hubble constant to be 7.6e24m/s. im struggling with how to interpret these findings.

#### naushaan

I guess you find the answer. I think that from the graph we can also assume that universe is homogenous and isotropic (?). But I am not sure ..
how would we know that?

#### davenn

Gold Member
i calculated the hubble constant to be 7.6e24m/s.
you have missed an important bit off the end of that

#### Arman777

Gold Member
so for redshift i got 0.53, and from then i calculated the hubble constant to be 7.6e24m/s. im struggling with how to interpret these findings.
Thats not quite right. First redshift value for what object ? Redshift is not some sort of a general property of space. You can find the corresponding redshift for an object, that has a distance d.

From the graph its clear that theres a linear relationship between $z$ and $d$. This was what Hubble observed in 1920.

Here the y axis, velocity, is just $cz$ since $z = v/c$. The x axis represents the distance in parsecs.

So the graph that you are given can be turned into Redshift vs Distance ($z$ vs $d$) graph. (You dont have to do that of course). And from that it can be turned into velocity vs distance graph which that is what Hubble did.

And you also know that, this linear relationship can be written as,

$cz = H_0d$

So from your graph we can find the value of the $H_0/c$ or the hubble distance by turning the scale factor into redshift for 2 points and measuring the slope of the line that connects these two points.

Now, if we assume that $c$ is the same in this imaginary universe as well, we can find the $H_0$ (If we cannot assume that $c$ is the same, then the only information you can get is $H_0/c$)

Do your calculations again. But be careful about the units. In your graph the distance is given as billion light years. However Hubble constant has units of $kms^{-1}Mpc^{-1}$.

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#### naushaan

Thats not quite right. First redshift value for what object ? Redshift is not some sort of a general property of space. You can find the corresponding redshift for an object, that has a distance d.

From the graph its clear that theres a linear relationship between $z$ and $d$. This was what Hubble observed in 1920.

View attachment 251791

Here the y axis, velocity, is just $cz$ since $z = v/c$. The x axis represents the distance in parsecs.

So the graph that you are given can be turned into Redshift vs Distance ($z$ vs $d$) graph. (You dont have to do that of course). And from that it can be turned into velocity vs distance graph which that is what Hubble did.

And you also know that, this linear relationship can be written as,

$cz = H_0d$

So from your graph we can find the value of the $H_0/c$ or the hubble distance by turning the scale factor into redshift for 2 points and measuring the slope of the line that connects these two points.

Now, if we assume that $c$ is the same in this imaginary universe as well, we can find the $H_0$ (If we cannot assume that $c$ is the same, then the only information you can get is $H_0/c$)

Do your calculations again. But be careful about the units. In your graph the distance is given as billion light years. However Hubble constant has units of $kms^{-1}Mpc^{-1}$.
this makes so much sense, thank you very much!

#### Bruce Wallman

I have looked at this question for some time. We are given redshift for SN1a. From that there is not a simple velocity calculation v=cz unless the SN1a is close by. Cosmology and the scale factor need to be taken into account. Does anyone have a simple way to go from redshift to recession velocity? Thanks.