# Interpreting a scale factor vs. distance graph

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• naushaan
In summary, the conversation discusses an imaginary universe with a different cosmology, where the distances and redshifts of Type Ia supernovae are measured to calculate the scale-factor of the universe. By analyzing a graph of distances and scale-factors, it is possible to find the Hubble distance and constant for this universe, assuming the speed of light is the same. However, there are concerns about interpreting the findings and assumptions made. The conversation also highlights the linear relationship between redshift and distance observed by Hubble and the importance of considering units in calculations.
naushaan
TL;DR 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:

So what have you come up with so far?

What is the meaning of scale factor here?

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 that's all I guess

naushaan
phinds said:
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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Arman777 said:
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 that's all I guess
will do that! thank you

Arman777
naushaan said:
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?

naushaan said:
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 ..

phinds said:
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. I am struggling with how to interpret these findings.

Arman777 said:
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?

naushaan said:
i calculated the Hubble constant to be 7.6e24m/s.

you have missed an important bit off the end of that

sophiecentaur
naushaan said:
so for redshift i got 0.53, and from then i calculated the Hubble constant to be 7.6e24m/s. I am 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 there's 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 don't 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, davenn and sophiecentaur
Arman777 said:
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 there's 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 don't 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!

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.

## 1. What is a scale factor vs. distance graph?

A scale factor vs. distance graph is a visual representation of the relationship between a scale factor and the corresponding distances on a graph. The scale factor is a constant that is used to determine the proportional relationship between two sets of measurements.

## 2. How do you interpret a scale factor vs. distance graph?

To interpret a scale factor vs. distance graph, you need to look at the slope of the line. The slope represents the scale factor, and the steeper the line, the larger the scale factor. You can also look at the distances on the graph to see how they relate to each other based on the scale factor.

## 3. What does the scale factor represent?

The scale factor represents the ratio of the corresponding distances on the graph. For example, if the scale factor is 2, it means that for every 1 unit increase in distance on the x-axis, there is a 2 unit increase in distance on the y-axis.

## 4. How does the scale factor affect the graph?

The scale factor affects the graph by determining the proportional relationship between the distances on the x-axis and y-axis. A larger scale factor will result in a steeper line on the graph, while a smaller scale factor will result in a flatter line.

## 5. What are some real-life applications of a scale factor vs. distance graph?

A scale factor vs. distance graph can be used in various fields such as engineering, architecture, and geography. For example, in engineering, it can be used to determine the scale of a blueprint or model compared to the actual structure. In geography, it can be used to represent the relationship between a map and the actual distances on the ground.

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