Accelerated expansion and Hubble plot

[GTFE]

Hello!

I have a question regarding the effect of the accelerated expansion of the universe on the Hubble plot (redshift over luminosity (or distance).
I understand that for relatively nearby galaxies, this appears to be a linear relationship but that because of the accelerated expansion of the universe this relationship does not appear linear anymore for distant galaxies and starts to lean towards lower luminosities (or greater distances) per redshift.
I do not understand why this is the case. Should the "additional expansion" due to the acceleration not have an effect of the same scale on the "stretching" of the light waves as it does on their intensity (or luminosity).

Thanks

Henning

 

[Chalnoth]

Hello!

I have a question regarding the effect of the accelerated expansion of the universe on the Hubble plot (redshift over luminosity (or distance).
I understand that for relatively nearby galaxies, this appears to be a linear relationship but that because of the accelerated expansion of the universe this relationship does not appear linear anymore for distant galaxies and starts to lean towards lower luminosities (or greater distances) per redshift.
I do not understand why this is the case. Should the "additional expansion" due to the acceleration not have an effect of the same scale on the "stretching" of the light waves as it does on their intensity (or luminosity).

Over long distances, the relationship between redshift and distance isn't linear under any physical model of our universe. What the accelerated expansion means is that objects which are at a given redshift are at higher distances than they would be without the accelerated expansion.

There is no effect of the accelerated expansion on distance/redshift observations beyond the expansion itself: if you were given the expansion rate over time, then even without any knowledge of the contents of the universe you could calculate what the distance/redshift relation for an object at any redshift should be.

 

[Chronos]

Acceleration of expansion is a tiny effect and difficult to detect without standard candles like Sn1a for calibration.

 

[GTFE]

Over long distances, the relationship between redshift and distance isn't linear under any physical model of our universe. What the accelerated expansion means is that objects which are at a given redshift are at higher distances than they would be without the accelerated expansion.

There is no effect of the accelerated expansion on distance/redshift observations beyond the expansion itself: if you were given the expansion rate over time, then even without any knowledge of the contents of the universe you could calculate what the distance/redshift relation for an object at any redshift should be.

Thanks for the reply Chalnoth!

OK, the relationship is not linear, got it! But still you say that, with accelerated expansion, distant objects at a given redshift are at a greater distance. So it appears to me that the "added" expansion seems to have a bigger effect on the distance (or luminosity) than on "redshifting". This is what confuses me.

Henning

 

[PeterDonis]

it appears to me that the "added" expansion seems to have a bigger effect on the distance (or luminosity) than on "redshifting". This is what confuses me.

What might be confusing you is that you are conflating "distance" with luminosity. In fact "distance" can have multiple meanings, and "luminosity distance" is only one of them.

Heuristically, "luminosity distance" means you take the observed luminosity of an object whose emitted luminosity is known (a "standard candle"), and calculate what distance that object would have to be from you for it to have its observed luminosity, assuming that luminosity decreases as the square of the distance. Note that this assumption has nothing whatever to do with "rate of expansion" or anything else; it's just a way of restating an observed luminosity in units of distance.

By contrast, the redshift (more precisely, $1 + z$ where $z$ is the redshift) tells you by what ratio the scale factor of the universe has increased between emission and reception; so for example $z = 1$ means the scale factor has doubled ($1 + z = 2$) from the emission of the light you are seeing now, to now. This factor is also the factor by which the "comoving distance" (the spatial distance between two objects in comoving, aka FRW, coordinates) has increased. But, as should be evident from the definition above, "comoving distance" (or equivalently redshift) and "luminosity distance" are not the same and do not have to change by the same factor. The observed relationship between them is in fact a key piece of data that helps us reconstruct the expansion history of the universe.

A good quick primer on the different "distances" and how they are related to each other and to the redshift is here:

http://www.astro.ucla.edu/~wright/cosmo_02.htm

 

[Jorrie]

This factor is also the factor by which the "comoving distance" (the spatial distance between two objects in comoving, aka FRW, coordinates) has increased

Peter, shouldn't this have been labelled "proper distance"? Doesn't 'comoving distance' remain constant for a given redshift? I may be slightly confused by the terminology.

 

[PeterDonis]

shouldn't this have been labelled "proper distance"?

Possibly. I've seen "comoving distance" used both ways, to denote the difference in comoving coordinates (which is constant) and to denote the physical distance associated with that difference in comoving coordinates (which increases as the scale factor increases). The terminology in this area could probably stand some improvement.d

 

[Chalnoth]

Thanks for the reply Chalnoth!

OK, the relationship is not linear, got it! But still you say that, with accelerated expansion, distant objects at a given redshift are at a greater distance. So it appears to me that the "added" expansion seems to have a bigger effect on the distance (or luminosity) than on "redshifting". This is what confuses me.

Henning

The cosmological redshift is the total amount of expansion that has occurred since the light was emitted. For example, if objects have on average moved twice as far away from one another, then the wavelengths of light will have doubled.

However, the distance to an object at a given redshift depends upon how the rate of expansion has changed since the light was emitted. An accelerated expansion leads to larger distances for far-away objects at a given redshift.

 

[GTFE]

Hi Peter!

A good quick primer on the different "distances" and how they are related to each other and to the redshift is here:

http://www.astro.ucla.edu/~wright/cosmo_02.htm

Thanks for the link! I will fight my way through it. Will take me a while though, I am afraid.

What might be confusing you is that you are conflating "distance" with luminosity. In fact "distance" can have multiple meanings, and "luminosity distance" is only one of them.

So let me try to rephrase my question once again without distances (and still without formulas) :

Why does the added expansion that the light experiences on it's way to us have a more pronounced effect on the "dimming" of the light than on the "stretching" of the lightwaves?

Best regards

Henning

 

[GTFE]

Hi Chalnoth!

The cosmological redshift is the total amount of expansion that has occurred since the light was emitted.

However, the distance to an object at a given redshift depends upon how the rate of expansion has changed since the light was emitted.

Maybe it is obvious if you know the Math behind it (I certainly don't), but to me it again sounds as if you describe the same thing with different words.
Isn't "how the rate of expansion has changed since the light was emitted" exactly what you need to know to determine the "total amount of expansion that has occurred since the light was emitted"?

Best regards

Henning

 

[vanhees71]

For a modern Hubble diagram, see p. 8 of this review of cosmology (from the particle data group review of particle physics 2016, i.e., quite up to date):

http://pdg.lbl.gov/2016/reviews/rpp2016-rev-bbang-cosmology.pdf

 

[PeterDonis]

Why does the added expansion that the light experiences on it's way to us have a more pronounced effect on the "dimming" of the light than on the "stretching" of the lightwaves?

Heuristically, because "dimming" (luminosity) is a quadratic relationship to distance, whereas "stretching" (redshift) is a linear relationship to distance.

 

[GTFE]

Heuristically, because "dimming" (luminosity) is a quadratic relationship to distance, whereas "stretching" (redshift) is a linear relationship to distance.

OK, but I understand this is also true if we assume a constant expansion. What changes here when the light experiences "additional expansion" due to the accelerated expansion?
Or why does the added expansion due to an accelerated expansion have a more pronounced effect on the luminosity, even if it has a quadratic relationship to distance, when we compare it to an added expansion that would result from the fact that is is simply further away?

 

[PeterDonis]

why does the added expansion due to an accelerated expansion have a more pronounced effect on the luminosity, even if it has a quadratic relationship to distance, when we compare it to an added expansion that would result from the fact that is is simply further away?

Remember that I said "heuristically". Have you looked at the actual math? We are at a point where it's not really possible to answer your questions without looking at the math. A good start would be to look at the page from Ned Wright's cosmology tutorial that I linked to in post #5.

 

[Chalnoth]

Hi Chalnoth!

Maybe it is obvious if you know the Math behind it (I certainly don't), but to me it again sounds as if you describe the same thing with different words.
Isn't "how the rate of expansion has changed since the light was emitted" exactly what you need to know to determine the "total amount of expansion that has occurred since the light was emitted"?

Best regards

Henning

No, they're different. The redshift is given by the ratio of average distances between the time the light was emitted and the time it arrived. So if distances have doubled, the wavelength of the light has doubled.

But there are lots of ways (in principle) that the wavelength could have doubled. There could have been a rapid expansion in the past that has slowed down over time. There could have been a slow expansion in the past that has sped up (note: this one is actually unlikely for a number of theoretical reasons that I won't get into). The expansion rate could have remained constant. All of these possibilities change the distance to the object, which depends not just upon the total amount of expansion, but how that expansion has changed over time.

 

[PeterDonis]

there are lots of ways (in principle) that the wavelength could have doubled.

Another way of putting this that might help the OP: the redshift gives you the ratio $a_{\text{now}} / a_{\text{emitted}}$, [Edit: should be $d_{\text{now}} / d_{\text{emitted}}$, see next two posts] between the scale factors [Edit: should be "distances"] now and when the light was emitted. But the luminosity distance depends on the actual values of $a_{\text{now}}$ and/or $a_{\text{emitted}}$ [Edit: $d_{\text{now}}$ and/or $d_{\text{emitted}}$] (since we know their ratio, either one's actual value is sufficient). And those actual values depend on the details of the expansion history.

 

[Chalnoth]

Another way of putting this that might help the OP: the redshift gives you the ratio $a_{\text{now}} / a_{\text{emitted}}$, between the scale factors now and when the light was emitted. But the luminosity distance depends on the actual values of $a_{\text{now}}$ and/or $a_{\text{emitted}}$ (since we know their ratio, either one's actual value is sufficient). And those actual values depend on the details of the expansion history.

That description is a little inaccurate, as $a$ a property of the universe as a whole, not of a particular object. Typically it's set so that $a = 1$ at the current time. Rather, it's the distance that matters, as that is object-specific.

If you replace every instance in your explanation with the distance, then it makes sense: the redshift depends upon the ratio of the distance now vs. then, while the distance itself is the actual value of the distance (either now or then, depending upon how you want to measure).

 

[PeterDonis]

That description is a little inaccurate, as aaa a property of the universe as a whole, not of a particular object.

Yes, you're right, a better expression would be $d_{\text{now}} / d_{\text{emitted}}$, to make clear that we are talking about the distances to the object. I've edited my previous post.