Something about calculating the Age of the Universe

[wabbit]

It's not really random - the basic assumption that simplifies everything and leads to a simple model, is that at large scale, space is homogeneous. This means a huge amount of symmetry, which when combined with the equations of General Relativity reduce the possibilities for how H can vary a lot - the Friedmann equations summarize that and they are quite simple, with just a few parameters in the "LCDM" version that is currently used.

H in the past is part modeled and part measured - for instance the luminosity-redshift relation for supernovae measures how H changes over the observed range - this picks the value of parameters, which in turn give predictions for earlier times.
This is a very rough picture, there are lots of observations in cosmology from a range of different methods, and even with all that there is no certainty - only a good model that works well.

 

[JohnnyGui]

It's not really random - the basic assumption that simplifies everything and leads to a simple model, is that at large scale, space is homogeneous. This means a huge amount of symmetry, which when combined with the equations of General Relativity reduce the possibilities for how H can vary a lot - the Friedmann equations summarize that and they are quite simple, with just a few parameters in the "LCDM" version that is currently used.

H in the past is part modeled and part measured - for instance the luminosity-redshift relation for supernovae measures how H changes over the observed range - this picks the value of parameters, which in turn give predictions for earlier times.
This is a very rough picture, there are lots of observations in cosmology from a range of different methods, and even with all that there is no certainty - only a good model that works well.

Ah, I kinda had the idea they were "reverse extrapolating" the relationship of H and t by observing H over the years.

I'm getting quite interested in the possible theoretical models of expansions that has been thought of, especially its possible shapes. Doesn't the shape of the expansion (for example a flat universe) influence the H over time and expansion rate as well? How are they so sure that it's flat? I bet there's only a mathematical explanation behind all this.

 

[wabbit]

Yes the spatial curvature plays a role in the expansion - you should look up the FRW (aka FLRW) model really and its motivation, at some point you need to look the equations in the eyes : )

I am not sure which reference to suggest, but there are many threads here in pf, and maybe @marcus can suggest a good starting point.

As to flatness, it is a conclusion from observations, not an a priori assumption. And strictly speaking, the universe is not known to be flat, only to have a very large radius of curvature (at least 100 bn lightyears, I can't remember the exact lower bound).

 

[marcus]

...of, especially its possible shapes. Doesn't the shape of the expansion (for example a flat universe) influence the H over time and expansion rate as well? How are they so sure that it's flat? ...

Hi Johnny, Wabbit advised some beginner reading and asked me for ideas. I would suggest tagging Brian Powell and George Jones, both are pros. They could answer any of your questions and also suggest reliable beginner reading.
What comes to my mind is a 2003 article that is free online, called "Inflation and the Cosmic Microwave Background" by Charley Lineweaver.
It covers a wide range of cosmology topics.

You could look at it, but it might not be "beginner" enough.

there are several ways to observationally check that the U is spatially nearly flat. Either absolute flat or with very slight curvature too small to be measurable by current instruments.

It comes down to measuring the angles of large triangles and checking that they add up to 180 degrees. And also you can do it by counting galaxies.
As a way of checking that the volume of a sphere increases exactly with R³.

I don;t know if you realize this but if space has some measurable positive curvature then larger triangles add up to more than 180,

and larger spheres volume start growing slower than the cube of the radius. counting galaxies gives a rough way to estimate volume.

 

[wabbit]

@JohnnyGui, I also came across this, which introduces general relativity and discusses cosmology. Maybe you could have a look ? I only briefly flipped through it so far but at first sight it looks quite good to me.
General Relativity Without Calculus

 

[marcus]

Looks good to me too. Thanks for finding it! I put the link to it in the A&C reference library. I liked the exercises that Natario made up for the chapter on Cosmology.

 

[marcus]

BTW Bob Dylan has a line in a song which goes
"Come mothers and fathers throughout the land--and don't criticize what you can't understand."

that's good advice not only for mothers and fathers: get to understand something first before you start doubting and skepticising.
This little book for HS students by Jose Natario can be very helpful to young people who want to know what it is they are questioning, at more than just a superficial verbal level. It gets into numbers (but in a very intuitive way) so it is not merely verbal.

 

[marcus]

Here's a variation on the "calculate the age of the universe" theme. Actually we should say "age of the expansion" because we don't know that the start of expansion was the beginning of the universe---it might have been contracting before that. We just know at some point the expansion we see and live in started and we can say how long THAT has been going on.

anyway imagine you are running for your life from a crowd of two-headed zombies and just as they are about to catch you you see a time machine. So you jump in and pull the lever. It lands you some unknown time in the future where you are welcomed by friendly natives who have no idea about cosmology.

You want to know how far you have been catapulted into the future, so you measure the temperature of the CMB, the background of ancient light.
It turns out to be EXACTLY 2.18 KELVIN.

You recall that for us, here and now, it was 2.725 kelvin. So how far in the future that that machine take you?

 

[wabbit]

Looks good to me too. Thanks for finding it! I put the link to it in the A&C reference library. I liked the exercises that Natario made up for the chapter on Cosmology.

Just looked at those, indeed they are really good - he says in the introduction the exercises are part of the book and should be done by the reader which is always good advice (though I am always tempted to skip that part...); they aren't difficult mathematically but they cover a lot of non trivial effects - many pf threads are nicely answered there : ) It is quite a feat he pulled off doing all this while keeping it accessible to his target audience. Actually I'm going to read it all, there's no reason high school students should be the only ones to use it : )

 

[Chronos]

There are plenty of college [an ex collegians] who come here to learn

 

[JohnnyGui]

Thank you so much @wabbit and @marcus for the sources. I'll dive into them for now and see how much I'll be able to comprehend

Btw, one thing I'm wondering about for a while now, is how we are so sure that the expansion of U should be negatively influenced by gravity to the extent that we have concluded that there must be dark energy that counteracts the effects of gravity (preventing a Big Crunch). Isn't there a possibility that the expansion ISN'T influenced by gravity at all? Or are the effects of gravity on expansion merely concluded from the fact that H or expansion rate changes accordingly to the density over time? (i.e. the expansion rate being slower when the density of the U was high early on and faster when the density decreased)

@Chronos : I haven't studied anything like astronomy or cosmology but am really fascinated by them to the extent of reading and following lectures on YouTube. Any time now until I consider this as a hint to begin such a study ;).

 

[wabbit]

Thank you so much @wabbit and @marcus for the sources. I'll dive into them for now and see how much I'll be able to comprehend

Don't hesitate asking questions here or in a new thread if you start diving into one of these - if you don't have questions and general relativity seems natural and obvious, read them again, you must have missed something : )

Btw, one thing I'm wondering about for a while now, is how we are so sure that the expansion of U should be negatively influenced by gravity to the extent that we have concluded that there must be dark energy that counteracts the effects of gravity (preventing a Big Crunch). Isn't there a possibility that the expansion ISN'T influenced by gravity at all? Or are the effects of gravity on expansion merely concluded from the fact that H or expansion rate changes accordingly to the density over time? (i.e. the expansion rate being slower when the density of the U was high early on and faster when the density decreased)

No, and the reason is very simple : gravity is attractive. If two bodies are flying apart from each other, their mutual gravitational attraction will tend to slow them down (and possibly bring them back together in a big crunch). This does not depend on the details of the model, only on the assumption that gravity behaves at large scales broadly similarly to how it does at smaller scales.

 

[JohnnyGui]

Hey wabbit, I'm reading the PDF you've given me (General Relativity without Calculus) and so far, I have survived.
There's a random question that popped up in my head when watching yet another Yale lecture on YouTube about calculating the velocity of an recessing galaxy or star using redshift.

Here's a drawing I made about this:
https://www.dropbox.com/s/z5gn2k5eldik63g/Redshift.jpg?dl=0

λEmit is the wavelength of the lightray right when the star sent it out. λObs (Observed) is the redshifted lightray after the star has recessed to distance D2.

Here's what the lecture covered about calculating the velocity of the star. Please correct me if I'm wrong in these steps:

1. The lightray the star is sending out to the observer comes from distance D1 and while that lightray was traveling to us, the star recessed to distance D2. One would then be able to calculate D1 by using the luminosity formula for this star.
2. Now that D1 is calculated, you can calculate the time duration that took for the star to recess to distance D2. This is done by using D1 / c (speed of light)
3. Now, the lightray the observer sees is the redshifted λObs. Furthermore, I read that D2 / D1 = λObs / λEmit.

Here's where I'm stuck; how can one measure λEmit to be able to know the ratio and calculate D2?? Do they just assume the star is sending out a particular wavelength value? If so, what are these assumptions based on? If they're assuming the star belongs to a particular class of stars with a particular wavelength λEmit, how would one be so sure the concerning star belongs to that specific class without knowing its original λEmit?

 

[wabbit]

Hi again, glad you're surviving that read : )

I'll have to check these calculations you mention, usually I don't look at D2 (we don't observe anything at D2, it is a distance "now" and as we know "now" is a matter of convention in relativity - even though this particular convention is not arbitrary).

But about this emission wavelength, yes, astronomers measure spectra of stars (supernovae) and/or galaxies, and these have shapes and emission and absorption lines which can be identified - at least that's my understanding, maybe someone better versed in these things can chime in.

Those spectra (and their evolution over time, for supernovae) are then used to classify the type of source - notably, in the case of supernova cosmology, in order to retain only those which meet the criteria for being (likely) "standard candles".

Note that there are uncertainties and noise in these measurements, for instance galaxies have proper velocities relative to the Hubble flow and this affects their redshift - so at a given distance we see a range of velocities, and it is the average that is assumed to be representative of overall expansion.

 

[JohnnyGui]

Hi again, glad you're surviving that read : )

I'll have to check these calculations you mention, usually I don't look at D2 (we don't observe anything at D2, it is a distance "now" and as we know "now" is a matter of convention in relativity - even though this particular convention is not arbitrary).

But about this emission wavelength, yes, astronomers measure spectra of stars (supernovae) and/or galaxies, and these have shapes and emission and absorption lines which can be identified - at least that's my understanding, maybe someone better versed in these things can chime in.

Those spectra (and their evolution over time, for supernovae) are then used to classify the type of source - notably, in the case of supernova cosmology, in order to retain only those which meet the criteria for being (likely) "standard candles".

Note that there are uncertainties and noise in these measurements, for instance galaxies have proper velocities relative to the Hubble flow and this affects their redshift - so at a given distance we see a range of velocities, and it is the average that is assumed to be representative of overall expansion.

Still have to finish the read though so I'm still yet to drown :P

I understand that D2 can't be seen, but it can't it be calculated?

About those spectra to classify a type, aren't these spectra also measured as λObs?? At the end, one would never really know the true emitted wavelength the moment those standard candles sent it out; it will always need time to reach us and therefore turn (even slightly) redshifted.

Regarding your last paragraph, I was exactly thinking about how they are able to distinguish the redshift caused by the expansion from the redshift by the proper velocites. It all sounds very roughly calculated.

 

[wabbit]

Yes offcourse D2 can be calculated, and it's used too - it's just that I don't pay attention, so I'm not sure about your formulas : )

As to the spectra, the spectrum at emission (which you do not see) is characterized by the chemical elements present, but the spectrum at reception (the one you measure) is just a shifted version which makes no sense as such - only when you shift it by the right amount does it match a meaningful spectrum, and how much you need to shift tells you the redshift.

To take a simple example say you re observing a nebula made of hydrogen alone - the hydrogen spectrum is well known, and you can tell by looking at all the lines together "this must be the 656 nm line", but it turns out you are seeing it at 800nm, so you know the shift is 144nm. I'm explaining this badly, it's hard without pictures - look at some spectra, check the wikipedia article about spectroscopy you'll see what I mean.

Disclaimer: I've never measured a redshift myself and I am just a layman with no qualification in the field, so take what I say with a grain of salt.

Edit: none of this is roughly calculated, these are precision measurements mades with great care, which are of course subject to various errors but those are modeled too and the measurements are given together with their uncertainty. Don't assume from a simplified broad description I give that those numbers are plucked out of thin air, they are not. Read some reference (*) articles about supernova cosmology, you'll see how detailed the analysis are, there are many more aspects covered than we discussed, studied in excruciating detail.

(*) I added "reference" because not all published articles are of the highest standard. But do visit http://supernova.lbl.gov if you haven't done so yet, it's a great resource.

 

[JohnnyGui]

Ah, reading your post and a wiki page on redshift (don't hit me for that) made me understand that they're looking at the pattern (i.e. the intervals between the absorbed spectral lines) of the observed wavelength. The only difference here is that it's redshifted but it's still the same pattern so they're able to compare it with known patterns from chemical elements to know what element the star is made out of. Is this correct?
There's apparently another way by looking if there's a same spectral line in both spectra but at a different wavelength and then calculate the redshift, only I don't know what they mean by "same spectral line" if it's not at the same wavelength. Doesn't a similar spectral line mean that it's at the same wavelength?

Regarding the formula I mentioned in post #103, I have compared its outcome with another redshift formula: v = c x (λΔ / λemit) and it gave me the same answer.

I will read the link you gave now.

 

[wabbit]

Yes that's what I was clumsily trying to explain : ) glad you found a better explanation in the wiki, I was struggling with how to formulate this.

A single line in isolation can't be used for redshift measurement, only a spectrum (pattern) can.

 

[JohnnyGui]

If it weren't for your many explanations in this thread, I would be still looking cluelessly at post #1 ;)

One thing I was wondering regarding the formulas:

- I read that redshift could be calculated by (λ_obs / λ_emit) - 1 = z.
- Since λ_obs / λ_emit = D2 / D1 (D1 being the distance of a star right when its light was emitted and D2 being the new distance after the redshift that we observe) I can say that (D2 / D1) - 1 = z
- For small distances, I read that z × c = v. I also know that v = H × D
- So I can say that ((D2 / D1) - 1) × c = H × D

My question here is, is D the D2 or the D1 of the star?? How would one be able to tell?

 

[wabbit]

Where does this z×c=v come from ? Not saying it's wrong but I can't make sense of it.

 

[JohnnyGui]

Believe me, I've been trying for half an hour to make sense out of that equation as well without result. There are several sources mentioning the formula, for example: http://astronomy.swin.edu.au/cosmos/c/cosmological+redshift as well as here: http://cas.sdss.org/dr5/en/proj/advanced/hubble/conclusion.asp

Apparently, the increase in wavelength by redshift compared to the emitted wavelength is the same fraction as the speed of the object compared to the speed of light but I don't get why.

EDIT: Ok, I found a thread from here where marcus discusses the formula, I'm reading it now: https://www.physicsforums.com/threads/prove-that-z-v-c.273160/

 

[wabbit]

Yes youre right it's the low velocity Doppler shift formula and it applies to recession velocities just fine as long as $v\ll c$

 

[JohnnyGui]

Knowing that, are you able to answer the question about D being D1 or D2 in post #109 and how one would be able to tell?

 

[wabbit]

Hmmm yes I was trying to avoid doing that

I was going to say D is neither D1 nor D2, it's the luminosity distance, but I'm not so sure now, I need to check, perhaps someone who knows this stuff better can help here.

 

[JohnnyGui]

Haha, nice tactic there ;)

Anyway, if D is based on the luminosity distance, then I would think that luminosity is based on the observed wavelength that was originally emitted when a star was in its first position and I would conclude that D is D1.

Perhaps @marcus could once again save the day here?

 

[Chronos]

It might be simpler to view z and D in terms of the scale factor of the universe - see http://www.physics.fsu.edu/users/ProsperH/AST3033/cosmology/ScaleFactor.htm for a brief discussion.

 

[JohnnyGui]

It might be simpler to view z and D in terms of the scale factor of the universe - see http://www.physics.fsu.edu/users/ProsperH/AST3033/cosmology/ScaleFactor.htm for a brief discussion.

Thanks, looks like a very good read to me. I'll dive into it. Isn't there a pm system on here so I can give @marcus a holler about my previous question?

 

[marcus]

Anyway, if D is based on the luminosity distance, then I would think that luminosity is based on the observed wavelength that was originally emitted when a star was in its first position and I would conclude that D is D1...

... Isn't there a pm system on here so I can give @marcus a holler about my previous question?

Hi JG, I haven't been following this thread and I'm not sure what D, and D1 and D2 are. I think luminosity is based on the RECEIVED wavelengths. I'm often mistaken at first and have to look things up. I'll look up "luminosity distance" and correct this if I'm mistaken.

You seem pretty well-informed and to be thinking clearly about this stuff so you probably realize already what I'm saying: distance reduces the received watts per cm^2 in two ways.
A. the energy is more spread out. It falls off as R^2 where R is the distance to the source by the time the light gets to us. Expansion contributes to the present distance to the source. So expansion contributes to R and reduces luminosity that way.
B. expansion also lengthens the wavelengths and so it drains energy from the light that way. You know the energy of a photon E =ħω so reducing the frequency (by wavestretch) reduces the energy of each photon AND expansion reduces the rate photons are arriving. So there is a kind of (z+1)² factor killing the energy FLUX.

So expansion reduces the watts per cm^2 on the detector surface TWO ways. By increasing the distance so things are more spread out, and by reducing the flux the other ways I mentioned.

I may be just confusing things by commenting like this without understanding what you are talking about : ^) I'll go look up "luminosity distance" and review that. Do you want to kind of summarize what your question is, so people just joining the thread can understand what it is about?

 

[marcus]

Damn! Astronomers have some pretty dumb conventions. Terminology is a proliferating accumulation of historical accidents.

Wikipedia on "luminosity distance" (D_L)indicates that the effect of redshift is NOT ALLOWED FOR in the definition of D_L
==quote==
Another way to express the luminosity distance is through the flux-luminosity relationship. Since,

where F is flux (W·cm−2), and L is luminosity (W),... From this the luminosity distance can be expressed as:

The luminosity distance is related to the "comoving transverse distance"

by the Etherington's reciprocity relation[citation needed]:

where z is the redshift.

is a factor that allows you to calculate the comoving distance between two objects with the same redshift but at different positions of the sky; if the two objects are separated by an angle [PLAIN]https://upload.wikimedia.org/math/6/7/7/677514f5ab1c654e3006964219831e7f.png, the comoving distance between them would be [PLAIN]https://upload.wikimedia.org/math/0/c/d/0cdb2ceca920b31a61e0ca85e430672e.png. In a spatially flat universe, the comoving transverse distance [PLAIN]https://upload.wikimedia.org/math/6/b/4/6b4edd602a37988d06321ebcc59d310e.png[B] is exactly equal to the radial comoving distance[/B] [PLAIN]https://upload.wikimedia.org/math/a/1/6/a16cff4686b360b3ddc1ab645852a051.png, i.e. the comoving distance from ourselves to the object.[1]
==endquote==
So you take the actual Watts output of the star, that's L. And you picture this in static Euclidean space and you naively say what radius R would be needed to spread this power out to get the observed Watts per cm^2. that is the first formula. It does not allow for redshift!

Now this obviously over estimates the distance. The absolute luminosity should not be L it should be L/(z+1)². so when you get it under the square root sign you get a correction factor of 1/(z+1)

So finally in the third equation they say "Oh drat we over estimated by a factor of (z+1)! So let's define a new distance D_M."

And that turns out to be the good, spatial flat, approximation of the right thing---the comoving or NOW distance that is actually the distance.

Astronomers can't go back and correct bad definitions that are time-honored and enshrined in the scholar literature because that would make the professional literature of the past unreadable. Words in the journals would mean different things before and after the reform/correction of the bad definitions. So we are stuck with something called "Luminosity distance" which is bigger than it should be by a factor of (z+1).

That is my dos centavos Señores. It may or may not be relevant and it may of course be incorrect. : ^)

 

[JohnnyGui]

Hi JG, I haven't been following this thread and I'm not sure what D, and D1 and D2 are. I think luminosity is based on the RECEIVED wavelengths. I'm often mistaken at first and have to look things up. I'll look up "luminosity distance" and correct this if I'm mistaken.

You seem pretty well-informed and to be thinking clearly about this stuff so you probably realize already what I'm saying: distance reduces the received watts per cm^2 in two ways.
A. the energy is more spread out. It falls off as R^2 where R is the distance to the source by the time the light gets to us. Expansion contributes to the present distance to the source. So expansion contributes to R and reduces luminosity that way.
B. expansion also lengthens the wavelengths and so it drains energy from the light that way. You know the energy of a photon E =ħω so reducing the frequency (by wavestretch) reduces the energy of each photon AND expansion reduces the rate photons are arriving. So there is a kind of (z+1)² factor killing the energy FLUX.

So expansion reduces the watts per cm^2 on the detector surface TWO ways. By increasing the distance so things are more spread out, and by reducing the flux the other ways I mentioned.

I may be just confusing things by commenting like this without understanding what you are talking about : ^) I'll go look up "luminosity distance" and review that. Do you want to kind of summarize what your question is, so people just joining the thread can understand what it is about?

Hey Marcus, I am indeed aware of this info but thanks a lot for the effort nonetheless :). Sure, I'll summarize my question for you:

I was talking with @wabbit about the following drawing that I made: https://www.dropbox.com/s/z5gn2k5eldik63g/Redshift.jpg?dl=0
λEmit is the wavelength of the lightray right when the star sent it out when it was at distance D1. λObs (Observed) is the redshifted lightray that we observe after the star has recessed to distance D2.

I concluded the following:
1. I saw a lecture from Yale and it covered the following formula: D2 / D1 = λObs / λEmit. I also read that redshift could be calculated by (λobs / λemit) - 1 = z.
2. Since λObs / λEmit = D2 / D1 (D1 being the distance of a star right when its light was emitted and D2 being the new distance after the observed redshift) I can say that (D2 / D1) - 1 = z as well.
3. Furthermore, for small distances, I read that z × c = v. I also know that v = H × D
4. So I can say that ((D2 / D1) - 1) × c = H × D

My question here is, is D the D2 or the D1 of the star and how would one be able to tell?

EDIT: Just noticed your second reply, but I'm not sure (yet) if this has answered my question or not. Sorry for that.