Something about calculating the Age of the Universe

[wabbit]

OK yes, this is because you assume constant velocity of 6 m/s from t=0 to t=1, then a jump to a velocity of 18 m/s at t=2, etc. This gives you the right qualitative picture, but to be more precise you need to look at much smaller time intervals over which the velocity is approximately constant, like
t=0s, d=3m, v=6m/s
t=0.01s, d=3.06m, v=6.12m/s
etc..
If you take very small intervals, at the limit you get the exponential.

 

[JohnnyGui]

OK yes, this is because you assume constant velocity of 6 m/s from t=0 to t=1, then a jump to a velocity of 18 m/s at t=2, etc. This gives you the right qualitative picture, but to be more precise you need to look at much smaller time intervals over which the velocity is approximately constant, like
t=0s, d=3m, v=6m/s
t=0.01s, d=3.06m, v=6.12m/s
etc..
If you take very small intervals, at the limit you get the exponential.

Aha, now it starts to make sense to me. Since ofcourse, velocity is changing in a continious way in the case of a constant H, the distance it would have traveled would differ from when I take too large time intervals of constant velocities on which my formula is based.

 

[wabbit]

Exactly. Use a spreadsheet to check that I am not making this up:)

 

[JohnnyGui]

Exactly. Use a spreadsheet to check that I am not making this up:)

Reading and looking at the amount of posts you've helped me with, I can reliably say that I don't need any objective source to verify your conclusions, good sir :P

Thank you so much for your time. I can now finally take on the challenge and look at the more complicated formulas that are posted on page 1 of my thread XD

 

[JohnnyGui]

Exactly. Use a spreadsheet to check that I am not making this up:)

Btw, I have been thinking about the relevance of the Hubble constant and I'm starting to think that it's nothing more but a byproduct of how the universe is expanding, since H is constantly changing over time.
I mean, if the expansion of the universe was happening in a vacuum without any matter or radiation whatsoever, who said that the expansion would increase exponentially according to a constant H over time? It might instead expand with a constant acceleration per time interval (such as 2m/s²) or in any other way of acceleration. H is merely "discovered" and calculated because of the stretching scenario the expansion has. However, when it comes to acceleration of that expansion, then the way/rate of the acceleration would be independent from H and H would be just merely a byproduct that is calculated based on that acceleration.

Am I making sense here?

 

[wabbit]

It might indeed, as you say H, both its current value and its (reconstructed/forecast) evolution, is derived from observations. However the equations of general relativity do put some constraints on how it can evolve depending on the contents (matter, radiation..) of the universe. And in our case the long term forecast is that this evolution will gradually look more and more like exponential expansion, though we're not there yet - marcus' thread about the simple model of expansion gives a good idea of what it looks like.

 

[JohnnyGui]

It might indeed, as you say H, both its current value and its (reconstructed/forecast) evolution, is derived from observations. However the equations of general relativity do put some constraints on how it can evolve depending on the contents (matter, radiation..) of the universe. And in our case the long term forecast is that this evolution will gradually look more and more like exponential expansion, though we're not there yet - marcus' thread about the simple model of expansion gives a good idea of what it looks like.

When you say exponential expansion, what kind of exponential expansion do you mean? Because aren't there different ways of exponential expansion other than according to a constant H over time? Such as a constant acceleration per time interval, etc.?
Another question would be, are there any theories or experiments done (if practical) about with what kind of acceleration the universe would expand if it didn't have any matter or radiation?

Sorry if these questions are answered by marcus's thread you pointed to.

 

[wabbit]

There are different possibilities for expansion, is just that "exponential expansion" means constant H (at least as I understand the term, maybe it's used in a broader sense by some), it refers to the exponential form above $e^{Ht}$

For a vacuum, the issue is delicate, because the same vacuum can be seen as expanding in different ways depending on what you chose as the "cosmic time" and comoving "observers" (if you have matter, this gives something to anchor the coordinates, but in a vacuum you are free to do as you please). But in FRW coordinates the answer is "vacuum = pure exponential expansion". There cannot be such striclty exponential expansion if matter is present, because matter generates gravity which slows down the expansion.

 

[JohnnyGui]

There are different possibilities for expansion, is just that "exponential expansion" means constant H (at least as I understand the term, maybe it's used in a broader sense by some), it refers to the exponential form above $e^{Ht}$

Has it actually been proven that the universe would expand that way (a constant H) if there was no matter or radiation or is this a kind of mathematical forecast for the future?

Having a constant H over time would result in that the acceleration itself of the expansion is being accelerated as well, thus an even "stronger" exponential expansion would take place. Is it ruled out that the expansion could have a constant acceleration instead? Note that I'm talking about expansion in a vacuum here.

 

[wabbit]

I don't know really, but
(a) vacuum solutions are fairly well understood I think, though I don't know them well myself ;
(b) they do not really represent a physical universe - what is "a universe containing nothing" ? spacetime "in itself" does not have concrete existence, which, as I understand it, is also one reason why they can be interpreted as expanding in various ways or even static.

 

[JohnnyGui]

I don't know really, but
(a) vacuum solutions are fairly well understood I think, though I don't know them well myself ;
(b) they do not really represent a physical universe - what is "a universe containing nothing" ? spacetime "in itself" does not have concrete existence, which, as I understand it, is also one reason why they can be interpreted as expanding in various ways or even static.

Quite interesting. Would spacetime even be created at the time of the Big Bang if there was no matter involved? How would time run in a vacuum without any matter?

I think if one would understand the way a vacuum would expand (as in determining the 1 way of expansion), one would understand the way how dark energy works a lot better since you're looking at its mechanism without any other influences of matter and whatnot. But since you already said that they are fairly well understood, I guess they have already passed that stage.

 

[wabbit]

I agree, and this prototype "exponentially expanding vacuum" characteristic of a cosmological constant is very interesting. It does represent a theoretical universe, one filled with "very fine dust" of infinitesimal density, i.e. test particles only, and this tells us how such particles behave far away from matter/energy sources and in the absence of gravitational waves.

As to a big bang, I don't think so : without matter or radiation, expansion lasts forever and starts in the infinite past.

Actually, this is something I am trying to understand too at the moment, I think it helps understand things better - there are many confusing statements around about expansion, and that simple case of "fine dust" in an otherwise empty universe is good to explore - it is the equivalent, for GR with a CC, of free Galilean motion far away from masses in a Newtonian universe.

 

[marcus]

Hi Johnny, hi Wabbit. The title suggests the thread is about calculating the Age (i.e. how long the U has been expanding according to the Friedmann equation model.)

Probably Johnny is interested not only in the Age but also in other things, so this could be a widening discussion--I haven't kept up.

But if the thread WERE just about the Age then it could be argued there is one obvious right answer about how to calculate it. In fact Wabbit showed us some of the steps in the argument. See if you find it persuasive. (Or perhaps calculating the Age isn't relevant at this point in thread? then simply ignore this.)

We measure the current and longterm Hubble constants, H₀ and H_∞ and we calculate the age from them. AFAIK there is essentially only one way to do that. Assuming space is to a good approximation flat, those two quantities uniquely determine the Friedmann age.

Measuring the Hubble constants is observational, empirical, basic to all cosmology. So whenever you calculate something you at least have those two quantities to start with. And in this case those two suffice.

I guess you can perform the calculation various equivalent ways. I would just take the ratio H₀/H_∞ = 1.201 (currently the best estimate I know)

Whatever units you like to use you can always take the ratio and have a number without units. And solving the Friedmann equation (which I assume we believe is a good enough approximation to reality and is essential to defining the Age) gives us a relation between time and expansion rate which we can invert so that we can calculate the time FROM the expansion rate.

Basically, inverting the H(x) function to solve for x(H) as a function of H, and plugging in 1.201, we have
$$x = \frac{1}{3}\ln(\frac{1.201+1}{1.201-1}) = \frac{1}{3}\ln(\frac{2.201}{0.201}) = 0.797$$

And then you just divide that x, which you calculated, by H_∞ to get the answer in whatever units you like to use, e.g. if you like billions of years as units for the Age, then you will get the answer 13.787 billion years, or so.

 

[JohnnyGui]

Hi Johnny, hi Wabbit. The title suggests the thread is about calculating the Age (i.e. how long the U has been expanding according to the Friedmann equation model.)

Probably Johnny is interested not only in the Age but also in other things, so this could be a widening discussion--I haven't kept up.

But if the thread WERE just about the Age then it could be argued there is one obvious right answer about how to calculate it. In fact Wabbit showed us some of the steps in the argument. See if you find it persuasive. (Or perhaps calculating the Age isn't relevant at this point in thread? then simply ignore this.)

We measure the current and longterm Hubble constants, H₀ and H_∞ and we calculate the age from them. AFAIK there is essentially only one way to do that. Assuming space is to a good approximation flat, those two quantities uniquely determine the Friedmann age.

Measuring the Hubble constants is observational, empirical, basic to all cosmology. So whenever you calculate something you at least have those two quantities to start with. And in this case those two suffice.

I guess you can perform the calculation various equivalent ways. I would just take the ratio H₀/H_∞ = 1.201 (currently the best estimate I know)

Whatever units you like to use you can always take the ratio and have a number without units. And solving the Friedmann equation (which I assume we believe is a good enough approximation to reality and is essential to defining the Age) gives us a relation between time and expansion rate which we can invert so that we can calculate the time FROM the expansion rate.

Basically, inverting the H(x) function to solve for x(H) as a function of H, and plugging in 1.201, we have
$$x = \frac{1}{3}\ln(\frac{1.201+1}{1.201-1}) = \frac{1}{3}\ln(\frac{2.201}{0.201}) = 0.797$$

And then you just divide that x, which you calculated, by H_∞ to get the answer in whatever units you like to use, e.g. if you like billions of years as units for the Age, then you will get the answer 13.787 billion years, or so.

Hey Marcus! I'm definitely still interested in this and am always open for new info. I'd have to look at this formula you gave and try to understand why it is formulated that way. My problem is that I'm kind of OCD about trying to figure out and concluding these formulas myself instead of just accepting them. You probably noticed that in my previous posts about concluding and making formulas up by myself :P

One question though, does this Friedmann equation take the slowdown of the expansion during the very early periods after the Big Bang into account, when the U was much more dense than now? Using the ratio of H0/H∞ somehow gives me the feeling that you're considering this ratio has been constant over the whole age of the U while it could have been different earlier on.

 

[wabbit]

Actually, the formula as quoted applies for the current age, when the expansion rate is H0 - at a different time t, the "H0" would be replaced by H(t)

So yes, that formula is based on a universe containing matter with decelerating expansion initially due to gravity.

In fact that formula expresses this : As marcus mentioned, knowing how fast the universe is currently expanding relative to its long term/vacuum rate, is what tells is how old the universe is. If that ratio $H_0/H_\infty$ is close to 1, it means the universe is already old. If it if high, the universe must be young. The exact quantitative relation between "how close to the vacuum rate" and "how old" is what marcus' formula gives, under the assumption that the universe contains mostly (slow moving) matter.

 

[JohnnyGui]

Actually, the formula as quoted applies for the current age, when the expansion rate is H0 - at a different time t, the "H0" would be replaced by H(t)

So yes, that formula is based on a universe containing matter with decelerating expansion initially due to gravity.

In fact that formula expresses this : As marcus mentioned, knowing how fast the universe is currently expanding relative to its long term/vacuum rate, is what tells is how old the universe is. If that ratio $H_0/H_\infty$ is close to 1, it means the universe is already old. If it if high, the universe must be young. The exact quantitative relation between "how close to the vacuum rate" and "how old" is what marcus' formula gives, under the assumption that the universe contains mostly (slow moving) matter.

I'm slowly starting to understand the formula from your good explanation. However, if the formula is using a ratio of 1.201 for the current age, doesn't that mean that the formula is considering that H0 has been constant all the time up till now? Or is there a function of H(t) for H0 hidden in the formula?

 

[wabbit]

No, the formula is just expressed for t=now, H(t)=H0; if you prefer you can write it $x= \frac{1}{3}\ln\left(\frac{\frac{H_t}{H_\infty}+1}{\frac{H_t}{H_\infty}-1}\right)$
A more explicit way, with units apparent, would be the equivalent form
$$t= \frac{1}{H_\infty}\cdot\frac{1}{3}\ln\left(\frac{\frac{H_t}{H_\infty}+1}{\frac{H_t}{H_\infty}-1}\right)$$
Try it. How old was the universe when it was expanding at ten times its long term rate?

 

[JohnnyGui]

No, the formula is just expressed for t=now, H(t)=H0; if you prefer you can write it $x= \frac{1}{3}\ln\left(\frac{\frac{H_t}{H_\infty}+1}{\frac{H_t}{H_\infty}-1}\right)$
A more explicit way, with units apparent, would be the equivalent form
$$t= \frac{1}{H_\infty}\cdot\frac{1}{3}\ln\left(\frac{\frac{H_t}{H_\infty}+1}{\frac{H_t}{H_\infty}-1}\right)$$
Try it. How old was the universe when it was expanding at ten times its long term rate?

Aha, I think I'm getting it now. In layman terms, the formula gives a plot of t (age) set out against different H0 values so it gives the corresponding age when a particular H0 is chosen.
I'm still amazed how there's an equation for this seeing that the expansion of the universe is independent from H and that i.a. matter could influence H in any random way. In other words, there are many factors that could influence H over time to the extent that there would be no equation/relationship possible between t and H. At least, that's what I would think.

 

[wabbit]

I agree, what is surprising about this model is that it is so simple. Mix matter and a cosmological constant and voilà, the history of the universe !

To be fair, things get more complicated early on, when matter wasn't dominating. But still.

One reason perhaps, is that it is a highly simplified view, valid only at very large scales (above galaxy supercluster or even higher), where we can say that the universe is homogeneous - so all that remains is the balance between two "forces" : gravity pulling everything together, and the cosmological constant pulling everything apart - and it turns out the possible solutions all look alike, when expressed in suitable units.

 

[JohnnyGui]

I agree, what is surprising about this model is that it is so simple. Mix matter and a cosmological constant and voilà, the history of the universe !

To be fair, things get more complicated early on, when matter wasn't dominating. But still.

One reason perhaps, is that it is a highly simplified view, valid only at very large scales (above galaxy supercluster or even higher), where we can say that the universe is homogeneous - so all that remains is the balance between two "forces" : gravity pulling everything together, and the cosmological constant pulling everything apart - and it turns out the possible solutions all look alike, when expressed in suitable units.

If there are so many factors that could influence H randomly, didn't they have to verify H over time in another way before being able to construct such a formula then? How were they able to determine the true H values in the past, while H could be randomly influenced by many factors, to be able to see its relationship with time and make such a formula?

 

[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.