Something about calculating the Age of the Universe

[Xyooj]

calculating the age of the universe is like pinpointing the position of an atom, by the time you thought you knew it's position it already changed?

perhaps it's not the universe that's expanding, but our own imagination that it is expanding so we develop the tools to design what we want to see?

 

[wabbit]

You are right, your formula $D_{Δt+t}= (H+1)^{Δt} • Dt$ is not correct, I looked at it too quickly.

$e\simeq 2.71828$ is http://en.m.wikipedia.org/wiki/E_(mathematical_constant)

It is a convenient basis for expressing exponential growth, precisely because it it the only number $x$ such that $x^{t+\delta }\simeq x^t (1+\delta)$ for small values of $\delta$

 

[wabbit]

calculating the age of the universe is like pinpointing the position of an atom, by the time you thought you knew it's position it already changed?

No, that age changes at the rate of one second per second, and this is not going to materially alter the result, which is currently estimated to be around 14 billion years.
It is not a figment of our imagination either, but a result of precise models of how things move. Of course these models can be superseded and the answer may change, as our understanding improves, but calling that an effect of imagination is a stretch.

 

[JohnnyGui]

You are right, your formula $D_{Δt+t}= (H+1)^{Δt} • Dt$ is not correct, I looked at it too quickly.

$e\simeq 2.71828$ is http://en.m.wikipedia.org/wiki/E_(mathematical_constant)

It is a convenient basis for expressing exponential growth, precisely because it it the only number $x$ such that $x^{t+\delta }\simeq x^t (1+\delta)$ for small values of $\delta$

I'm baffled... How come my formula isn't correct while it gives me correct answers in my examples? Is my given example wrongly executed?

 

[wabbit]

I think the reason is that H is (currently) a very small number, and $(1+H)^{\Delta t}\simeq e^{H\Delta t}\simeq 1+H\Delta t$ in this case.

So actually I was wrong again, your formula is OK (but only when $H$ is small, it doesn't work at all for $H=1$ for instance)... Sorry, should think more before writing:)

 

[JohnnyGui]

I think the reason is that H is (currently) a very small number, and $(1+H)^{\Delta t}\simeq e^{H\Delta t}\simeq 1+H\Delta t$ in this case.

So actually I was wrong again, your formula is OK (but only when $H$ is small, it doesn't work at all for $H=1$ for instance)... Sorry, should think more before writing:)

The thing is, even when I give H a small number such as 2 and Δt for example 3, (1 + H)^Δt would give me 27 while e^HΔt would give me a value of ≈403

 

[wabbit]

Right, and this is the case where your formula breaks down. "Small value of $H$" here means much smaller than one (I was assuming you used $H$ is SI units, in which case the current value is very small)

Note that your formula has a unit problem, is you change the time unit, the formula changes since 1 is dimensionless but H has units of inverse time.

 

[JohnnyGui]

Right, and this is the case where your formula breaks down. "Small value of $H$" here means much smaller than one.

Note that your formula has a unit problem, is you change the time unit, the formula changes.

Sorry for being so stubborn but wouldn't a value of H=2 in my formula in that case then give me a wrong answer than shown in my example? GIving H a value of 1 also give me correct answers according to my example.

 

[wabbit]

Yes it should. I need to look at your example again, this cannot work for H of order one or larger.

Edit:: actually I do not see a numerical examle in your post, so I don't know what is telling you that your formula works.

 

[JohnnyGui]

Yes it should. I need to look at your example again, this cannot work for H of order one or larger.

Edit:: actually I do not see a numerical examle in your post, so I don't know what is telling you that your formula works.

I was referring to my post #60:

Suppose there is a constant H of 2m/s/m over time.
An object that starts at 3m distance, and thus a velocity of 6m/s (H x D), will have the following distances in the following 3 seconds:
Start distance is 3m at t=0, at which it will have a velocity of 6 m/s
Distance is 9m at t=1, at which it will then have a velocity of 18m/s (H x D)
Distance is 27m at t=2, at which it will then have a velocity of 54 m/s (H x D)
Distance is 81m at t=3
Regarding units, I don't think (and correct me if I'm wrong on this) this is a unit problem since only very large values give me wrong answers. I randomly gave H a value of 1500, made an example of that, and my formula indeed gave me wrong answers. When giving H a value of 50 and making an example out of that, the anwers my formula gives are still correct.

I think one should plot my formula against your mentioned formula and see at which value of H they start to deviate from each other..

 

[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?