Simple no-pressure cosmic model gives meaning to Lambda

[marcus]

Hi Wabbit, good to see you! This post is not in reply to yours. Our posts crossed, what I'm doing here is continuing from post #49 where two questions are mentioned:
1) discover the time-scale (the "Uday" will turn out to be 17.3 billion years) and
2) discover what the present is, on that scale (how many Udays has it been since start of universe expansion).
If you assume an answer to the first question, the second is easy. That's because the present day fractional growth rate is directly measurable from the wave-stretch of light from nearby things we know the distances to.
Think about what it means to say the present Hubble time T₁ = 14 billion years.
It means that distances are growing about 7% per billion years. That is what you get if you take 1 over 14 billion years. You get a fractional growth of 1/14 per billion years.

That is the kind of growth we can actually SEE and measure, more precisely 1/14.4 per Gy, or to say it another way T₁ = 14.4 Gy.

So if we had already chosen our time SCALE to be 17.3 Gy, measured in Udays the present Hubble time would be 14.4/17.3

Now we know that T(x) = 14.4/17.3, and the one basic model equation says that T(x) = tanh(1.5 x), so all we need to do is find the cosmic time x that solves tanh(1.5 x) = 14.4/17.3.

It amounts to looking up the inverse function of tanh, applying tanh to 14.4/17.3 in REVERSE. Wikipedia has ample information of that sort about the hyperbolic trig functions. It says how to UNDO tanh:
$$\frac{3}{2}x = \frac{1}{2} \ln \frac{1+ \frac{14.4}{17.3}}{1 - \frac{14.4}{17.3}}$$ Or, in slightly different notation introduced earlier:$$x_{now} = \frac{1}{3}\ln \frac {T_\infty + T_{1}}{T_\infty - T_{1}}$$

 

[marcus]

...
One thing that seems special with ours, is that (our) life started out at about the time of cc-matter equality. I can't see a reason for that, a lower cc would have made no difference to anything, and we would still be in a matter era then. So just a weird coincidence it seems.

As I recall the inflection point in the distance growth curve comes about year 8 billion. That is when deceleration changes over to acceleration.
I would guess that life appeared on Earth about 4 billion years ago, so if this is year 13.8 billion that would be around year 9.8 billion.

As a coincidence it is about 2 billion years off. Still, it's a coincidence of sorts. I think you are right that it is a meaningless one though.

Terrestrial protozoa had to wait for the solar system to form, and for this particularly nice planet to form, and even then they had to get pretty lucky.
I guess such critterlets could have arisen much earlier elsewhere in this and other galaxies, and may yet arise elsewhere in future. Avi Loeb of Harvard Smithsonian has a paper about this, if I remember correctly .

 

[marcus]

The theme I'm working on at the moment is something that Wabbit paraphrased in post #50.
There is only ONE equation basically. All the rest follows from the way the fractional growth curve evolves: T(x) = tanh(1.5x).
and things we can measure. The present day Hubble time T₁ is straightforward, Hubble measured it already circa 1930s. The time-scale T_∞ requires more work---fitting curve to wavestretch-distance data.

ooops barely got started on this, have to go help with something.
If you care about the outcome, it takes two people to crank pasta for pot-stickers
back now.

I'm thinking of someone who fears calculus or at least tends to avoid it, but who likes the universe.
Here is a chance to review the power rule and the chain rule in a pleasant context.
remember you put the power out front and subtract one from the power
x^p ⇒ px^p-1
and by the chain rule if it is another function raised to the power then you first do that and then differentiate what was inside being raised to that power
f^p ⇒ pf^p-1f'
The chain rule says f(g(x)) ⇒ f'(g(x))g'(x) and
f(g(h(x))) ⇒ f'(g(h(x)))g'(h(x)) h'(x) so that eventually everybody gets their turn at getting differentiated

If the power is 2/3 and you subtract 1 you get -1/3
So if we start with sinh^2/3(1.5x) we'll get a factor of 2/3, and then sinh^-1/3(1.5x) (something in the denominator) times the derivative of what was inside, namely sinh(1.5x).
But that is cosh(1.5x) times 1.5 (chain rule again, slope of 1.5x is 1.5.

The factors of 2/3 and 1.5 cancel and we get cosh(1.5x)/sinh^1/3(1.5x)

Now if we DIVIDE by another copy of the scale factor (to get the FRACTIONAL growth, the increase as a fraction of the distance itself) then we have a full sinh(1.5x) in the denominator.
The fractional growth rate is cosh(1.5x)/sinh(1.5x).

By definition the reciprocal of that fractional growth rate is the Hubble time T(x) and so we have the desired result T(x) = tanh(1.5x)

The growth of the unnormalized scale factor sinh^2/3(1.5x) is exactly what is required by the basic model equation T(x) = tanh(1.5x)

the model is defined by one equation (wherein one has to discover, by observational measurements) what the right time-scale is and what the present time is on that scale

In a sense T(x) = tanh(1.5x) is both simpler and more fundamental because we already used it to find
x_now from the two key Hubble times 14.4 and 17.3 by running this equation in reverse and solving 14.4/17.3 = tanh(1.5x) for x.

 

[Jorrie]

... - there will be a characteristic time at which cc overtakes matter, and an ultimate Hubble radius or time, and those two tell the whole story.
One thing that seems special with ours, is that (our) life started out at about the time of cc-matter equality.

There are two possible meanings and hence two times involved here: "cc overtakes matter" could be the inflection point, where the deceleration went over to acceleration, around T=7.6 Gy (S~1.65); and the "cc-matter equality", when the matter density equaled the effective "Lambda energy density", which happened at about T=10 Gy (S~1.33). The latter is around life's appearance on Earth, but as you said, likely to be just an interesting coincidence.

PS: It is the ~30% present matter density that determines the ~17.3 Gy timescale. I'm not sure if we can call this density 'an observable', but if so, and read with the flatness of space, the 17.3 Gy is totally based on observables. Or am I too optimistic?

 

[wabbit]

Right, my paraphrase was quite incorrect.

Trying again now :

There is exactly one matter-lambda solution to the FRW equation. It has an intrinsic clock, which defines a natural time and distance unit.

Asuming our universe looks like that, some special things for us are :
- we are currently (and life arose) at a time of order unity (a presumed coincidence)
- our intrinsic timescale (human life span or other) is of order 10^-9, but this probably a number more or less derivable from more fundamental time and distance scales, so more relevant is for instance the ratio of the Planck time to the universe time, which is key to how complex structures can evolve here. (this ratio is just a rephrasing of the value of the CC in Planck units)

 

[Jimster41]

QUOTE="marcus, post: 5069127, member: 66"]To recap, in this thread we are examining the standard LambdaCDM cosmic model with some scaling of the time variable that simplifies the formulas, and makes it easy to draw curves: curves showing how the expansion rate (percent per billion years) changes over time, how a sample d

I have some catching up to do here. But I did just run into something interesting in this.

Sorry, Chaisson again.
http://arxiv.org/abs/1410.7374

He's all into the Universe as a non-equilibrium phenomenon (my paraphrasing is not helpful probably)

On page 6 he describes the "sigmoidal" complexification curve of some evolutionary processes, in contrast to others which are exponential

Sigmoid function
From Wikipedia, the free encyclopedia
Plot of the error function
A sigmoid function is a mathematical function having an "S" shape (sigmoid curve). Often, sigmoid function refers to the special case of the logistic function shown in the first figure and defined by the formula

Other examples of similar shapes include the Gompertz curve (used in modeling systems that saturate at large values of t) and the ogee curve (used in the spillway of some dams). A wide variety of sigmoid functions have been used as the activation function of artificial neurons, including the logistic and hyperbolic tangent functions. Sigmoid curves are also common in statistics as cumulative distribution functions, such as the integrals of the logistic distribution, the normal distribution, and Student's t probability density functions.

I can't help notice a) the curve of growth rate over time (in post #27), appears somewhat sigmoidal. b) functions of the form above are popping up in the development of this model. Not trying to make something out of it, I just think coincidences are worth observing, and I had been wondering about that curve in #27

 

[wabbit]

PS: It is the ~30% present matter density that determines the ~17.3 Gy timescale. I'm not sure if we can call this density 'an observable', but if so, and read with the flatness of space, the 17.3 Gy is totally based on observables. Or am I too optimistic?

Not sure what you mean - it is in fact observable, from supernova etc., or we wouldn't be discussing it ? Or do you mean, observable without a model ? But even then, I think fitting the luminosity/redshift relation is enough to measure the (local) CC in principle, as something related to some second derivative read off that curve.

Edit : barring possible errors including stray signs,
$$H_\infty=\frac{1}{D'_0}\sqrt{1+\frac{2}{3}\frac{D''_0}{D'_0}}$$
Where the derivatives are taken with respect to z and evaluated at z=0.

Edit 2 : this formula is however derived from a matter-lambda FRW model.

 

[wabbit]

@marcus I do not think it is a coincindence that life arose when it did, in relation to a matter-only universe expansion- it arose when the temperature for that was right more or less (after the nucleosynthesis temperature then other critical temperatures where passed of course - why those temperatures are arranged as they are however I have no idea except that it must relate to how the different forces are hierarchised). But since the whole expansion would be essentially the same with or without CC, the fact that this was around x of order unity, it at least unexplained if not a sheer coincidence- unless we find that the CC is related to other fundamental constants.

 

[Jorrie]

Edit : barring possible errors,
$$H_\infty=\frac{1}{D'_0}\sqrt{1+\frac{2}{3}\frac{D''_0}{D'_0}}$$
Where the derivatives are taken with respect to z and evaluated at z=0.

Interesting! Cannot recall having seen it before. How is this relation derived?

I'm under the impression that a lot of observational data are needed in order to find the 'best-buy' solution for matter density and Lambda as a combination.

 

[wabbit]

Interesting! Cannot recall having seen it before. How is this relation derived?

Combining $H(z)=H_0\sqrt{\Omega_{\lambda}+(1-\Omega_{\lambda})(1+z)^3}$ with $D'(z)=1/H(z)$ and taking derivatives at z=0.
Actually what this gives first is the simpler
$\Omega_m=\frac{2}{3}\frac{H'_0}{H_0}=-\frac{2}{3}\frac{D''_0}{D'_0}$ , the clumsier relation follows.

I'm under the impression that a lot of observational data are needed in order to find the 'best-buy' solution for matter density and Lambda as a combination.

I am certainly not claiming that the relation above is a smart way to estimate - clearly, fitting the whole curve is more reliable. It's purpose was only to exhibit in principle an explicit formula for measuring the CC from a set of comoving standard candles.

But it does rely on the FRW model. I must retract my incorrect suggestion above that this might be a model free formula. It think it might be possible to do that in principle but I don't know how.

 

[marcus]

My PoV is not necessarily decisive in this thread, of course, but I'll tell you my impression about matter density. I don't think we can estimate it at all accurately, what with dark matter clouds and gas and all kinds of stuff besides stars. Even the estimates of luminous matter in galaxies are rather uncertain. So I think the matter density estimate just comes from observing the lack of overall curvature and calculating the (matter+radiation only) CRITICAL.

As far as concerns me, Lambda is not an energy and does not contribute to flatness the way matter density does. In any case not for the model discussed here.
Friedmann equation , for our purposes in this thread, has Lambda on the left hand side as reciprocal square time.
$$H^2 - \frac{\Lambda}{3} = \frac{8\pi G}{3c^2}\rho^*$$
Friedmann equation inherits that Lambda, on the LHS, directly from 1917 Einstein GR equation.
*Reminder: as I just said, ρ* is a matter&radiation density. It does not contain any "dark energy" component. The curvature constant Λ is explicitly on the left side. This equation must be satisfied for there to be overall spatial flatness.
By definition
$$H_\infty^2 = \frac{\Lambda}{3}$$
Therefore the Friedmann can be written this way:
$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho^*$$

EDIT: I deleted a reference to "ρ_crit" when it was pointed out in a helpful comment that this might be confusing. As additional guard against confusion I put an asterisk on the density as a reminder that, as the density of matter and radiation, it doesn't involve a Lambda component. The equation must be satisfied for spatial flatness and so, in that sense, ρ* is critical for spatial flatness once the two expansion rates H and H_∞ have been determined.

A few posts back, Wabbit pointed out a useful version of the Friedmann equation (for matter-era on into indefinite future, since there is no "Lambda era") that saves a fair amount of bother, writing density, and constants like π and G etc. I'll write it using the wavestretch factor that Jorrie introduced in the Lightcone calculator. S=1 denotes the present.

$$H(s)^2 - H_\infty^2 = (H(1)^2 - H_\infty^2)s^3$$

For me, in this thread, the main topic is this model in which Λ, or more precisely T_∞ serves as a time scale. So to proceed we should evaluate the terms in the equation. Obviously the present value of the Hubble constant is 173/144 = 1.201... and its square is 1.443. Obviously, in the timescale we are using, H_∞ = 1 and its square is 1.
The RHS of the Friedmann equation evaluates to:
$$H(s)^2 - H_\infty^2 = (H(1)^2 - H_\infty^2)s^3 = 0.443s^3$$
And in our time scale the Friedmann simplifies to:
$$H(s)^2 - 1 = (H(1)^2 - 1)s^3 = 0.443s^3$$

 

[Jorrie]

But it does rely on the FRW model. I must retract my incorrect suggestion above that this might be a model free formula. It think it might be possible to do that in principle but I don't know how.

Interesting equation, thanks.
Matter density can be obtained from other independent observations, perhaps most importantly from grav. lensing. If that's accurate enough, Lambda is indirectly available for the flat space case. Still never quite model-free, I guess...

 

[Jorrie]

Since we observe spatial flatness, near enough to it anyway, that defines the (matter-radiation) critical density ρ_crit
$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho_{crit}$$

One must be careful not to confuse here, because isn't the present (matter-radiation) critical density only 30% of the 'standard' (quoted) critical density?

If so, shouldn't you give it a different subscript?

 

[marcus]

It is the combined density of all known forms of energy that is required for spatial flatness. Do you have any suggestions?
How about "rho_flatness" or "rho flat"?
See how you think these would work:
$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho_{flat}$$
$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho_\flat$$

Not using LaTex involves using the word as in ρ_flat
or having the symbol available to paste (since I don't know how to type it, with a Mac)
ρ_♭

 

[marcus]

Referring back to post #61:
In our time scale the Friedmann simplifies to:
$$H(s)^2 - 1 = (H(1)^2 - 1)s^3 = 0.443s^3$$ Referring also to a post or two on previous pages:

So adapting your equation in post#35
$$ D(a)=\int_a^1\frac{da}{a^2H(a)}=\int_0^z\frac{dz}{H(z)} $$we can write:
$$ D(S) = \int_1^S T(s)ds = \int_1^S (( (\frac{17.3}{14.4})^2 - 1) s^3 +1)^{-1/2}ds$$

This is where we got the equation (with help of Wabbit's post #35) for the present distance to a source we are now receiving light from that is stretched by factor S
$$ D(S) = \int_1^S T(s)ds = \int_1^S (0.443 s^3 +1)^{-1/2}ds$$
This formula is the basic tool that allows astronomers to directly tell the cosmological time-scale constant Λ from wavestretch-distance data. To consider an example such data could for instance consist of pairs of numbers (s, D) each giving the stretch factor of some light received and the standard candle estimate of current distance to its source.

The procedure basically relies on assuming near spatial flatness, which is supported by a variety of evidence. Given that, and that you have independently determine the present Hubble time 14.4 billion years, you choose some alternative time-scales to try out: 16.3 billion years, 17.3 billion years, 18.3 billion years. Each one will change the 0.443 number somewhat.
Then for each observed wave-stretch factor in your sample you compute the D(S) distance that light should have covered (don't forget to multiply by the distance scale). And you see if that matches the "standard candle" distance that was also part of the data.

It turns out that the expansion time and distance scale 17.3 billion years gives the best fit to the wavestretch-distance data, at least so far. The point I'm emphasizing is the sense in which it is directly observable without having to know the value of the matter density or assuming any model specifics. Sure it depends on General Relativity (from which the Friedmann equation is derived) and on the assumption of near spatial flatness, but those are widely accepted general assumptions.

 

[marcus]

So if we want to compare two assumptions H_∞ = 1/17.3 per billion years, or = 1/20 per billion years, using (s, D) data, we calculate
(20/14.4)² - 1 = 0.929
(17.3/14.4)² - 1 = 0.443
And we evaluate these integrals which give the distances in billions of lightyears:
$$ D(S) = 17.3 \int_1^S T(s)ds = \int_1^S (0.443 s^3 +1)^{-1/2}ds$$
$$ D(S) = 20 \int_1^S T(s)ds = \int_1^S (0.929 s^3 +1)^{-1/2}ds$$

I've tried it using an online definite integrator and the latter (the "20" one) gives noticeably smaller distances, especially in the higher wavestretch range such as S > 1.5 and even more so for S > 2.

The H_∞ = 1/20 per billion years is after all closer to zero than 1/17.3 per billion years.
So the "20" case is more like having zero cosmological constant. What woke people up to the fact of a positive cosmological constant was that measured distances to standard candle supernovae were distinctly larger than theoretically predicted assuming zero Lambda.
H_\infty = \sqrt{\Lambda/3} is the operative form of the cosmological curvature constant here.
And I find its reciprocal, the longterm Hubble time T_∞ = 1/H_∞ ≈ 17.3 billion years, is its most useful, easiest-to-remember quantitative expression.

 

[Jorrie]

It is the combined density of all known forms of energy that is required for spatial flatness. Do you have any suggestions?
How about "rho_flatness" or "rho flat"?
See how you think these would work:
$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho_{flat}$$

This may still confuse readers, because "flat" is normally associated with total energy density being critical. I noticed that The Perlmutter et al. paper of 1998 (http://arxiv.org/abs/astro-ph/9812133) made use of a super- and subscript to indicate the "matter density component of a spatially flat cosmos", i.e. \Omega^{flat}_M, so \rho^{flat}_M may be a good solution for clarity here.

 

[wabbit]

This may still confuse readers, because "flat" is normally associated with total energy density being critical. I noticed that The Perlmutter et al. paper of 1998 (http://arxiv.org/abs/astro-ph/9812133) made use of a super- and subscript to indicate the "matter density component of a spatially flat cosmos", i.e. \Omega^{flat}_M, so \rho^{flat}_M may be a good solution for clarity here.

I think the discussion of flatness and its impact on interpreting things can be confusing here - would it not be better to separate this as successive independent steps such as :
(a) we know that to a good approximation the universe is spatially flat and was so during the period concerned,
(b) with that in mind (and also perhaps an observational argument for stating that radiation is negligible over that period), we can think of the universe as made of matter, with a CC, and nothing else
(c) on this basis, follow with marcus' presentation of how we can measure the CC/matter proportion etc.

These steps may not be really that independent, but as a first introduction it still seems a fair simplification to me. Maybe not though, not quite sure here.

 

[Jorrie]

And we evaluate these integrals which give the distances in billions of lightyears:
$$ D(S) = 17.3 \int_1^S T(s)ds = \int_1^S (0.443 s^3 +1)^{-1/2}ds$$
$$ D(S) = 20 \int_1^S T(s)ds = \int_1^S (0.929 s^3 +1)^{-1/2}ds$$

I've tried it using an online definite integrator and the latter (the "20" one) gives noticeably smaller distances, especially in the higher wavestretch range such as S > 1.5 and even more so for S > 2.

While the derivation of this approximation is interesting and educational, the result is more easily (and probably more accurately) obtained by Lightcone 7. To simulate a "no Lambda" flat universe, just copy and paste the max allowable R_\infty (999999) into the box, set S_upper to 2 (or whatever). Calculate and look at the value of D_now at (say) S=2 or lower. If you make R_\infty just marginally larger than R₀ (say 14.41), you get a near-Lambda-only, 0.1% matter flat universe. The calculator is not designed for matter closer to zero than that.

 

[marcus]

This may still confuse readers, because "flat" is normally associated with total energy density being critical. I noticed that The Perlmutter et al. paper of 1998 (http://arxiv.org/abs/astro-ph/9812133) made use of a super- and subscript to indicate the "matter density component of a spatially flat cosmos", i.e. \Omega^{flat}_M, so \rho^{flat}_M may be a good solution for clarity here.

Thanks Jorrie,
I went back and edited post #61, eliminating the notation ρ_crit. I shall use the notation ρ* and put in frequent reminders that the cosmological constant term is here on the left side, so there is no Lambda contribution to the energy density.

 

[marcus]

While the derivation of this approximation is interesting and educational, the result is more easily (and probably more accurately) obtained by Lightcone 7. To simulate a "no Lambda" flat universe, just copy and paste the max allowable R_\infty (999999) into the box, set S_upper to 2 (or whatever). ...

Thanks. I'm glad you approve of the exercise. Interest and hands-on learning are the main aims, in fact. The simple model here (I think George Jones first suggested it to us, it may be well-known) seems to approximate Lightcone numbers fairly well as long as one stays away from the radiation dominated era ( e.g. don't go earlier than, say, year million). Basically AFAICS Lightcone is the online gold standard. I'm constantly checking number calculated with this simple "by hand" method to see how well they agree. I see the two things working together in the same learning experience.

I hadn't planned on including a "no Lambda" calculation. The "20" calculation was just a sample of a substantially smaller Lambda. But I see one could get the required distances as you describe---using a large Hubble radius.

 

[Jorrie]

I shall use the notation ρ* and put in frequent reminders that the cosmological constant term is here on the left side, so there is no Lambda contribution to the energy density.

I'm comfortable with this as a valid interpretation of the standard model, assuming that Lambda is really constant. If proved otherwise, then "simple, no-pressure" may have to be abandoned, but that's surely not for beginners. I think this is a very useful introductory approach...

 

[George Jones]

A related pedagogical paper: "A new standard model of the universe" by Gron,

http://arxiv.org/abs/0801.0552

 

[marcus]

Øyvind Grøn of Oslo University. Neat! I see he has some of the same-shaped curves! coth for H(t) and sinh^2/3 for the scale of a sample distance growing over time. Crucial differences--I wouldn't want to mix discussion of the two models--but interesting. Thanks GJ, I'm glad to know of Grøn's paper

 

[marcus]

I'm exploring ways to present the flat LambdaCDM as it works any time after the early years when radiation was the dominant content. As soon as matter takes over from radiation as the main component of their combined energy density ρ* , the standard cosmic model equations simplify and there is a reasonable facsimile of LambdaCDM that we can work by hand.
Here we use the same two basic inputs used in the Lightcone calculator: present and future Hubble times 14.4 and 17.3 billion years, and the same independent variable S = z+1=1/a, the wave and distance stretch factor. It is the reciprocal of the normalized scale factor, so S=1 denotes the present and its definition extends into the future.

I assume that the two Hubble growth rates H_now and H_∞ have been determined to be 1/14.4 and 1/17.3 per billion years.
The essential point here is that if we make 17.3 billion years and 17.3 billion light years our units of time and distance then the Friedmann equation (the main cosmic model equation) takes a remarkably simple form.

First notice that the present Hubble growth rate is simply 17.3/14.4 = 1.2014 per unit time. If you call our new unit of time a "Uday" then the present rate is 1.2 per Uday, and the eventual Hubble growth rate is exactly one. H_∞ = 1 per Uday.
An important term that appears in the Friedmann equation involves the square growth rates:
H_now² - H_∞²
And that turns out to be 1.443 - 1 = 0.443

To make a short story even shorter, when we use these units of time and distance the Friedmann equation basic to cosmology simplifies to:

H(s)² - 1 = 0.443s³

I've described earlier in the thread how this is derived from the more recognizable form of Friedmann:
H² - H_∞² = [const.] ρ*
I'll review that derivation later, and it's not hard so you may see for yourself how it goes. But now let's just look at the extremely simple version of the cosmic equation.
Something to notice is how close this equation is to observation---observation of actual wavelengths of light.
If you are visiting an observatory at night and the telescope is trained on a galaxy and you are told the stretch factor is 2, the wavelengths coming in are TWICE what they were when the light was emitted and headed our way,
then you can almost do in your HEAD the calculation of what the Hubble expansion rate was back then when the light was emitted. 2 cubed is 8.

And 8 times .443 is about 3.5.
Plus 1 is 4.5, whose square root is 2.1.

So you can remark to your astronomer friends: "I know what the Hubble growth rate was back then when the light was emitted! It was a little over twice the longterm rate."
If you wanted to spell it out in terms of years instead of the longterm rate unit, it is 2.1 times the longterm rate of 1/17.3 per billion years, so 2.1/17.3, or about 1/8.1, per billion years.
So you could add: "The Hubble time must have been around 8.1 billion years back then!"

 

[marcus]

To take another example, suppose your friend is observing a galaxy and says the incoming light has stretch S = 3.
3 cubed is 27, times 0.443 is about twelve.
Add one and take the square root. The square root of 13 is about 3.6.
So the Hubble growth rate back then, when the light was emitted, was 3.6 times the longterm rate.

The longterm expansion rate H_∞ is our natural UNIT for picturing growth rates, and I am trying to think in terms of that. So I won't bother to work out what that would be in "per-billion-Earth-year" terms. But you could, if you want, divide 3.6 by 17.3.

To review a bit. We look at the standard cosmic model, ΛCDM, focusing on the spatial flat matter-dominated case. and we let an important feature of that model, namely Λ, determine our scale of time and distance. The square root of Λ/3 is the longterm growth rate H_∞, in the standard model. And the corresponding Hubble time, 1/H_∞ is generally estimated to be around 17.3 billion years. We treat that as one "universe day", or Uday.

so the longterm growth rate H_∞ is now our unit growth rate. As it happens the current rate of distance growth H_now is about 20 percent larger. It is 17.3/14.4 = 1.2014. That number and its square, 1.443, help to characterize the present for us.
In terms of our new time scale, H_now = 1.201 per Uday and H_now² is 1.443 per Uday²

The Friedmann equation, the basic cosmic model equation, simplifies on our time-scale to:
$$H(s)^2 -1 = 0.443s^3$$
where s is the stretch factor, s = z+1. In the Lightcone calculator the stretch factor is capital S. I may have to switch over to that, or alternate s and S.
S = 2 denotes a time in the past when distances were 1/S = 1/2 their present size.

The wave stretch of light received today tells us the distance it has traveled---how far it is now from its source matter. This includes the effect of distance expansion while the light was in transit.
$$D(S) = \int_1^S \frac{ds}{\sqrt{0.443s^3 + 1}}$$

 

[Jorrie]

The Friedmann equation, the basic cosmic model equation, simplifies on our time-scale to:
$$H(s) -1 = 0.443s^3$$
where s is the stretch factor, s = z+1.

You seem to have missed a squaring in the above: should be \small H(s)^2 -1 = 0.443s^3 (?)

 

[marcus]

Corrected. Thanks for catching that.

 

[Jorrie]

Corrected. Thanks for catching that.

Not wanting to be pedantic, but why not just stick the constant 17.3 in front of the proper distance integral and let the answer come out in Gly, like we are used to on this forum? You are using the constant 0.443, which comes from it anyway...

 

[marcus]

I know what you mean, Jorrie. Tried that earlier. This is exploratory. I want to see what the approach looks like when you go all the way over to the new units. Time scale 17.3 Gy and distance scale 17.3 Gly. If I was doing a practical calculation where I wanted to compare results with your calculator, I'd certainly stick a 17.3 out in front and save a step in the calculation as was done here.

...
And we evaluate these integrals which give the distances in billions of lightyears:
$$ D(S) = 17.3 \int_1^S T(s)ds = \int_1^S (0.443 s^3 +1)^{-1/2}ds$$
...

BTW if you have the time and want to help explore this and post some equations/derivations/explanations here in thread, it would be very welcome.

 

[marcus]

To review a bit. We look at the standard cosmic model, ΛCDM, focusing on the spatial flat matter-dominated case. and we let an important feature of that model, namely Λ, determine our scale of time and distance. The square root of Λ/3 is the longterm growth rate H_∞, in the standard model. And the corresponding Hubble time, 1/H_∞ is generally estimated to be around 17.3 billion years. We treat that as one "universe day", or Uday.

so the longterm growth rate H_∞ is now our unit growth rate. As it happens the current rate of distance growth H_now is about 20 percent larger. It is 17.3/14.4 = 1.2014. That number and its square, 1.443, help to characterize the present for us.
In terms of our new time scale, H_now = 1.201 per Uday and H_now² is 1.443 per Uday²

In this scale of time and distance the Friedmann equation simplifies to $$H(s)^2 -1 = 0.443s^3$$
and a flash of light's distance from its source simplifies to $$D(S) = \int_1^S \frac{ds}{\sqrt{0.443s^3 + 1}}$$

I guess my observation about the number 0.443 is that it doesn't depend on Earth years or any particular human units. Rather it depends on this present moment in universe history. Imagine people on a different planet with a different year, perhaps not even using years to measure time but some "second" defined using their own atomic clock.

They could still measure H_{now} and H_\infty

and they would still find that
$$\frac{H_{now}}{H_\infty} = 1.201...$$ $$(\frac{H_{now}}{H_\infty})^2 = 1.443...$$

 

[Jorrie]

In this scale of time and distance the Friedmann equation simplifies to $$H(s)^2 -1 = 0.443s^3$$
and a flash of light's distance from its source simplifies to $$D(S) = \int_1^S \frac{ds}{\sqrt{0.443s^3 + 1}}$$

I guess my observation about the number 0.443 is that it doesn't depend on Earth years or any particular human units. Rather it depends on this present moment in universe history.

I agree, but am still unsure about the wisdom of scaling the equations to the so-called "long term Hubble time (or radius)" 1/H_∞. The conventional method is to scale to 1/H₀, which is an LCDM-model-independent observable (AFAIK). I don't think 1/H_∞ is.

The penalty for doing so is (perhaps) a slightly messier equation, but then it correlates directly with what everyone reads in the textbooks.
Starting with your $$H(S)^2 - H_{\infty}^2 = (H_0^2 - H_{\infty}^2)S^3 $$ (which is 'cool') and normalizing for H₀=1 instead of (your) H_∞, we get $$H(S)^2 - \Omega_\Lambda = (1-\Omega_\Lambda)S^3$$ or $$H(S)^2 - \Omega_\Lambda = \Omega_m S^3$$
Any student will recognize this as the matter + cosmological constant 'first Friedmann' for the spatially flat LCDM model.

 

[marcus]

Hi Jorrie, thanks for pointing out that alternative. I think they are both interesting ways to present the standard LambdaCDM cosmic model in a simple post-radiation-era spatial flat version.
I think you could argue that they each have advantages and it's somewhat a matter of taste. Maybe both should be developed in tandem and both should be available for teacher and student inspection.

It's clear that in our lifetimes the quantity 14.4 billion years is not going to change (except to be determined more precisely) so even if in principle it is an accident special to the present moment in time one can and does use it as a a time-scale! I think that's the essence of what you suggest here.
The current Hubble growth rate 1/14.4 per billion years is not in a strict sense a "constant" but it is practically speaking constant enough to provide a convenient timescale. So why not use it? At least it is less arbitrary than "years" the orbit period of this one particular planet in this one little solar system! I can see the point. It also makes the equations nice. Probably we should explore both.

Also we don't KNOW that Einstein's curvature constant Lambda is actually a constant of nature's geometry. We are admittedly in the dark about that. I think what I am doing is testing something out. We don't know that the speed of light is constant, but so far evidence supports the idea. We don't know that Planck's hbar is a constant but so far it seems to be. And so far it looks like there's this curvature constant Lambda in nature. Let's suppose it really is a constant and tailor a bit of expository writing around that---treat it as such IOW.

 

[marcus]

So, Jorrie, the premise here (at least within the confines of this thread) is that there is this growth rate H_∞ whose square is Λ/3, a third of Einstein's curvature constant. And that this growth rate H_∞ is a universal constant of nature which is not going to change "as long as the rivers run to the sea" (as the European colonists and the Native Americans used to write forever in their treaties)

I don't know if you take much interest QG threads in BtSM forum---you may or may not have noticed this development: Several approaches to formulating quantum geometry/gravity use simplexes to represent quanta of geometry (triangles, tetrahedra, pentachora).
It's been found interesting and possibly useful, instead of using flat ones, cut out of flat space, to use simplexes with a constant curvature Lambda. It's conceivable that nature's simplexes are actually not flat, but very slightly curved.

This is one motivation for wanting to try treating Lambda as a geometric constant of nature, which, as such, defines a natural time scale.

 

[Jorrie]

This is one motivation for wanting to try treating Lambda as a geometric constant of nature, which, as such, defines a natural time scale.

I have no problem with Lambda as a constant - both approaches assume that. My concern is more that we may confuse students with this unconventional 'natural time scale', while the Hubble time is well known and documented. AFAIK, it is also the more directly observable one; the value of 1/H_∞ is still more uncertain.

'Pushing the boundaries' may not be appropriate for beginners courses, but more advanced students will certainly find this stimulating. So the 'tandem approach' that you mentioned may be a good option.

 

[marcus]

Just have time to begin this post, company just arrived. Will edit these quotes from earlier posts down to essentials as soon as I have time
==revised excerpts==
I think the matter density estimate just comes from observing the lack of overall curvature and calculating the (matter+radiation only) CRITICAL. As far as concerns me, Lambda is not an energy and does not gravitate the way matter density does. It's just the original Einstein curvature constant. In the model discussed here, the Friedmann equation , for our purposes in this thread, has Lambda on the left hand side as reciprocal square time (the square of a a distance growth rate).
$$H^2 - \frac{\Lambda}{3} = \frac{8\pi G}{3c^2}\rho^*$$Friedmann equation inherits that Lambda, on the LHS, directly from 1917 Einstein GR equation.
*Reminder: as I just said, ρ* is a matter&radiation density. It does not contain any "dark energy" component. The curvature constant Λ is explicitly on the left side. This equation must be satisfied for there to be overall spatial flatness.
By definition
$$H_\infty^2 = \frac{\Lambda}{3}$$Therefore the Friedmann can be written this way:
$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho^*$$This shows that as the matter&radiation density thins out the growth rate H must approach a longterm limit growth rate H_∞. I put an asterisk on the density as a reminder that, as the density of matter and radiation, it doesn't involve a Lambda component. The equation must be satisfied for spatial flatness and so, in that sense, once the current and longterm expansion rates H and H_∞ have been determined, ρ* is critical for spatial flatness.
A few posts back, Wabbit pointed out a useful version of the Friedmann equation (for matter-era on into indefinite future, since there is no "Lambda era") that saves a fair amount of bother, writing density, and constants like π and G etc. I'll write it using the wavestretch factor that Jorrie introduced in the Lightcone calculator. s=1 denotes the present.$$H(s)^2 - H_\infty^2 = (H(1)^2 - H_\infty^2)s^3$$The main novelty in this thread is the way in which Λ, or more precisely T_∞ = 1/H_∞ serves as a time scale. So to proceed we should evaluate the terms in the equation. Obviously the present value of the Hubble constant is 173/144 = 1.201... and its square is 1.443. Obviously, in the timescale we are using, H_∞ = 1 and its square is 1.
The RHS of the Friedmann equation evaluates to:
$$H(s)^2 - H_\infty^2 = (H(1)^2 - H_\infty^2)s^3 = 0.443s^3$$
And in our time scale the Friedmann simplifies to:
$$H(s)^2 - 1 = (H(1)^2 - 1)s^3 = 0.443s^3$$Since the Friedmann equation simplifies to $$H(s)^2 -1 = 0.443s^3$$a flash of light's distance wave stretch factor s = z+1 which we can read directly from it tells us how far it has gotten from its source, how far away its source now is:
$$D(S) = \int_1^S \frac{ds}{\sqrt{0.443s^3 + 1}}$$This can be used to DETERMINE H_∞ from standard candle distance redshift data as discussed earlier. So in that sense this is a universal form of the Friedmann equation at this point in the history of the cosmos.
My observation about the number 0.443 is that it doesn't depend on Earth years or any particular human units. Rather it depends on this present moment in universe history. Imagine people on a different planet with a different year, perhaps not even using their planet's years to measure time but some other natural cycle. They could still measure H_{now} and H_\infty and as long as they are our contemporaries they would still find that
$$\frac{H_{now}}{H_\infty} = 1.201...$$ $$(\frac{H_{now}}{H_\infty})^2 = 1.443...$$==end of revised quotes==

 

[marcus]

The distance growth rate (Hubble parameter) as a function of time is given by an unusually simple equation. I'll use x to denote time in units of
1/H_∞ = T_∞ = 17.3 billion years. Exploring the use of that scale is basically what we are doing in this thread. On that scale the present epoch is 0.8 , more precisely 0.797 but 0.8 is close enough. It is just the usual age in billions of years, 13.8, divided by 17.3.
Nature seems to like that scale because in those terms the growth rate at any given time in history is simply the hyperbolic cotangent$$H(x) = \coth(\frac{3}{2} x) $$The scale factor, the size of a generic cosmological distance as it grows over time, is also give by a hyper-trig function, the hyperbolic sine$$u(x) = \sinh^\frac{2}{3}(\frac{3}{2} x) $$For some purposes it's nice to have the scale factor NORMALIZED to equal one at the present epoch. To normalize this we just find out its value at x = 0.8 and divide by that so that the new scale factor a(x) is forced to take the value 1.$$a(x) = u(x)/u(0.8) = u(x)/1.311$$

 

[marcus]

There was an earlier curve that also belongs to this model and that I want to recall. It is the INVERSE of the normalized scale factor a(x) as a function of time x. Remember we are measuring time in "Universe days" or "Udays" of 17.3 billion years so the present is x=0.8.
What the normalized scale factor does , given a time x, is tell you the size of distances back at time x, compared with now. The formula is
$$a(x) = \sinh^\frac{2}{3}(\frac{3}{2}x)/1.311$$The 1.311 is just a normalization factor put into make sure that a(x) = 1 at the present epoch.
We might want to solve that in reverse. For instance some light comes in at the observatory and we put it through a grating to spread out the various colors/wavelengths and we recognize a hot sodium line but the wavelength is THREE TIMES that of the yellow light that hot sodium makes in the lab.
So while that light was traveling to us, wavelengths and distances were stretched three-fold. Distances were 1/3 their present size. So a(x_then)=1/3.
And we wonder WHEN WAS THAT? When was it when that galaxy emitted the light I just put through the grating? What were things like? What was the distance expansion rate back then? How far away is that galaxy now?
So we want to find the time x, for which a(x) = 1/3. Want the reverse or inverse function of the normalized scale factor a(x)
Call it x(a) if you like, the time x that corresponds to a scale factor a.
$$x = \frac{2}{3}\ln(\sqrt{(1.311a)^3} + \sqrt{(1.311a)^3+1})$$It's basically just the inverse of the hyperbolic sine, sinh. Here's the plot. It should look sort of like one half of that earlier ("antelope horns") a(x) picture turned over on its side:

Notice that it gives the right age of the universe. Light with scale factor a = 1 is emitted and received in the present, without any stretch. For a=1 the curve says the time x = 0.8. That is 0.8 Udays or 80% of our time unit 17.3 billion years---namely 13.8. Here's some earlier discussion which gives some other examples.

To recap, in this thread we are examining the standard LambdaCDM cosmic model with some scaling of the time variable that simplifies the formulas, and makes it easy to draw curves: curves showing how the expansion rate (percent per billion years) changes over time, how a sample distance grows over time, and so on.
Time is measured in units of 17.3 billion years. If it makes it easier to remember think of that as a day in the life of the universe: a Uday.
I've been using the variable x to denote time measured in Udays,
Our unnormalized scale factor u(x) = sinh^2/3(1.5x) shows the way a generic distance grows.
Normalizing it to equal 1 at present involves dividing by u(x_now) = 1.311.
The normalized scale factor is denoted a(x) = u(x)/1.311.
The fractional distance growth rate H(x) = u'(x)/u(x) = a'(x)/a(x) = coth(1.5x)

Note that the normalized scale factor a is something we observe. When some light comes in and we see that characteristic wavelengths are twice as long as when they were emitted, we know the light began its journey when a = 0.5, when distances were half the size they are today.

So by the same token the unnormalized u = 1.311a is observed. We just have to measure a and multiply it by 1.311.

Knowing the wavelength enlargement we can immediately calculate the TIME x (measured in Udays) when that light was emitted and started on its way to us. Here is x as a function of the number u.
$$x = \frac{2}{3}\ln(\sqrt{u^3} + \sqrt{u^3+1})$$Since u = 1.311a, we can also write time as a function of the more readily observed number a.$$x = \frac{2}{3}\ln(\sqrt{(1.311a)^3} + \sqrt{(1.311a)^3+1})$$Here's a plot of the time x(a) when the light was emitted as a function of how much smaller the wavelength was at emission compared with the wavelength on arrival.
View attachment 81741
You can see, for example, that if the wavelength at emission was 0.2 of its present size, then that light was emitted at time x = 0.1. since time unit is 17.3 billion years, that means emitted in year 1.73 billion.
On the other hand if the wavelength was originally 0.8 of its present size, which mean it has only been enlarged by a factor of 1.25, then the light was emitted more recently: at time x = 0.6.
Multiplying by our 17.3 billion year time unit we learn it was emitted around year 10.4 billion.

You might think the curve is useless to the right of a = 1 (which denotes the present). But it shows the shape and, come to think of it, tells something about the future. If today we send a flash of light which later is picked up in a distant galaxy by which time its wavelengths are 40% larger---1.4 of their present size---what year will that be?
You can see from the curve that it will be time x=1.1
which means the year will be 17.3+1.73=19.03---about year 19 billion.

And the curve also tells how long the universe has been expanding, just evaluate the time at a=1.

 

[marcus]

So just for fun (or for hands-on practice) let's do an example. I realize I didn't put in the curve for a(x) yet, I only put the curve for the Unnormalized scale factor u(x), where we didn't divide by 1.311 to make it equal one at the present day. So here is the normalized scale factor a(x). This is the most basic curve of the model. It shows how the size of the universe (or a sample distance) changes over time:

You can see how a(x) equals one at the present time (0.8 Uday).
I think of this as the "antelope horns" picture. : ^)
So imagine we are at an observatory and some light comes in that has been wave-stretched 5-fold. What does it tell us? It comes from a time when distances were 1/5 their present size, so 0.2 on the vertical scale.
When was that? We can see from the red curve that the time was x = 0.1, one TENTH of a Uday, one tenth of 17.3 billion years. But we are using the Uday time scale so we don't bother to convert to billions of Earth years, we just think of it as time x = 0.1

What was the expansion rate H(x) back then? We just put x=0.1 into our tanh formula (that gives the Hubble time, one over that gives the Hubble rate)
tanh(1.5*0.1)
Google says:
tanh(1.5 * 0.1) = 0.14888503362
IN OTHER WORDS GROWTH BACK THEN WAS ABOUT SEVEN TIMES THE LONGTERM GROWTH RATE
The eventual longterm growth rate is a good unit to think in terms of. We are used to it by now in many guises: one per 17.3 billion years, 1/173 percent per million years. 1/17.3 per billion years, about 0.06 per billion years. It is the eventual growth rate the universe is heading towards. and 1/0.1488 is about 7.
So when that light was emitted which we just analyzed, the growth rate was 7 times eventual.

If you want Hubble time in billions of Earth years, just multiply the 0.1488.. by 17.3 and you get 2.5757...billion years. I would read the Hubble growth rate from that as 1/26 of a percent per million years. It's somewhat a matter of taste and habit what you find the most comfortable way to express and think about the Hubble distance growth parameter.

 

[marcus]

How far is that galaxy from us now? We were thinking of an example were some light came in with stretch factor 5, from a galaxy back in the time when distances were 1/5 what they are today. And we used our model to say WHEN that was (time x = 0.1, or year 1.73 billion)
and also used our model to say what the growth rate H(x) was, back then (about 7 times the universe's longterm rate).

So suppose now we ask how far that galaxy is now? How far has this flash of light traveled from its source? (on its own and aided by expansion).

This is a job for the "number empire" integrator. We integrate out to the incoming stretch factor s=5.
$$\int_1^5 \frac{ds}{\sqrt{.443s^3 + 1}}$$
You google "number empire definite" and you get
http://www.numberempire.com/definiteintegralcalculator.php
and you just type in the integrand and the limits 1 and 5. It's easy.
The integrand is (0.443*s^3+1)^(-1/2). You press "compute"
It gives 1.38, that is the distance to the galaxy in light Udays our distance unit when we use this time scale. So if you like, mulitiply the 1.38 by 17.3 billion light years to get it in those Earth year terms.

And we can tell right away what the distance to the galaxy was THEN, when the light was emitted and started on its way to us. Distances were 1/5 their present size back then so divide 1.38 by 5.

Basically the light you receive tells you all these things by showing you how much it was stretched.

 

[marcus]

In terms of this model, how do we think of today's distance growth rate?
Well just as our time unit is the longterm Hubble time T_∞ = 1/H_∞ [ = 17.3 billion years]
our unit growth rate is H_∞

So we measure the current growth rate by comparing it with the eventual rate that growth is tending towards.
And it turns out that current rate is 20% larger. H_now = 1.20 H_∞
or, with excess precision, 1.2013

That is where the number 0.443 in the distance integral came from, in the previous post.
1.2013² = 1.443
In our terms, the number 0.443 is H_now² - 1

today's date: 0.8
today's expansion rate: 1.2
In a sense what we're doing is describing the universe in its own terms, or in terms which, as far as we can tell, are eternal.
The current growth rate is always changing, declining actually, albeit slowly, so it seems awkward to use it as a unit of measurement. On the other hand, as far as we can tell the longterm growth rate is a constant. It is where the current rate is tending. So on that account it is better adapted for use as a unit.

So we measure the present growth rate in terms which have nothing to do with our planet's year, or human time-keeping conventions, but which take their meaning from the overall behavior of the universe.
Furthermore we state the present age in unchanging terms, which conceivably might be understood in the distant future and perhaps even cross culturally. This moment, the time since the start of expansion, we express in terms of a unit which we have reason to believe is universal

 

[Jorrie]

Marcus, I have just posted my conclusion to the "new jacket" for your simplified model and I now fully concede that what you are doing here is far superior. I would suggest that you consider writing this up in a paper, because I think it is also superior to the published Grøn simplified model.

 

[marcus]

Thanks so much for the encouragement, Jorrie! What we are doing in this thread is part of an informal unplanned PF collaboration involving you, George Jones, and Wabbit, as well as myself.

I can't think of any place better to post a brief summary of this simple cosmic model than your website (Relativity-4-Engineers) alongside the Lightcone cosmic histories calculator. I'm a big fan of Lightcone, as you know, and I think the two go together because you can check the results of the simplified formulas given in this thread against the tables and curves generated by standard LCDM cosmology embodied in Lightcone.

If you have the inclination and energy to post a description at R-4-E don't hesitate. I can't imagine that would preclude anybody from writing the model up later in some more publishable form. You are welcome to copy selectively and edit from any of my writings here if you want to patch something together for R-4-E.

My feeling is that examining and weighing alternatives (some of which you were doing in the other thread) is part of a rational development process and contributes to perspective, so I'm glad you were covering that angle.

 

[Ken G]

I'm coming into this late, but it seems to me that what you are saying is that the Friedmann equation without radiation has a simple enough form to allow three levels of simplification:
1) a trivial level that since the only unknown for a flat universe is the ratio of dark matter to dark energy, all relevant a[x] solutions form a one-parameter family of solutions based on that ratio, expressed via the Lambda parameter
2) a more interesting level that since the form of the mass density and dark energy density are simple monomials, the one-parameter family of a[x] solutions admits a rescaling of a and x, based on Lambda, such that the rescaled a[x] is a single universal function.
3) a particularly lucky level that the universal function can be written closed form.
That is pretty neat indeed-- nature has been kind to offer these simplifications, we would be foolish not to take advantage of them.

Then there is also the deeper question of whether or not there is physical significance to these simplifications, or whether it is just a happenstance mathematical convenience that nature has been so nice. After all, we did have to neglect radiation, though note a similar simplification is possible in early times, when there is no dark energy to worry about, and then the universal function also has a closed form, it looks like 1 - Sqrt[1+x]+x*Sqrt[1+x]/2 in some appropriately scaled way. So at early times, it looks like we have one appropriately scaled universal function, which transitions before recombination into your universal function, though the latter is scaled in a different way which depends on the ratio of dark matter to dark energy rather than the radiation-to-dark-matter ratio. Does the fact that we have these universal functions reveal some deeper physical insight?

It seems that you like to think of it as an insight in the sense that we can think of dark energy entirely in terms of how it scales that universal function to bring it into contact with our sense of distances and times. Also, we can think of the dark matter density in similar terms, entirely about the scaling of the early-time universal function to bring it into contact with our concept of space and time in that early period. Since we are restricting to flat universes, which is a reasonable thing to do, this is a potentially powerful way to think about those two parameters. Instead of thinking about dark matter as a mass density, we can think of it as a rescaling of the early-period universal function into space and time, and instead of thinking of dark energy as an energy density, we can think of it is a rescaling of the late-period universal function into space and time. These two rescaled universal functions merge into each other in a transition regime that can be approximated by a kink where they cross, though would be awkward to treat accurately. That does seem like a potent re-interpretation of cosmological dark matter and dark energy to me.

 

[slatts]

Looks like I overlooked LambdaCDM in my Lame "Cantor Dust" post earlier today...a lot of good that c-minus in trig 50 years ago did me...

 

[marcus]

This is a question for anyone who has been reading this thread and has ideas about it. Do you think portions of the material here would be suitable for introductory presentation in the new "PF Insights" context? What would you cover? How would you title it? What would you not include?

Should it be co-authored? It's the result of an accidental unplanned interaction involving George Jones, Jorrie, Wabbit, myself, and others.
Would anyone like to produce a draft or outline for how an "Insight's" piece could go? Comments on any of my writing here, and suggestions, are welcome. We can draft and edit here, if desired, and only submit it later, if it seems appropriate.

The basic idea is this (but this doesn't have to be stated explicitly, just understood). In several branches of physics it is normal to use units where natural recurrent quantities are set equal to 1. E.g. in cosmology often G=c=1, or 8πG=c=1. In other research you might see Planck's ħ = 1, or Boltzmann's constant k=1. It makes the equations simpler and can actually aid intuition. You are using units of measurement and scales that nature "likes".

So in cosmology we study the distance growth rate H(t) and how it changes over time, and how you can test that with observations. I'll restrict this to the spatially flat case (all the evidence supports overall near spatial flatness.) Overall spatial curvature might not be exactly zero but it is so close to zero that this flat model gives an excellent approximation.

And as far as we can tell the growth rate H(t) has behaved (and will behave) like this:

when you look at this what stares you in the face is a natural unit---of growth per unit time. The curve has a limit, call it H_∞, which in Earth year terms (not the natural unit, but a familiar one) is about 0.06 per billion years.
...
If we make that our unit rate then its reciprocal becomes our unit of time and according to the most recent measurements that quantity 1/H_∞ of time which becomes our unit is in a narrow uncertainty range around 17.3 billion years. For convenience we can use a name for this unit of time. Unless there's strong objection or someone has a better idea let's call it one universe day, or Uday. Maybe we could just call it a Day, with a capital letter. For definiteness let's fix our unit time to be a Uday of 17.3 billion Earth years. Or is a Day that equals 17.3 billion years?

So that's the longterm distance growth rate. In the long-range future a distance will grow by a millionth of its size in a millionth of a Uday. (or should I say in a millionth of a Day?)

And what about the present rate of growth? H_now has been measured quite a lot and is known to be around 1.20 H_∞
More exactly H_now = 1.201 (to avoid round-off error I might sometimes use 1.2014).
The main thing is that the square H_now² = 1.443
The square growth rate is what enters in the main cosmology equation, the Friedmann equation.

 

[marcus]

In our terms the Friedmann equation (in the spatial flat case) takes this form:
$$H^2 - H_\infty^2 = [constant] ρ^*$$
Where ρ* is the combined energy density of radiation and matter (including both ordinary and dark matter)

The density now satisfies $$H_{now}^2 - H_\infty^2 = [constant] ρ_{now}^*$$ and based on observations we know that in our units the left hand side is 1.443 - 1 = 0.443
But matter density increases as the cube of the stretch factor s. If wavelengths and distances in some era are being stretched by a factor of 2 between then and now, then distances were 1/2 their present size back then and matter density was 8 times its present value. Except in the very early universe matter is the main component and we neglect radiation in the next equation.
So $$ρ^* = ρ_{now}^*s^3$$ The density at any other time is just the density now multiplied by the cube of the s-factor corresponding to that time. $$H(s)^2 - H_\infty^2 = [constant] ρ_{now}^*s^3 = (H_{now}^2 - H_\infty^2)s^3$$
In our units $$H(s)^2 - 1 = 0.443s^3$$ Incoming light, by its wavelengths, tells you its stretch factor. So by the same token it tells you the distance expansion rate H(s) at the time it was emitted.
$$H(s) = \sqrt{0.443s^3 + 1}$$

 

[marcus]

So imagine you are visiting some astronomer friends at an observatory and they are doing spectroscopy (wavelength measurement) on the light coming in from a distant galaxy. One of them mentions that the redshift is about 1. That means a stretch factor of two.
So ...two cubed is eight.
And 8 times .443 is about 3.5.
Plus 1 is 4.5, whose square root is 2.1.

So you can casually remark to your astronomer friends: "The Hubble growth rate back then when the light was emitted must have been a little over twice the longterm rate."

Or on another occasion they may have been studying a cluster of galaxies and happen to mention that the incoming light had redshift of about 2. That means a stretch of three.
And you think...three cubed is 27.
And 27 times 0.443 is about twelve.
12+1 = 13, and the square root of 13 is 3.6.

I gather you are amazingly good at doing arithmetic in your head! So then you casually remark to one of your friends, "hmmm...when the light from that cluster was emitted expansion must have been at between 3 and 4 times the longterm rate."

 

[Ken G]

I like the simple form of what you have, but I think you should use the traditional scale parameter a = 1/s. It's what people will recognize, though the equation comes out a little messier. Interestingly, H = (ds/dx)/s = -(da/dx)/a, so that part doesn't change form much when you choose between s and a. The next question is, does one simply use s or a as their independent variable, or switch to the more conventional use of x? Since the equation you write has a closed-form solution when you switch to the x variable, there's probably not much lost in doing that, and the gain is being able to talk about time instead of scale. Not that this is automatically a good thing-- as you point out, scale (or stretch) actually has a more direct connection with observed redshifts, whereas time is always awkward! I actually think we should talk about scale and not time (so we don't say the recombination era was at 400,000 years, we say it was at a scale parameter of 1/1000, or a stretch of 1000), so it might even be better to not change the independent variable.

I don't know how the PF Insights section works, so I can't comment on that. Maybe I should have a look at that section, I'm not familiar with it.

 

[marcus]

Thanks for sharing your thoughts, Ken! It's really helpful to have some feedback. I'll keep your comments in mind and hope others will respond as well.

Ken, occasionally we need an informal name for the unit time (17.3 billion years in Earth terms) and I have gotten used to thinking of it as a universe Day. How do you think that would work, and should the abbreviation be Uday or just Day with a capital? Maybe the context will be enough to distinguish that time unit from ordinary 24 hour days. Here's an example where the corresponding distance lightDay would come up in discussion:
==quote from an earlier post==
We were thinking of an example where some light came in with stretch factor 5, from a galaxy back in the time when distances were 1/5 what they are today. ...
... suppose now we ask how far that galaxy is now? How far has this flash of light traveled from its source? (on its own and aided by expansion).

This is a job for the "number empire" integrator. We integrate from s=1 as far as the incoming stretch factor s=5.
$$\int_1^5 \frac{ds}{\sqrt{.443s^3 + 1}}$$
You google "number empire definite" and you get
http://www.numberempire.com/definiteintegralcalculator.php
and you just type in the integrand and the limits 1 and 5. It's easy.
The integrand is (0.443*s^3+1)^(-1/2). You press "compute"
It gives 1.38, that is the distance to the galaxy in light Udays our distance unit when we use this time scale. So if you like, mulitiply the 1.38 by 17.3 billion light years to get it in those Earth year terms.
...
Basically the light you receive tells you all these things by showing you how much it was stretched.
===endquote==
You just touched on the "change of variable" idea, I think, and so may have been expecting this: the distance traveled formula, with a change of integration variable
$$\int_{x_{em}}^{x_{now}} s(x)cdx = \int_1^5 \frac{ds}{H(s)} = \int_1^5 \frac{ds}{\sqrt{.443s^3 + 1}}$$

I don't know if it fits in a PF Insights piece, but ds/dx = (-a'/a²) = -s(x)H(x)
so that dx = -ds/(sH)
and the integrand changes from s(x)dx to -ds/H(s)
Maybe that step is worth including.