Simple no-pressure cosmic model gives meaning to Lambda

[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.

 

[Jorrie]

Marcus, I will comment more fully later, but your Udays and especially light-Udays raise some concerns. A light-day has an established meaning in Physics and there is some confusion potential. Secondly, your Uday is a very long, non-cyclic time and perhaps a word that has such connotations would have been better; something like a Universal Time Constant. This specific one would not abbreviate well, because UT and UTC have already been established. Perhaps $U \tau$?

Us engineers are used to $\tau$ for time constants. :)

 

[Ken G]

Incidentally, I have a suggested minor change of form for your main equation for H(s). Instead of the coefficient 0.443, which occupies a somewhat arbitrary place, take its inverse cube root, which is 1.31, and place it under the s in your equation. Then 1.31 has the interpretation of "the stretch factor when the effective mass density of dark energy equals that of actual mass", so is a number that itself carries a meaning. Or, if you switch to a, take the inverse of 1.31 and that's the scale parameter when dark energy took over the lead. It's in the same spirit as your Uday parameter, just a meaningful scale for the stretch factor instead of a scale for the time parameter. I wouldn't recommend redefining the scale of s just to get a universal form for the equation, because s (or a) already has a unique meaning that relates directly to redshifts as you know, but this new form at least explicitly indicates the scale for s.

 

[marcus]

Marcus, I will comment more fully later, but your Udays and especially light-Udays raise some concerns. A light-day has an established meaning in Physics and there is some confusion potential. Secondly, your Uday is a very long, non-cyclic time and perhaps a word that has such connotations would have been better; something like a Universal Time Constant. This specific one would not abbreviate well, because UT and UTC have already been established. Perhaps $U \tau$?

Us engineers are used to $\tau$ for time constants. :)

Hi Jorrie, you could say that we don't have much NEED for a time unit name like Day or Universe Day because we can always talk around it and use a symbol for Hubble time, like 1/H_∞, or refer to the quantity as "our time unit 17.3 billion years".

Even if we do arrive at a unit name we like, we still might not make a lot of use of it. Might only need to use it occasionally.

Still, I'd like to try out some name. You've expressed reservations about Day and Uday. Maybe this other idea would work:

Aeon is a word associated with long spans of time. Roger Penrose has used that and I think also Paul Steinhardt may have used it. Anyway it is used.

So how about a magazine-style article on this simple version of the standard LambdaCDM where the title is

"From Aeon to Zeon"

The title would be just kidding, no need to discuss other people's "aeons", the whole point of the article (and this thread) is to explore introducing a time scale based on the cosmological curvature constant Λ and the eventual longterm distance growth rate H_∞ that we get from the curvature Λ.

In effect, the whole thing is about the zeon (whatever we happen to call it).

So the present age of expansion is 0.8 zeon, that's our way of saying "now".

And our distance unit is LZ instead of LY, or lz instead of ly.

And the current expansion rate is 1.2 per zeon, or more precisely 1.201 per zeon

and the longterm distance growth rate is 1 per zeon, which is our unit of expansion rate.

I hope that seems OK to you.

Also wondering how it sounds to Ken G, and others who might be reading the thread!

 

[Ken G]

I like the idea of using some version of "eon" for your time unit, you are talking about the single most fundamental long-time unit there is in our universe. Perhaps a "cosmological aeon", rather than aeon or zeon, because it could be abbreviated ca, not a or z. a is the scale parameter, z is redshift, and Z is metallicity, all important cosmological topics.

 

[Jorrie]

I like the idea of using some version of "eon" for your time unit, you are talking about the single most fundamental long-time unit there is in our universe. Perhaps a "cosmological aeon", rather than aeon or zeon, because it could be abbreviated ca, not a or z. a is the scale parameter, z is redshift, and Z is metallicity, all important cosmological topics.

Yea, what about "cosmaeon", still abbreviated ca and lca for distance? It is pronounced just like "cosmeon", but the latter is a trademark in the cosmetics industry. AFAIK, "Cosmaeon" has no established meaning. Aeon and Zeon both have known meanings, the latter in liturgy (http://en.wikipedia.org/wiki/Zeon_(liturgy)).

I'm busy with an article-style write-up, so maybe it could be titled "From Aeon to Cosmaeon".

 

[Ken G]

I do like cosmaeon. Perhaps a title like "How many eons in a cosmaeon?"