What can you learn about cosmology from Google calculator?

[wabbit]

Yes I'm reading another one now on inflation which was suggested by Chalnoth - Lesgourgues' lecture notes. Quite good, but the latter part on high energy physics is beyond me. Might switch to Lineweaver, he seems more gentle.

I saw in Wilson-Ewing that he refers to an earlier paper for a fuller LQC treatment of the bounce (just bounce plus radiation era)
http://arxiv.org/abs/1404.4036
T. Pawlowski, R. Pierini, and E. Wilson-Ewing, “Loop quantum cosmology of a radiation-dominated flat FLRW universe"

 

[marcus]

If you look at x = -0.45 on the Desmos plot in post #88 you see an inflection where deceleration changes to acceleration.
Up to that point the contraction is decelerating (as if under the influence of Lambda ) and after that point, contraction is accelerating (as if under the influence of matter.)

But that is just if you want to view the whole curve as being of cosmological interest (it shows the Cai&Wilson_Ewing picture, in broad outline.) Otherwise we just refer to the right half (the positive x side).

On the right, keep in mind that the present day corresponds to x_now = 0.8.
For simplicity we are using the H_∞ time scale
H_∞T_now = 1.832 attohertz*13.787 billion years = 0.797 ≈ 0.8
We know that on the conventional timescale the inflection (in distance growth) where acceleration kicks in, happens around year 8 billion, perhaps slightly before.
x_infl = H_∞T_infl = 1.832attohertz*8 billion years ≈ 0.45.

If you look on the expansion side of the picture, at x = 0.45, it is the same story. There is an inflection. As long as matter dominates, expansion decelerates, and then as soon as Lambda dominates it begins to accelerate.

The natural sinh^2/3(1.5x) time x = 0.45 corresponds to year 8 billion, just as x = 0.8 corresponds to the present year 13.787 billion.
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You get the same story in Lineweaver's Figure 14, but the point on the x-axis to look for is "-6 billion" that is 6 billion years before present. IOW around 8 billion.
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I'm thinking of how a newcomer might best be introduced to cosmology. The first question is the relation of time to distance: the expansion history. As so many very naturally ask, what about this ACCELERATION I've heard about?

That Figure 14 shape IS the most characteristic thing about our universe expansion history. It shows growth of scale factor over time. And the
$$a(x) = \sinh^{2/3}(1.5x)$$
shape is the most characteristic thing about it.
The fact that the distance growth curve has an inflection point is what the cosmology folks mean when they talk about "acceleration".

that curve could be step one of an exposition. And the timescale H_∞ = 1.832 attohertz could be step two, because that is how you translate conventional year numbers (as we humans measure time) into the x time that the sinh^2/3 function likes. Google calculator helps here:
1.832 attohertz*13.787 billion years ≈ 0.8
1.832 attohertz*8 billion years ≈ 0.45
Step three might be the idea of NORMALIZING the a(...) function so that it equals one at present. E.g. by deciding to always divide a(x) by a(.8)

Or maybe one just leaves it as it is, un-normalized, since with the scale factor all we really care about is RATIOS, of scales at various times. It could be argued that normalizing it to unity at the present is somewhat "presento-centric". But it's both convenient and conventional so maybe we should normalize.

 

[wabbit]

About that distance to a past light emission event - while we can't solve it explicitly for a matter-lambda mix, there are formulas for lambda-only / de Sitter space and for matter-only expansion that might perhaps be useful for exposition, the actual behavior falling somewhere in between.

Let's call $D(t,t_0)$ the distance from the light emission at t, as measured by the observer at $t_0$ .

No expansion :$$ D(t,t_0) =c(t_0-t)=_{def}d $$
De Sitter expansion :$$ D(t,t_0) =\frac{c}{H}\left(e^{H(t_0-t)}-1\right)$$
Matter expansion : $$ D(t,t_0) = c t_0\cdot 3\left(1-\left(\frac{t}{t_0}\right)^{1/3}\right)
=\frac{3c(t_0-t)}{1+\left(\frac{t}{t_0}\right)^{1/3}+\left(\frac{t}{t_0}\right)^{2/3}}$$

 

[wabbit]

A couple more - let's compute the ratio $\xi=D(t,t_0)/d$

No expansion :$$ \xi=1 $$
De Sitter :$$ \xi=\frac{e^{Hd/c}-1}{Hd/c}$$
Matter :$$ \xi=\frac{3}{1+\left(\frac{t}{t_0}\right)^{1/3}+\left(\frac{t}{t_0}\right)^{2/3}}$$

 

[marcus]

"Nothing exists except fields and geometry; everything else is opinion."

 

[wabbit]

I still find it difficult to figure what $D(t_0,t)$ really is - I said before we're measuring a spatial distance between two events from the point of view of the comoving observer at t0, but what is his point of view ? i.e. which coordinates is he using and in what sense is this a distance ? It is not the distance to the current position of the comoving light source, and it is only defined as an integral...

Well it is the comoving distance at the time of emission between the source and us. Whatever that means. Or the comoving distance to the emission event. Still doesn't register.

 

[wabbit]

"Nothing exists except fields and geometry; everything else is opinion."

But geometry is nothing but fields, so nothing exists except fields.

... actually, nothing exists except information, everything else is opinion - and that is information too, so :

Nothing exists except information.

So we did make some progress over 2500 years

 

[marcus]

We seem to be getting the gist of Jorrie's Lightcone calculator just using Google calculator to evaluate some fairly simple formulas. There is some satisfaction at being able to "do it yourself" or "cook from scratch" and maybe I understand the standard cosmic model a little better because of it.
I still think Lightcone tables are a great hands-on way of getting acquainted with LambdaCDM model and past+future cosmic history.

In this approach, we are using the eventual Hubble radius (a version of cosmological constant) as a time and distance scale.
It makes the formulas much neater and easier to type to measure time in multiples of 17.3 billion years and distance in multiples of 17.3 billion light years. So our time variable, x = T/17.3 billion years, is just our conventional time divided by R_∞ = 1/H_∞

I want some sample times to use in trying out our formulas so I'll list some, along with normalized scale factors S = 1.311 sinh(1.5x)^(2/3)

x-time  billion years   S
.1         1.73               4.632
.2         3.46               2.896
.3         5.19               2.183
.4         6.92               1.771
.5         8.65               1.494
.6        10.38               1.288
.7        12.11               1.127
.797      13.787              1.000

These should check with Lightcone (to within a percent or so) in the sense that if you type an upper S value into Lightcone, like S = 1.494 you should get the time T = 8.65 billion years. Or close to it. These S values are google-calculator ones.
Now if you put S=1.494 into Lightcone, for example, you get a bunch of other information. Like D_now(S). This is the proper distance NOW of a galaxy whose light was emitted back in year 8.65 billion and comes to us today stretched out by a factor of 1.494. wavelengths are one and a half times longer than they were when the light was emitted.
Our formula for the distance is
$$D_{now} = 1.311 \int_{x_{em}}^.797 |\sinh(\frac{3}{2}x)|^{-2/3} dx$$
That gives it to us in our terms and to convert to years we just multiply by 17.3 billion light years.
The result should agree with the D_now from Lightcone calculator. I'll use my laptop's integration utility. I put a couple of Lightcone values in for comparison.

x-time  billion years       S            Dnow billion light years
.1         1.73               4.632      23.03 (23.014)
.2         3.46               2.896      16.80
.3         5.19               2.183      12.47
.4         6.92               1.771       9.08
.5         8.65               1.494       6.26
.6        10.38               1.288       3.87
.7        12.11               1.127       1.78  (1.759)
.797      13.787              1.000

Hey the Mac laptop integration utility is really easy to use! It's hardly any more trouble than just typing in the formula and the upper and lower limits of integration. Scarcely any more bother than using the google calculator. You just have to go to it (applications --> utilities --> grapher). It makes a graph of what it's integrating.

 

[wabbit]

Dnow(S). This is the proper distance NOW of a galaxy whose light was emitted back in year 8.65 billion and comes to us today stretched out by a factor of 1.494.

But is it ? I've been struggling with this. If $d_{em}$ was our distance to that galaxy at the time of emission, its distance now is just $d_{em}\frac{a}{a_{em}}$ but the light traveled $D=\int \frac{cdt}{a(t)}$ and I don't see why the two should be equal - I read D as the spatial distance to the emission event but this doesn't make much concrete sense to me. So far I can only understand it as "how much the light travelled" but even that is a bit hairy...

Edit : no I see it now you're right sorry, $D=d_{em}$ since $D=\int_0^d dx$. And you must have included the factor $\frac{a}{a_{em}}$ in your definition of D. Is that right, your D is actually $\frac{a}{a_{em}}\int \frac{cdt}{a(t)}$?

 

[marcus]

Hi Wabbit,
You probably recognize the 1.311 factor as sinh^2/3(1.5* 0.797). When we drew the curve for how a(x) increases over time, the present was around .8 and its a(x) = a(.8)was around 1.3. that lovely antelope-horns graph.

So S(x) the factor the wavelength of bit of light will get stretched if it is emitted at time x and received today, is the ratio
1.311/sinh^2/3(1.5* x)

and if a bit of light is traveling already and between time x and x+dx it travels a certain amount cdx (we can ignore c = 1) then that little interval will ALSO get expanded along with everything else by the ratio S(x) by the time it arrives.

So what the integral formula does is simple. It just adds up all the little bits of distance the light traveled and magnifies each one by the appropriate stretch factor.

We are using the same definition and concept of Dnow as Jorrie's calculator. We just have a kind of naive basic formula for calculating a (fairly good) approximation to it. I took the numbers in parens from the Dnow column of the Lightcone calculator.

I picture Dnow as the distance light travels between time x_em and time x_now=.797.

the formulas can be fairly light, easy to type because scaled for simplicity. If you come in with a conventional time T in years,
you have to divide by 17.3 billion years to get x.
If your formula gives you a distance you then have to multiply by 17.3 billion light years to get the conventional distance in light years.

 

[marcus]

I forget if you have a Mac. You may have said. The numerical integration utility is so easy to use! It's refreshing.

You don't even have to go through the formality of typing an integral sign. You just type in the function you want integrated and type in the limits of integration. It draws a picture of the function with a shaded area between the upper and lower limits, and tells you the answer.

 

[wabbit]

To be honest I'm having a hard time following now, with the change of units and coefficients. Each time I see x I need to remind myself of what it is, and same for 1.31 or so.

 

[wabbit]

I forget if you have a Mac. You may have said. The numerical integration utility is so easy to use! It's refreshing

Nope, no Mac. I integrate with pen and paper Sir, like our forebears : )(Oh if I really need to, I use a lowly PC )

 

[marcus]

To be honest I'm having a hard time following now, with the change of units and coefficients. Each time I see x I need to remind myself of what it is, and same for 1.31 or so.

Thanks for the heads-up! It's time to pause, straighten out the notation, summarize. We've been discovering relations and possibilities as we went along. Tine to review, simplify and make consistent. I think presented in orderly fashion none of it will be very hard, and it will match the standard model (e.g. Jorrie's calculator) pretty well.

 

[wabbit]

About that, to me at least, it's much easier when the units are explicit (i.e. everything is expressed as a ratio or similar, like $H_{\infty}T$ or (easiest) $\frac{T}{T_\infty}$) rather than implicit like $x$. Of course the formulas becomes lighter with just x but in any case to plug them into a calculator you need to substitute with a ratio. Another possibility is also to formulate with x but frequently add a reminder "(where $x=\frac{T}{T_\infty}$)".

 

[marcus]

That's a good idea. T_∞ = 17.3 billion years, is a time scale and something we use constantly.

If you walk in the door with an ordinary time T in years the first thing you do is divide by 17.3 billion years and get T/ T_∞
and if the formula gives you a distance like Hubble radius R at that time, then to get back into conventional terms you multiply by cT_∞ = 17.3 billion light years and walk out the door with the answer in billions of light years.

T_∞ and cT_∞ are how you translate between conventional scales and the world seen through hyperbolics like sinh and tanh.
So I'll try writing the formulas making that explicit.

 

[wabbit]

Sounds good to me : ) also, what's nice about large scale units is that c=1 light-year per year, so that's one constant we don't need to worry about much.

Edit : which makes me realize that c>>1 is really a very anthropocentric view. Cosomology-wise, c=1 is very natural, the universe has comparable dimensions in time and space. It's just that we are very very slow ourselves, perhaps just because we have cooled down to a very low temperature since the radiation era when everybody thought c=1 was kinda obvious. So in that sense c>>1 is a measure of how cold this universe has become, and we might be able I dare speculate to express c as some normal looking constant times some power of the ratio of the CMB temperature to the Planck temperature... Or something:)Hmm this isn't quite true our local temp is driven by the Sun not the CMB - well, something like that, just don't look too closely:)

 

[marcus]

... Another possibility is also to formulate with x but frequently add a reminder "(where $x=\frac{T}{T_\infty}$)".

That might be an even better approach! best of both ways. remarking and getting used to the change of scale when you enter and exit the hyperbolic model, but also keeping the equations light and trim.

We also should have a distinct notation for the un-normalized scale factor. It was confusing when I wrote a(x) because the scale factor is usually normalized to equal 1 at present. Let's try u(x) = sinh^2/3(1.5 x)
mnemonically, u stands for "unnormalized" so it might be easy enough to adjust to.
Keeping in mind that x_now = T_now/T_∞ = 13.787/17.3 = 0.797, we have the normalized scale factor:

a(x) = u(x)/u(x_now) = u(x)/u(.797) = u(x)/1.311

And this simplified cosmic model's most elaborate equation so far is for the present distance spanned by light emitted at time x_em:

$$D_{now}(x_{em}) = 1.311 \int_{x_{em}}^{x_{now}} \frac{cdx}{u(x)}$$

where c = 1

 

[wabbit]

About x just one more thing - i know that time=space but still, there are conventions in the back of our mind ; : ) so perhaps $\theta$ or $\tau$ could be a better name.

Regarding a(x) vs u(x) I'm mixed... I see the advantage of u(x) and I find the convention a=1 now to be confusing at times but there is also the issue of relating what's written here to what's elsewhere, and a() has that advantage of being commonly used, so is the gain in switching to u enough to balance the need to memorize one more name and how the two relate ? I just can't tell, it might even depend on who reads it and in what context.

 

[marcus]

In this thread we employ time and distance scales, provisionally dubbed T_∞ and R_∞, which are the longterm limits of the present-day Hubble time and Hubble radius scales often used in cosmology. Our scales are, in fact, the eventual these quantities are tending towards, and are based on the cosmological constant.
It turns out that the current Hubble time, is 5/6 of T_∞ , and the current Hubble radius is 5/6 of R_∞. On the other hand, the present age of universe expansion is only about 4/5 of T_∞.
T_∞ and R_∞ are estimated at 17.3 billion years and 17.3 billion light years.

On these scales of time and distance, the speed of light is one, and the present-day expansion age = x_now = 13.787/17.3 = 0.797 ≈ 4/5
Provisionally at least, we will denote times measured on this scale by x, where x = time in years divided by 17.3 billion years.
x = T/T_∞

With time scaled this way, it turns out that the size of the universe, u(x), as tracked by the size of a generic distance, follows a simple curve:

$$u(x) = \sinh^{2/3}(\frac{3}{2} x)$$

If we want to normalize this size function so that it equals one at present, we just have to compute its value at present, and divide by that.
the normalized scale factor can be called a(x)
$$a(x) = \frac{u(x)}{u(x_{now})} = \frac{u(x)}{1.311}$$

 

[marcus]

Continuing in mock tutorial form, to try out notation:
"Another nice formula in the closely approximate model we get with this scaling of time and distance is the one for H(x) the fractional growth rate at any given time x. Keep in mind that the scaled time variable x is years divided by 17.3 billion years.

By definition, the Hubble time Θ is the reciprocal of the fractional growth rate. It changes inversely as the growth rate of distances changes.
I want a distinctive notation for the Hubble time, that won't be confused with the age of the universe or the time-variable itself. I'm going to try the capital letter Theta.

At any time x, the distance growth rate H(x) and the Hubble time Θ(x) are related by:
Θ(x) = 1/H(x)
for example if the Hubble time is 10 billion years, what that means is that any given distance (between stationary points) is growing by 1/10 of its length per billion years. Or 1/10,000 of its length every million years. Or in more familiar growth rate terms, growing by 1/100 of one percent per million years.
The Hubble time is a convenient way of encoding that growth rate.
Our formula for Hubble time, showing how it grows with the expansion age of the universe, is quite simple:
$$Θ(x) = \tanh(\frac{3}{2}x)$$
To do an example, remember that x_now is about 4/5, more exactly 0.797
If we calculate tanh(1.5*.797) in google we get 0.83227 (about 5/6). We can interpret that either in time or distance terms.
To get the answer in conventional distance terms, we multiply by the eventual longterm Hubble radius R_∞ = 17.3 billion light years, and get 14.398... which rounds to 14.4 billion light years.
===================================
Earlier I had misgivings about the T notation for Hubble time. Here's part of what I wrote when I was still undecided. What follows will probably be deleted, when the issue is resolved.
In an earlier post you suggested a different symbol for a time quantity, Hubble time perhaps.
Should one use Θ = 1/H?
I just now wrote
$$T(x) = \tanh(\frac{3}{2}x)$$
the formula for Hubble time. Should that instead be
$$\Theta(x) = \tanh(\frac{3}{2}x)$$
The uppercase T risks being overused.---temperature,...---and Theta starts with "T". The uppercase Theta has what looks like a small capital H in its middle, H for Hubble time, H for hyperbolic tangent...
Maybe it would be mnemonic to define Θ(x) = 1/H(x) at time x.

 

[marcus]

I think it's mathematically most convenient to use S as the independent variable, as in Lightcone tables, or its reciprocal the normalized scale factor. But time is intuitive and familiar. So newcomers may like to see a table generated by this simplified model cosmos where the x-time (time scaled by 17.3 billion years) serves as the driving variable. (Just a sketch. I might add some other columns later.)
a(x) is the normalized scale factor at time x: sinh^2/3(1.5x)/1.311
S(x) is the stretch since that time, 1/a(x) = z+1
Θ(x) is the reciprocal growth rate, the Hubble time (equiv. Hubble radius), namely tanh(1.5x)

x-time  (Gy)    a(x)    S       Theta   (Gy)     Dnow     Dnow (Gly)
.1      1.73    .216    4.632   .149    2.58    1.331       23.03
.2      3.46    .345    2.896   .291    5.04     .971       16.80
.3      5.19    .458    2.183   .422    7.30     .721       12.47
.4      6.92    .565    1.771   .537    9.29     .525        9.08
.5      8.65    .670    1.494   .635   10.99     .362        6.26
.6     10.38    .776    1.288   .716   12.39     .224        3.87
.7     12.11    .887    1.127   .782   13.53     .103        1.78
.797   13.787  1.000    1.000   .832   14.40    0            0

Everything except Dnow was calculated by google
for example a(.1) was sinh(1.5* .1)^(2/3)/1.311
Theta(.1) was tanh(1.5* .1)
and its billion year (Gy) value was 17.3*tanh(1.5* .1)

The exception Dnow required numerical integration, which is easier than one might think.
In the Mac "Grapher" utility one simply types in 1.311 (sinh(1.5 x)^-2/3 using ^ for superscript,
clicks "integration" in the "equation" menu, and enters the limits, for instance .1 and .797.