What can you learn about cosmology from Google calculator?

[marcus]

Before I say, I just want to mention that the Danish words for 15 and 18 are femten and atten
Atten sounds a bit like the English word "eighteen".
So that is where we get our metric prefixes femto- for 10^-15 and atto- for 10^-18.

When you do cosmology the most common quantity, or one of the most, that you encounter is the present-day value of the Hubble parameter---something around 67.9 km/s per Mpc.

If you type that into the google box and press return you get:
67.9 km/s per Mpc = 2.20...x 10^-18 hertz.

In other words, the google calculator thinks that Hubble growth rate is 2.20 attohertz.

In other words, in one second of universe time, the distance between two objects at cosmic rest increases by a small fraction of itself, namely 2.20 billionths of a billionth or 2.20 x 10^-18

Can we learn anything from the google calculator, in this case? Is there any point to taking that seriously for a moment, or do we just shrug it off as the calculator's quirky behavior?

One thing we learn is that google thinks hertz is the metric term for "per second". It doesn't have to be anything in particular per second. It doesn't have to be wave-cycles per second, it can be other kinds of counting. Radians per second, fractional growth per second, rotations per second.
OK, we can reject this and insist that hertz can only mean cycles per second. Or we can take a suggestion from the calculator and broaden our perspective a little---so we can take hertz as a metric term for seconds^-1. A synonym for "per second" generally. Either way seems reasonable enough. I'll pick the latter.

So I'm thinking of the Hubble parameter (at this point in universe standard time) as 2.20 attohertz.
What happens if I want to convert back?
Try it yourself. Type in [2.20 attohertz in (km/s per Mpc)] without the brackets.
Google will convert back into the old units and give you 67.9 km/s per Mpc.
The google calculator understands the word attohertz even though it may prefer to say "10^-18 hertz."

There's more that we can learn. I'll make another post of it so this one doesn't get too long.

 

[marcus]

Another thing you encounter in cosmology a lot is the cosmological curvature constant Lambda. This is an inverse area quantity or an inverse time-squared quantity that appears on the LEFT hand side of the GR equation (the geometry/curvature side).
If you look up "general relativity" in Wikipedia you see the main equation in a box near the top of the page, like this:

( http://en.wikipedia.org/wiki/General_relativity )

One way of writing the value, according to recent measurement, of this constant is Λ = 1.007 x 10^-35 seconds^-2

In other words, in a form that google calculator might understand, Λ = 10.07 square attohertz = 10.07 attohertz²

Let's check that, to make sure the calculator gets it. Type in [10.07 attohertz^2] without the brackets, press return, and google gives you
10.07 (attohertz^2) = 1.00700 × 10^-35 s^-2

Lambda gives its name to the standard cosmic model LambdaCDM or "LCDM" for short. The way the cosmological constant makes itself known in cosmology is through the longterm Hubble growth rate H_∞
This growth rate is related to Lambda by the equation Λ = 3H_∞²
If you solve that for H_∞ you find it equals 1.83 attohertz.
As a check, type this in (10.07/3)^.5, you should get 1.83

There is some confusion surrounding the word "acceleration". What actually we see is the Hubble rate H(t) having declined as if tending towards a longterm positive limit H_∞ instead of towards zero. This indicates an intrinsic spacetime curvature Λ, which persists after all other sources of curvature have dissipated. An innate residual curvature. There is so far no scientific evidence that it arises from anything we would normally call an "energy". What we see is a slight curvature, "dark energy" is more in the realm of conjecture and unnecessary complication, not to say myth.

Anyway, the Hubble rate has been acting over time as if its decline is going to level out at 1.83 attohertz. And here is the standard cosmology equation, the spatial flat Friedmann, that shows this.

H(t)² - H_∞² = [Friedmann constant] ρ(t)

ρ is the combined energy (equivalent) density of radiation and matter (dark and ordinary). Its present-day value is 0.24 nanojoule per m³. As density thins out and goes to zero, obviously the difference between H and H_∞ has to go to zero! That is the leveling out "flight path" that H(t) appears to be on, the observations tell us.
The Friedmann constant [8πG/3c²] converts energy density on the right to square attohertz (or whatever squared growth rate unit we're using) on the left.
In these units the Friedmann constant is 6.22 attohertz² per (n J/m³)
I know 6.22 is correct because if i put this in the box: 8 pi G/(3c^2) in square attohertz per (nJ/m^3)
Google gives me back:
(8 * pi * G) / (3 * (c^2)) =
6.2208967 (square attohertz) per (n J / (m^3))

Here n J/m³ is the density unit. And the present day combined density is 0.24 n J / (m^3)
So to check the Friedmann equation for the present day, we have to verify that:

2.20^2 - 1.83^2 = 6.22⋅ 0.24

Both sides are 1.49
or, in units, 1.49 square attohertz

 

[wabbit]

So the end is near ! Or less dramatically, the current value isn't very far from the long term rate. You posted before a chart of the expansion, but looking around I didn't find one that shows specifically H(t) as a function of t, or maybe I just missed it - would you happen to know of a source for that?

 

[marcus]

Maybe someone else will think a good plot of the H(t) curve. Or I will think one. But right off the top, the best curve I can think of is the curve of the RECIPROCAL 1/H OR c/H which is the so-called "Hubble time" or "Hubble radius". I can make a curve of that over any time-range you want, using Jorrie's calculator ("Lightcone") very easily.
As H(t) goes down, and levels out, so the reciprocal, say the Hubble radius, must ascend and level off.
It should level off at c/H_∞
Let's make google find that for us in billions of lightyears
We type in [c/(1.83 attohertz) in light years] without the brackets and google gives back
c / (1.83 attohertz) = 1.73162648 × 10¹⁰ light years
That is 17.3 billion light years---it is the expected longterm Hubble radius

So I have to plot a curve of the Hubble radius R(t) say from year 1/2 billion to year 50 billion or thereabouts.
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
First a rough table:
$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly) \\ \hline 0.100&10.000&0.5454&0.8196\\ \hline 0.126&7.943&0.7707&1.1568\\ \hline 0.158&6.310&1.0886&1.6308\\ \hline 0.200&5.012&1.5362&2.2939\\ \hline 0.251&3.981&2.1646&3.2127\\ \hline 0.316&3.162&3.0412&4.4626\\ \hline 0.398&2.512&4.2500&6.1052\\ \hline 0.501&1.995&5.8828&8.1349\\ \hline 0.631&1.585&8.0151&10.4035\\ \hline 0.794&1.259&10.6685&12.6018\\ \hline 1.000&1.000&13.7872&14.3999\\ \hline 1.259&0.794&17.2572&15.6486\\ \hline 1.585&0.631&20.9561&16.4103\\ \hline 1.995&0.501&24.7888&16.8364\\ \hline 2.512&0.398&28.6942&17.0630\\ \hline 3.162&0.316&32.6380&17.1800\\ \hline 3.981&0.251&36.6015&17.2395\\ \hline 5.012&0.200&40.5748&17.2696\\ \hline 6.310&0.158&44.5532&17.2847\\ \hline 7.943&0.126&48.5341&17.2923\\ \hline 10.000&0.100&52.5163&17.2961\\ \hline \end{array}}$$
oops the timer in the kitchen just went off. have to make the chart later

 

[ChrisVer]

If we learn something about Hertz vs (km/Mpc s)
I'd say it's a nice choice when you want to compare the Hubble parameter H with the different momentum modes k...? when eg you make a Fourier transform of a scalar field.

 

[wabbit]

Wow thanks ! This table is just what I was looking for - H, 1/H - graph, table - tomato, potato.

Edit : Very interesting, I expected to see something more sedate after the early shenanigans.

Edit : Saw that site before but found it a bit intimidating, I guess I should have a second look.

 

[marcus]

This is the plot that Lightcone does for that table, where I selected time to be the x-axis and eliminated all the other columns besides R(t)

At the moment I can't think of a plot of H(t) itself, only the reciprocal.

 

[wabbit]

This is really good. Another thing I get from this table is a direct comparison of how the two lengths a and R evolve, something I'd been wondering about - I assume a is in Gly as well - they give units everywhere but for this one.

 

[wabbit]

We are living at a special time - R(t) starts linear, then curves, then flattens to its limit ; and we are right at (the beginning of) the curve part (~10 to ~25 Gly or thereabout)

 

[marcus]

This is really good. Another thing I get from this table is a direct comparison of how the two lengths a and R evolve, something I'd been wondering about - I assume a is in Gly as well - they give units everywhere but for this one.

a(t) the so-called "scale factor" is a pure number, unit-less. It is normalized to equal one at the present day.

so you take some large distance between two things, not bound gravitationally, each at cosmic rest (or CMB rest) and you DIVIDE by its value at present:

a(t) = X(t)/X(now)

So whatever units X(t) had are canceled out. a(t), being normalized, is "dimensionless" , a pure number.

the conventions of Jorrie's Lightcone table are that S = 1+z, this is the actual enlargement factor by which distances and wavelengths get enlarged.

Redshift z = 1 means that the wavelength is now, when we receive it, TWICE what it was when it was emitted by the star and began traveling towards us. And distances have doubled while the light was en route. that means that S = z+1 is in many ways more useful than z.

Also S = 1 denotes the present (no redshift, wavelengths not enlarged)

And the scale factor a is just the reciprocal of the stretch-or-enlargement factor: a = 1/S

a is a common notation in cosmology. S is here just a local convenience notation. AFAIK there is no regular conventional symbol for the reciprocal 1/a of the scale factor.

 

[wabbit]

Ah OK thanks... Yes of course I suppose one can pick at will the point where a=1 and why not today hmm... For some reason i was thinking there was a natural choice but that doesn't make so much sense . In any case so when i compare their evolution i.just need to rescale a little.

Never saw so many numbers about cosmology before this makes it weirdly concrete for such a lofty suject, like a kind of tabletop experiment :-))

Btw thanks for the explanations , the site is great but the reason i found it intimidating is that such explanations weren t easy to find (plus the fact that i don't now where to start as far as input values, afraid to break something : ) ) - as you can see my prior acquaintance with cosmology was highly superficial

 

[BiGyElLoWhAt]

This is actually really interesting. I've been really curious as to how the wavelength changed for light as it curved around the sun. Now I think I have enough info to start googling around. Thanks!

 

[ChrisVer]

This is actually really interesting. I've been really curious as to how the wavelength changed for light as it curved around the sun. Now I think I have enough info to start googling around. Thanks!

How did this thread help you in that? The talk here is about a different metric/spacetime.
The wavelength of the light doesn't change afterall from sun's gravitational potential...

 

[BiGyElLoWhAt]

I'm talking about the lensing, I have a few things that I can look up now that could help me understand the "different metric/spacetime" better, and to see what actually is happening there.

 

[wabbit]

The wavelength of the light doesn't change afterall from sun's gravitational potential...

Nitpicking here - but I believe it's redshifted as it travels away from the Sun and "climbs the potential" - not that this is particularly relevant. But if you meant that the effect from appoaching cancels out with the effect from moving out, I wouldn't argue with that.

 

[BiGyElLoWhAt]

Shouldn't it's wavelength only be the same at equipotential surfaces? So if it's emitted at a location of potential A w.r.t the sun, travels some path through the potential, and it ends up here on Earth so we can measure it, at potential B w.r.t. the sun, there should still be a net shift from it's emitted state, which, assuming A != B, will result in a non-zero frequency shift. I believe*.

 

[wabbit]

Agreed, I was thinking of the case where emission and detection are both far away enough from the Sun that we can neglect that difference : I suspect the Earth is far enough that there is at least no "obvious" shift for a source at infinity - but I haven't done the calculation.

 

[BiGyElLoWhAt]

Ahh yes. I've actually been really curios about this lately. I just simply do not like the fact that people model lensing as bosonic interactions, and I think the key to settling the debate will lie in the consecutive measurements of redshift of the ambient spectrum of some particular cross section at various locations behind (or not so much) the sun. The resulting system of equations will (or should, in my opinion) show the discrete differences between the two trains of thought via experimental evidence and GR + Particle Physics explanations = Fight to the death.

 

[marcus]

But what other things related to cosmology can we learn from the google calculator?

Remember it taught us that the Hubble rate of distance growth is measured in hertz?
If you type in [67.9 km/s per megaparsec] it simplifies that and gives you back:
2.20 x 10^-18 hertz

which is 2.20 attohertz

We learned a form of the Friedmann equation that depends on knowing the energy density of the universe ρ(t) at a particular cosmic time t.
H(t)² - H_∞² = [Friedmann const] ρ(t)

this ρ is very important. As the geometry expands the density (in energy terms) of radiation, ordinary and dark matter thins out, understandably, and that tells us how the growth rate H(t) changes. The equation gives us a grip on the process. So what is the present-day density ρ(now)?

It is ρ(now) = 0.24 nanojoule per cubic meter

Let's see what google calculator makes of that! Put in [0.24 nanojoule per cubic meter] and google gives back:

(0.24 nanojoules) per (cubic meter) =
2.4 × 10^-10 pascals
It says it emphatically like that, in large type.

So apparently the calculator thinks force per area is algebraically equivalent to energy per volume and Joule per m³ is the same as Newton per m² is the same as Pascal. The ratio between a system's energy density and its pressure is a dimensionless (unit-less) number. They are two physically different quantities but measured with the same unit. That's very strange. Maybe we should use a different TYPEFACE to keep it straight. Our Physicsforums "Arial" for pressure? and TIMES NEW ROMAN for energy density?
So a pressure of 0.24 nanopascal would be abbreviated the usual way 0.24 nPa
and an energy density like that of the universe at present, would be 0.24 nPa
have to go to supper, back soon. I wonder if this is a good idea.

 

[PeterDonis]

the calculator thinks force per area is algebraically equivalent to energy per volume and Joule per m3 is the same as Newton per m2 is the same as Pascal.

That's because these are the same unit; if you break both of them down to the basic units of mass, distance, and time, they come out the same: both come out to $kg \cdot m^{-1} \cdot s^{-2}$ .

The ratio between a system's energy density and its pressure is a dimensionless (unit-less) number.

At a given point in spacetime, yes. If you consider a region of spacetime, you have a function relating the two, usually called the equation of state, and it will have dimensionless coefficients.

 

[marcus]

Thanks Peter, just saw your post. It's good to be explicit about this equivalence of pressure and energy density units, and the fact that the EoS ratio of the two is dimensionless (unit-less).
The reason I liked the type-face mnemonic thing is that Google calculator can't tell the difference between the two type-faces when you paste something into the window that you want calculated, in the same way that it can't tell the difference between the metric pressure unit and energy density unit. They are the same unit, just used to measure different things. Like it's the same capital letter P, just in a sans-serif font and a serif font.

 

[marcus]

As Peter explained, the metric units for pressure and energy density both reduce to the same basic elements so essentially the same unit works for both. The pascal (N/m² = Nm/m³ = J/m³) is a well-defined unit of energy density as well as pressure. So I'm making too much fuss about this. Instead of using a different type-face I'll just temporarily keep track by underlining the pascal abbreviation Pa when it is applied to energy density.
The Friedmann equation constant is what takes an energy density like presentday 0.24 nPa and converts it to a distance growth rate squared---e.g. in square attohertz
H(t)² - H_∞² = [Friedmann const]ρ(t)
For maximum transparency the Friedmann constant can be expressed so as to make explicit from what to what it is converting: for example in the metric context "square attohertz per nPa"

Let's see if Google will calculate it that way. Paste in
[8 pi G/(3c^2) in square attohertz per nPa] without the brackets.

Google calculator comes back with it's usual typographic emphasis:
((8 * pi) * G) / (3 * (c^2)) =
6.2208967 (square attohertz) per nPa
which we know (although the calculator doesn't make the distinction) is really
6.22 square attohertz per nPa
To repeat, this factor is what we multiply the density of the universe ρ(t) by, in the Friedmann equation,
H(t)² - H_∞² = [Friedmann const]ρ(t)
H(t)² - H_∞² = (6.22 attohz²/nPa) ρ(t)
to get the reduced square growth rate.

 

[marcus]

I don't want to give the impression that this way of treating the basic cosmic model, the Friedmann, is the only kind of cosmo/astro thing we can learn from using the Google calculator. You can challenge yourself calculating stuff with it (and at the same time challenge the calculator, since you are finding out how much it knows: does it know the speed of light, does it know the Boltzmann constant, does it know the mass of the electron or of the Earth?)

For example suppose you don't know how fast the Earth is going in its orbit around the sun. So you challenge yourself to find out. Put this in the window (G*mass of sun/ 1 AU)^.5
Or better, tell it to give the answer in km/s. Paste this in.
(G*mass of sun/ 1 AU)^.5 in km/s

that uses the formula for circular orbit speed (the Earth's orbit being nearly circular) and you can check by putting in
2 pi AU/(1 year) in km/s

 

[marcus]

Suppose you are observing a galaxy with redshift z = 1 and you think "I wonder what the Hubble parameter is for THEM". How would you go about it? Maybe google calculator can help. Or maybe we can almost do it in our heads.
z=1 means distances and wavelengths doubled while the light was en route to us. So densities were 8 times what they are now. So ρ(them) = 0.24 x 8 = 1.92 nanopascal or 1.92 nPa, for short.

And we just have to multiply that by the Friedmann constant 6.22 square attohz per nanopascal

You can see it's like 6x2, it'll come out roughly 12 square attohz. actually 6.22*1.92=11.94

And then we have the friedmann, remember H_∞ = 1.83 attohz and we have to solve for H
H² - (1.83 attohz)² = 11.94 attohz²
H² = 11.94+ 1.83²...
H = (11.94 +1.83^2)^.5 attohz = 3.91 attohz.

For comparison recall that the present-day Hubble rate is 2.20 attohz, so Hubble rate for them, back then, was a bit under twice today's growth rate.

There wasn't much to this. But you can see how to find past expansion rates. The one fine point is that radiation energy density increases as the fourth power of the linear shrinkage, while matter (equivalent) energy density just increases as the third power. Matter has been the predominant component for a long time so unless you go way back to early universe you may as well use the third power.

this is just one sample calculation and different people probably have different preferences about how to carry it out.

If you like the units "km/s per Mpc" you can always convert, just type in:
[3.91 attohertz in (km/s per Mpc)]
without the brackets.
It works,

The only place we really needed the google or some other calculator was for:
(11.94 + (1.83^2))^.5 = 3.91010230045

 

[cptstubing]

You all realize that once we learn the inner workings of the universe, it will cease to exist (according to Douglas Adams), so please go easy on this discussion. Tax time is coming up and I'll likely get a refund :)

 

[marcus]

This discussion is only playing around with minor details. I'm intentionally holding back from revealing my Big Insight so as to prevent just what Douglas Adams mentioned from happening. Have no fear, your tax refund is safe!

You all realize that once we learn the inner workings of the universe, it will cease to exist (according to Douglas Adams), so please go easy on this discussion. Tax time is coming up and I'll likely get a refund :)

 

[marcus]

The google calculator reminds me of a trampoline. You can bounce on it. And it conserves your energy in a sense because it knows things that you then don't have to look up. Say you know that the orbit periods of the 4 main Jupiter moons are 1.77, 3.55, 7.15, and 16.69 days. And just for kicks you want to know their SPEEDS. There's a "cube root formula" . (2piGM/P)^1/3 where P is the period. So you paste in:
(2pi G*mass of jupiter/1.77 days)^(1/3) in km/s
(2pi G*mass of jupiter/3.55 days)^(1/3) in km/s
(2pi G*mass of jupiter/7.15 days)^(1/3) in km/s
(2pi G*mass of jupiter/16.69 days)^(1/3) in km/s

and suppose you want to know their ORBIT RADII as well, so you multiply each of the speeds by P/2pi (the time it takes to travel a radian of orbit)
or simply paste in:
(2pi G*mass of jupiter/1.77 days)^(1/3)*1.77 days/(2pi)
(2pi G*mass of jupiter/3.55 days)^(1/3)*3.55 days/(2pi)
(2pi G*mass of jupiter/7.15 days)^(1/3)*7.15 days/(2pi)
(2pi G*mass of jupiter/16.69 days)^(1/3)*16.69 days/(2pi)

If you remember that the moon's orbit period is around 27 days, the calculator can get you the distance to the moon
(2pi G*mass of earth/27 days)^(1/3)*27 days/(2pi)
It says a bit over 380,000 km which I believe is about right. The point is that it KNOWS the Newton constant G and the mass of the Earth, so you don't have to look those things up and get them all in the right units to calculate with and so on. You just type in, or paste in, the formula.
And if you happen to be curious about how fast the Moon is traveling in its orbit, then
(2pi G*mass of earth/27 days)^(1/3)
By coincidence it comes out right around 1000 m/s. A kilometer a second.

 

[marcus]

What rate were distances expanding at when the first stars and galaxies were forming?
Well some of the earliest stars and small protogalaxies are from around redshift z = 9. The factor that matters is always 1+z so that is 10.
Distances and wavelengths have been enlarged by a factor of 10 while their light was on its way to us.

So let's calculate the Hubble expansion rate for that early star-forming era. One shortcut is to multiply the Friedmann constant by TODAY'S density:
6.22 x 0.24 = 1.49 attohz². If you remember that, or that it is approximately 1.5 attohz², then you can mentally estimate past and future Hubble rates easily.

The reduced square growth rate H²-H_∞² is 1.49 attohz² now, so when the energy density is 1000-fold greater it must have been 1490 attohz². It's as simple as that.
The constant term on the left is only 3.35 attohz² so it hardly matters. To find H(back then) you are taking the square root of either
1490 or 1493.35 and there is not much difference.
Either way the value of H rounds off to 38.6 attohz.
Recall that the present distance growth rate is 2.20 attohz.
So in a rough order of magnitude way, distance growth was almost 20 times more rapid back when the first stars and protogalaxies were forming.

If you like "kilometers per second per megaparsec" and that is more meaningful to you, you can always paste in [38.6 attohertz in (km/s per Mpc)] without the brackets and get the growth rate expressed in traditional units.

Some exceptionally early star formation has been observed at higher z, like 1+z = 11. If curious, one can redo the calculation for an even earlier era. The factor (1+z)³ is good as long as matter is predominant over radiation, which is certainly true say as far back as 1+z = 20. Farther even. Radiation is the dominant component of energy density before around 1+z = 3300

 

[marcus]

As I see it, part of getting acquainted with cosmology is understanding the standard (spatial flat) LCDM cosmic model that is embodied in a hands-on way in Jorrie's calculator. It makes tables of the past and future history of the cosmos based on parameters that you control.
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

In Jorrie's implementation, the two main handles you have on the LCDM model are the two LENGTHS you see at the top, for which the default values are 14.4 billion light years and 17.3 billion light years. With those two handles you control almost everything about the model.

You can change them, play around with them, and see how the past and future history of the universe changes. The only other significant parameter controls the balance between matter and radiation. That's the one with default value S_eq= 3400. It's mostly of interest in the early universe when radiation becomes an important component of the overall energy density, which is typically well before year 1 million. The fourth parameter, as long as it reads Ω=1 simply signifies spatial flatness, which I'm assuming throughout.

Let's look at the TIME versions of these two main handles: 14.4 billion years and 17.3 billion years. And let's see what google calculator says about their reciprocals
Paste in [1/(14.4 billion years)] and google gives back 2.20 x 10^-18 hertz, in other words 2.20 attohz
Paste in [1/(17.3 billion years)] and google gives back 1.83 x 10^-18 hertz, in other words 1.83 attohz.

To put it another way. These two attohz quantities H(now) and the constant growth rate H_∞ are the two main handles on the standard cosmic model. Disguised as reciprocals

We already know how to get the current energy density ρ(now) from them. Just take the difference of their squares and divide by the 6.22 constant
(2.20² - 1.83²)/6.22 = 0.24
which is the present-day average energy density in nanojoules per cubic meter aka in nPa

 

[marcus]

In his "wikidot" explanation that goes along with the calculator, Jorrie mentions that the stretch factor S = z+1 is more useful in calculation than the redshift z itself, so he describes the equations of the standard cosmic model in terms of S. It is slightly neater. And a bit more intuitive since S is the actual factor by which distances and wavelengths get enlarged while the light is on its way to us.
http://cosmocalc.wikidot.com/advanced-user

As explained there, to calculate the cosmic time corresponding to a given spread factor S takes numerical integration---building the number up bit by bit. But there is a hairy formula that gives a pretty good quick approximation. I'll show it in the default case where the two main model parameters are 14.4 and 17.3 billion light years.
The formula gives the cosmic time as a fraction of 17.3 billion years. What you have to plug in is the 3/2 power of your chosen S.

So imagine you are observing a galaxy whose light comes in with a stretch S = 4, the wavelengths are 4 times what they were when emitted.
And you think "I wonder how old the universe was when that light started on its way to me."

Well first you have to raise 4 to the 3/2 power. S^3/2 = 8
and then you plug 8 into the hairy formula. I'll write it in a form that can be pasted into google.
ln(1.5/8 + (1 + (1.5/8)^2)^(1/2))*2/3*17.3 billion years

Google gives back 2.150... billion years. That's correct to three significant figures, pretty good accuracy,
So the universe was only 2.15 billion years old when the light started out, and now it's year 13.8 billion, or so. Long time traveling.

For good measure, I'll write the hairy cosmic time formula with the variable S^3/2 you have to plug in, instead of the 8. Here's what it looks like in googlish form.

T_S = ln(1.5/S^3/2 + (1 + (1.5/S^3/2)^2)^(1/2))*2/3*17.3 billion years

Here's what it looks like in conventional form.
$T_S = \frac{2}{3} ln(\frac {1.5} {S^{3/2}} + (1+(\frac {1.5} {S^{3/2}})^2)^{1/2})$17.3 billion years
The number that 17.3 billion years (the eventual Hubble time) gets multiplied by is:
$$ \frac{2}{3} ln(\frac {1.5} {S^{3/2}} + (1+(\frac {1.5} {S^{3/2}})^2)^{1/2})$$

EDIT:The 1.5 which appears as numerator twice in this expression depends on the model parameters
(H₀²/H_∞² - 1)^-1/2 = (R_∞²/R₀² - 1)^-1/2= ((17.3/14.4)^2 - 1)^(-1/2) = 1.502..

 

[wabbit]

Heh! Something new to me again... Is that 3/2 power an empirical fit or does it relate to some general formula for FLRW spaces or something?

 

[marcus]

Wabbit, glad to know you took a look. The 3/2 power is not an empirical fit, comes out of the algebra/calculus of trying to solve Friedmann equation in closed form. But the numerator 1.5... is derived from the two adjustable parameters 14.4 and 17.3 billion years. Now and future Hubble times. Or radii, or growth rates. Fitting data involves adjusting those two which would change that coefficient 1.49...1.5...1.51.
((173/144)^2 - 1)^(-1/2) = 1.502...

(H₀²/H_∞² - 1)^-1/2

 

[wabbit]

Thanks. I suppose at some point I should bite the bullet and actually learn the FLRW metric, you certainly make it sound less intimidating with this concrete perspective.

 

[marcus]

Encouragement, thanks! I'm trying to explore how doing some hands-on calculation with google or other calculator could help get comfortable with quantitative cosmology and be responsive to problems about it people often come to PF with. Don't know if it will work, just have to see.

Not uncommon for someone to come here wondering "how do scientists calculate the size of the universe at any given year?" how do they plot the growth of distance over time?

It takes numerical integration (as Jorrie explains) if you want to account for the energy density being a variable mix of radiation and matter, but since matter greatly dominates over radiation for almost the entire history you can use a formula that is almost right. Given T what we want is S , then you can get redshift as S-1 or scale factor a=1/S which is a good size marker. So we want a function S(T). Google can compute it for you!
Try these times: T = year 2.15 billion, 0.64 billion, and 0.269 billion.
1.5/sinh(0.269/17.3*3/2)
1.5/sinh(0.64/17.3*3/2)
1.5/sinh(2.15/17.3*3/2)
If you paste these in you get back, not S, but rather S^3/2
To get the stretch factor S you can mentally raise each of the four numbers to the 2/3 power. That gives S and you can for example say what the redshift z = S-1 is for that time in history if that's what you want.

As before the 3/2 is a pure number, model independent and exact. But that 1.5 depends on the two model parameters, so if you are varying the 14.4 and 17.3 say to fit some new data then that would cause the 1.5 to vary.
$$S(T) = (1.5/sinh(\frac{3}{2}\frac{T}{17.3 billion years}))^{2/3}$$

 

[marcus]

Wait a minute. Google likes to see the two model parameters as 2.20 attohz, and 1.83 attohz . Instead of dividing T by 17.3 billion years we could multiply it by 1.83 attohertz which is the eventual growth rate H_∞ and a form of the cosmological constant namely (Λ/3)^1/2.

$$S(T) = (1.5/sinh(\frac{3}{2}H_\infty T))^{2/3}$$

and that 1.5 number depends on the two main parameters via

$(H_0^2/H_\infty^2 - 1)^{-1/2}$ = (2.20²/1.83² - 1)^-1/2 ≈ 1.5