1. Mar 16, 2015

### 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.

Last edited: Mar 16, 2015
2. Mar 16, 2015

### 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 = 3H2
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 - H2 = [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)
(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

Last edited: Mar 16, 2015
3. Mar 16, 2015

### 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?

4. Mar 16, 2015

### 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

Last edited: Mar 17, 2015
5. Mar 16, 2015

### 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.

6. Mar 16, 2015

### 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.

Last edited: Mar 16, 2015
7. Mar 16, 2015

### 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.

8. Mar 16, 2015

### 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.

9. Mar 16, 2015

### 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)

10. Mar 16, 2015

### marcus

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.

Last edited: Mar 16, 2015
11. Mar 16, 2015

### 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 doesnt 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 dont 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

Last edited: Mar 16, 2015
12. Mar 17, 2015

### 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!

13. Mar 18, 2015

### ChrisVer

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

14. Mar 18, 2015

### 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.

15. Mar 18, 2015

### wabbit

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.

16. Mar 18, 2015

### 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*.

17. Mar 18, 2015

### 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.

Last edited: Mar 18, 2015
18. Mar 18, 2015

### 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.

19. Mar 18, 2015

### 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 - H2 = [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 n
Pa

have to go to supper, back soon. I wonder if this is a good idea.

Last edited: Mar 22, 2015
20. Mar 18, 2015

### Staff: Mentor

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 $kg \cdot m^{-1} \cdot s^{-2}$ .

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.