Effort to get us all on the same page (balloon analogy)

[marcus]

Ray, you claim to be non-mathy but this you can get for sure, Einstein in 1915 was driving at a relation between geometry (which he put on the left side of the equation) and matter (which he put on the right). describing their influence on each other.
I'm saying that Lambda is a feature of geometry and belongs on the LEFT. If you see an equation that puts Lambda on the right, then all one can say is it shows a deplorable lack of judgment and good taste. Something fishy about it: A fictitious energy, a spurious term--something that doesn't belong, pretending to be part of the matter side.

Lambda is an inherent minimal growth rate that nature's geometry has a built-in tendency towards. For most of history because it got such a terrific kickoff at the start, the growth rate has been much bigger. but now it is settling down. As the density thins out, H² is getting closer and closer to H_∞²
That's what the Friedmann equation says. The amount it has left to go is proportional to the density
H² - H_∞² = (8πG/3)ρ
and as distances (and volumes) enlarge, the density gets less and less.

Sorry if you think the proportionality constant 8πG/3 is a bit clunky and elaborate. On Planet Gizmo they probably write it with a single symbol K. We humans, by a series of historical accidents, just happen to write it 8πG/3. It's a proportionality between density and the square of growth rates. If you have a density (mass per unit volume) and you multiply by K what you get is the square of some percent per unit time growth rate. So the proportionality looks a bit clunky but please don't be put off by that! It's really very nice that there is such a clean simple relation between matter conditions and the changing geometry features.

 

[RayYates]

I'm pretty mathy but your a couple orders of magnitude ahead of me.

If I asked, "What is ∏?" One could say = approximately 3927/1250. That would be a more precise answer than "the ratio between a diameter and circumference" but doesn't say what it is.

Not to beat the dead horse.

its what you get when you multiply it by c2 and divide by 3.

I've seen it defined as "represents the energy of empty space", but I know you agree with that so since its geometry and not energy... it would be... called... a relationship between...

 

[marcus]

Comment welcome. I'm working on improving this presentation of basic cosmology for newcomers. It assumes elementary differential calculus: chain rule, product rule...for derivatives. Otherwise very basic.

For definiteness I use the key model parameters from the 2010 WMAP7 report by Komatsu et al, namely 0.272, 0.728, and 70.4 km/s per Mpc.
The current Hubble growth rate of 70.4 km/s per Mpc means distances between stationary observers are currently increasing by 1/139 of a percent per million years. By the same token, the Hubble radius (a kind of threshold for incoming photons, within which distances are expanding slower than c) is currently 13.9 billion lightyears.
The table shows how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther.

 Standard model with 2010 numbers ( % is per million years)
time(Gyr)   z    H-then   H(%)  Hub-radius(Gly)  dist-now   dist-back-then 
   0     0.000     70.4   1/139      13.9
   1     0.076     72.7   1/134      13.4
   2     0.161     75.6   1/129      12.9
   3     0.256     79.2   1/123      12.3
   4     0.365     83.9   1/117      11.7
   5     0.492     89.9   1/109      10.9
   6     0.642     97.9   1/100      10.0
   7     0.824    108.6   1/90        9.0 
   8     1.054    123.7   1/79        7.9
   9     1.355    145.7   1/67        6.7
  10     1.778    180.4   1/54        5.4
  11     2.436    241.5   1/40        4.0
  12     3.659    374.3   1/26        2.6
  13     7.190    863.7   1/11        1.1

You can see that, according to the standard cosmic model, 13 billion years ago the Hubble rate of distance expansion was 1/11 of a percent per million years. The Hubble radius then was only 1.1 billion lightyears. Expansion had, by then, been in progress for 0.7 billion years and light emitted at that time would be received by us redshifted by about z = 7.19. This means that distances and wavelengths from that time have increased by a factor of 8.19.
The factor is always 1+z, one more than the stated redshift.
A key tool here is the scalefactor curve a(t) which tracks proportional increase in distances over time. It is a rising curve which is arbitrarily set equal to one at the present time: a(now) = 1. The Hubble rate is the fractional rate of increase of the scalefactor:
By definition H = a'/a.

You can see that the Hubble rate has been declining more and more slowly over the millennia. This is the most important point: according to our best understanding (especially since the supernova studies reported in 1998) the decline is leveling off towards a constant limiting rate of about 1/163 of one percent per million years.

This is the expected longterm expansion rate which is here denoted H_∞.
The Hubble expansion rate H, which we can measure and infer past values for, is declining ever more slowly and in the distant future will approach H_∞ as its limit.

This was implicit in the GR equation early on but it was assumed by most students of cosmology that this limit was ZERO. It was only in 1998 that it became generally accepted that the limit is non-zero. H_∞ is one guise of a constant Λ that appears naturally in the GR equation and is called the "cosmological constant"*.

The simple model of the universe which generates these numbers, producing a remarkably good fit with observation, is called the Friedmann equation.
H² - H_∞² = (8πG/3)ρ
It's a simplification of the 1915 equation of General Relativity obtained by assuming overall uniformity of the universe--an assumption that so far has proven to be quite reasonable and makes the model a lot easier to use.
I want to explain the terms in this equation and help see how it works. (To start with we're focusing on the spatially "flat" or k=0 version. It's widely used because at large scale space does seem to have little or no overall curvature.)

We'll use ρ to stand for the combined mass density of dark matter, ordinary matter and radiation. In the early universe radiation played a dominant role but for most of expansion history the density has been matter-dominated with radiation making only a very small contribution to the total. Because of this, ρ goes as the reciprocal of volume. It's equal to some constant M divided by the cube of the scalefactor: M/a³.
Differentiating, we get an important formula for the change in density, namely ρ'.
ρ' = (M/a³)' = -3(M/a⁴)a' = -3ρ(a'/a) = -3ρH
The last step is by definition of H, which equals a'/a

The idea now is to work with the Friedmann equation and get it to tell us some things. First let's differentiate it--the constant term will drop out:
H² - H_∞² = (8πG/3)ρ
2HH' = (8πG/3)ρ'
Then we can use our formula for the density change:
2HH' = (8πG/3)(-3ρH) = - 8πGρH, and we can cancel 2H to get the change in H, namely H':

H' = - 4πGρ

Again by definition H = a'/a, so we can differentiate that by the quotient rule and find the change in H by another route:
H' = (a'/a)' = a"/a - (a'/a)²
H' = a"/a - H²
Now we have two different expressions for H', this one and the highlighted one, so we can write:

a"/a - H² = - 4πGρ

The Friedman equation tells us we can replace H² by H_∞² + (8πG/3)ρ. So we have
a"/a - H² = a"/a - H_∞² - (8πG/3)ρ = - 4πGρ

We group geometry on the left and matter on the right--then, noticing that 8/3 - 4 = - 4/3, we get:
a"/a - H_∞² = (8πG/3)ρ - 4πGρ
a"/a - H_∞² = - (4πG/3)ρ

This is the socalled "second Friedmann equation" in the matter-dominated case where radiation pressure is neglected. In modeling the early universe, when light contributed largely to the overall density, a radiation pressure term would be included and, instead of just ρ in the second Friedmann equation, we would have ρ+3p.

The second Friedmann equation is also called the "acceleration Friedmann equation" because it yields information about a". We would like to use it to discover at what moment in history an INFLECTION occurred in the scalefactor a(t) distance growth curve. When did the slope of the scalefactor curve stop leveling out and begin to get steeper? This is the moment when a" = 0. It marks when a" changed from negative to positive and actual acceleration of distance growth began.

The way we do this is to rearrange the "second Friedmann" slightly:
a"/a - H_∞² = - (4πG/3)ρ
a"/a = H_∞² - (4πG/3)ρ
and use the main Friedmann equation to replace (4πG/3)ρ by (H² - H_∞²)/2.

a"/a = H_∞² - (H² - H_∞²)/2
a"/a = (3H_∞² - H²)/2

Now to find the inflection time, all we need to do is discover when it was that
H² = 3H_∞²
since then their difference will be zero, making a" = 0.
That means H = √3 H_∞ = √3/163 percent per million years
H = 1/94 percent per million years.
As one sees from the table, that happened a little less than 7 billion years ago. In other words when expansion was a bit less than 7 billion years old. You can see that from the table. The 1/94 fits right in between 6 billion years ago and 7 billion years ago. In between 1/100 and 1/90.

*The relation is H_∞² = Λc²/3

 

[marcus]

One thing that you might wish to try is actually CALCULATING something using the current Hubble expansion rate of 1/139 percent per million years. Often that means pasting or typing something into the google window (which doubles as a scientific calculator able to convert units and supply physical constants.)

As a reminder, here's the Friedmann equation.
H² - H_∞² = (8πG/3)ρ

Here's a sample calculation: try pasting this into the google window. It will give the CRITICAL MATTER DENSITY required for spatial flatness at this time in history, expressed as an energy density (joules per cubic meter).

c^2*(1/139^2 - 1/163^2)*(percent per million years)^2/(8pi*G/3)

Basically what you are doing is solving for ρ, to get the mass density, and then multiplying by c² to turn that into the equivalent energy density.

If you look closely at what is to be pasted into the calculator you will see that
(1/139^2 - 1/163^2)*(percent per million years)^2
is just a version of the familiar H² - H_∞², the lefthand side of the Friedmann equation.
To solve for the density ρ all we need to do is divide by (8pi*G/3).

So what I'm suggesting you paste into the window should make sense in terms of the preceding discussion. You should get 0.23 nanopascals. That is 0.23 nanojoules per cubic meter. We know that the matter density of our universe is pretty close to that, because spatially it's pretty close to flat.

One way to think of it is to translate the energy density into 0.23 joules per cubic kilometer. It's easy to get an idea of a joule of energy: just drop a conventional physics textbook (one-kilogram) from a height of 10 centimeters. For 0.23 joules, drop it from 2.3 centimeters. It makes a little thud. That thud is the energy equivalent of how much mass a cubic kilometer of today's universe, on average, contains.

 

[marcus]

I've added to the table in post #318. The first few columns show how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther. The new columns show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving.

 Standard model with 2010 numbers ( % is per million years)
time(Gyr)   z    H-then   H(%)  Hub-radius(Gly)  dist-now   dist-then 
   0     0.000     70.4   1/139      13.9        0.0          0.0
   1     0.076     72.7   1/134      13.4        1.04         0.97
   2     0.161     75.6   1/129      12.9        2.16         1.86
   3     0.256     79.2   1/123      12.3        3.36         2.68
   4     0.365     83.9   1/117      11.7        4.67         3.42          
   5     0.492     89.9   1/109      10.9        6.10         4.09
   6     0.642     97.9   1/100      10.0        7.66         4.67
   7     0.824    108.6   1/90        9.0        9.39         5.15
   8     1.054    123.7   1/79        7.9       11.33         5.52
   9     1.355    145.7   1/67        6.7       13.53         5.74
  10     1.778    180.4   1/54        5.4       16.08         5.79
  11     2.436    241.5   1/40        4.0       19.16         5.58
  12     3.659    374.3   1/26        2.6       23.13         4.97
  13     7.190    863.7   1/11        1.1       29.15         3.56

You can see that, according to the standard cosmic model, 13 billion years ago the Hubble rate of distance expansion was 1/11 of a percent per million years. The Hubble radius then was only 1.1 billion lightyears. Expansion had, by then, been in progress for 0.7 billion years and light emitted at that time would be received by us redshifted by about z = 7.19.

The table also shows that when we observe a galaxy as it was 10 billion years in the past we know that the light we are getting was emitted while the galaxy was receding faster than c. This is revealed by the fact that the then-distance (5.79 Gly) exceeded the then-Hubbleradius (5.4 Gly). This would be true for any galaxy observed to have redshift z > 1.64 or thereabouts. This means most of the galaxies we can see---the distance to any such galaxy was increasing faster than c when it emitted the light we're getting and has continued to increase faster than c all the while the light has been traveling to us.

The redshift z = 1.64 is also interesting because it marks the angular size minimum. Objects with the same physical size will look bigger (take up a larger angle in the sky) if their redshift is greater than about 1.64 and if it is less. Then-distance peaks right around z = 1.64

Incidental information: According to Peeble's Cosmic Inventory ( http://arxiv.org/abs/astro-ph/0406095 ) ordinary matter makes up about 16% of the total matter of the universe. Roughly a sixth, in other words.

 

[marcus]

In another thread someone had a question concerning the PRESENT rate of distance expansion in real terms--namely how much was it currently accelerating. So I wanted to show an easy way to address the question of acceleration and come up with a definite number.
Of course to get a definite speed one has to specify some particular distance between two stationary observers and see how fast that is growing and how much the growth is speeding up.

What distance one chooses to look at is somewhat arbitrary---I picked 13.9 billion lightyears because it makes the numbers simple. It is a bit less than a third of the current radius of the observable region.

For definiteness I use the key model parameters from the 2010 WMAP7 report by Komatsu et al, namely 0.272, 0.728, and 70.4 km/s per Mpc.
The current Hubble growth rate of 70.4 km/s per Mpc means distances between stationary observers are currently increasing by H = 1/139 of a percent per million years. And according to the standard model the limit that H is tending to is H_∞ = 1/163 of a percent per million years*

The Friedmann equation model in the spatially flat case is
H² - H_∞² = (8πG/3)ρ
where ρ is the density of all kinds of matter and radiation (excluding the cosmological constant, which I'm taking to be simply that: the cosmological constant.)
In the case where the contribution of radiation to ρ is small compared with that of dark and ordinary matter, the acceleration equation takes this form:
a"/a - H_∞² = - (4πG/3)ρ

So then we have:
a"/a = H_∞² - (4πG/3)ρ
and using the main Friedmann equation to replace (4πG/3)ρ by (H² - H_∞²)/2, we have:
a"/a = H_∞² - (H² - H_∞²)/2
a"/a = (3H_∞² - H²)/2, and factoring out H² we get:
a"/a = [(3(H_∞/H)² - 1)/2]H²
a"/a = [(3(139/163)² - 1)/2]H²
a"/a = 0.59 H²

Since we are asking about acceleration at the present time and by convention the scalefactor a(now) = 1 we can just write a" = 0.59 H²
and if we choose, as mentioned earlier, the distance R = 13.9 billion lightyears to be the present separation between the pair of stationary observers or objects then the acceleration is just gotten by multiplying on both sides by R:
a"R = 0.59 H²R

Now HR = c, because that's how R was chosen, and so
a"R = 0.59 Hc
This means that the current acceleration is 0.59/139 = 1/236 of a percent of the speed of light per million years.

I like this example because it gives an idea of how slow the acceleration is. The distance itself is currently increasing at the speed of light. And that rate is scarcely changing at all! Indeed after a million years it will still only be just slightly (a small fraction of a percent) larger than the speed of light.*The relation to the cosmological constant is H_∞² = Λc²/3

 

[marcus]

Since we've turned a page I'll bring forward the table from post #318, to which more columns have been added. The Hubble rate is shown both in conventional units (km/s per Mpc) and as a fractional growth rate per d=10⁸y. The first few columns show how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther and farther. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving.

 Standard model -- WMAP parameters (distances in Gly)
time(Gyr)   z    H(conv)   H(d[SUP]-1[/SUP])  Hub-radius  dist-now   dist-then 
   0     0.000     70.4   1/139      13.9        0.0          0.0
   1     0.076     72.7   1/134      13.4        1.04         0.97
   2     0.161     75.6   1/129      12.9        2.16         1.86
   3     0.256     79.2   1/123      12.3        3.36         2.68
   4     0.365     83.9   1/117      11.7        4.67         3.42          
   5     0.492     89.9   1/109      10.9        6.10         4.09
   6     0.642     97.9   1/100      10.0        7.66         4.67
   7     0.824    108.6   1/90        9.0        9.39         5.15
   8     1.054    123.7   1/79        7.9       11.33         5.52
   9     1.355    145.7   1/67        6.7       13.53         5.74
  10     1.778    180.4   1/54        5.4       16.08         5.79
  11     2.436    241.5   1/40        4.0       19.16         5.58
  12     3.659    374.3   1/26        2.6       23.13         4.97
  13     7.190    863.7   1/11        1.1       29.15         3.56
 13.6   22.22    4122.8   1/2.37      0.237     36.69         1.58

To illustrate: according to the standard cosmic model with final WMAP parameters, 13 billion years ago the Hubble rate of distance expansion was 1/11 of a percent per million years. For brevity this can also be written as a fractional growth rate of 1/11 per d (= 10⁸y). The Hubble radius then was 1.1 billion lightyears. Expansion had, by then, been in progress an estimated 0.757 billion years and light emitted at that time is now being received by us redshifted by about z = 7.19.

The redshift z = 1.64 marks the angular size minimum. Objects with the same physical size will look bigger (take up a larger angle in the sky) if their redshift is greater than about 1.64 and also if it is less. "Distance-then" peaks right around z = 1.64

Incidental information: According to Peeble's Cosmic Inventory ( http://arxiv.org/abs/astro-ph/0406095 ) ordinary matter makes up about 16% of the total matter of the universe. Roughly a sixth, in other words.

The timescale d=10⁸ years turns out to be convenient to work with so to get an intuitive feel for it as an interval of time here's a geological timeline:
http://www.ucmp.berkeley.edu/help/timeform.php/
For example the Paleozoic Era = about 3 of this unit
and is divided into 6 roughly equal Ages (Cambrian, Ordovician, Silurian, Devonian, Carboniferous, Permian ) each of these lasting approximately 1/2 unit.
In this context the names don't matter, only the idea that geological ages tend to be on the order of one of these d=10⁸y. It is a length of time during which something can happen which is distinctive enough in terms of geology or biological evolution so that the relevant professionals decide to give it a name.

It is also a period of time during which distances between pairs of observers at rest wrt background can grow by some fractional amount, such as 1/139 at present, or 1/11 much earlier.

 

[marcus]

Key quantities in cosmology are fractional distance growth rates, and average density. The basic equation of cosmology, the Friedmann equation, relates (the square of) the fractional growth rate to overall matter density ρ.

H² - H_∞² = (8πG/3)ρ

Here ρ is the mass-equivalent density of all kinds of matter and radiation (not the cosmological constant, which I'm taking to be simply that: the cosmological constant.)
Using the convenient time unit d=10⁸ years, the present and eventual values of the distance growth rate can be written 1/139 per d, and 1/163 per d.
It's convenient to solve for the energy-equivalent form of the density: ρc² which will come out in nanopascals, that is nanojoules per cubic meter.

[3c²/(8πG)](H² - H_∞²) = ρc²

It turns out that the coefficient 3c²/(8πG) = 16144 nanopascal d².

So it's a straightforward calculation to find the density (in energy-equivalent form). The d² cancels and we have:

16144 nanopascal(1/139² - 1/163²) = 0.228 nanopascal

As basic arithmetic, this works in the google calculator. Pasting in
16144(1/139^2 - 1/163^2)
gives 0.228

Expressed in energy-equivalent form, the average matter density in the universe today is presumably close to 0.228 nanojoule per m³. Or in other words 0.228 joule per cubic kilometer. About 16% of this is ordinary matter and most of the rest is dark matter.

What we calculated there is actually the critical matter density---that necessary for overall spatial flatness. Since it continues to be found that the cosmos is nearly flat---at large scales the overall spatial curvature is at least very close to zero---the current critical matter density is a good estimate for the actual one.
For simplicity the version of the Friedmann equation used here assumes spatial flatness.

The essential takeaway message here is that if you know two fractional growth rates (the present and the future target rate), namely 1/139 and 1/163 per d then this simple arithmetic:
16144(1/139² - 1/163²) = 0.228
gives you the estimated current matter density (expressed as energy equivalent per unit volume.)
A joule of energy (dropping a kilo textbook from about 10 cm) is easy to imagine. Or think 2.28 centimeters to get the 0.228 joule figure. And that amount of work has to be contained in a cubic kilometer.
The 3c²/(8πG) = 16144 nanopascal d² thing can be thought of as a constant of nature---relating fractional expansion rate to density. You can get the 16144 nanopascals for yourself, from google calculator, just by pasting in
3c^2/(8pi*G)/(10^8 year)^2

 

[Jorrie]

Hi Marcus; I found your calculations very interesting and I agree with your values. Just one thing that I do not quite follow is this statement:

What we calculated there is actually the critical matter density---that necessary for overall spatial flatness. Since it continues to be found that the cosmos is nearly flat---at large scales the overall spatial curvature is at least very close to zero---the current critical matter density is a good estimate for the actual one.

Does the matter density required for critical mass not depend on what value we experimentally find for the cosmological constant? If we have found the contribution of Lambda to be higher, say 90%, would the matter requirement for flatness not have been less? Or am I missing something?

I guess my comment is more on the semantics, i.e. is it correct to call it the critical matter density, or should it rather be "present matter requirement for critical total density"?

 

[marcus]

...
I guess my comment is more on the semantics, i.e. is it correct to call it the critical matter density, or should it rather be "present matter requirement for critical total density"?

You are right to stress "present". In all treatments of Friedmann cosmology the critical density is time-dependent because it depends on the rate of expansion and that keeps changing.

To achieve spatial flatness the matter density must somehow balance the current expansion rate.

BTW I'm so glad you found the calculations here interesting! Thanks for the comments.

You are also right to call attention to the SEMANTICS issues.

I am not treating the cosmo constant as part of the total density because I consider it to be a curvature constant of nature as it appears in the Einstein field equation.

One can always multiply a curvature by some stuff and get a fictitious "energy" or pseudo-energy. That amount which would have caused the curvature if there were no cosmo constant already. But I don't bother with that line of approach. It's like attributing the tendency to fly off a merry-go-round to a fictitious "force".

So the critical matter density calculated here is the actual current matter density that is critical for flatness.

This approach is influenced by the Bianchi Rovelli paper which refers to Lambda as "vacuum curvature" rather than "vacuum energy".
http://arxiv.org/abs/1002.3966/
They remind us that, back in 1917 or so, Einstein's Lambda was in fact a curvature and it still is that in the EFE of regular GR. It is a curvature constant which arises naturally (somewhat like a constant of integration) in the GR equation and which for many years was assumed by most people to be zero. Then in 1998 it was found out to not be zero.

Of course the accuracy of the calculation depends on the accuracy with which one knows H² and H_∞².
So if you changed either estimate significantly you would change the crit matter density we calculate.

Note that H_∞² is just the cosmological constant in a different guise.
H_∞² = Λc²/3

Because Λ is a curvature which means it has units of reciprocal area, you have to multiply it by c² to change it into a reciprocal time² to make the units agree.
But except for the c² factor H_∞² is essentially the same constant Lambda that Einstein put in his equation way back when.

 

[marcus]

Hi Marcus; I found your calculations very interesting and I agree with your values...

Actually, as I'm sure you realize, the values of the various times and distances corresponding to different redshifts came from your calculator. I should acknowledge your help, it's an excellent tool. I keep the link in my signature so as to have it handy.
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm

There's also a pedagogical value connected with newcomers to cosmology being able to calculate stuff for themselves and get hands-on experience with the standard cosmic model.
I'd like to encourage others to use your calculator and also the google scientific calculator.

Here's another thing. Your calculator has the parameters set to the final WMAP estimates.
Those are the ones I assume in these posts. The two most important numbers are
the current Hubble rate 70.4 km/s per Mpc
and the "dark energy" number 0.728
I hope that everybody reading gets to the point where they can use those to calculate the two key fractional growth rates 1/139 and 1/163 per d.
Then if they want to try this with other values, say 71 km/s per Mpc and 0.74 and see how much the results differ they can do that.

0.728 is just the ratio of H_∞² to H² so once one gets H one can easily get H_∞. It's just given by:
H_∞ = sqrt(0.728)H

So we have to get H. To do that you just put this in google:
1/(70.4 km/s per Mpc)
and google will say 13.889.. billion years. I round that off to 13.9, so the answer is 1/139.
You can see how those numbers are related in every row of the table.

Then continuing, put this in google:
139/sqrt(0.728)
and google will say 162.91... which I round off to 163. So the answer is 1/163.

 

[Jorrie]

Actually, as I'm sure you realize, the values of the various times and distances corresponding to different redshifts came from your calculator.

I'm glad that the humble calculator is of use to you. Maybe another version with simplified outputs would be a good idea, perhaps with some of your parameters included - it is an interesting way of looking at things...

BTW, when I looked at your table from that POV, it struck me that your first column heading may be a bit confusing. You labeled it 'time', while it is actually the look-back time (how long it took the light to travel to us).

I will have a closer look as time allows.

 

[marcus]

The numbers in this table were gotten with the help of Jorrie's calculator, as mentioned in the past two or three posts. The calculator gives multidigit precision and I've rounded off. Hubble rates at various times in past are shown both in conventional units (km/s per Mpc) and as fractional growth rates per d=10⁸y. The first few columns show lookback time in billions of years, and how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving. The numbers in parenthesis are fractions or multiples of the speed of light showing how rapidly the particular distance was growing.

 Standard model with WMAP parameters 70.4 km/s per Mpc and 0.728. 
Lookback times shown in Gy, distances (Hubble, now, then) are shown in Gly.
The "now" and "then" distances are shown with their growth speeds (in c)
time      z     H(conv)   H(d[SUP]-1[/SUP])    Hub      now          back then 
   0     0.000     70.4   1/139    13.9      0.0          0.0
   1     0.076     72.7   1/134    13.4      1.0(0.075)   1.0(0.072)
   2     0.161     75.6   1/129    12.9      2.2(0.16)    1.9(0.14)
   3     0.256     79.2   1/123    12.3      3.4(0.24)    2.7(0.22)
   4     0.365     83.9   1/117    11.7      4.7(0.34)    3.4(0.29)          
   5     0.492     89.9   1/109    10.9      6.1(0.44     4.1(0.38
   6     0.642     97.9   1/100    10.0      7.7(0.55)    4.7(0.47)
   7     0.824    108.6   1/90      9.0      9.4(0.68)    5.2(0.57)
   8     1.054    123.7   1/79      7.9     11.3(0.82)    5.5(0.70)
   9     1.355    145.7   1/67      6.7     13.5(0.97)    5.7(0.86)
  10     1.778    180.4   1/54      5.4     16.1(1.16)    5.8(1.07)
  11     2.436    241.5   1/40      4.0     19.2(1.38)    5.6(1.38)
  12     3.659    374.3   1/26      2.6     23.1(1.67)    5.0(1.90)
  13     7.190    863.7   1/11      1.1     29.2(2.10)    3.6(3.15)
 13.6   22.22    4122.8   1/2.37    0.237   36.7(2.64)    1.6(6.66)

Abbreviations used in the table:
"time" : Lookback time, how long ago, or how long the light has been traveling.
z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original.
H : Hubble expansion rate, at present or at times in past. Distances between observers at rest grow at this fractional rate--a certain fraction or percent of their length per unit time.
H(conv) : conventional notation in km/s per Megaparsec.
H(d^-1) : fractional increase per convenient unit of time d = 10⁸ years.
"Hub" : Hubble radius = c/H, distances smaller than this grow slower than the speed of light.
"now" : distance to object at present moment of universe time (time as measured by observers at CMB rest). Proper distance i.e. as if one could freeze geometric expansion at the given moment.
"then" : distance to object at the time when it emitted the light.

 

[marcus]

It occurs to me that to a large extent what this discussion boils down to is the heavy solid curve on this graph:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg

That is the distance* growth curve for the Standard Model cosmos with parameters practically the same as what we are using here, namely 71 km/s per Mpc and 0.73.
I just happen to be using 70.4 km/s per Mpc and 0.728 because those figures came out more recently (the 2010 WMAP report) and Jorrie uses them in his calculator. But a small difference in parameters like that makes almost no difference in the results. So essentally that curve is what we are talking about.

Lineweaver calls it the R(t) curve because he uses R to stand for the scalefactor, a number increasing with time that is normalized so that R(now) = 1.
In my posts I've been using the letter a to stand for the same thing. So we would call it the a(t) curve, or simply the scalefactor curve.

Almost the whole business with this curve is that it is generated by a special growth equation, a differential equation that uses the symbol H to stand for a'/a.
H² - H_∞² = (8πG/3)ρ
This equation is a simplification of the 1915 Einstein field equation of GR. Once you understand about the two constants in it, namely H_∞² and 8πG/3,
it is really very simple.
All it says is that a certain fractional growth rate, squared, is proportional to the matter density ρ. So naturally as the matter density declines, the fractional rate of growth of distance must also decline.

Notice that a'/a can be thought of as the increase in any distance divided by the distance itself, so it is a fractional growth rate. Like the interest rate on bank savings account. And the equation just says that this fractional growth rate has to decline as the density of matter decreases (which it must do as distances grow.)

So that simple idea of a declining fractional growth rate is what generates the curve. In a sense, the curve is the real thing and the rest is just a mixture of words and numbers.
That curve is the scalefactor of our universe and what we really want to do is understand that curve.

*We keep in mind that distance here means distance between motionless observers (those at rest with respect to background) at a given moment of universe time (i.e. time as clocked by observers at universal rest.) This is the type of distance in terms of which Hubble law expansion is formulated.

 

[Jorrie]

It occurs to me that to a large extent what this discussion boils down to is the heavy solid curve on this graph:
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg

Yes, that's a very interesting graph. One of the intriguing things is that the empty universe (0.0,0.0) curve has virtually the same t_0 as the solid (0.27,0.73) LCDM curve. This is because the Hubble time (13.9) is so close to the present age of the universe (13.7), but that's probably just a coincidence(?).

Another thing that may intrigue beginners is the fact that all the curves have the same slope at the 'now' crosshatch. This is no coincidence, because the slope of each curve at any point reflects the variable Hubble constant H(t) for the specific curve and time - and the curves have all been drawn for the same present H(t)=Ho.

H2 - H∞2 = (8πG/3)ρ
...
All it says is that a certain fractional growth rate, squared, is proportional to the matter density ρ. So naturally as the matter density declines, the fractional rate of growth of distance must also decline.

This is broadly so, but I do not think this is quite correct, because matter density will approach zero on the long term, while the H(t) will approach a constant non-zero value. So they can't really be proportional.

That curve is the scalefactor of our universe and what we really want to do is understand that curve.

Despite the interesting math relations discussed, there is still something to say for the idea of negative pressure of the cosmological constant that causes the curve to swing upwards.
Radiation and matter (normal and dark) dilute to a point where they have no further influence on the slope of the curve, but the vacuum energy density remains constant and the curve becomes exponential.

-J

 

[marcus]

...while the H(t) will approach a constant non-zero value. So they can't really be proportional...

Heh, heh. I know and I was accordingly careful in my wording, Jorrie. I did not say that H(t) squared was proportional to density. It obviously is not because of the constant.

What I said was that a certain fractional growth rate squared was proportional, namely
H²-H_∞². This is the square of some fractional growth rate and it is proportional to density, and it does indeed go to zero as the density does.

My aim was to give the basic gist of the equation, stripped of detail: a square-of-fractional-growth-rate quantity on one side and a density on the other, connected by the proportionality constant.

Thanks much for reading and your many comments! It's a real encouragement.

 

[Jorrie]

What I said was that a certain fractional growth rate squared was proportional, namely
H²-H_∞². This is the square of some fractional growth rate and it is proportional to density, and it does indeed go to zero as the density does.

Oops, you were right!

It reminds me of the emergent gravity, that was discussed in this thread. Seems like the universe strives to minimize H²-H_∞². This means a constant Hubble radius in the future. I'm still trying to understand what a constant Hubble radius means observationally.

It seems that up to about z=1, we observe things that were inside of our Hubble radius at the time of emission. Farther than z=1, those objects were outside of our Hubble radius, not so?

 

[marcus]

Sorry about the confusing wording. I will have to rewrite some. Your reactions are a real help.What you're asking about here has several interesting facets. Considering what is observable now and will be in future when Hubble radius is almost constant.I think this part of your question is about presentday obseration, is that right?

...It seems that up to about z=1, we observe things that were inside of our Hubble radius at the time of emission. Farther than z=1, those objects were outside of our Hubble radius, not so?

To answer your question put z = 1.64 in your calculator.

You will see that the object was just slightly inside our Hubble radius at the time of emission. So anything we observe with redshift less than 1.64 was inside our Hubble radius at the time. (By definition because it's recession speed at the time was less than c.)

I'm not sure if you were asking about conditions now, though. It is interesting to look ahead to when the Hubble radius is more nearly constant, at (assuming the 2010 parameters are right) 16.3 Gly. Then the Hub radius essentially coincides with the cosmic event horizon.
All the galaxies initially within that range will eventually drift out beyond 16.3 but we will never see them cross the line. Their images will seem pinned to the horizon and just get redder and redder until the wavelengths get so long it isn't practical to try to see them.

this is what I think. does that square with how you imagine it?

 

[Jorrie]

To answer your question put z = 1.64 in your calculator.

You will see that the object was just slightly inside our Hubble radius at the time of emission. So anything we observe with redshift less than 1.64 was inside our Hubble radius at the time. (By definition because it's recession speed at the time was less than c.)

OK, I see - we have to compare the proper distance 'then' to the Hubble radius 'then'. What makes things more complex is that due to the early deceleration of expansion, we observe a lot of stuff today that were originally outside of our 'then' Hubble radius. The extreme example is the present CMB photons that originated 42 million light years from us, while our 'then' Hubble radius was a mere 650 thousand light years. Those photon were receding from as at some 65c at the time of emission, yet they caught up with us.

I agree with the rest of your summary. It is good to keep the difference that you pointed out in mind - between Hubble radius and cosmological event horizon, where our observed redshift will tend to infinity.

 

[cepheid]

marcus,

I've been a bit puzzled by your equations, in particular, about what this "H-infinity" business is all about. The Friedmann equation (in a flat universe) is just $$(\dot{a}/a)^2 = (8 \pi G /3)\rho_{\textrm{tot}}$$where a is the scale factor, and ρ_tot is the total mass-energy density of the universe, taking into account all constituents. Since the Hubble parameter is defined as $H \equiv \dot{a}/a$, we have$$H^2 = (8 \pi G /3)\rho_{\textrm{tot}}$$ That's it. Now if you assume that the only constiuents that are important (i.e. able to affect the dynamics of the expansion) are matter (ρ_m) and dark energy (ρ_de), you can write$$H^2 = \frac{8\pi G}{3}\rho_m + \frac{\Lambda}{3}$$where we have defined $\Lambda \equiv 8\pi G\rho_{de}$ and assumed that ρ_de = const. This is the Friedmann equation in the form that I'm used to seeing.

THEN it hit me. You don't like dark energy. You've been going on and on () all around the site about how $\Lambda$ should just be accepted as another fundamental constant that appears in GR, just like G, and it is a purely geometric term, all based on this one paper (that I admittedly haven't read). So all you did was move the Lambda term from the "this stuff is mass-energy" side of the Einstein field equation to the "this stuff is geometry" side of the equation, and then define $H_\infty \equiv (\Lambda/3)^{1/2}$. This makes sense, because H_∞ is then the value that H approaches asymptotically as t → ∞ (since ρ_m → 0). I'm on to you marcus!

Actually I've been meaning to take this up with you for a while. I don't know, just moving things around and saying "it's just a part of the geometry" seems a bit contrived to me. You can clearly show from the second Friedmann equation that a component with negative pressure is required to produce accelerated expansion, and if the pressure is exactly the negative of the energy density, then the energy density will be constant with time, which lends itself naturally to a physical interpretation as "vacuum energy" or energy of empty space (I know that there are huge problems with this right now). It seems like some sort of physical interpretation or explanation is called for here, for what exactly this negative pressure component is. Not only that, but I haven't personally seen any trend amongst the cosmologists I've talked to of moving away from the interpretation of Lambda as being due to some mysterious dark energy. On the contrary, missions are gearing up to try to measure or constrain w, the equation of state of dark energy, and it seems like many people are seriously considering a time-variable equation of state w(a), which would not correspond to a simple cosmological constant term in the Friedmann equations.

I assume that the argument you are advocating goes something along the lines of, "well, 'G' does not require any sort of physical interpretation, so why should $\Lambda$?" So, what are you saying, that because the theory admits a fundamental constant, and because that constant's value is positive in our universe, the expansion of the universe just naturally tends to accelerate (in the absence of matter), because, "that's just the way it is?"

 

[marcus]

Right, except you suggest that I moved Lambda over to LHS. That is where Einstein originally had it. And his Lambda was a curvature, a vacuum curvature, not an "energy".

My attitude is conservative in this respect. I see no scientific or physical grounds for moving Lambda over to right and converting it to an "energy". I await with interest some positive evidence that it is NOT simply a constant. So far all the observational evidence is tending to confirm simple constancy.

So the Ockham viewpoint is "don't make up stuff when you don't need to."

When you write a physics theory you put in the terms allowed by the symmetries of the theory. Diffeo sym or "general covariance" allows just those two constants. So you put them in and let Nature tell you what their values are.

I hope you read the Bianchi Rovelli paper. They are certainly not the only advocates of the idea that the ball is in the quantum relativist's court to explain why this value of Lambda emerges and what its significance is. That is, it is a feature we didn't realize about our geometry and if it has an explanation it most likely will come from a deeper understanding of geometry.
http://arxiv.org/abs/1002.3966/

 

[marcus]

Here's a question (or request or mild challenge) for anyone reading, especially Cepheid and Jorrie
It would be nice to have a simple verbal intuitive explanation of the following "coincidence".

There is exactly one redshift (which with Jorrie's parameters comes out z=1.64) for which the recession speed when the light was emitted is c.
Galaxies with less redshift were receding slower than c when they emitted the light.
Galaxies with z>1.64 were receding > c when they emitted the light we are getting from them.

Now, this is ALSO the redshift where the galaxy has the smallest angular size. Why is that?

In other words equal size galaxies make a bigger angle in the sky if they are either farther away than z=1.64 or nearer than z=1.64.
Redshift 1.64 is where the angular size minimum comes. Why should that correspond to where the distance, at emission-time, is growing exactly at rate c?

The problem is one of finding the right intuitive words to explain something at beginner or wide audience level, not to give a mathematical proof. There should be a simple explanation everybody can understand.

 

[cepheid]

I hope you read the Bianchi Rovelli paper. They are certainly not the only advocates of the idea that the ball is in the quantum relativist's court to explain why this value of Lambda emerges and what its significance is. That is, it is a feature we didn't realize about our geometry and if it has an explanation it most likely will come from a deeper understanding of geometry.
http://arxiv.org/abs/1002.3966/

Okay, I read the paper (the whole thing), and I must admit that it was extremely interesting and well-argued. I think I understood most of the first two arguments (secs II and III), with the exception of this statement about the "coincidence" problem:

First, if the universe expands forever, as in the stan- dard ΛCDM model, then we cannot assume that we are in a random moment of the history of the universe, because all moments are “at the beginning” of a forever-lasting time.

To be honest, I'm not sure if I understand the implications of that statement, and I would have to think about it further. But I understood the general argument that follows that this is not as "special" a time in the history of the universe as people claim, and that the strict cosmological principle that proponents of the "coincidence" argument are trying to invoke is just observationally false anyway.

I'm not going to claim that I understood much of sec. IV, since I don't have much of a grounding in field theory, but this statement, in particular, stood out for me:

To trust flat-space QFT telling us something about the origin or the nature of a term in Einstein equations which implies that spacetime cannot be flat, is a delicate and possibly misleading step. To argue that a term in Einstein’s equations -is “problematic” because flat-space QFT predicts it, but predicts it wrong, seems a non sequitur to us. It is saying that a simple explanation is false because an ill-founded alternative explanation gives a wrong answer.

I changed their emphasis from italics to bold, since quoted text on PF is entirely in italics.

 

[marcus]

Thanks so much! It's great to have a second pair of eyes looking over these things!

My interest it it is primarily pedagogical. How best to introduce the cosmo constant Λ to complete beginners.
I think the most important quantity in cosmology is H and they need to get an intuitive grasp of H. What is it? It is like the interest on your bank savings account (fractional increase per unit time) but for *distances* instead of savings accounts.

The present value of H is a very slow rate of growth: 1/139 of one percent per million years. Or if we introduce a convenient time unit d = 10⁸ years then a small fractional increase 1/139 per d.

I think/hope beginners can grasp the idea of a distance growth rate without immediately jumping to pictures of galaxies whizzing this way and that. And 1/139 is possible to visualize.

So cosmology is about this distance growth rate H and how it changes over time. Now I want to introduce the asymptotic longrange limit of H namely H_∞.

That's going to be intuitive because they already have the basis, understanding what H itself is. So the message is that H changes, it gradually declines (like the bank slowly lowering the savings account interest rate) and we used to think it would decline eventually to zero. But no! It turns out the limit is a positive rate H_∞ = 1/163 per d.

Like an airplane landing on a raised platform instead of at ground level.

Then we can say what the cosmological constant Λ is. It is related to the asymptotic distance growth rate by:
H_∞² = Λc²/3

Lambda just happens to be a reciprocal area, units wise, the way Einstein originally put it in the equation governing how geometry (lengths etc) evolves. Beginners won't be familiar with what reciprocal areas are used for, spacetime curvature sounds mysterious. But a reciprocal area is the square of one over length.
And multiplying that by c² makes it a square of reciprocal time.
The square of a fractional growth rate.
So Einstein's constant Λ, looked at this way, comes very close to being a quantity of a familiar sort we are all used to--interest rate--except squared.

That's how i think the most readily intuitive beginner's introduction goes. I want to develop this approach to explaining Λ. Any comments or suggestions would be most welcome!
Thanks to you and Jorrie for your reactions so far.

 

[Jorrie]

Galaxies with less redshift were receding slower than c when they emitted the light.
Galaxies with z>1.64 were receding > c when they emitted the light we are getting from them.

Now, this is ALSO the redshift where the galaxy has the smallest angular size. Why is that?

I think the balloon analogy provides a reasonably intuitive answer to this. Here is my attempt.

Photons that left the source from closer than the (then) Hubble radius had a shrinking proper distance to us, while photons that left from farther than that were first moving away from us. As the Hubble radius increased due to the deceleration, those photons later started to make headway towards us (from a proper distance p.o.v).

The paths of photons from a distant galaxy coming from the left side and the right side respectively, were driven apart (diverged) by the expansion, until such time as the Hubble radius caught up with them. Hence, we 'see' them at a greater angle. Photons from observed galaxies closer than the (then) Hubble radius never diverged, so there is no 'magnification' by the expansion (in flat space, at least).

I should have made an accompanying sketch, but I do not have the time right now. Maybe later.

How does it sound?

 

[marcus]

Jorrie, I can't right now give an intuitive simple explanation. I appreciate you having the gumption to try, but I don't understand your explanation. I'm inclined to think it is doesn't quite work.
There's something curious here. It is a different "horizon" that we don't normally hear about.
This 5.8 billion lightyears is the maximum distance we can see things in the sense that it is the farthest away they could be at time the light was emitted.

It is the distance THEN maximum. Small angular size corresponds to far away at the time of emission. And smallest angular size necessarily has to correspond to greatest THEN distance.

We are used to the "particle horizon" of 45 to 46 billion ly which is the farthest away NOW distance, of things we can get light from. But as you know that matter which is out there was only 41 or 42 million ly when it emitted. So it is certainly not the farthest matter in then-distance terms. This is a different idea of farthest. It's strange.
To repeat the key thing: small angular size corresponds to far away at the time of emission. Large angular size (other things being equal) corresponds to being close, at the time of emission.

Maybe essentially what you are saying is that light that was emitted more than 5.8 billion lightyears away from us simply has not yet had the time to get here! The light that was emitted exactly at the max, exactly at 5.8 billion ly distance, has taken 9.7 billion years to get here and is only just arriving. I'm not sure, still thinking about this.

Here's something to think about: have a look at this figure from Lineweaver's paper.
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
Look at the middle graph which has comoving distance but ordinary time. It looks to me as if the lightcone and the Hubble radius cross right at an expansion age of 4 billion years. That would correspond to a lookback time of 9.7 billion years. It is exactly the moment we are talking about. A galaxy with redshift 1.64 that we see today is both on our lightcone AND on the Hubble sphere because receding exactly at speed c. So the intersection of those two curves is what we are talking about. Maybe this figure can help us understand why 5.8 billion ly is the farthest then-distance we can see.

If you look at the top graph of that same figure, which has proper distance, you notice that the lightcone has a teardrop shape, there is a point where it is fattest, and its tangent is vertical. that is the place where it is widest and the point we are talking about. Its diameter is 5.8 billion ly there. It is also where the Hubble radius crosses. I think you can see that in the figure. It is interesting. There is also an intersection about the same time level, between the particle horizon and the cosmic event horizon.

 

[marcus]

As an engineer you've had plenty of college calculus and there is a calculus explanation which might be worth mentioning. A continuous differentiable function on a compact interval must have a max.
Suppose we define a function of redshift z by saying f(z) = then-distance. Well we know that f(0) = 0 ly,
and f(1088) = 42 million ly which is just a pittance. A million ly is hardly anything.
So f must have a maximum somewhere in the interval [0, 1088]. And it just happens the max comes at z = 1.64. The max value is f(1.64) = 5.8 Gly. But that is so unintuitive!.

I think I should bring forward the earlier table, to have it handy. It shows the then-distance maximum around 5.8 billion ly. To remind anyone who happens to be reading, the numbers in this table were gotten with the help of Jorrie's calculator. The calculator gives multidigit precision and I've rounded off. Hubble rates at various times in past are shown both in conventional units (km/s per Mpc) and as fractional growth rates per d=10⁸y. The first few columns show lookback time in billions of years, and how the Hubble rate has been declining, while the Hubble radius (reciprocally) has extended out farther. The columns on the right show the proper distance (in Gly) of an object seen at given redshift z both now and back when it emitted the light we are currently receiving. The numbers in parenthesis are fractions or multiples of the speed of light showing how rapidly the particular distance was growing.

 Standard model with WMAP parameters 70.4 km/s per Mpc and 0.728. 
Lookback times shown in Gy, distances (Hubble, now, then) are shown in Gly.
The "now" and "then" distances are shown with their growth speeds (in c)
time      z     H(conv)   H(d[SUP]-1[/SUP])    Hub      now          back then 
   0     0.000     70.4   1/139    13.9      0.0          0.0
   1     0.076     72.7   1/134    13.4      1.0(0.075)   1.0(0.072)
   2     0.161     75.6   1/129    12.9      2.2(0.16)    1.9(0.14)
   3     0.256     79.2   1/123    12.3      3.4(0.24)    2.7(0.22)
   4     0.365     83.9   1/117    11.7      4.7(0.34)    3.4(0.29)          
   5     0.492     89.9   1/109    10.9      6.1(0.44     4.1(0.38
   6     0.642     97.9   1/100    10.0      7.7(0.55)    4.7(0.47)
   7     0.824    108.6   1/90      9.0      9.4(0.68)    5.2(0.57)
   8     1.054    123.7   1/79      7.9     11.3(0.82)    5.5(0.70)
   9     1.355    145.7   1/67      6.7     13.5(0.97)    5.7(0.86)
  10     1.778    180.4   1/54      5.4     16.1(1.16)    5.8(1.07)
  11     2.436    241.5   1/40      4.0     19.2(1.38)    5.6(1.38)
  12     3.659    374.3   1/26      2.6     23.1(1.67)    5.0(1.90)
  13     7.190    863.7   1/11      1.1     29.2(2.10)    3.6(3.15)
 13.6   22.22    4122.8   1/2.37    0.237   36.7(2.64)    1.6(6.66)

Abbreviations used in the table:
"time" : Lookback time, how long ago, or how long the light has been traveling.
z : fractional amount distances and wavelengths have increased while light was in transit. Arriving wavelength is 1+z times original.
H : Hubble expansion rate, at present or at times in past. Distances between observers at rest grow at this fractional rate--a certain fraction or percent of their length per unit time.
H(conv) : conventional notation in km/s per Megaparsec.
H(d^-1) : fractional increase per convenient unit of time d = 10⁸ years.
"Hub" : Hubble radius = c/H, distances smaller than this grow slower than the speed of light.
"now" : distance to object at present moment of universe time (time as measured by observers at CMB rest). Proper distance i.e. as if one could freeze geometric expansion at the given moment.
"then" : distance to object at the time when it emitted the light.

 

[Jorrie]

Maybe essentially what you are saying is that light that was emitted more than 5.8 billion lightyears away from us simply has not yet had the time to get here! The light that was emitted exactly at the max, exactly at 5.8 billion ly distance, has taken 9.7 billion years to get here and is only just arriving.

Yes, that's essentially true, but not all that useful.

Lineweaver's teardrop lightcone in the top diagram shows what I meant by "As the Hubble radius increased due to the deceleration, those photons later started to make headway towards us (from a proper distance p.o.v)." This happens at the fattest part of the teardrop, as you said.

I'm trying to get the balloon analogy ("cosmic balloon") worked in, because inside its applicability zone it makes many things intuitive, especially since it gives us two spatial dimensions to work with. If one gives Lineweaver's teardrop a second proper distance dimension, then a constant time-slice through it is represented by a circle on the cosmic balloon, centered on us. Now we can put a two dimensional galaxy or cluster on the circumference of the circle, at various time-slices (i.e. also various balloon radii). Identical galaxies observed from emissions while the teardrop was growing, will be magnified in angular size, when compared to ones from where the teardrop was fattest and just started to shrink, I think. (We must obviously keep the proper size of the galaxies the same at all times, it is just the photon paths that diverge when from outside the Hubble radius).

I will concoct a sketch sometime...

Does this not answer your puzzle: "In other words equal size galaxies make a bigger angle in the sky if they are either farther away than z=1.64 or nearer than z=1.64"?

 

[marcus]

Hi Jorrie, I'm getting the idea. The "teardrop" lightcone consists of geodesics in 4d spacetime. It has a kind of "equator" round its biggest diameter. Something sending light from below the "equator" has its rays spread out until they cross the equator and then they start to come together.

the same thing, if it was on the equator, would look smaller because its rays would not have spread out. I think I understand your explanation of why a object looks smallest when it has z=1.64.

"In terms of proper distance the teardrop lightcone has a max radius of 5.8 Gly, so we cannot presently see any galaxies that were originally farther than that, corresponding to z=1.64". (My original wording was poor and Jorrie suggested this clearer version, so I just substituted it in. Much better.)

anything with z>1.64 comes from "below the equator of the lightcone" and was actually nearer than 5.8 Gly when it emitted the light, and so it has a bigger angular size.

Yeah! I think I've understood your expanantion and I think its right. What I'm calling the lightcone's "equator" is actually a sphere not a circle, I'm thinking in terms of Lineweaver's schematic picture which is dimensionally reduced. Basically its all about the teardrop shape lightcone. Thanks for working this out!

I'll bring forward the link to that Lineweaver graph of the teardrop lightcone and other stuff.
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
It's useful, maybe I should swap "einstein-online" out of my signature and swap that picture in.
It'd be nice to have handy.

 

[Jorrie]

Yes, I think the 'equator' analogy is a good one.

I think one must however be careful with wording like this:

... so we cannot see any galaxies farther than z=1.64.

The prequalifier of "proper distance" makes it sort-of correct, but it will confuse many novices, since we observe galaxies up to almost z=10.

Perhaps better to say: "In terms of proper distance the teardrop lightcone has a max radius of 5.8 Gly, so we cannot presently see any galaxies that were originally farther than that, corresponding to z=1.64".

 

[marcus]

Thanks. My original sentence was confusing and your proposed rewording much clearer, so I simply adopted it (blue quotes post #344)

 

[Jorrie]

...
anything with z>1.64 comes from "below the equator of the lightcone" and was actually nearer than 5.8 Gly when it emitted the light, and so it has a bigger angular size.

Hi Marcus. Good as your equator analogy is for easy comprehension, one must still be a bit careful. I think the equator only works for closed spatial models, while the effect is also present for flat and open spatial cases, provided there is a positive rate of expansion. A flat or 'open' Earth surface without expansion would not work.

I'm still pondering an intuitive way to present it on the surface of an expanding balloon, without facing the challenges of 4D spacetime, but I haven't found it yet. The equator must be replaced by the Hubble radius (R_H), i.e. proper recession speed = c.

Maybe we can build on the equator idea, but also bring in expansion of a flat space as a stepping stone, before going the whole hog with the open case. Any ideas?

 

[marcus]

Hi Jorrie,
I swapped in the Caltech Lineweaver graphs, to have them handy in signature. I believe they refer to the spatially flat case. In the original article there is a long paragraph of explanatory material right below.

In case anyone hasn't read the original Lineweaver article, and wants to:
http://arxiv.org/abs/astro-ph/0305179
The figure is on page 6
The same figure is used in the Lineweaver-Davis article that is often referred to:
Look on page 3 of http://arxiv.org/pdf/astro-ph/0310808.pdf
The graphic quality is better there---plots show up larger plus there's explanatory text as well!

People always use the word "teardrop" to describe the shape of the lightcone plotted in proper distance. I would rather say an entirely convex pearshape, like this Anjou pear
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
The "equator" we are referring to is analogous to a belt around the widest part of the pear.

Regret to say: no helpful ideas about the exposition at the moment. Maybe some will come.

 

[Jorrie]

Hi Marcus.

Rather than trying to find a "better explanation" for the angular diameter max, I have spent the time more fruitfully (I hope) to update my cosmo-calculator to include values on your latest table (plus some presentational enhancements). Have not substituted it on my website yet, but here is a temporary link for testing purposes. I have opted for a more conventional value for your $H(d^{-1})$, namely 'Time for 1% proper distance increase' in Gy, since it fits in better with my calculator's units and style. I hope I have the conversion correct?

I would appreciate comments from yourself and any other interested parties. In time I should also add some more descriptive notes/links.

 

[George Jones]

People always use the word "teardrop" to describe the shape of the lightcone plotted in proper distance. I would rather say an entirely convex pearshape, like this Anjou pear

Ellis and Rothman, in their Am.J.Phys. paper "Lost Horizons", use the term "onion", and I think that I have seen this term used in a few other places.

From Lost Horizons:

How do we explain the shape of the past light onion?