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

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/M...es/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.
 Recognitions: Gold Member Science Advisor 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. Code:  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-1) 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.

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 Quote by marcus 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"?
 Recognitions: Gold Member Science Advisor 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/M...es/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.

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

I think one must however be careful with wording like this:
 Quote by marcus ... 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".
 Recognitions: Gold Member Science Advisor Thanks. My original sentence was confusing and your proposed rewording much clearer, so I simply adopted it (blue quotes post #344)

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 Quote by marcus ... 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?
 Recognitions: Gold Member Science Advisor 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-conte...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.
 Recognitions: Gold Member 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.

Mentor
 Quote by marcus 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?

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 Quote by George Jones 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:
I'm a big fan of George Ellis but I think he made a mistake in the produce department. Lightpear has fewer syllables and is more accurately descriptive than "lightonion".

Onions tend to be altogether too round, and how could one resist this depiction of an Anjou pear:
http://carrotsareorange.com/wp-conte...pear-anjou.jpg

However in any case one should never say lightcone, so I would approve switching to either vegetable of terminology.

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 Quote by 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.
Jorrie, as long as we are in a lighthearted mood and you will take any comments on calc as friendly intended, I'll make a few comments just from personal PoV. I think it is a great calculator and extremely useful.

However I would change "distance traveled by light" to "lookback time" and write My instead of Mly. I just think it is the more conventional term. What you are really talking about is the "light transit time"---the time it took for the light to get here. And people customarily call that the lookback time. I guess you could also call it "light transit time". or "light travel time". Not sure which is best.

I know what you mean by "distance traveled by light"---it's the distance it would have traveled on its own (without the help of expansion) but it's a bit confusing to refer to lookback time in those terms and also not conventional.

And also it would be more conventional (and slightly more correct mathematically) to say
"1% of Hubble time" and give units (as you do) in My (instead of time needed for 1% increase...).

Hubble time (defined as 1/H) is standard terminology and I think it's really nice to have the calculator give 1% of it in millions of years, because it is a good reminder of how I am always thinking of the instantaneous distance growth rate. So convenient! You just take the number that shows up in the box, e.g. 139.xxx, and write one over it, and bingo you have 1/139 % per My. A great way to visualize H as a distance growth rate!

That one excellent convenience outweighs my quibbles of terminology so I would be glad to see you make the changes "as is" based on that alone!

However since you asked for comment, i am quibbling that it would be conventional and slightly more mathly correct to say "1% of Hubble time" or maybe use the asterisk and label it "1% of Hubble time*"

where down below your footnote says something like "*approximate time needed for a 1% growth of proper distance"

You realize that a bank account that grows at the instantaneous rate of 1% per year (continuously compounded) will therefore grow slightly more than 1% in the course of a year. Strictly speaking you have to say "approx." because the reciprocal of an instantaneous rate, which is what the Hubble constant is, slightly understates the amount of growth in the given unit of time due to continuous compounding.

I hope you do write a few notes to accompany the calculator.

EDIT: I just put e^.01 into the google calculator and got 1.01005017
which is so close to 1% that I feel foolish making the distinction. If something grows at an instantaneous rate of 1% per million years, then if you wait 1 million years then (even with continuous compounding which is a feature of instantaneous rates) it will to any reasonable person look like it has grown 1%.
Why should I fuss about the difference between 1% and 1.005%. OK OK. No objection to the new version of your calculator. Go with it.

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 Quote by marcus I'm a big fan of George Ellis but I think he made a mistake in the produce department.

I hope that you will forgive me for straying a little further off topic. George Ellis, Roy Maartens, and Malcolm MacCallum have coauthored a new advanced cosmology text,

http://www.amazon.com/Relativistic-C...283155&s=books

I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.

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 Quote by George Jones I hope that you will forgive me for straying a little further off topic. George Ellis, Roy Maartens, and Malcolm MacCallum have coauthored a new advanced cosmology text, http://www.amazon.com/Relativistic-C...283155&s=books I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.
http://www.amazon.com/Relativistic-C...dp/0521381150/
Nice cover illustration! Two little blobs of overdensity in the microwave skymap giving birth to a cluster of galaxies! Picture worth many words.

April 2012 Cambridge U.P. and browsing allowed at the Amazon page. I will have a look at the ToC. thanks for the pointer!

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 Quote by marcus However I would change "distance traveled by light" to "lookback time" and write My instead of Mly. I just think it is the more conventional term.
I fully agree with lookback time as more conventional, but I thought the distance interpretation to be more intuitive than lookback time, which for beginners has to be explained. I guess that with more notes/footnotes, this requirement may however be met with the conventional term.

 Quote by marcus ... it would be more conventional and mathly correct to say "1% of Hubble time" or maybe use a footnote and label it "1% of Hubble time*" where down below your footnote says something like "*approximate time needed for a 1% growth of distance"
Excellent idea! Gives us "two for the price of one" in terms of info.

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 Quote by George Jones I hope that you will forgive me for straying a little further off topic. George Ellis, Roy Maartens, and Malcolm MacCallum have coauthored a new advanced cosmology text, http://www.amazon.com/Relativistic-C...283155&s=books I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.
I took a peek at pages 526-530, the section called "20.4 Loop quantum gravity and cosmology"
Page 537: " Like string theory, loop quantum gravity is still in its infancy--and either or both of these candidate quantum gravity theories could fail as a result of further discoveries...
...Given the uncertain status of all current attempts to develop quantum gravity, it is also useful to have competing paradigms."

Starting on page 537 you get section 20.4.1 "Basic features of quantum geometry" which is a thumbnail sketch of LQG with its main results (discrete area spectrum, Immirzi parameter..)
On page 528 begins section 20.4.2 "Loop quantum cosmology"
followed by section 20.4.3 "Loop quantum cosmology resolution of the big bang singularity
ending on page 530 with Figure 20.4 showing the evolution of the scalefactor during the LQC bounce,
and giving the semiclassical modified Friedmann and Raychaudhuri equations (equations 20.44 and 20.45)

It's highly condensed but all in all pretty good!
Eqn 20.41 gives the density range where LQC differs from classical, namely
ρ ≥ ρPlanck.
Eqn 20.42 gives an equation for the the critical density ρcrit, the max density achieved at bounce, and says that under usual assumptions works it out to about 0.4ρPlanck.
Eqn 20.43 indicates that an inflationary epoch would begin after a large expansion resulting from the bounce itself which reduces the density initially by a factor of 10-11.
(ρ/ρcrit)infl~10-11.

These are consequences of 20.44 and 20.45 which are the familiar Friedmann and Raychaudhuri equations with an addiitional term ρ/ρcrit which is suppressed except at densities near Planck scale. The authors cover the basic LQC stuff that researchers working on LQC phenomology use regularly. Roy Maartens has written some Loop cosmology pheno papers as I recall. The treatment is brief but impresses me as thoroughly solid/knowledgeable. Glad to see it in a major advanced cosmology text like this!

To take this peek (in case anyone wants to) you just go to the Amazon page and click on "look inside" and enter "loop quantum gravity" in the search box. It will give you a choice of clicking on page 513 or 526. I happened to choose page 526. The other passage seems more general overviewy, so less interesting.
 Recognitions: Gold Member Science Advisor Jorrie, I didn't see your post #355 and edited my post #352 to remove a minor objection. So far everything you are proposing looks fine from here!