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

[marcus]

My first reaction is (cheers!) we can throw out my google-calculator approximations for anything like z>> 10. They become too inaccurate for z > 100.

But I don't have microsoft Excel on this computer and have never installed a spreadsheet in my life (our son, who visits now and then, may help with that). So I don't have much of an idea how the new "front-end" will look and work.

I have the XLS file on my desktop, waiting, but so far have only opened it as text. I think I need Excel to open it as an actual spreadsheet.

This is exciting, I picture that the three inputs to the front end are (y_now, y_∞, z_eq) and that it outputs perhaps single values of stuff (like a, t, z, y_a or y_t, D_now, D_then...)
Or perhaps, if not now then possibly in future, a table. Assuming the user has specified a sequence of values of a, or values of t, to run down the first or lefthand column of the table.

That seems pedagogically beautiful, to me. It says to the beginner "all you do is specify two percentage growth growth rates of distance: the present and the eventual future one"
and the model does the rest.
So attention is focused on percentage growth rate instead of "speed".
And the cosmo constant is no mystery, but simply manifest in the eventual percentage growth rate.

Looking forward to seeing the actual front-end, this is just how I picture it.

 

[marcus]

In the meantime, I'm getting a routine down for setting up the 13.9/16.3 model in Jorrie's calculator.

First go to google and say Mpc/(km/s)/13.9 billion years

that outputs the number 70.3463274​

so you copy to clipboard and paste into the calculator's H_now box.

Next go to google and say 1 - .7272 - .0000812

that outputs the number 0.2727188​

so you copy that to clipboard and paste in the Omega Matter Now box.

Then you are ready to go! You have almost exactly the right parameters loaded for the 13.9/16.3 model. And if you want to try variations with different y_now and y_∞ you can use this same format. The only difference is the second step you say
1 - (y_now/y_∞)² - .0000812

Because all the .7272 is is the square of the ratio of the two Hubbletimes.

======================
Extra, probably unneeded explanation: the second step is to ensure perfect flatness.
Premultiplying 1/(13.9 billion years) by the factor Mpc/(km/s) is simply to get rid of the units km/s/Mpc so that what you get out is a pure number 70.346... to paste into the H box. In this approach
1/(13.9 billion years) actually is the Hubble growth rate, expressed in "per time" terms.

 

[Jorrie]

...
But I don't have microsoft Excel on this computer and have never installed a spreadsheet in my life (our son, who visits now and then, may help with that). So I don't have much of an idea how the new "front-end" will look and work.

It is possible that Firefox or MS IE may open the spreadsheet for you. I will also see if I can save the spreadsheet as a Google doc.

In the meantime I have attached the spreadsheet data as a .pdf, where you can see the 'front-end', but unfortunately not manipulate it. I could not get my pdf writer to print headers on each page; sorry about that, but it will give you a good idea.

The spreadsheet is primitive, but you will see some pointers like "equal", CMB for the green hi-lighted rows...

PS: I have also attached the graph for some values on the sheet.

 

[Jorrie]

And if you want to try variations with different y_now and y_∞ you can use this same format. The only difference is the second step you say
1 - (y_now/y_∞)² - .0000812

Because all the .7272 is is the square of the ratio of the two Hubbletimes.

I found the extra precision gained for 'perfect flatness' by the process you described to be absolutely negligible, even in the precise calculator. That's why the default Omega values in my calculator does not add up to precisely one, but to 1.0000812. This gives a very-very slight positive spatial curvature and hence a large, but finite cosmos (which I always like best :-)

 

[marcus]

More to my taste too. So be it.

The graph is handsome. Nice to see the Hubbleradius and the CEH coverging at 16.3 but having a temporary gap. Beautiful curves.
Also the red curve, if you flip the graph over, exchanging x and y axes, so time is on the horizontal and scalefactor is up the side, you get this Lineweaver figure #14
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg

The nice thing is you can see the inflection in the curve, where it changes from convex to concave:
from decelerating to accelerating.

So in your figure you can see the same inflection, a bit below the 10 billion year line, we know it is about 7 billion, but it's nearly linear for a stretch so it's hard to spot the exact point of inflection by eye.

 

[marcus]

I should summarize where I see this going as a result of the work and discussion in this thread. We have two resources shaping up. On one hand there are Jorrie's calculator(s) which AFAICS deliver professional-grade accuracy and implement the standard cosmic model. One reason they are interesting is that they let you input different model parameters and they output quite a variety of information about the universe: distances to galaxies then and now, distance expansion speeds, percentage expansion rates, and so forth.

On the other hand we've got a simple "do it yourself" cosmic model that delivers reasonably good precision back over the past 12 or 13 billion years. It is more trouble to use because it is essentially just based on a single formula. This is the formula for the expansion history of a generic distance: the scalefactor a(t) as a function of time.

This is arbitrarily pegged at the present value of 1, so a(now) = 1, and it shows how any cosmological distance has grown---it also extends into the future to show expected future growth.

This simple DIY cosmic model is is more trouble to use because essentially given the growth history you have to figure everything else from that, for yourself. But it's not as bad as it sounds and one picks up some understanding along the way.

My posts on the last several pages can be summed up streamlined as follows.
If you go to the online calculator http://web2.0calc.com/ (the first hit if you do a search for "online scientific calculator")
and paste this formula in:

(((16.3/13.9)^2 - 1)/((tanh(1.5*t/16.3))^-2-1))^(1/3)

you will see that as soon as it is pasted in, the calculator displays a neater, easier-to-read form.$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}$$What this computes, given a time t (in billion years) is the scalefactor a(t)---in other words it tells you the expansion history of a generic distance over time.
To use it, just replace the "x" by a time expressed in billions of years (such as 1, or 2, or 13.759 which is the current expansion age) and press the equals sign.
The answer will appear in the window and you can copy it to clipboard if you want. Then to repeat the calculation, click on the neat version of the formula as it appears above the window, and you can substitute something else in for the variable.
========================
1--- 0.1471433... (when the universe was 1 billion years old, distances were 14.7% what they are now)
2--- 0.2342347... (at 2 billion years, distances were about 23% what they are now)
...
...
13.759--- 0.9999836... (at the present age of 13.759 billion years they are of course 100% of their present lengths.:-)
...
20--- 1.5235746... (at age 20 billion years, distances will be 52% bigger than they are today.)
==================
A modified version of this same formula gives you the reciprocal scalefactor, 1/a(t), which turns out to be quite useful
The model's formula for 1/a(t) is
$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{-1/3}$$
and the single-file version, that you paste into the calculator if you are working with it that way is:
(((16.3/13.9)^2 - 1)/((tanh(1.5*t/16.3))^-2-1))^(-1/3)

Adding up successive values of 1/a as you work back in time is actually the way the present distance to a source is calculated!
========================
Some examples of the reciprocal scalefactor 1/a(t) for various times.
1---6.80... (distances and wavelengths of traveling light have expanded by a factor of 6.8 since the universe was 1 billion years old
2--- 4.27... (distances and wavelengths of traveling light have expanded by a factor of 4.27 since the universe was 1 billion years old)
...
...
13.759--- 1.000... (this is the present, distances and waves are their present lengths :-)
...
==================
It's conceivable you might sometime want to find the time t that gives a particular scalefactor, IOW invert the above formula for a(t). In that case paste this in
(16.3/1.5)atanh((((16.3/13.9)^2 -1)/a^3+1)^-.5)
which the calculator will display as $$\frac{16.3}{1.5}atanh\left(\left(\frac{(\frac{16.3}{13.9})^2-1}{a^3}+1\right)^{-.5}\right)$$
==================
The main equation in this model is this one. It gives the scalefactor a(t) at each time, going back pretty far into the early history of expansion where it gets a bit off track (because when you get back to the first few 100 million years much of the density in the universe was radiation rather than particles of matter, and radiation behaves differently in expansion , so the physics is not as simple. Anyway the main model equation is this:

$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}$$

The coefficient 1.5 and the exponent 1/3 both reflect the fact that we're in a matter dominated era, and have been since the first few 100 million years, and matter density falls off as volume increases---as the cube of distance. That is where the 3 and the 1.5 come from. In a radiation dominated world they would be 4 and 2. I derived some equations earlier in this thread and can go back to that later if there's interest.

But the most significant parameters in that "expansion history equation" are the Hubbletime parameters 13.9 and 16.3 billion years.

THOSE TWO TIME QUANTITIES *SHAPE* THE GROWTH CURVE.

If you change them the scalefactor curve a(t) showing the growth of a generic distance will change.

These two quantities are worth understanding. They express the CURRENT percentage growth rate of distance and the eventual longterm LIMIT growth rate that the present one is slowly tending towards.

I'll pick up and continue from here in my next post. This is enough for now.

 

[fat f...]

nobody replied to me___________________http://www.astro.ucla.edu/~wright/CMB-MN-03/FRL-28Oct08clean.pdf (last pic)

#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#

and do they see that milky way and andromeda are moving away from each other and/or away from observer's direction (like observers on Earth see how galaxies are moving away in the distant space(like in picture))................

here it looks like implosion(inside of a sphere), because dark age can be observed around the universe(in each direction)

and this looks like surface of a sphere model

is reality in between these models? then it would be like a donut model. if there is no centerpoint which serves as center of universe then it would support what i put into "# #"

................this doesn't help stuff because if universe would be this way then there one could also look back(away from the center):
http://en.wikipedia.org/wiki/File:Embedded_LambdaCDM_geometry.png

..............

was explosion(big bang) this enormous that space expands with 70mpc / s? is inflation responsible for this? and is speed the same as it was billions of years ago or does it become faster(expansion)? and does it expand now with speed of 2.16 trilliards(number with 21 characters) km/s?...............does somebody understands it trully? i follow scientifical concept to understand something and if i take away one step before another i can't precede, understanding starts from simple and gets more complicated once something is understood but i read stuff where people start from the middle and then its complicated, then i must ask things(puzzle parts) and if they don't contradict each other, i get the picture from answers

did knowledge came(to cosmologists) from observing mathematical equations which don't contradict each other and was it then applied to understanding(imagination) which then trully understood how universe works/looks like?

 

[twofish-quant]

The are different ways that try to show the same thing.

The first diagram is a two dimensional diagram in *time*. What it shows is what you see if you look back in time.

The second diagram is a three dimensional diagram in both space and time.

To get from the second diagram to the first, imagine a cone that ends at now, and shows the path of light that is arriving at you at this very moment. The intersection between the second diagram and the "light cone" will get you the first diagram.

 

[marcus]

Hi Twofish! Glad to see you! Let's see if we can get F.F. to start his own thread(s) with these questions.He has so much to ask and learn about I fear it would overload this thread.

http://www.astro.ucla.edu/~wright/CMB-MN-03/FRL-28Oct08clean.pdf (last pic)

#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#
...
...

does somebody understands it trully? i follow scientifical concept to understand something and if i take away one step before another i can't precede, understanding starts from simple and gets more complicated once something is understood but i read stuff where people start from the middle and then its complicated, then i must ask things(puzzle parts) and if they don't contradict each other, i get the picture from answers

did knowledge came(to cosmologists) from observing mathematical equations which don't contradict each other and was it then applied to understanding(imagination) which then trully understood how universe works/looks like?

You have many questions--many things that you don't understand and want to talk about.
Too many for this thread. You should start your own thread. Start off with one clear question. Don't ask everything all at once.

Like start a thread with this question (it is a good one)
"#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#"

That is a really good question. If you start a thread with just that, I would certainly answer. Other people would also. You might get several hours of people's time discussing that. Clarifying confusions about how distance is measured in cosmology.

BTW personally I think the first cosmological knowledge did not come from equations.

One of the first bits of knowledge came to a man named Anaxagoras in 250 BC (before there was equation-solving as we know it) by carefully reasoning about the distances to things.. He figured out that the sun is more than 10 times farther than the moon.
This enabled him to deduce that the width of Earth shadow (at the distance of the moon) was nearly as wide as the Earth itself. Then he observed that the Earth shadow when cast on moon during eclipse, was about 3 times greater than the width of the moon.
So he could estimate that the Earth itself (being slightly wider than its shadow at that distance) is slightly more than 3 times wider, maybe something around 4 times wider, than the moon. Figuring that out from scratch is no small achievement!

Since the time of Anaxagoras, most new knowledge about cosmos has evolved by careful reasoning about distances. What Hubble did was not so different from Anaxagoras. He learned a new way to estimate distances to galaxies, and when he surveyed them he found that distances to most galaxies were increasing a certain tiny percentage each year. Discovering expansion this way came, for him, before understanding and believing the Einstein equation--although that equation has geometric expansion as one of its most likely solutions.

From Anaxagoras to Hubble cosmologists have gained knowledge primarily by reasoning carefully about and devising smart ways to estimate the distances to things. So if you want to understand, a good way is to begin at the beginning and ask yourself how do you know that the sun is more distant than the moon.

Or start a thread with that ONE question you asked. Get people to explain the answer to that one question. But not in this thread, it would get too far off the current topic.

 

[twofish-quant]

1) Does somebody understands it trully?

As far as the parts for the diagram that you are showing, people understand it pretty well. Part of the reason is that there is very little "weird physics"

 

[marcus]

About the general question "how do we know", I'll add to what I said a couple of posts back:

From Anaxagoras to Hubble cosmologists have gained knowledge primarily by reasoning carefully about distances and angles (often as seen from another observer's viewpoint) and by devising ways to estimate distances to things.

It's not a bad idea to go over some of the steps in that long human history of accumulated insight, and in effect re-experience. For instance Anaxagoras (the name means "kings market") had the idea to visualize the angle between Earth and sun that someone on moon would see (when we see a half-moon) and realize that it was a right angle.

So he could sketch a right triangle, with the square corner at the moon, and realize that the sun was much farther from us than the moon (because of the near-right angle between them that WE see from earth, at half moon time.)

Now remember from that "much farther" and watching an eclipse he could tell that Earth is something like 4 times wider than moon. But the angle the moon makes in the sky is only 1/120 of a SIXTH of a circle! (Greeks learned from Babs that it's sometimes smart to judge angles as fractions of a SIXTH of a circle rather than of a whole circle.) Which means its distance from us is 120 TIMES ITS WIDTH! That would mean, if Earth is 4 times wider than moon, that distance to moon is 30 Earth diameters.

These complicated chains of reasoning about distances and angles are still at the heart of cosmology. If you practice on Anaxagoras it might make it easier to overcome confusion about the temperature map of the microwave background (the most ancient light we can see.) The angular sizes of its fluctuations are the analogs of the angles and proportions Anaxagoras perceived in the sky.

 

[marcus]

To get back to discussing the simple one-formula model cosmos, here is what I was saying a few posts back:
==================
The main equation in this model is this one. It gives the scalefactor a(t) at each time, going back pretty far into the early history of expansion where it gets a bit off track (because when you get back to the first few million years much of the density in the universe was radiation rather than particles of matter, and radiation behaves differently in expansion , so the physics is not as simple. Anyway the main model equation is this:

$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}$$

The coefficient 1.5 and the exponent 1/3 both reflect the fact that we're in a matter dominated era, and have been since the first few million years. Matter density falls off as volume increases---as the cube of distance---and that is where the 3 and the 1.5 come from. In a radiation dominated world they would be 4 and 2. I derived some equations earlier in this thread and can go back to that later if there's interest.

But the most significant parameters in that "expansion history equation" are the Hubbletime parameters 13.9 and 16.3 billion years.

THOSE TWO TIME QUANTITIES *SHAPE* THE GROWTH CURVE.

If you change them the scalefactor curve a(t) showing the growth of a generic distance will change.

These two quantities are worth understanding. They express the CURRENT percentage growth rate of distance and the eventual longterm LIMIT growth rate that the present one is slowly tending towards. Each of the (rather long) intervals of time is the reciprocal of a (rather slow) instantaneous distance expansion rate.

For example the presentday Hubbletime 13.9 billion years can be understood intuitively by thinking that ONE PERCENT of it, namely 139 million years, is the time a distance would take to increase by one percent.
(Continuing steadily at its present speed of growth.) The two hubbletimes 13.9 and 16.3 billion years are convenient to handle and rememeber, and we use them as the two main parameters in the model, but the actual growth rates we are concerned with are their RECIPROCALS which you can write as 1/13.9 per billion years and 1/16.3 per billion years---fractional rates of growth. These are ridiculously (I should say "astronomically") slow rates of fractional growth. So it's easier to work with the times than with the rates.

Going back to the earlier post, here are some sample outputs from our model's main formula: scalefactors a(t) calculated for various times t;
==================
1--- 0.1471433... (when the universe was 1 billion years old, distances were 14.7% what they are now)
2--- 0.2342347... (at 2 billion years, distances were about 23% what they are now)
...
...
13.759--- 0.9999836... (at the present age of 13.759 billion years distances are of course 100% of their present lengths.:-)
...
20--- 1.5235746... (at age 20 billion years, distances will be 52% bigger than they are today.)
==================

What I want to explain is how we can use the output from this one formula to find out other things:
DISTANCES to sources which emitted the light we're getting from them at various times in the past.
DISTANCES to sources whose light comes to us with wavelengths expanded by some factor
SPEEDS that the distances computed as above are now increasing, and were increasing when the light was emitted.

I'll try to get to that in the next post.

 

[marcus]

To begin to deal with distances in terms of this simple model I need to add up this cumulative sum of terms t_s where s is the reciprocal scalefactor 1/a and run thru some range like [1,2] in steps of 0.1, say. Let's start it at s = 1.1 and go 1.2, 1.3,... and at each value of s we will evaluate this formula
atanh((.375136* s^3+1)^-.5)
using http://web2.0calc.com
to get t_s the time (expressed in billions of years) when the light was emitted that we now see wave-lengthened by a factor of s.

1.14678...(starting with 1.1)
2.1893904284755
3.14075638766957
4.01193171154709
4.81239653369235
5.55030772362728
6.2327011396146
6.86566003274671
7.45445670103674
8.00367220130405 (this was for 2.0)
8.51729768096363
8.99882016696824
9.4512951759886
9.87740815857389
10.2795265021777
10.65974356945189
11.0199160299271
11.36169555160973
11.68655575226624
11.99581516649566 (the last one was for 3.0)

So how to use this cumulative sum to get distances? Well suppose a galaxy's light comes in with a scaleup s=2.0 (wavelengths twice as long as when emitted.)
The number from the list we use is the one for 1.9 namely 7.45445670103674, and that gets multiplied by the Δs, the step size, which is 0.1

In addition there are two other things to do: add (1+Δ/2)*t₁
which is 1.05*1.2661864372681=1.329495759131505
and subtract (2-Δ/2)*t₂ = 1.95*0.54921550026731=1.0709702255212545
The difference is 0.2585255336102505, so that's what gets added:
0.745445670103674+0.2585255336102505 = 1.0039712037139245
Finally multiply that by 16.3/1.5 and get 10.9098... Gly.
Ned Wright's calculator says 10.901 Gly (with equivalent model parameters). So the accuracy is not so bad.
that's the current distance: 10.91 billion lightyears.

All those extra decimals are ridiculous but it is too much trouble to be rounding off all the time so I just take what the calculator gives and finally round it off to something sensible at the end.

the corresponding thing for s=3
1.168655575226624+1.329495759131505-0.91231527197679=1.585836062381339
and then again finally multiply by 16.3/1.5, to get 17.23275...billion lightyears
Ned Wright says 17.220, so again we are off by 1 in the fourth digit.

 

[marcus]

I didn't get around to editing the previous post until after the deadline for changes and it needs some clarification.
The key formula in the toy version cosmic model I'm working with is
(16.3/1.5)*atanh((.375136* s^3+1)^-.5)
This gives the time (expansion age in billions of years) at which the reciprocal scalefactor was a particular value s.
Another way to say it: t_s is the time (expressed in billions of years) when the light was emitted that we now see wave-lengthened by a factor of s.
s can be thought of, if you like, as the "scale-up factor": because since the time t_s, distances and wavelengths have been scaled up by a factor of s. Back at time t₂ distances were 1/2 their present size, back at t₉ distances were 1/9 their present size, t₁ is the universe's present expansion age, and so on. I won't be using the traditional notation for s, which is "1+z", since it makes the formulas even messier than they are already.

The Hubbletime parameter 16.3 billion years represents the cosmological constant (asymptotic distance growth rate.) The other Hubbletime parameter, giving present growth rate, is 13.9 Gy and since that most often appears in combination as (16.3/13.9)²-1=0.375136... I have, for simplicity, packaged it in that number.

When I have to do a lot of calculating, e.g. numerical integration, I leave off the coefficient (16.3/1.5) and factor it in only at the end.

It turns out that we can do a pretty good job of estimating the distances to sources at various scaleups by essentially just adding up a long string of arctanh values, with s advancing from 1 to s in steps of some smallish stepsize Δ. If we take Δ = 0.1, this amounts to:

1.05t₁+ 0.1( t_1.1+ t_1.2+...+ t_s-0.1) - (s-0.05)t_s

This is the bare bones of a numerical integration for c∫ s dt.
The idea is that at each interval dt of time in the past, the light from an object travels a distance cdt and this gets scaled up by a the appropriate factor s. So at present the distance to the object is the sum of all those scaled-up segments and equals c∫ s dt.

It looks messy but seems to work out all right.
Here's a cumulative sum of atanh((.375136* s^3+1)^-.5)
using http://web2.0calc.com

1.14678...(starting with 1.1)
2.1893904284755
3.14075638766957
4.01193171154709
4.81239653369235
5.55030772362728
6.2327011396146
6.86566003274671
7.45445670103674
8.00367220130405 (this was for 2.0)
8.51729768096363
8.99882016696824
9.4512951759886
9.87740815857389
10.2795265021777
10.65974356945189
11.0199160299271
11.36169555160973
11.68655575226624
11.99581516649566 (this was for 3.0)
12.29065686210409
12.57214523554962
12.84124042899438
13.09881073851359
13.34564332215593
13.58245346592667
13.80989262370986
14.02855541222707
14.23898571313266
14.44168201025018 (for 4.0)
14.63710206991032
14.82566705565418
15.00776515463345
15.18375478139281
15.3539674149637
15.51871011700947
15.67826777187171
15.83290508355766
15.98286835979816
16.12838710914517 (for 5.0)

So suppose we evaluate 1.05t₁+ 0.1( t_1.1+ t_1.2+...+ t_s-0.1) - (s-0.05)t_s
to find the distance now
to a source with s = 4 (its light comes in wavelengthened by a factor of 4).
As long as we are using the stepsize Δ=0.1, the first terms is always 1.329495759131505,
and the sum, multiplied by the stepsize, can be read off that list: 1.423898571313266.
The term at the end, that gets subtracted, is 3.95*0.202696297=0.800650373

1.329495759 +1.423898571 - 0.800650373 = 1.95274396

And then at the end the whole thing gets multiplied by the cosmological constant term 16.3/1.5,
to give 21.2198≈ 21.22 billion light years.
Sorry about all the meaningless extra digits but it is too much trouble to be rounding off every time I take a result from the calculator, so I just round off at the end.
Let's compare this with Wright's calculator.
Well, Wright's says 21.204 Gly. So as usual we are OK for three significant figures and off in the 4th place.

Notice that since this numerical summing procedure gives us the NOW distance to the source, all we need to do is multiply by the scalefactor a, or alternatively divide by the scaleup s, and we get the THEN distance---how far from our matter or galaxy the thing was when it emitted the light.

So this primitive model already does quite a bit that one expects from serious cosmology calculators. Given a scalefactor a (or the reciprocal 1/a = s) it can give the corresponding expansion age---the time when the source galaxy emitted the light. And it can give the Now and Then distances to the source (proper distance, as if you could halt expansion at the given moment and measure directly).

The THEN distance is essentially the ANGULAR SIZE distance (our model is spatial flat) so that's taken care of.

It still might be nice to be able to calculate the HUBBLETIME corresponding to a given scalefactor a, or its reciprocal s. That is hour handle on the rate of expansion going on at the time the light was emitted.

Notice that the light itself, when it arrives, tells us the scalefactor, or equivalently its reciprocal s, which we focus on here, so the other things we want to know should be calculated from s.

 

[marcus]

It looks as if the Hubbletime Y_s corresponding to a given scaleup s should be given in billions of years by:

16.3(0.375136s³+1)^-.5=16.3/sqrt(.375136*s^3 + 1)

so using the calculator let's try that for s=3

It gives Y₃ = 4.8861403 ≈ 4.886 billion years.

Let's see if I've made a mistake.

Apparently not, Jorrie's calculator (with the corresponding parameters) gives 4.885 billion years.


Remember that the two parameters we're using in the model's formulas, namely 16.3 and 0.375136, are just an equivalent form of the two Hubbletimes which determine two key expansion rates, now and in distant future.
Y_now = 13.9 billion years

Y_∞ = 16.3 billion years

The number .375136 is simply what (Y_∞/Y_now)² - 1 = (16.3/13.9)² - 1 works out to be.
You can think of 1.375136 as a ratio of two expansion rates, squared. It is simply (H_now/H_∞)² so it tells you how much more the percentagewise expansion rate is now than it will be in the longterm future.

When people talk about "acceleration" what they mean is what you see when the H expansion rate is declining only very slowly or is steady at some given value. As long as H is not declining too rapidly, if you watch a particular distance it will grow by increasing annual amounts as the principal grows. Not terribly dramatic, given the very low "interest rate" but there is acceleration in a literal sense.

In the previous post we calculated that a galaxy we see with scaleup factor s = 4 (wavelengths quadrupled) is now at a distance of 21.22 Gly.
How fast is that distance now growing?
That is very simple to calculate. We just divide 21.22 Gly by 13.9 Gy.
D_now/Y_now = 21.22/13.9 = 1.53c.
That means it is growing at 1.53 times the speed of light. Calculation easy with these quantities

For comparison and a bit more practice, back in post #433 we found that the distance NOW for s=3 was 17.23275 billion lightyears, which means distance THEN was 17.23275/3=5.744 billion lightyears.

However we just found that also for s=3 we have Hubbletime Y_then = 4.886 Gy.

So for a s=3 galaxy, whose distance THEN at time light was emitted was 5.744 Gly, how fast was that distance then growing?
Well obviously D_then/Y_then = 5.744 Gly/4.886 Gy = 1.18 ly/y = 1.18c.

The notation is still far from perfect, but I hope some of this is comprehensible
==============
referring back to post #434 the last increment was 0.14551874934701
and to find the now distance to an s=5 source D₅(now) one would take 4.95*0.14551874934701=0.720317809 off at the end, so it looks like
1.329495759 +1.598286836 - 0.720317809 = 2.207464786
which then gets multiplied finally by 16.3/1.5 to give 23.987784 billion lightyears.
So unless I've made a mistake that's the distance now to an s=5 source. I'll compare with what Wright's says.
It says 23.970 Gly. So we are still OK for three digits.
D₅(now) = 23.99 Gly
D₅(then) = 23.987784/5 ≈ 4.798 Gly
Y₁ = 13.9 Gy
Y₅ = 16.3/sqrt(.375136*5^3 + 1)= 2.35535 ≈ 2.355 Gy

So we can say that for an s=5 galaxy, when the light we are now getting was emitted, the distance was expanding at a speed 4.798 Gly/2.355 Gy = 2.037 c, over twice the speed of light. The then and now speeds of expansion are given by:
D₅(then)/Y₅ and
D₅(now)/Y₁

 

[Jorrie]

It looks as if the Hubbletime Y_s corresponding to a given scaleup s should be given in billions of years by:

16.3(0.375136s³+1)^-.5=16.3/sqrt(.375136*s^3 + 1)

It appears to me that your 'scaleup factor' (s = 1/a = z+1) is a useful one, since it does not go negative for future times, but just goes smaller than 1. I would caution against the terminology though, as it is too close to the conventional 'scalefactor' and may cause confusion. Maybe something like 'upscale ratio' or 'expansion ratio'? I would prefer 'upscale ratio of distances' rather than 'scaleup factor of wavelengths', so as to not also cause potential confusion with Doppler effects.

I'm working on a variant of the cosmo-calculator that will give a table for a range of z (or s?), with some useful values in the columns. Not quite there yet, but it looks practical.

 

[marcus]

How about calling it "stretch factor"?
Or "extension ratio"?
Another idea, similar to something you suggested is to call the s number the "enlargement"
because it is the ratio by which distances are enlarged during the time the light is on its way
somewhat reminiscent of photographs being blown up.
I'd welcome more suggestions.
I'm glad to hear your new calculator is looking practical! I think the idea of generating tables for a range of z (or for s !) is a good one. Even fairly short tables with 5 to 10 lines can give someone extra perspective and intuition about how things are evolving. Just being able to compare two or three lines can be informative. After trying different things I do agree that the reciprocal scalefactor (whatever you call it and whether or not you subtract 1 from it) is the most useful handle on the situation.

 

[marcus]

Here's one way to think about it: say we number the stages of expansion history according to how much distances have been enlarged since then.
It means that earlier slices of spacetime have larger s numbers, which at first seems turned around, but in fact it's cleaner formula-wise to do the numbering backwards that way.

To illustrate, suppose we are watching a galaxy as it was when distances were 1/4 of present size. We can denote that stage of expansion history by saying s = 4. While the light was on its way, distances and wavelengths have been enlarged by that factor. So that galaxy, as we see it, is in "slice 4" of expansion history.

In that way of denoting stages of expansion, the present is s=1, because enlarging by a factor of 1 is the identity.

In the simplified toy model, the expansion age t_s associated with a stage s is given (in billions of years) by:
$$t_s = \frac{16.3}{1.5}arctanh \left( \left( 0.375136 s^3 + 1\right)^{-1/2}\right)$$
And the corresponding Hubbletime at stage s, also in billions of years, is:
$$Y_s = 16.3 \left( 0.375136 s^3 + 1\right)^{-1/2}$$
These are the two basic equations of the model--the other usual quantities such as distances and expansion speeds can be derived from these two. There is one peculiar thing to notice, which is that with expansion stages numbered this way, not only do we have the present tagged s = 1 but also future infinity is s = 0.
So the eventual, or longterm value of the Hubbletime (a key parameter in the model) is:
$$Y_0 = 16.3 \left( 0.375136 \times 0^3 + 1\right)^{-1/2} = 16.3 Gy$$
while the present Hubbletime is:
$$Y_1 = 16.3 \left( 0.375136 \times 1^3 + 1\right)^{-1/2} = 13.9 Gy$$
I keep having to write this number 0.375136, which is kind of like a parameter of the system being the square ratio of our two Hubbletimes, less one. (Y₀/Y₁)² - 1.
So I will call that number capital Theta Θ. The two basic equations of the model are then:

$$t_s = \frac{2}{3}Y_0 arctanh \left( \left( \Theta s^3 + 1\right)^{-1/2}\right)$$
$$Y_s = Y_0 \left( \Theta s^3 + 1\right)^{-1/2}$$
People who don't like greek letters should just remember it is a shorthand for an ordinary number ≈ 0.375 that essentially says something about the amount bigger current expansion rate is than the eventual longterm rate. (their ratio is about sqrt(1.375)

 

[marcus]

I want to try out some terminology in part suggested to me by Jorrie's comments and which he might be puttng to use in another project.But I can try several ideas out, tentatively, in connection with this simple cosmic model. The main variable could be called the "stretch" because it is the factor by which distances from a past slice of spacetime are enlarged (and wavelengths too) between then and now.

The idea is that if you start back to some earlier stage in expansion history and the enlargement of distances (and wavelengths) from then to now is a stretch factor of four (say S = 4)
then the scale back then, relative to now is 1/4, or 0.25.
So the stetch and scalefactor are reciprocals, like 4 and 1/4.

It just turns out that the stretch is a convenient variable to run the model, or the calculator, on. You get the simplest formulas that way, of the various things I've tried.

So I'm using S to stand for the stretch and the conventional letter a (= 1/S) to stand for the scale factor. The lineup of numerical information could (tentatively) go like this:
Stretch---Scale factor---Expansion age---Hubble time---Distance now---Distance then

and then again, this time showing the symbols that might be used to denote these quantities:

Stretch (S=1/a)---Scale factor (a)---Expansion age (t_S)---Hubble time (Y_S)---Distance now (D_S[now])---Distance then (D_S[then])

The idea is, we observe a galaxy and its light tells us the stretch factor S, say it is 4. Wavelengths 4 times what they were at the start of the trip. The galaxy is living back when distances were 1/4 present size. Then D₄[now] tells us proper distance to the galaxy NOW, and D₄[then] tells us distance back then, when light was emitted, from our matter (that became us) to the galaxy.

If we want to know the SPEED of distance growth, you simply divide the distance by the Hubble time belonging to that slice. Back then when the distance was D₄[then], it was growing at speed D₄[then]/Y₄.

The present is denoted S=1 and the present Hubbletime is Y₁ = 13.9 billion years. So the present distance is expanding at speed D₄[now]/Y₁.
Today's distances, if you want to know what speed they are expanding, you just divide them by 13.9 billion years.

So that's a provisional idea for a list of 6 related numbers that the model, or a calculator, can give you, that seems like enough to work with and get a picture of the expansion history from.

 

[marcus]

I don't want to forget that simple model, although Jorrie has now put an excellent online tabulator on line.
Here is the single-line formula calculator we were using, see post #434
http://web2.0calc.com

Here is the calculator formula to compute the time given the stretch S:
(16.3/1.5)atanh((.375136* S^3+1)^-.5)

There's also a more complicated version for it given the scalefactor, but we probably won't use it.
(16.3/1.5)atanh((((16.3/13.9)^2 -1)/a^3+1)^-.5)

I just had a kind of exciting look into the future. I went to web2.0calc and put in exactly what I mentioned, namely
(16.3/1.5)atanh((.375136* S^3+1)^-.5)

And decided to see when distances would be 100 times what they are today
which means scalefactor a=100 and reciprocal S = 1/a = 0.1

So I put 0.1 in place of S, in the formula and pressed = and it said that would happen
in year 87.92 billion.

So that is kind of cool. When expansion has been going on for about 88 billion years distances can be expected to be about 100 times what they are today.

Let's try another. when will it be that distances are FIFTY times what they are today? Put in S=0.02 to the web2.0calc.
Bingo. It says that will happen in year 76.625 billion.

And just as a side comment with continuous compounding the Hubbletime 16.3 billion years corresponds to a doubling time of 11.3 billion years. That is the natural log(2) times 16.3. So it seems right that you go from scale 50 to scale 100 in something a little over 11 billion years.

 

[marcus]

In post#434 I used a crude numerical integration of Sdt to find the distance now to a galaxy in the past in era S
It turned out that when you rearrange an Sdt integration to make it easy to add up you get what LOOKS like a tdS integration (with extra terms at either end). this is just algebraic rearrangement. Then the steps can be of S rather than time and we can use the formula t_S
(16.3/1.5)*atanh((.375136* S^3+1)^-.5)
which gives the time (expansion age in billions of years) when the reciprocal scalefactor (stretch) was a particular value S

In the earlier post we had S advancing from 1 to S in steps of some smallish stepsize Δ. If we take Δ = 0.1, this amounts to:

1.05t₁+ 0.1( t_1.1+ t_1.2+...+ t_s-0.1) - (s-0.05)t_s

This is what the numerical integration for c∫ S dt boiled down to.
The idea was that at each interval dt of time in the past, the light from an object travels a distance cdt and this gets scaled up by the appropriate factor S. So at present the distance to the object is the sum of all those scaled-up segments and equals c∫ S dt.
====================

So I decided to look into the future with the same technique and I found that if a galaxy is going to pass thru your forward lightcone at S=0.5, that is when distances are TWICE what they are today, then the distance NOW to it is 7.5 Gly.
Where are the galaxies NOW which you could hit with a flash of light you send today and which arrives wavestretched to double length? They are 7.5 billion lightyears from here.
It's like a time reverse image of the earlier game when we asked things about a galaxy whose light comes in wavestretched by a factor of 2, where was it when it emitted the light, where is it now etc.

The numerical integration boiled down to:
.55t_.5+ 0.1( t_.6+ t_.7+...+ t_.9) - .95t_1.0

And I evaluated that and got 7.5 billion lightyears.

So then we can say that when the signal we send arrives the distance to the target galaxy will be 15.0 billion light years. Because we know the expansion of scale between now and that time in the future.

 

[marcus]

In the previous post I did a rough numerical integration based on toy model and got the estimate that a galaxy we can send a message to which will arrive when distances are TWICE today (namely an S=0.5 galaxy) is currently at distance 7.5 Gly and when the message arrives it will be at S=15 Gly.

Now I can confirm that with the A20 calculator, which sees the shape of expansion history in the future as well as the past
http://www.einsteins-theory-of-relativity-4engineers.com/CosmoLean_A20.html

I just make a small table running from present S=1 out to S=.1 in future, in steps of 0.1.

So it covers the S=0.5 case but also gives me a little context (to help grow intuition/feel for the expansion process.)

===quote===
Hubble time now (Ynow) 13.9 Gy Change as desired (9 to 16 Gy)
Hubble time at infinity (Yinf) 16.3 Gy Change as desired (larger than Ynow)
Radiation and matter crossover (S_eq) 3350 Radiation influence (inverse: larger means less influence)
Upper limit of Stretch range (S_upper) 1.0 S value at the top row of the table (equal or larger than 1)
Lower limit of Stretch range (S_lower) 0.1 S value at the bottom row of table (S_lower smaller than S_upper)
Step size (S_step) 0.1 Step size for output display (equal or larger than 0.01)

Stretch (S) Scale (a) Time (Gy) T_Hubble (Gy) D_now (Gly) D_then (Gly)
1.000 1.000 13.756 13.900 0.000 0.000
0.900 1.111 15.250 14.444 -1.417 -1.575
0.800 1.250 16.981 14.929 -2.887 -3.608
0.700 1.429 19.004 15.342 -4.401 -6.287
0.600 1.667 21.396 15.677 -5.952 -9.921
0.500 2.000 24.279 15.930 -7.533 -15.066
0.400 2.500 27.856 16.108 -9.135 -22.839
0.300 3.333 32.507 16.218 -10.752 -35.840
0.200 5.000 39.097 16.275 -12.377 -61.886
0.100 10.000 50.388 16.297 -14.006 -140.059
===endquote===

So the quick and dirty estimate I did earlier worked OK. For an S=.5 galaxy (where our message reaches when distances are TWICE) the present distance really is 7.5 and the distance then when message arrives really is 15 Gly.

the minus signs have to do with the direction the light is going, from us to them.
whereas in the past the distances have positive sign because the light is coming from them to us----itself a kind of nice feature.

Also as an extra bonus the A20 tells me that the message that we send today (expansion age 13.75 billion years) will arrive when expansion age is 24.3 billion years. So it will take around 11 billion years to get there.

That makes sense: when it arrives at destination the message will be 15 billion lightyears from us, and will have been traveling 11 billion years---you have to allow for some expansion of distances so naturally 15 > 11.

 

[marcus]

The last 5 pages or so have been largely devoted to discussing the development of what I think is a really fine cosmology teaching/learning calculator, by Jorrie, and also working out some simplified equations that approximately reproduce part the expansion history (after the radiation energy density stopped being a major factor in the early U.) I should stress that Jorrie's calculator is what I would call professional grade--it reproduces the standard cosmic model--whereas the other thing we were working on is different, more of a "toy model".

Now the calculator (currently version A25) has its own sticky thread "Look 88 billion years into the future..." and I'd like to find a way to get this thread back into the groove of helping to "get us all on the same page."

One thing that could be highlighted, that we haven't discussed much here so far, is that if a massive particle or object is given a kick so that it has its own individual motion relative to the universe rest frame---the "Hubble flow"---CMB rest, it will gradually slow down relative to CMB rest and given enough time will REJOIN the "Hubble flow", or come approximately to a STOP relative to the ancient light.

This is rather un-Newtonian and could be unintuitive to newcomers. It violates conventional conservation notions. But it is really basic to understanding so we should talk about it. It is analogous to the redshifting of light. the light loses energy and momentum as it travels across cosmological distances, although its speed doesn't change.
With a massive object, the mass doesn't change but the speed does, so there is the same loss of energy and momentum.

A thing's individual velocity relative to the ancient light is called its peculiar velocity (meaning "special to itself", not weird). The basic message is if a massive object is given a kick so it acquires some peculiar velocity (relative to CMB rest) then over a long period of time that velocity will tail off to zero and it will asymptotically come to rest.

This figures in discussions of the "tethered galaxy problem". We were discussing that in another thread and Jorrie plotted some informative curves . For now, at least, I'll just give a link to his post:
https://www.physicsforums.com/showthread.php?p=4082096#post4082096

BTW when thinking about the balloon analogy it's good to remember that fixed points on the balloon surface represent points at CMB rest. A galaxy at some fixed point like that sees the CMB the same temperature in all directions, instead of having a doppler hotspot caused by its own peculiar motion. In Ned Wright's balloon animation all the galaxies are at rest (no peculiar motion) shown by their staying always at the same latitude and longitude on the balloon surface. The photons on the other hand have motion. Each moves at constant speed in constant great-circle direction. However if you watch carefully will see their wavelengths enlarge to symbolically show redshift.

 

[Jorrie]

One thing that could be highlighted, that we haven't discussed much here so far, is that if a massive particle or object is given a kick so that it has its own individual motion relative to the universe rest frame---the "Hubble flow"---CMB rest, it will gradually slow down relative to CMB rest and given enough time will REJOIN the "Hubble flow", or come approximately to a STOP relative to the ancient light.

This is rather un-Newtonian and could be unintuitive to newcomers. It violates conventional conservation notions. But it is really basic to understanding so we should talk about it. It is analogous to the redshifting of light. the light loses energy and momentum as it travels across cosmological distances, although its speed doesn't change.
With a massive object, the mass doesn't change but the speed does, so there is the same loss of energy and momentum.

Actually, there is a way in which the balloon analogy can make cosmic particle momentum decay intuitive. Simply consider a massive, frictionless particle that moves along the surface of the spherical balloon as a Kepler orbit around the center of a balloon. This particle must conserve angular momentum relative to the center of the balloon, i.e.

$L = r^2 m d\phi/dt$ = constant. Since $v = r d\phi/dt$, it means that for non-relativistic speeds, the particle speed scales with $1/r$.

If the balloon is being inflated, the particle must lose surface speed, just like a Kepler orbit that is losing orbital speed at larger radius. If the increase in balloon radius is kept up, the particle’s surface speed will eventually approach zero, as radius tends to infinity. In cosmology, this is usually described as 'joining the Hubble flow'.

The analogy seems to hold even for relativist particles. The relativistic Keplerian equation for the conservation of orbital angular momentum is:

$L = (1-v^2/c^2)^{-0.5} r^2 m d\phi/dt$ = constant (e.g. MTW eq. 25.18).

This simple scheme can be shown to reproduce the curves of figure 3.5 obtained by Davis (2004) [http://arxiv.org/abs/astro-ph/0402278] (with the exception of $v=c$).

I think that the $v=c$ case can also be handled by the analogy; the particle’s momentum must then be expressed in terms of the de Broglie wavelength.

Edit:
Relativistic de Broglie wavelength is given by: $\lambda=\gamma h/(m v)$, where $\gamma$ is the Lorentz factor.

If we write the angular momentum of the 'balloon particle' in terms of surface velocity, it is simply $L = \gamma r m v$. Taking $\gamma$ from the de Broglie wavelength, gives the conservation of angular momentum as

$L = r h / λ =$ constant, valid for photons and matter.

 

[PeterJ]

I have no ability to understand a sphere with nothing off the surface. I'd rather have Plotinus's hypersphere. This is more or less a balloon analogy, with the spacetime universe on the surface as usual, but it has a centre and at least one extra dimension. Then we have the ability to include an extra dimension (or bundle of them) at each point in spacetime, since this would be represented as a connection to the centre point. Plotinus was not talking physics exactly, but his model seems more sophisticated than a 3D balloon representing a 2D spacetime.

But I am way out my depths here.

 

[marcus]

I have no ability to understand a sphere with nothing off the surface. I'd rather have Plotinus's hypersphere. This is more or less a balloon analogy, with the spacetime universe on the surface as usual, but it has a centre and at least one extra dimension. Then we have the ability to include an extra dimension (or bundle of them) at each point in spacetime, since this would be represented as a connection to the centre point. Plotinus was not talking physics exactly, but his model seems more sophisticated than a 3D balloon representing a 2D spacetime.

But I am way out my depths here.

Instructive example! A mystic will postulate additional details (like the "edge" of the universe, or extra spatial dimensions) because they appeal to his imagination. Or even that they are "required" by his imagination.

the type of person we could call pragmatic or perhaps "Occamite" will avoid adding features which lack an operational meaning---i.e. some way to experience, even if very tenuous or indirect.

I would say try to think of the EXPERIENCE of being 2D and living in a 2D sphere. Don't picture the sphere as if you are a God, outside and looking from outside at the sphere. Using some new type of lightrays that travel in 3D rather than 2D. Think of a sphere as the experience of living in it. And also think of a hypersphere that way.

Let's say that you and your brothers discover a remarkable fact about the world namely that there is a special area K which you have determined experimentally which allows you to reliably predict the area of any triangle!
You just have to sum the angles, subtract π, and multiply by that area K!
this always turns out to give the area (if you take the trouble to measure the area carefully.

The rule used by Euclid, namely 1/2 the base times the height does not work for you, it is only approximately right for small area triangles and gets progressively wronger for larger ones.

That's part of what I mean by the experience. It would apply also to living in a hypersphere. It does not involve postulating an extra dimension which we don't experience and cannot access. It just involves experimenting with triangles and determining the value of the area K.

Circumnavigating is another aspect of the experience which you (as creature living in sphere or hypersphere) might have. You can think of various others.

 

[PeterJ]

No it's okay. I'm happy with my way of thinking about it. I'd say that the topography of the universe is unrepresentable as a visual model, and so within quite wide limits it would be a matter of personal preference how we do it. A Klein bottle or Necker cube would also be relevant images.

I'm not sure what you mean about 'mystics' and the stuff that appeals to their imagination. It has nothing to do with imagination. If Plotinus is to believed he is trying to describe what he is seeing, and he was not seeing any edges, nor any inside or outside. Perhaps he is not to be believed, but his model does at least allow for the idea that distance is arbitrary, which seems to make it useful, and it's only one more dimension, making it more economical than string theory. He even adds the proviso 'it is as if'.

I'm not suggesting that his description is 'true', just something to consider.

 

[marcus]

New narrowed-down values of the cosmological parameters, coming out of the SPT (south pole telescope)
http://arxiv.org/pdf/1210.7231v1.pdf
Scroll to Table 3 on page 12 and look at the rightmost column which combines the most data:

Ω[SUB]Λ[/SUB]     0.7152 ± 0.0098
H[SUB]0[/SUB]     69.62 ± 0.79
σ[SUB]8 [/SUB]    0.823 ± 0.015
z[SUB]EQ[/SUB]    3301 ± 47

Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ω_k

For that, see equation (21) on page 14
Ω_k =−0.0059±0.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ω_total = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.

So the idea which is traditionally favored of perfect flatness and spatial infinite is hanging on by its 2 sigma fingernails. It is still "consistent" with the data at a 2 sigma level.

But the hypersphere ( abbreviated S³ for the 3D analog of the 2D surface of a ball ) is looming realer and realer as a kind of ignored elephant in the room. It could still go away of course. We avert our eyes and hope it will have the politeness to do so.

For Jorrie's A25 calculator the important parameters as estimated by the SPT report are
current Hubble time = 14.0 billion years
future Hubble time = 16.6 billion years
matter radiation balance S_eq = 3300

or with more precision put these into Google calculator:
1/(69.62 km/s per Mpc)
1/(69.62 km/s per Mpc)/.7152^.5

 

[hagendaz]

Can I go light speed riding the expansion of space?

On the subject of space expansion...
If I built a spaceship with a machine (an engine if you will) on the tail end of the ship that eliminated all effects of gravity at the tail end of the ship , so as to counter act any gravitational pull on the tail end of the ship, would my spaceship thus be capable of traveling at light speed as I ride the expansion of space in what ever direction my ship is pointed?
Or can I only travel away from the center of the universe?

Would I instantly be moving at the same speed as the space is expanding the moment I turn on my spaceship, but feel no acceleration?

 

[marcus]

On the subject of space expansion...
If I built a spaceship with a machine (an engine if you will) on the tail end of the ship that eliminated all effects of gravity at the tail end of the ship , so as to counter act any gravitational pull on the tail end of the ship, would my spaceship thus be capable of traveling at light speed as I ride the expansion of space in what ever direction my ship is pointed?
Or can I only travel away from the center of the universe?

Would I instantly be moving at the same speed as the space is expanding the moment I turn on my spaceship, but feel no acceleration?

The expansion of distances doesn't GO anywhere. No person or object approaches any destination. Simply put: Things that aren't held together by their own gravity or molecular forces just get farther apart.

There is no "center" that anyone can point to.

So you cannot "ride" the expansion of space in any direction. Since there is no center you cannot " travel away from the center" as you say, either.

Typical very largescale distances grow several times faster than the speed of light but nothing travels anywhere.

The balloon analogy is intended to illustrate those things to make them easy to visualize. You might try studying the brief animation movie of it, reading some of this thread, or the FAQs.

You could start your own thread with this question, since it does not fit in so well in this balloon analogy thread.

 

[Larkus]

I would say try to think of the EXPERIENCE of being 2D and living in a 2D sphere. Don't picture the sphere as if you are a God, outside and looking from outside at the sphere. Using some new type of lightrays that travel in 3D rather than 2D. Think of a sphere as the experience of living in it. And also think of a hypersphere that way.

Let's say that you and your brothers discover a remarkable fact about the world namely that there is a special area K which you have determined experimentally which allows you to reliably predict the area of any triangle!
You just have to sum the angles, subtract π, and multiply by that area K!
this always turns out to give the area (if you take the trouble to measure the area carefully.

The rule used by Euclid, namely 1/2 the base times the height does not work for you, it is only approximately right for small area triangles and gets progressively wronger for larger ones.

That's part of what I mean by the experience. It would apply also to living in a hypersphere. It does not involve postulating an extra dimension which we don't experience and cannot access. It just involves experimenting with triangles and determining the value of the area K.

Circumnavigating is another aspect of the experience which you (as creature living in sphere or hypersphere) might have. You can think of various others.

This comic does a good job visualising it in an entertaining way:

'The Adventures of Archibald Higgins: Here's Looking at Euclid'

 

[marcus]

This comic does a good job visualising it in an entertaining way:

'The Adventures of Archibald Higgins: Here's Looking at Euclid'

That's a really nice piece of work! Thanks for the link, Larkus.

I hope you keep us informed about more clever cosmology stuff from Petit and his "learning without boundaries" project.

Today in the "How to prove the stretching of space" thread, I noticed a neat explanation by Brian Powell of how the wavelengths of light get stretched out as distances expand.

From general relativity (specifically, the geodesic equation), it is seen that the momentum of a particle is inversely proportional to the expansion (the scale factor, a(t)). From de Broglie, this becomes a statement about the wavelength of photons -- as space expands, the wavelength of light must increase.

Timmdeeg's reaction says it:
"...your explanation why λ goes with a(t) is very convincing and new to me, thanks."
I think this is an especially nice way to look at it, which doesn't exclude others as well.

 

[marcus]

The South Pole Telescope (SPT) has given us new narrowed-down ranges for the cosmological parameters.

At the highest confidence level these correspond to a cosmos which is NOT "Euclidean flat" and NOT spatially infinite but is the 3D hypersphere analog of the 2D spherical balloon surface model.

The SPT curvature estimates translate into an estimated range of the "radius of curvature" namely from 140 to 320 billion light years.

This may not be right, the U may not be spatially finite, or it might be finite and these numbers might subsequently be revised. But let's take them at face value and see. After all it is a fine instrument, a respected team, and these are the most recent published estimates. Here's what I posted earlier about it:
==quote post #448==
http://arxiv.org/pdf/1210.7231v1.pdf
Scroll to Table 3 on page 12 and look at the rightmost column which combines the most data:

Ω[SUB]Λ[/SUB]     0.7152 ± 0.0098
H[SUB]0[/SUB]     69.62 ± 0.79
σ[SUB]8 [/SUB]    0.823 ± 0.015
z[SUB]EQ[/SUB]    3301 ± 47

Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ω_k

For that, see equation (21) on page 14
Ω_k =−0.0059±0.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ω_total = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.

For Jorrie's A27 calculator the important parameters as estimated by the SPT report are
current Hubble time = 14.0 billion years
future Hubble time = 16.6 billion years
matter radiation balance S_eq = 3300
==endquote==

Since I posted that, Jorrie upgraded calculator from A25 to A27, so I made that change in the quote.

2 pi ≈ 6
so you can, if you wish, estimate the CIRCUMFERENCE of the universe simply by multiplying the "radius of curvature" figures by 6. The smallest it could be is 140 x 6 billion lightyears
and the largest it could be is 320 x 6 billion lightyears.
So if you could stop the expansion process, to make circumnavigation possible, you would have to travel in a straight line for six times 140-320 Gly before you'd be back at starting point.
If you sent a laser flash off in some direction it would be six times 140-320 billion years before it came back at you from the opposite direction.

This is just a way of understanding equation (21) on page 14 of the SPT report.

Ω_k =−0.0059±0.0040.

It's a way to get an intuitive feel in your imagination for what it means.
Here, again, is the link to the technical paper itself:
http://arxiv.org/abs/1210.7231

 

[RayYates]

In non technical language.. OMG the scale is mind-blowing!

 

[TalonD]

That's a very disappointing development. Infinite would have been much more aesthetically pleasing to me. Oh well.