To foster an engaging cosmology forum, participants should balance mental freedom with a foundational understanding of the Lambda-cold-dark-matter (LCDM) model. The discussion emphasizes the importance of the balloon analogy, which helps visualize the expansion of the universe and the relationship between galaxies, redshift, and distance. It aims to clarify misconceptions surrounding the analogy, particularly regarding the nature of space and the movement of galaxies relative to the cosmic microwave background (CMB). Participants are encouraged to explore intuitive concepts without heavy reliance on mathematical jargon or abbreviations. Ultimately, a shared understanding of these foundational ideas will enhance discussions and learning in the forum.
...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.
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?
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?
Jorrie said:
...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?
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.
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 ρtot 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 (ρm) and dark energy (ρde), 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 ρde = 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 ρm → 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?"
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/
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.
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.
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.
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.
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.
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[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.
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"?
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.
I think one must however be careful with wording like this:
marcus said:
... 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".
...
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?
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.
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.
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?
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-content/uploads/2010/05/pear-anjou.jpg
However in any case one should never say lightcone, so I would approve switching to either vegetable of terminology.
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.
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,
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,
I like Ruth Durrer's review. I have ordered a copy, which I should receive on Monday.
Looks to be an important book! Some additional information on this page: https://www.amazon.com/dp/0521381150/?tag=pfamazon01-20
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!
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.
marcus said:
... 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.
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,
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.
Since we've turned a page, I'll 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[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.
Remember that "proper" distance, the distance used in Hubble law to describe expansion, is "freezeframe". The proper distance at a given moment in Universe time is what you would measure (by radar or string or whatever usual method) if at that moment you could stop the expansion process long enough to make the measurement.
The Hubble law describes the expansion of distances between observers at rest with respect to the background of ancient light and the process itself. Observers who see the ancient light and the expansion process approximately the same in all directions, e.g. no Doppler hotspots.
The field of an observer's view is not conical, but rather it is pear-shape because distances were shorter back then. Here is a picture of an Anjou pear.
http://carrotsareorange.com/wp-content/uploads/2010/05/pear-anjou.jpg
Here is Lineweaver's spacetime diagram: http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure1.jpg
The upperstory figure, with horizontal scale in proper distance, shows the lightpear outline.
Here is Lineweavers plot of the growth of the scalefactor R(t), which models the growth of all distances between observers at universe-rest (at rest with respect to background.) http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid line is according to standard model parameters. Various other cases are shown as well.
... 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.
The "Hubble time" 1/H is an interesting variable. Normally we write the core cosmology equations in terms of H, but "time needed for a 1% growth of distance" is intuitively appealing. Let's try, as an experiment, writing the Friedmann eqn (flat case, matter era) in terms of Y = 1/H, instead of in terms of H.
I will show that the Friedmann equation translates into a very simple differential equation for Y:
Y' = (3/2)[1- (Y/Y∞)2]
A nice feature is that Y is a time so its derivative (ΔY/Δt) is a pure number.
We can easily see that the present value of this number is 0.4
In other words it doesn't matter what unit of time we use. The units cancel. So for example let me use the cosmologically convenient time unit d = 108 year.
Y = 139 d
Y∞ = 163 d
Y' will be the increase in Y per unit time.
Y' = (3/2)[1- (139/163)2] = 0.41... ≈ 0.4
So over the next 100 million years we can expect an increase from 139.0 to 139.4.
In other words, if a one percent increase in distance NOW takes 139 million years, looking ahead it will then take 139.4 million years.
Or if you think of the Hubble growth rate as now being 1/139 percent per million years, at that time in the future it will have decreased to 1/139.4 percent per million years.
Of course in the very long run we know that H will settle down to 1/163 percent per million years, but it's nice having a simple differential equation for Y so one can how it is changing at present, on what timescale.
For people who would like to see the (elementary calculus) way the equation for Y' is derived:
Y' = (1/H)' = - H'/H2 = 4πGρ/H2 = 4πGρY2
All this uses is H' = - 4πGρ, which we know from a previous post. And then we use the Friedmann equation to get an expression for ρ, and substitute it into the above.
ρ = (3/8πG)[1/Y2 - 1/Y∞2]
Y' = (3/2)[1/Y2 - 1/Y∞2]Y2
= (3/2)[1- Y2/Y∞2]
The square of the ratio 139/163 is a familiar model parameter that is often quoted, namely 0.728.
Here we give it a new significance as determining the current rate of increase of the Hubble time.
One minus 0.728, namely 0.272, multiplied by 3/2, is this number 0.41... we're talking about.
The current value of the Hubble time is increasing 0.4 year per year. Or 0.4 century per century. Or 0.4 Gy per Gy. That is (if the rate were steady it would result in) an increase from 13.9 billion years to 14.3 billion years in a billion year interval. find it more convenient to think of time in units of d. So I say the Hubble time is increasing 0.4 d per d---or from 139.0 to 139.4 d in 100 million years.
Since the equations here are based on introductory work in post#313, which is several pages back, I will bring forward part of that post:
=====quote post#313======
By definition H = a'/a, the fractional rate of increase of the scalefactor.
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
Next comes the Friedmann equation conditioned on spatial flatness.
H2 - H∞2 = (8πG/3)ρ
Differentiating, the constant term drops out.
2HH' = (8πG/3)ρ'
Then we 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
====endquote====
That's all we needed from the earlier post, but I'll quote the rest of the passage to have it handy. What follows is extra: This is a derivation of the so-called "Raychaudhuri" or "Second Friedmann" equation. Also called the "acceleration Friedmann equation" because it gives a handle on the second derivative of the scalefactor a(t).
===continuation===
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 = a"/a - H2
Now the Friedman equation tells us we can replace H2 by H∞2 + (8πG/3)ρ. So we have
H' = a"/a - H2 = a"/a - H∞2 - (8πG/3)ρ = - 4πGρ
We group geometry on the left and matter on the right, as usual, and get:
a"/a - H∞2 = (8πG/3)ρ - 4πGρ = - (4πG/3)ρ
using the arithmetic that 8/3 - 4 = - 4/3
This is the socalled "second Friedmann equation" in the matter-dominated case where radiation pressure is neglected.
a"/a - H∞2 = - (4πG/3)ρ
In the early universe where light contributes 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.
===endquote===
I did not say that H(t) squared was proportional to density. It obviously is not because of the constant.
) 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 ρm → 0). I'm on to you marcus! 