Thanks for the answer, I did not expect my universe to crunch itself so soon. Regarding the Hubble constant, 72.5 was a random value I found on wiki and 68 serves the thought experiment just as well. Here are some complementary questions I have.

I like your big bounce scenario too, would it be possible without negative radiation density? Tweaking the w parameter might be the most plausible way, since it is the mysterious dark energy, which in my opinion may yet surprise us with unusual properties.  

I am also thinking if the big bounce scenario could be survived by life on planets, or at least objects in the universe, like galaxies, intact more or less, since on your graph, the scale factor does not drop all the way to zero. I also noticed that your bouncing universe does not expand from exactly zero either. If we were in a bouncing universe, how would we see the far away objects red/blue-shifted? What would the graphs for densities and light cones look like? It all probably depends on how you tweak the parameters. 

The galaxy ... has always had a recession velocity faster than light. Yet we are receiving photons from this galaxy today!  I think this is one of the most counter-intuitive features of the expanding universe.

I can stomach galaxies moving faster than light, but just how is it possible for us to observe something that has always been moving at a superluminal velocity away from us? How can it even be in our past light cone?

I am definitely following the "Answers bring up more questions" rule here.

An'shur wrote:

> I can stomach galaxies moving faster than light, but just how is it possible for

>  us to observe something that has always been moving at a superluminal velocity away

>  from us? How can it even be in our past light cone?

Well, that's where you need to understand relativity and the universe expansion well. Galaxies are not "moving", they are being pulld away by space expansion of the universe. In fact saying that the galaxies are recessing faster than light is a shortcut. What's happening is that all 3 dimensions of space are expanding between us and the galaxies, and it is thus a rate of expansion, a volume gain per unit of time and space. The hubble constant values you mentionned are close to the estimated value around 70-71 and the unit of this constant in km per Megaparsec and per second. Which means if you consider a cube of 1 mega parsec (around 3.26 light-years) around us (this cube can easily inlcude Andromeda Galaxy), its sides are expanding by 70-71 km every second.
Projected in one dimension and multiplying by how many seconds there have been since the light of the galaxy has left and by the number of megaparsecs between us and the galaxy, this where you eventually get figures higher than 300.000km/s. Again, the galaxy is not moving at that speed, Proper motions of galaxies are aound hundreds of km per second, but space is expanding between us, so no violation to relativity. What happens is that because of that strecth of space light gets redshifted, which we interpret as a speed of  recession. This is why the effect of space expansion are only to be taken into account at huge distance, more than 1Gly, and over a certain limit the present light of these galaxies will never be seen but we still see their past light.
A good page to read about distance-time concepts is the following that I use a lot to explain this to my students.
[url=http://www.atlasoftheuniverse.com/redshift.html]Atlas of the Universe[/url]
There you can see that the distance by Light travel is only one (and not the best) way to think distance, and there are two better ways, like the comoving distance (how far is the galaxy now) and the angular diameter distance (how far was the galaxy when the light we see left). 
[img]http://www.atlasoftheuniverse.com/redshift.gif[/img]
Beyond around 10Gly in light travels, galaxies who were 5.8 Gly then, are now exiting our horizon and we wil never see there present light. 

I see we have more staff for our SE Q&A topic .

astroclu, good explanation!  Recession velocity indeed is not and should not be thought of as a velocity through space, which often causes a lot of confusion.  Expansion also causes quite a headache in terms of our calculations of things like distances, for which there are many definitions that account for its effects in different ways.

I think though that the heart of the question, "How can we observe galaxies that have always had recession velocities greater than c?", is still a bit mysterious, and I'd like to try to elaborate on it.  This is something I struggled with for a while as well when I first learned it, even after I was already quite comfortable with the idea that there is no upper limit to recession velocity because it is the space in between that is expanding.  I have found that even some professional cosmologists have had difficulty with it.

An'shur, I think the trick to understanding this is to realize that the galaxy and the photon it emits are exploring different regions of space as the universe expands.  The galaxy stays basically still (as astroclu noted they usually have velocities in the 100s to 1000s of km/s range), while the photon streams away at (measured locally) the speed of light.  Then the next thing to be aware of is that the distance to where the recession velocity equals the speed of light is also changing with time, and is generally increasing, because the Hubble constant is generally decreasing.  (This might be another source of confusion: constant expansion means a decreasing Hubble constant, and a constant Hubble constant means exponential expansion.  I'll explain this as well in a moment.)

So even if the galaxy is always in a region of space which is receding from us faster than light, a photon it emits may find itself in a region where the recession velocity is slower than light, and eventually reach us.  The "v<c" recession region reaches the photon but not the galaxy.  

We can see this perhaps more clearly with the earlier space-time diagram, which I'll blow up again here.  

[center][/center]

When the universe was 2 billion years old, a galaxy just over 5 billion light years away (proper distance) from us emits a photon.  We mark this event's location with the large black dot.  At that moment, the cosmic expansion is causing that galaxy to recede from us at nearly twice the speed of light.  The photon it emits is moving in our direction, but the expansion is dragging it farther away from us. Its effective velocity (more precisely: its change in proper distance from us over time) is away from us, at slightly less than 1c.

For about the next 2 billion years, that photon continues to be pulled farther away from us by the expansion.  But the Hubble constant is decreasing, so the distance to where the recession velocity equals c is increasing.  When the universe is about 4 billion years old, that "v=c" recession distance meets the photon.  The photon's proper velocity relative to us is now zero.  Someone at the photon's location will say it's heading in our direction at c, but we say its distance from us is not changing, because there is exactly the right amount of expansion of space between us and it to cancel the progress it makes.  

But it is not trapped like that for long.  As the v=c recession line sweeps past the photon, the photon's proper distance from us begins decreasing, and eventually it meets Earth at the present day.  Meanwhile, the galaxy it came from has only been receding farther away, "superluminally".  The v=c boundary never reached it, and never will.

That, in a nutshell, is the journey the photons took from galaxies that we see when the universe was less than about 2 billion years old, with a redshift of z greater than about 3.2.  The light we see from them was originally getting farther from us even as it moved in our direction, and light being emitted by those same galaxies today will never reach us because they are now outside of our cosmic event horizon.



Going back to the relationship between Hubble constant "H" and expansion: A constant value of the Hubble constant does not mean constant expansion.  It actually means exponential expansion.  If H=70km/s/Mpc, and stays that way, then a galaxy 1Mpc away recedes at 70km/s, but when it reaches 2Mpc away, the constant value of H must mean it is then receding at 140km/s.  So a constant H means the distance to objects accelerates, and the size of the universe grows exponentially.

On the other hand, if the expansion is constant (galaxies maintain the same recession velocity as they get farther away), then H must decrease.

In our real universe, H has been decreasing (Hubble "constant" is rather a poor name for it in many ways...), and will asymptotically approach about 56km/s/Mpc as dark energy dominates.

[center][/center]

I am definitely following the "Answers bring up more questions" rule here.  :D

I love it!   The most interesting questions are those whose answers raise even more questions.  I will love also to explore the properties of the aforementioned "bouncing" universe.  Consider it another entry on my unofficial "TODO list".

I have a question Wats : have any of your students suffered in their physics studies due to poor spatial-orientation skills? I mean yes, the science in of itself does challenge some of our assumptions of what is logical, especially if you dive deeper into the quantum level, but just based on your excellent write-up above, a non-geometrically inclined mindset would seem to limit a student's learning capacity.

Have any of your students suffered in their physics studies due to poor spatial-orientation skills? 

Yes!  I was (and still often am) one of them.  I can have a hard time thinking about and accurately representing 3D geometry on paper, for example if I need to determine relationships between lengths and angles.  It especially gets tough in different coordinate systems (e.g. spherical coordinates).  And in my experience this has been a common feeling among students.

I'd like to say that this gets easier with time and practice, which I think it does, but still for some it comes easier than others.  I think there are also a lot of different aspects of "doing physics" that can be difficult for different people, whether it be the concepts, the techniques for approaching and solving a problem, or just the mathematics itself.

At the end of the day, the tried and true method for getting better at it is to do more of it, even if it hurts.  I view it as more of a skill that can be learned than as an innate talent that a person either has or does not.


Now, I'd like to get back to Gargenox's question:

More questions arise with every answer, but out of all scenarios what is the soonest and latest the universe would end? And how could the Big Rip happen without the heat death of the universe happening first?

The most likely situation (by physics and by data) is that w=-1 exactly, meaning that dark energy acts like a cosmological constant and does not dilute as space expands.  If that is the case, then the Big Rip never happens.  Instead the universe suffers "heat death" (for which I think a better name really should be "entropic death"), meaning that thermodynamic equilibrium is reached and there is no more useful energy available.  It is the end of all "non-boring" processes.  

Heat death itself is a kind of asymptotic process, never really striking at a particular instant.  But for a rough figure, we can say it will happen after the supermassive black holes have all evaporated, and their Hawking radiation spread out uniformly, which takes something on the order of 10¹⁰⁰ years.  (I do not pull this number completely from thin air -- there is a nice paper by Egan and Lineweaver which calculates the entropy contents of the universe, and how it changes with time.)  


For a Big Rip to occur, the parameter w must be less than -1.  The further below -1 it is, the sooner the Big Rip happens.

According to the Planck 2015 results, w is constrained to be -1.006, plus or minus 0.045  (1-sigma or 68% confidence level).  This means we are 2-sigma (95%) sure it is within .09 of -1.006, and 3-sigma (99.7%) sure it is within 0.135 of it.

Let's suppose that the true value is a 2-sigma outlier of -1.1.  How soon will the Big Rip happen then?  Let's plug this value into the simulation:

[center][/center]

[left]With w=-1.1, the Big Rip happens about 114 billion years from now.  The scale factor begins blowing up super-exponentially, becoming infinite in finite time.  This is the sign of a Big Rip happening.  The universe grows so fast that any object that freely expands with the universe expands suddenly to infinite size.  In my opinion this sounds mildly uncomfortable.  [/left]

[left]Let's check the space-time diagram:[/left]

[center][/center]

[left]Less than 10 billion years from now, the accelerating expansion takes over and the distance to where the recession velocity exceeds the speed of light starts decreasing, eventually to zero.  In fact, the distance to where the recession velocity is any speed goes to zero.  Now this sounds even more uncomfortable...[/left]

[left]What if we make w even more negative?  How about -1.2?  In that case, the Big Rip happens in about 60 billion years.  [/left]

[left]If w=-1.3?  That's 6 sigma = 99.9999998% chance that it isn't that small, but if it is then it makes it happen in about 40 billion years.  And so on.  A more negative w makes the Big Rip happen sooner, but with the odds being exponentially less likely.[/left]

[left]Meanwhile, heat death doesn't occur for ~10¹⁰⁰ years, which is a stupidly long time.  So in the event that there is a Big Rip, it probably happens much sooner.  Even w=-1.05, which we can't exclude with better than 70% confidence, would lead to a Big Rip in ~250 billion years, which is much sooner than heat death.  Stars would still be shining.  Planets would probably still exist.  Perhaps even life, to witness the ultimate destruction of our space-time.  The distant galaxies would recede from their view first, and then the stars pulled away, then their solar systems disassembled, and finally every thing, down to the atoms, ripped apart and stretched to infinity.[/left]



[left]Am I worried about such a thing happening?  Not even remotely.   Dark energy is easily one of the most mysterious things in the universe, but I think there are very good reasons, which include ever more precise observational constraints, for thinking that the true value of w is -1 exactly, so that the Big Rip doesn't even happen at all.  But it sure is fun to think about.[/left]

How do we know that w is constant?

How do we know that w is constant?

The short answer is that we expect it to be constant, but we check the possibility that it might not be, and observations are consistent with the expectation that it is.  Here's the run-down:

If dark energy behaves as the cosmological constant introduced by Einstein, then w=-1 always, by virtue of being independent of the scale factor.  Again what w=-1 means is that the density remains constant despite expansion.  w=0 describes matter, and w=1/3 describes radiation.  And in cosmology, this description of "how much is the thing diluted by expansion" relates directly to how it affects the expansion rate.  w=-1/3 leads to no acceleration, while w<-1/3 accelerates and w>-1/3 decelerates.

But what if w does depend on the scale factor?  Then to first order we can write it as a function,

[center][/center]
[left]Essentially this means we put extra knobs onto the standard Lambda-CDM model, and then map where observations lie in that new parameter space.  In this case we say that to first order, w may change linearly with the scale factor a, depending on a new parameter w_a which describes how sensitive it is to the scale factor, with w₀ being the initial value.  Then if the cosmological constant best describes reality, we should find that w₀ = -1 and w_a = 0.  [/left]

Here are what the data show:

[center][/center]

[left]So the observations are consistent with dark energy being described by a cosmological constant, with w₀ = -1 and w_a = 0. (Or just w=-1).[/left]
[left]Of course, we can never have perfect precision -- there will always be uncertainty ellipses.  But these uncertainty ellipses continue to grow smaller with improved observations, and as they have shrunk we have not yet seen any evidence that a more complicated model is required to describe how dark energy behaves.  So for the time being this is what we use in Lambda-CDM cosmology.[/left]

[left]If future observations end up excluding w₀ = -1 and w_a = 0, then that would be very exciting, and a great deal more strange than dark energy already is. [/left]

Thanks.  When we learn what w is, dark energy, etc, the most exciting part is likely not what it is, but how we know it, why we didn't see it before, and the new questions we discovered.

Ah good, more questions are discovered!

I don't know if the Big Rip or the spontaneous decay of a false quantum vacuum would be more thrilling! I suppose we can rest assured that the mass of the Higgs Boson is indeed 125 gigaelectron volts and there are no other particles along the Planck Scale that would jeopardize our current position in a metastable vacuum.

Also I suppose the expansion of the universe is solely dependent on the dark energy that fills the void. Is this some sort of vacuum energy too? I think of it as simply an emergent property of entropy and the result of energy spreading out as the 2nd law of thermodynamics dictates.

The density of Dark Energy always remains the same for a specific volume but as the volume increase so does the amount of energy, keeping the density the same, yet matter and radiation fall behind as the amount of Dark Energy increases, determining our fate. So we are dealing with two parameters here, expansion rate and density. I hear rumors that the expansion rate has changed over time. I have no idea why, and that the density has always remained the same, again no idea why.

Thank you too, your explanations always hit right to the core of the topic and in easy to understand human language! I can easily see how expansion allows the Big Rip to occur before the heat runs out. Is our measurements of 'w' really a consequence of our age? Are we sure that the density of Dark Energy can not change? I could type question marks all day long but I'll hold off right here.

Ha! I see that was answered

I like your big bounce scenario too, would it be possible without negative radiation density? Tweaking the w parameter might be the most plausible way

Yes, let's take a closer look at it! To make the universe collapse, I'll give it much greater than the critical density of matter.  But to make it rebound before collapsing to zero size, I'll have to fill it with something that causes acceleration, and that thing will need to dominate the total mass-energy density when the universe is a very small size, and be negligible when the universe is larger.  This means it must dilute with expansion faster than matter.  Actually the most natural thing that dilutes in the required way is radiation.  But we want to keep radiation in the model and keep the radiation gravitationally attractive (positive mass density).

So we'll change the dark energy, and give it w=1/3 so that it dilutes like radiation and dominates at small scale factor.  But there's another problem if we do that: any substance with a positive energy density and with equation of state w>-1/3 will cause deceleration.  And what we want is acceleration.  (Dark energy does not accelerate the expansion because it is "repulsive".  It accelerates the expansion because it does not dilute!)  So to make the dark energy dilute like radiation and still cause acceleration for a rebound, I must change its density to negative.

In this way, what we're describing is exactly equivalent to repulsive photons, anyway.  But we'll just call it dark energy with a negative energy density and a different equation of state w=1/3.

Let's run the model.  Well set Ω_m=8, Ω_r=9.2x10^-5, Ω_Λ=-0.5, and w=1/3.  Maybe we'll call this "An'shur's Cyclic Universe".   Here's the scale factor over time:

[center][/center]

[left]This universe has no clear beginning or ending.  It just endlessly oscillates between near-crunches.  I have to have some sort of label for the initial simulation time though, and I set it to be t=0 at the moment of a bounce.  Then I choose the present time to be when the universe is collapsing, not the first time, but the second time, 25.6 billion years later.  The contraction rate at this moment is -100km/s/Mpc:[/left]

[center][/center]

How do the matter, radiation, and "dark energy" densities change with time?  

[center][/center]

[left]The dark energy density (green) is negative, so I plot the negative of it so that it shows up on a logarithmic scale.  The absolute value of the dark energy density is always less than the matter, except for the brief moments that the universe is rebounding, where its density spikes and causes the sudden acceleration to re-expansion.  [/left]

[left]Plotting the densities in terms of their ratios to the critical density makes quite a pretty graphic as well.  Here Ω_k is 1 minus the sum of all the others and can be thought of as a "curvature density".  The effect of this curvature density will become more clear in a moment.[/left]

[center][/center]

[left]And now, the thing which I think is the most interesting of all:  the space-time diagram:[/left]

[center][/center]


[left]What the heck is happening here?!  [/left]

[left]Well, here I am showing something which I did not show earlier when we looked at the closed, collapsing universe model.  That time I had treated the geometry of space as being flat (Euclidean), which is actually not true when the total density is greater than the critical density.  This time I include the effect of the spatial curvature.  Because this universe has such a high density, space curves back on itself, like the surface of a sphere, but in 3 dimensions.  Travel far enough in any direction in this universe, and you will return to your origin!  The distance to this boundary where space has curved back on itself and distances start diminishing again is shown with the solid white curve labelled "curvature horizon".  Its size is related to the value of the curvature density, Ω_k.[/left]
Because of the curvature, there is no space beyond that horizon.  This universe is "closed", and its volume is finite.

[left]Another odd feature of the curvature is that proper distances are greater than we would expect from a flat geometry.  The 2D analogy with the curvature of a sphere helps here too: the distance to the opposite side of the Earth is greater as measured over the surface than if you tunneled straight through the middle.  Near the antipodal point, distances over the surface grow rapidly, for just a small change in direct distance through the Earth.  This property of the curvature explains why the light paths curve the way they do, with sharp cusps near the curvature horizon.[/left]

[left]Light rays which are currently arriving at our location during this second contraction phase have had a crazy journey.  Because of the closed spatial geometry, the light rays have circled around the universe multiple times during each bounce.  We will see multiple images of the same galaxies!  Also, because the universe bounces at a finite size rather than going full crunch to singularity, we can trace the paths of those light rays through the bounce!  Previous bounces of the universe are visible!  This would be a very trippy universe to live inside of, like the inside of a spherical mirror, but where moving toward the edge leads you back to where you started.[/left]

[left]Would it be survivable?  Maybe!  But it sounds rather dangerous.  If it has stars and galaxies and planets and so forth, then the rates of galaxy mergers must become extraordinarily high during the bounces.  The energy density of radiation will also get more intense, with all the photons being brought together and blueshifted to higher energy.  But in principle, there's nothing making a bounce unsurvivable to an observer who is well equipped to pass through it.  I imagine it would be an amazing spectacle, if you lived long enough to watch the change.[/left]

[left]A close-up of the galaxies and light rays through the bounce:[/left]


[center][/center]
[left]Wild... I think this is one of the more fascinating "types" of universes that can come out of the equations (and the physics).[/left]
[left]Of course, the material requirements for it are quite unphysical.  We know of no substance that dilutes like photons but has negative energy, and we are so confident that dark energy does not have w as high as 1/3 that it's insane.  This is more of an academic exercise of "let's play with the parameters to make neat things happen."  A deeper problem is that it's very ambiguous as to how such a universe would get started.  Our universe has a clear evolution from the Big Bang, but what was the initial state of a cyclic universe?  [/left]

[left]Perhaps we must conclude it has existed like this forever.  [/left]
[left]Or maybe it is a simulation on a computer somewhere... [/left]

Watsisname, the time and effort you put into these explanations and presentations for us cro-magnons is exemplary! Truly we are fortunate to have such a patient and wise teacher such as you on this forum. You should honestly publish this stuff, it's so good and easy to understand (well, most of it  ).

Anyway, I have another question that might be a bit 'out there': Is it possible there are other fundamental forces at work in the universe in addition to visible matter, dark matter and dark energy? This is assuming of course the last two are comprised of roughly congruent and relatively unknown physical laws working in tandem (which seems to be general consensus so far) of which we only see their effects? I'm NOT talking about paranormal or supernatural woowoo, just the possibility of additional 'dark'.... stuff in the universe. Or are dark matter and dark energy the true frontiers of our current understanding of the universe?

Or maybe it is a simulation on a computer somewhere...

Whats this :shock:? An admission to metaphysics here?!?  .

 Is it possible there are other fundamental forces at work in the universe in addition to visible matter, dark matter and dark energy?

There are more of them than just gravity.  There's a Wikipedia article giving an overview of the fundamental "forces".  These four "forces" may be manifestations of a single underlying phenomenon.  We don't know.

 I have another question that might be a bit 'out there': Is it possible there are other fundamental forces at work in the universe in addition to visible matter, dark matter and dark energy?

Yes, that is certainly possible.   To be clear, as Midtskogen said "fundamental forces" in physics means gravity, electromagnetism, and nuclear forces, but I think what you're asking about are the things that can modify the expansion rate of the universe.

The three main things that we know of which affect the expansion are matter (both regular and dark, which in cosmology we lump together because their gravitational effects and the way they are diluted by expansion are the same), radiation (photons), and the mysterious dark energy.  But there could be more than that.  It's possible the universe contains some additional unknown sort of mass-energy, and it could affect the expansion rate.

If there is any additional mass-energy, then whatever it is, there can only be so much of it at the present time.  Actually, there must be quite little.  We know this because the total amount of mass-energy in the universe determines the spatial curvature -- how much space bends back on itself or not.  We know the spatial geometry is very close to flat, which means the total density must be quite close to the critical density.  And we also know that the best fit to the cosmological parameters gives us a sum of matter, radiation, and dark energy density very close to the critical density.  So there is very little room for there to be anything else (unless there is a fortuitous combination of things with both positive and negative energy density, such that the total is still close to the critical density, but that would be both very odd and quite a coincidence indeed.)

A more plausible way for there to be more substance in the universe without us noticing it yet is for it to have a very different equation of state, such that its effects are small now but become stronger in the future as the universe gets larger.  Dark energy is a good example of that already.  If there were astronomers around when our universe was 5 billion years old, it would have been extremely difficult for them to detect the effects of dark energy, because back then its energy density was much less than the matter density and so it did not yet have much effect on the expansion of the universe.  We only detect dark energy now because its density stays the same while the matter and radiation get diluted, so as the universe gets bigger the dark energy becomes more important.  

Maybe there is something in the universe with a very low density now, but with an equation of state w<-1 such that its energy density gets larger over time.  Then its effects would be small now and in the past, so we could not detect it, but it would eventually become stronger even than dark energy, so astronomers in the distant future would notice it.  (Also as we saw with the discussion of the "Big Rip", anything with positive energy density and w<-1 eventually leads to a Big Rip, so those future astronomers would notice it by noticing a trend toward faster-than-exponential expansion rate.  They would probably be very worried about that.)



So, long story short, for the moment we do not see any evidence that there are additional, cosmologically important types of mass-energy in the universe.  But we are constantly looking, by obtaining ever more precise measurements of the expansion history of the universe, and of the cosmological parameters.

Thank you all by the way for your kind feedback!  Cosmology is a challenging, counter-intuitive subject, and it is difficult to write about it in an approachable way.  So it's nice to see the interest in it.  I am also thinking of splitting the discussion out at some point to a dedicated thread, similar to FFT's thread for understanding solar system dynamics.

Also I suppose the expansion of the universe is solely dependent on the dark energy that fills the void.

Not quite.  The expansion itself started in the Big Bang (for reasons we don't yet know).  A good way to think about it is by the analogy to throwing a rock off of a planet.  The throw is like the Big Bang, and the trajectory of the rock (altitude over time) is like the evolution of the universe (its scale factor over time).  The speed of the rock changes because of the gravity of the planet pulling back on it, and the expansion rate of the universe changes depending on what the universe contains.  An empty universe will have a constant expansion rate, just like a rock thrown with no gravity will have a constant velocity.

Right now dark energy is the biggest factor for how the expansion rate changes, but matter is a close second (and was more important up to about 4 billion years ago), and radiation affects it too but is quite unimportant now (much more important in the very early universe).

The manner in which the contents of the universe change the expansion rate (speed it up or slow it down) depends on what I've sometimes called "the equation of state", which is that parameter "w", representing how those contents are diluted by the expansion.  I think this is a very interesting thing, and your guess,

I think of it as simply an emergent property of entropy and the result of energy spreading out as the 2nd law of thermodynamics dictates.

is close to the mark!  The way the expansion rate changes can be worked out by statistical mechanics, through the relationship between the work done on expanding the substance, and the change in the pressure and density of that substance.  This is deeply connected to the 2nd law, and the thermodynamic identity.  


I plan on going over the details of this more rigorously in the near future, because it's an important step in deriving the Friedmann equations, and I think it is a very beautiful piece of physics.