The value of w is interesting, because in one of the cyclic models you actually need a value of w close to -1 for the universe to be cyclic. 

No, they need w for dark energy to be close to -1 near the present time, in order for it to be consistent with observations.  

Having w be close to -1 does not make it cyclic.  To make it cyclic, they assume that w is slightly less than -1, so that it eventually causes a Big Rip.  But a Big Rip is still not a cyclic universe.  So they then further assume that, somehow, the Big Rip is reversed at the last moment.  And that Big Rip must be reversed at the last moment and no sooner or else it would lead to predictions that would already have been falsified.

So basically that's adding in a whole bunch of assumptions about how dark energy behaves, in order to force a cyclic solution from it.  Nothing wrong with it per se, as an exercise of showing what's possible with math and a long list of assumptions, or for investigating interesting alternative cosmologies.  But for us to believe it we need some convincing predictive successes from that model.  Right now, there is no reason for us to believe that dark energy or the universe behaves that way.

As I showed here, observations are consistent and increasingly well constrained for dark energy having a steady equation of state with w=-1, and not being a function of time or scale factor.  In other words, the simplest working model for dark energy is that it behaves as the cosmological constant.  Which is also how we expect it to behave, because there is a simple physical interpretation for w=-1, which you do not get for any other value (besides w=0 exactly for matter, and w=1/3 exactly for radiation).

It has been argued that the ΛCDM model is built upon a foundation of conventionalist stratagems, rendering it unfalsifiable in the sense defined by Karl Popper.

This is not all that uncommon of an opinion.  Many people think it is all based on ad hoc tweaks to models to make the numbers come out right and avoid being falsified.  When dark matter and dark energy were first proposed (with dark energy having an even longer and interesting history going back to Einstein and the cosmological constant), that view was even justified.  But it is very incorrect now, especially through the last few decades of observations and higher precision cosmology.  The key is that Lambda-CDM does make predictions, beyond what it was originally formulated to explain, and those predictions act as tests which can potentially falsify it.  It has even already passed many of those tests.  So, I'd like to dedicate this post to explain how Lambda-CDM and its components are testable, and how cosmologists (who by their nature are a rather skeptical bunch), became convinced to use it as the current standard model to describe the universe.


[center]The Testability or Falsifiability of the Lambda-CDM Model[/center]

Let's start with cold dark matter: the "CDM" in Lambda-CDM.  This started out as a proposal of additional, unseen matter to explain observed galaxy rotation curves.  How can we be sure it's not just a misunderstanding of how gravity works on large scales, or something else?  How is dark matter potentially falsifiable?  For that I can refer to an earlier post where I describe the observations of the Bullet Cluster, along with some additional insights on the following page.  The whole discussion is actually worth reading if you have the time.  

The short version is that if you propose the presence of a lot of additional, weakly interacting, invisible matter to explain rotation curves, then this naturally predicts that we should also see the excess gravitation go straight through a collision between galaxies or galaxy clusters, while the majority of the known matter, which is in the form of gas, gets left behind in the pile up.  In other words we should see situations where dark matter can be separated from regular matter.  In fact, that's exactly what we have observed!  


An example of how the dark energy component (the Lambda in ΛCDM) is testable is that it affects the "CMB angular power spectrum".  Angular power spectrum a fancy term for the distribution of sizes of the fluctuations seen in the cosmic microwave background radiation.  Dark energy affects this because any form of mass-energy contributes to the universe's spatial curvature, and spatial curvature magnifies or demagnifies distant objects, including the CMB fluctuations.  Increasing the amount of dark energy has the effect of shifting the positions of the peaks of the angular power spectrum.

The CMB angular power spectrum acts as another test of dark matter, too.  The amount of dark matter shifts the heights of the acoustic peaks.

As it turns out, the required amount of dark energy needed to fit the expansion history, plus the required amount of dark matter needed to explain galaxy rotation curves, fits that spectrum.  This is actually rather remarkable.  There's no real reason to expect these things to also be consistent with something so different and so much farther away (or younger in age) than what they were originally made to explain, if they did not have a physical reality to them.  That it does correctly fit the CMB is a fantastic predictive success.  This is one of the key measurements that persuaded cosmologists to use Lambda-CDM.

Are there any other reasons?  Yes.  The same amount of dark matter and dark energy also does a good job predicting the formation of structure, as seen in the Millenium Simulation.  It and other simulations are an excellent testing ground for comparing the predictions of Lambda-CDM to observations.

There are still other ways in which Lambda-CDM and its components are testable, from more to less technical.  To drive the point home, here's one more example, and a bit less technical.  Let's look at the age-redshift relation.  That is, how redshifted will the light be from sources we see at different distances (by light travel time) in the universe?  Here I plot it for the Lambda-CDM model, and for a model where we do not include any additional matter or dark energy.  As a reference point, I put blue intersections for the Cosmic Eyelash, which has a redshift of z=2.3.

[center][/center]

In the Lambda-CDM model, its age is about ~2.8 billion years for that redshift.  Removing dark matter and dark energy increases it to ~3.6 billion years.  This is a pretty large shift!  So you can see the dark energy and dark matter has a profound implication on the history of star formation vs. redshift.  That is something that can be tested with the next generation telescopes like the James Webb.

You might have also noticed in the legend that the current age of the universe (t_now) changes.  Without dark matter or dark energy, the universe would be younger by about 300 million years, for the same observed value of Hubble constant today, because the expansion rate would have dropped to that value more quickly without the dark energy acting on it.  (Not having dark matter would also not slow it down as fast, but a lack of dark energy turns out to be more important).


I hope by now I've convinced you, or other readers, that Lambda-CDM is in fact very testable and potentially falsifiable.  It is as much as any good scientific model should be.

The value of w is interesting, because in one of the cyclic models you actually need a value of w close to -1 for the universe to be cyclic. 

No, they need w for dark energy to be close to -1 near the present time, in order for it to be consistent with observations.  

Having w be close to -1 does not make it cyclic.  To make it cyclic, they assume that w is slightly less than -1, so that it eventually causes a Big Rip.  But a Big Rip is not a cyclic universe.  So they then further assume that, somehow, the Big Rip is reversed at the last moment.  And that Big Rip must be reversed at the last moment and no sooner or else it would lead to predictions that would already have been falsified.

So basically that's adding in a whole bunch of assumptions about how dark energy behaves, in order to force a cyclic solution from it.  Nothing wrong with it per se, as an exercise of showing what's possible with math and a long list of assumptions, or for investigating interesting alternative cosmologies.  But for us to believe it we need some convincing predictive successes from that model.  Right now, there is no reason for us to believe that dark energy or the universe behaves that way.

As I showed here, observations are consistent and increasingly well constrained for dark energy having a steady equation of state with w=-1, and not being a function of time or scale factor.  In other words, the simplest working model for dark energy is that it behaves as the cosmological constant.  Which is also how we expect it to behave, because there is a simple physical interpretation for w=-1, which you do not get for any other value (besides w=0 exactly for matter, and w=1/3 exactly for radiation).

I know, I just like looking at different theoretical models and trying to find some kind of idea how likely any of them are.   What do you think about Penrose's Conformal Cyclic Cosmology for which he says he has evidence from Planck satellite data?   I think I really like the exercise in math lol and how mentally stimulating all of these many theories are.  I find it much more exciting than a political argument like I see on other socalled science forums where some come in to debate the idea that there is a mass extinction going on right now or anthropogenic climate change (ugh, it's been obvious for decades it's happening.).  Honestly, I dont believe we have the right answer and likely wont have it in our lifetime about these theories.  But at least we can strive to get closer to the truth.  Looking at all the unanswered questions and unsolved problems, not the least of which is the value of the cosmological constant, as we continue to explore the universe further, I think we're in for lots of surprises.
I wonder about other universes with other physical laws and if we'll be able to detect signatures of them in the cosmic background too, and if there is ever any interaction between universes even subtle.  I'm sure you're familiar with black hole cosmology and Poplawski's idea which would unify relativity and quantum mechanics.  It's not a new one, but if a solution like that is close to reality, then I can imagine both space and time being cyclic across the entire omniverse.  In the sense that if you go enough in any direction, you'll end up right back where you started.

Dark matter and dark energy definitely exist, we just need to find them.  I had thought that the LHC would have detected them, but perhaps supersymmetry isn't as correct as we thought it to be.  What do you think about the prospects of sterile neutrinos and other particles making up the majority of dark matter?  Or is it possible that dark matter could for the most part be a large abundance of things like brown dwarf stars and other objects we haven't been able to detect yet?

By the way, is it possible that we can combine all of these models and there are parts of all them that are correct?  That is, the Lamba-CDM model is correct and the universe is also cyclic?  And perhaps the entire omniverse?

Also, if I could, for the next mode of discussion I would like to broaden our discussion to include this

https://en.wikipedia.org/wiki/Multiverse

I find Tegmark's classification vs Greene's very interesting.  In some ways I'm partial to Greene because I took a class of his and I like that he stopped eating meat and dairy when he saw how animals get treated on farms, but I also really like Tegmark's system.  Also some of the possible evidence we might have or could possibly obtain in the future about this.

This video really helps get your head around light in an expanding universe.

[youtube]OM9KepKsg6U[/youtube]

Frankly, I can't wrap my head around the idea that an ant _wouldn't_ reach the end of an expanding rope.  This "paradox" was silly.

Well it is if you just look at the 2 figures. The expanding faster and increasing distance compared to a slower steady rate of the ant at first glance makes it seem like the ant would fall further and further behind. That is exactly why it's paradoxical.

I can't free my mind from imagining myself as the ant on the rope.  From that position I can always see progress as I move from one strand to the next.  Progress will be slower and slower as the distance between the strands and fibers increases, but there is no way progress will completely halt.  If it could be proven somehow that it would be impossible to reach the end, that would indeed be a paradox.

Perhaps if we invoke light-speed and flip the problem we can make an apparent paradox out of this:

Knowing that travel faster than light is impossible, but suppose that I observe that the end of the rope appears to be moving faster than light which is ok since the light-speed limit doesn't apply to expansion only to travel along the rope.  Yet, I always make progress and no new rope is being added, so I must reach the end of the rope eventually, but how can that be if the end is already moving faster than light away from me?

Knowing that travel faster than light is impossible, but suppose that I observe that the end of the rope appears to be moving faster than light which is ok since the light-speed limit doesn't apply to expansion only to travel along the rope.  Yet, I always make progress and no new rope is being added, so I must reach the end of the rope eventually, but how can that be if the end is already moving faster than light away from me?

Yes, I think that's the better way to apply the "ant on a rope" analogy to an apparent paradox in cosmology.  If the expansion of the universe happened at a constant speed (i.e. if galaxies moving away from us maintained the same recession velocity as they move away, which is what Kevin refers to as constant expansion of the rope), then all photons will -- eventually -- reach us from arbitrarily far away.  Even if the galaxies they came from are being drawn away faster than the speed of light by the expansion.

The analogy is also good for visualizing how the light we receive now from distant galaxies was actually being drawn farther away from us by expansion at early times.  The ant starts out some distance away and crawls towards us, but the stretching rope makes the distance from the ant to us increase at first.  I don't think it's immediately obvious that this doesn't stay that way, that the stretching does not continue to pull the ant ever farther away.  But if the stretching rate is constant, then the ant eventually does reach us.  There is a "turnaround point", where the ant reaches a position on the rope where the speed that part of the rope is being pulled away is equal to the speed the ant can crawl, and after that it finally makes progress in our direction, not just along the rope but also in terms of distance away from us.

If we trace the path of light that is now reaching us from the early universe, we can see this same effect.  For the first ~4 billion years the expansion drew that light farther away from us, but then the distance where the expansion velocity drops below the speed of light caught up to the light, and then for the next ~10 billion years the light was able to make progress in our direction and reach us today.  It was really moving in our direction all along, but it took some time for it to reach a point where it could move more distance in our direction than the expansion could pull it away.

If it could be proven somehow that it would be impossible to reach the end, that would indeed be a paradox.

Later in the video Kevin makes an important point that the expansion of the universe is actually not constant.  It was slowing down early on due to the gravitational attraction of the matter and radiation, and now it is speeding up due to dark energy.  In the ant-rope analogy we must then make the stretching be a function of time, and that can greatly affect the result.  With accelerating expansion, the ant might not ever be able to reach the end.  Or light that started out too far away in the universe will never reach us.  The argument that "there isn't any more rope being added" no longer works.  The problem is not that there is more rope being added, but that there is more space being added (for example, the distance between the atoms in the rope is increasing), and the rate at which that space is being added (or how that rate itself changes), can make a difference for whether the ant, or light, can reach the destination.

In cosmology we say that the distance from which any signal (light beam or whatever) can ever reach us is the cosmological "event horizon".  If the expansion rate was constant, then the size of this horizon would be infinite (everything in the universe would eventually be able to see everything else), but with accelerating expansion it can only be a finite size.  Right now it lies about 16 billion light years away, meaning light being emitted now from objects that are currently more than that distance away will never reach us.  

Knowing that travel faster than light is impossible, but suppose that I observe that the end of the rope appears to be moving faster than light which is ok since the light-speed limit doesn't apply to expansion only to travel along the rope.  Yet, I always make progress and no new rope is being added, so I must reach the end of the rope eventually, but how can that be if the end is already moving faster than light away from me?

Yes, I think that's the better way to apply the "ant on a rope" analogy to an apparent paradox in cosmology.  If the expansion of the universe happened at a constant speed (i.e. if galaxies moving away from us maintained the same recession velocity as they move away, which is what Kevin refers to as constant expansion of the rope), then all photons will -- eventually -- reach us from arbitrarily far away.  Even if the galaxies they came from are being drawn away faster than the speed of light by the expansion.

The analogy is also good for visualizing how the light we receive now from distant galaxies was actually being drawn farther away from us by expansion at early times.  The ant starts out some distance away and crawls towards us, but the stretching rope makes the distance from the ant to us increase at first.  I don't think it's immediately obvious that this doesn't stay that way, that the stretching does not continue to pull the ant ever farther away.  But if the stretching rate is constant, then the ant eventually does reach us.  There is a "turnaround point", where the ant reaches a position on the rope where the speed that part of the rope is being pulled away is equal to the speed the ant can crawl, and after that it finally makes progress in our direction, not just along the rope but also in terms of distance away from us.

If we trace the path of light that is now reaching us from the early universe, we can see this same effect.  For the first ~4 billion years the expansion drew that light farther away from us, but then the distance where the expansion velocity drops below the speed of light caught up to the light, and then for the next ~10 billion years the light was able to make progress in our direction and reach us today.  It was really moving in our direction all along, but it took some time for it to reach a point where it could move more distance in our direction than the expansion could pull it away.

If it could be proven somehow that it would be impossible to reach the end, that would indeed be a paradox.

Later in the video Kevin makes an important point that the expansion of the universe is actually not constant.  It was slowing down early on due to the gravitational attraction of the matter and radiation, and now it is speeding up due to dark energy.  In the ant-rope analogy we must then make the stretching be a function of time, and that can greatly affect the result.  With accelerating expansion, the ant might not ever be able to reach the end.  Or light that started out too far away in the universe will never reach us.  The argument that "there isn't any more rope being added" no longer works.  The problem is not that there is not more rope being added, but that there is more space being added (for example, the distance between the atoms in the rope is increasing), and the rate at which that space is being added (or how that rate itself changes), can make a difference for whether the ant, or light, can reach the destination.

In cosmology we say that the distance from which any signal (light beam or whatever) can ever reach us is the cosmological "event horizon".  If the expansion rate was constant, then the size of this horizon would be infinite (everything in the universe would eventually be able to see everything else), but with accelerating expansion it can only be a finite size.  Right now it lies about 16 billion light years away, meaning light being emitted now from objects that are currently more than that distance away will never reach us.  

It's interesting when terms like comoving distance are used though, in which case the furthest objects we see are further away than the 16 billion light year distance, I believe in comoving distance, the radius of the observable universe is somewhere around 38 billion light years.  I wonder if one could travel to those objects that we see that are that far away, what they look like today?  To them, we'd be the ones who were extremely redshifted, so the rate of expansion is relative to location in the universe.

A-L-E-X, the comoving distance to the edge of the observable universe is about 46 billion light years.  This is also called the distance to the "particle horizon".  It is defined as the distance from which light emitted at the Big Bang is reaching us today, so it acts as the fundamental limit to how far we can currently see.

In contrast, the distance to the event horizon is the maximum distance from which light emitted today can ever reach us in the future.  It acts as the fundamental limit to what we can ever get a signal to or from if that signal was sent now.  You can also think of the particle horizon as being traced out (forwards) by light rays we emit at the Big Bang, and the event horizon as being traced out (backwards) by light rays we emit at the end of the universe.  They are our future and past light cones, emitted at t=0 and t=infinity, respectively.

The particle horizon is currently farther away, and always growing farther.  That's because light signals emitted at the Big Bang are always moving outward.  So as time goes on, more and more of the past universe comes into view.  Meanwhile, the comoving distance to the event horizon grows smaller with time.  The reasoning is mirrored: as time goes on, you have less time to send a signal somewhere, or for someone else to send a signal to you.


There is also proper distance, which is defined to be the same as the comoving distance today.  The difference is that comoving distances stay the same with expansion while proper distances do not.  The particle horizon grows larger even faster with time in terms of proper distance (because expansion drags the light rays away even more).  The proper distance to the event horizon also grows with time, but not as quickly.  As we learned from the ant on a rope analogy, ants, or photons, could travel arbitrarily far given enough time in a constant-expansion-rate universe.  But with accelerated expansion this is no longer true, and with the dark energy driven exponential expansion in the future, the proper distance to the event horizon gets "stuck" at around 16 billion light years.  Meanwhile the proper distance to the particle horizon will grow to infinity.

To them, we'd be the ones who were extremely redshifted, so the rate of expansion is relative to location in the universe.

The rate of expansion is the same everywhere.  Any two objects that are 1Mpc apart are receding from each other by the rate given the Hubble constant, regardless of where they are in the universe compared to us.  Of course all observers are also equally valid to choose their own frame of reference as their "rest frame", saying that all other distant galaxies recede from them. 

Astronomers in a galaxy at the edge of our observable universe say that they are at the center of their observable universe.  When they look toward us, they say we are at the edge of their observable universe, and receding from them at the same rate that we say they are receding from us.  It's a symmetry, as it should be to obey the Cosmological Principle.  This also implies that what those galaxies we see at the edge of our observable universe look like, if you could visit them "now", is basically the same as any nearby galaxies.  The distribution of matter and energy are the same here as there, and they evolved with the universe in the same way.

I should add that we cannot prove this, because the slice of space-time that we can actually see is quite small compared to the entire space-time.  But if we imagine that it was not true -- that distant patches in space or in time were very different -- then it would affect how the universe evolves, making the evolution different than what the Friedmann equations predict.  That in turn would lead to observable consequences.

Some speculate that this might actually be the case (breakdown of the cosmological principle) and explain weird observations like the dark flow.  It's an interesting speculation, but further research and better data are required before we can reliably reach such a conclusion.

Yes!  You read my mind, I was thinking about dark flow, because that's something we have yet to explain, whether it's changes in our fundamental constants at the "edge" (or beginning/end) of the universe or if it's the gravitational force of other universes.  

Things like large clumps of galaxies separated by wide voids (superclusters and supervoids) make interesting patterns in the universe.

Thanks for the explanation of the different horizons and distances.  It makes a lot of sense when you think of space in such a way like a balloon that is always expanding, ever faster. A toy model can be constructed (minus one spatial dimension) if we imagine 2 dimensional ants on a balloon as air is blown into it ever faster (the air represents dark energy) and the surface area of the balloon grows at an accelerated rate.   The outer surface of the balloon represents the universe as we know it (minus that one spatial dimension of course).  Where did the big bang occur?  At every point simultaneously since we were all part of it.  But also at no point because the universe has expanded well beyond that point (analogous to the center of the balloon no longer being part of the surface of the balloon since the balloon has rapidly inflated since it occurred.)  I do wonder if it would grow to infinity though, or if some kind of interaction (with whatever is on the "outside") would put a halt to that in the far future.

I guess this is somewhat related to Zeno's paradox, from which he concluded that motion is an illusion (but that seems like a false extrapolation from what he really showed, that motion at an instant of time is meaningless).  Paradoxes are often resolved by uncovering false assumptions.

It is very similar to Zeno's paradox.   I think it ultimately speaks to the need for calculus, or at least some method of determining whether the sum of an infinite series converges or diverges.  You can add an infinite number of positive numbers together and get either infinity or a finite number, depending on how quickly the numbers in the list grow or shrink.

Another example that comes to mind is the idea of escape velocity.  It's something most of us here probably understand quite well, but many people struggle with it at first.  Knowing that the gravitational force has infinite range and is always attractive, it can be tempting to conclude that something hurled from a planet (and ignoring any other objects to pull on it), should always be slowing down and therefore eventually turn around and come back.  (Also many people have heard the old saying "whatever goes up must come down.")  But not so!  The gravitational pull drops off as 1/r², so there is some speed you can launch it so that even though it is always slowing down, it never turns around and never comes back.


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A-L-E-X, the balloon analogy is great for illustrating how metric expansion works.  Choose any two points on the surface of an inflating balloon, and the distance, measured over the surface, increases at a rate which is proportional to the distance between them.  From this we get a "Hubble Law" for the balloon, just like the Hubble Law for the universe.   

However, like all analogies, this breaks down if taken too far.  In many ways the balloon is not a good analogy for the expanding universe, and we must be careful to understand its limitations.  

For example, an inflating balloon expands into the space around it.  The 2D surface of it grows larger because it is "expanding into the third dimension".  But this is not true for an expanding universe. There need not be any higher spatial dimension for the universe to expand into!  This is extremely difficult to visualize or comprehend.  How can we argue that it is true, then?  Because the expansion of the universe, according to general relativity, only depends on what the universe contains.  It is an intrinsic property, not dependent on anything that lies outside of the universe.  Unlike the balloon, it need not be embedded within a higher dimensional space at all.

While we cannot visualize how something can expand without having something outside of it to expand into, we can visualize something related to it.  Let's use an example of curvature:

The surface of a sphere is curved.  Obviously.  You can tell just by looking at it.  But how can you rigorously prove that it is curved?  Well, you can draw a triangle on it and add up all the angles in the triangle.  It will turn out to be more than 180°, which is a proof that the surface is curved.  Alternatively, you can cut open the sphere and try to lay it flat.  You'll discover that you can't, not without distorting it.  This is why all 2D maps of the Earth are distorted.  These properties show that a sphere has "intrinsic" curvature: the curvature is a property of the surface itself, not how it is embedded in 3D space.

How about a cylinder?  It is also obviously curved.  Right?  Well, no, actually.  Try drawing the triangle on it and measure the angles.  Or even easier: cut it open and try laying it flat.  It is perfectly flat!  The only reason that a cylinder "looks curved" is because of how it was rolled up inside of three dimensions.  But the surface itself was not curved at all.  So the cylinder's curvature is an extrinsic property rather than intrinsic.

Another problem with balloons: to continue to inflate a balloon, we must continue to add air to it.  (Or decrease the pressure outside of it, I suppose).  A universe does not work that way.  A universe containing nothing (no matter, no radiation, no dark energy) will expand forever at a constant rate.  No outside help required!  In this sense, a better analogy for expansion might be inertia.  It is more similar throwing a rock and tracing its trajectory than it is to inflating a balloon.  The rock analogy can actually be taken pretty far and still be useful, even in mathematical rigor, to the true cosmic expansion, and I'll explore this again in more detail soon enough.



We are creatures stuck on a tiny planet and whose lifetimes are very short compared to the age of the universe.  So the expansion of the universe is not something we understand intuitively from every day experience.  There is no perfect every-day analogy for it, though we can at least get some useful insight in a few ways.  Otherwise, we require physics and mathematics to accurately describe it.