Thanks for the clarification, Wat.  The way I have heard it explained is that the universe creates its own space which it expands into.  There are various unifying theories that conjecture higher dimensions, but they would exist adjacent to the universe and not be a part of it- even Einstein conjectured on other dimensions being necessary to unify quantum mechanics and relativity.

The funny thing about expansion is that it is accelerating.  Inertia could work well with a constant rate of expansion, but to have accelerated expansion we need to have dark energy.


Haha you got me thinking about the shape of the universe.  Is it a torus?  If you go enough in one direction will you end up right back where you started (like you could do on the Earth)?

One of the problems of trying to picture this is a sphere isn't a good analogy for the universe either.  A sphere has a two dimensional surface area that curves in a third dimension.  What does the universe curve into?  Itself?

...is a sphere isn't a good analogy for the universe...

It is, due to the fact that light and matter expands in an unidirectional way - i.e. no particular x, y or z coordinate is favored (for whichever reason) in expansion, so it expands spherically. It does lack some geometric traits specific to sphere of course, due to relativistic properties as Watsisname explained. It isn't a perfect analogy because really none could be when it comes to the entire universe, but it serves for at least the observable universe.

The way I have heard it explained is that the universe creates its own space which it expands into.

It's not a bad explanation, though I do try to discourage taking it too seriously for a few reasons.  It raises some difficult questions that do not have meaningful or testable answers.  Like, "where does that new space come from, or where does it vanish to if the universe is collapsing?"  I think it also can produce the wrong visual intuition for what's happening.  When I hear it, I think "okay, imagine the universe is a sphere expanding into nothingness, and it is creating space at the edge of itself so that it can expand into it."  And that is 100% not how it works.

It is better, at least in my own opinion, to refer to it as "metric expansion" and then explain what metric expansion means and where it comes from.  Or if metric expansion is too technical then we can simply say expansion, or stretch, or whatever.  They're just words.  The observable effect is that the distances between a collection of free particles increases.

Of course the question still stubbornly persists: "from where does this extra distance come from?"  Well, it comes from the same thing that causes the distance between something you drop and the ground to decrease.  You don't invoke space being "destroyed" to explain why an object falls towards Earth.  In Newtonian mechanics you explain it with a gravitational force.  Then in General Relativity you throw out the force and refer instead to space-time.

General relativity's description of space-time is the key to understanding cosmic expansion (we can also still usefully refer back to Newtonian mechanics and forces to describe it, though it won't be perfect).  In the general relativistic context, the presence of matter and energy change the shape of the space-time, determining the path that you will follow if you are in freefall.  Space-time tells mass how to move, and mass tells space-time how to curve.  For something dropped near the Earth, the freefall path is towards the center of the Earth.  Inside a black hole, every allowed path goes inward.  And for a universe uniformly filled with matter and energy, everything spreads apart by the inertia from the Big Bang, but the gravitational attraction of matter tries to bring them together, and dark energy drives them faster apart.  The expansion of the universe can be understood as the path things take through space-time in exactly the same way that a dropped object falls to Earth because of space-time geometry around the Earth.

Haha you got me thinking about the shape of the universe.  Is it a torus? 

Nope!  The universe can be closed on itself, but it cannot be like a torus.  (A shame; it would be very cool if we lived in a donut universe).   The reason is that on a torus, the distance you must travel to return to your origin is different depending on which direction you go in.  But that would contradict the principles of isotropy and homogeneity for the universe -- the space cannot be shaped that way because it does not match the symmetries for how matter is distributed.  If the universe is closed, then the distance to where space has curved back on itself (what I have loosely called the "curvature horizon" in previous posts) must be the same in any direction you pick, or from any starting location.  

Because the universe is homogeneous and isotropic, general relativity only permits three possible shapes for it:

• Spatially Flat:  If the total mass density equals the critical density (which it does, to within current precision of measurement), then the space is everywhere 3D Euclidean (what we call "flat").  In this case straight parallel lines remain parallel forever when extended, and the sum of angles in a triangle is always exactly 180°.  Such a universe is also infinite in spatial extent, never curving back on itself.  You can travel in any direction infinitely far and never return to your origin.
• Positively Curved:  If the total density is greater than the critical density, then the space curves back on itself.  Move far enough in any direction, and you will return to your origin (ignoring the effects of expansion which could prevent that.)  Sum of angles in any triangle is greater than 180° and straight, initially parallel lines will converge.  It is exactly the 3D analogy to the surface of a sphere.  We could call it a hypersphere.
• Negatively Curved:  Final case, if the total density is less than the critical density, then space is warped in such a way that the sum of angles in a triangle is less than 180°, and straight, parallel lines diverge from one another.  A visual analogy is the region around a saddle point, where the surface bends upward along one axis and downward in the other.  Except now try to visualize that property everywhere and in 3D.  We can describe this as being a hyperbolic geometry.

The 2 dimensional analogues are illustrated here.  Again a universe being "flat" doesn't mean it is like a pancake, but rather flat in 3 dimensions, which is simply the 3D Euclidean space that we are familiar with.  Curved 3D spaces are hard to visualize.




Is it possible for us to measure the true shape of the universe?  Yes, actually!  The triangles in the above sketch illustrate the idea. By measuring the angular size of the fluctuations in the CMB, we are essentially "making very big triangles" through the universe, which in turn tells us its geometry.  Specifically, this is the measurement of the "CMB angular power spectrum" I described on the previous page.  The observations indicate that the space is very close to flat, with a precision of something on the order of 10%.  Another way to test for curvature is to check the total mass density of the universe and compare to the critical density.  Again if they are the same, then space is flat.  If greater than critical, then positively curved, and if less than critical, negatively curved.  Constraints on the mass density tell us the universe is flat to within about 2%.

What do these constraints tell us about how much space can curve back on itself (or away from itself, in the case of negative curvature)?  The Planck 2015 observations indicate the total density parameter is Ω_total = 1.0023, with 1-sigma error bars of about .0055.  So let's look at the values 1.0023 (best estimate), 1.0078 (68% confidence it's less than that), 1.0133 (95% confidence), and 1.0188 (99.7% confidence).  For each of these cases, what is the proper distance to where space starts curving back and distances start decreasing again?

For Ω_total = 1.0023, it comes out to 471GLY.  If you could freeze the expansion and travel that far in any direction, you will then start to get closer to Earth again, just as if you went half-way around the world.

For Ω_total = 1.0078, this distance is reduced to 256GLY.

For Ω_total = 1.0133 it is 196GLY,

and finally for Ω_total = 1.0188 (99.7% confidence it isn't that large), it becomes 165GLY.


All of these distances are several times larger than the proper distance to the edge of the observable universe.  So if the universe does curve back on itself, it must do so over much greater distances than what we can see.  (Otherwise, of course, we would see it already.)

It is also over much greater distances than you can ever hope to travel (unless somehow we learn to go faster than the speed of light, and I am not holding my breath for that.)  The distance to the cosmic event horizon was only ~16GLY!  So we can safely conclude that even if space is curved in this way, we still cannot hope to travel out in a straight line and get back to where we started.  Sad.


Now, I focused on the possibility that the total density was greater than the critical density.  If it is instead equal or less, then space isn't closed on itself at all.  Better future measurements of the density of the universe and the curvature may constrain it to be closer and closer to the critical density (no curvature), but can never exclude the possibility that it is curved at least a little bit.  It's sort of like how if we lived on a perfectly smooth sphere, then the larger the sphere is, the harder it is to measure the curvature.  If it's really large, then it just looks flat.  Hence the Flat Earther's out there... (maybe they need to update to being Flat Universer's). 

It's pretty interesting that the universe is so close to being flat!  I wonder if that has to do with inflation somehow, that the universe expanded so quickly and suddenly in the beginning that it really didn't have time to develop a curvature.

I could envision the universe being flat in 3 dimensions but still taking on the shape of a hypersphere or glome (a sphere with a three dimensional surface), if its size is so huge that it seems perfectly flat to us because our perspective is not large enough to notice its curvature.  That's why people used to think the earth was flat- the human perspective just isn't enough to notice the slight curvature over the distances which humans can perceive.  But based on those calculations, a hypersphere solution isn't likely because the universe would have to be extremely large way beyond anything we have estimated for it to be for the curvature to be small enough for us not to detect.  Of course if there really is that much universe beyond what we can observe, then there is no way for us to know, all we can do is more precisely make measurements to get a more exact value and to narrow down the constraints.  It's just really interesting that the universe would be that close to being perfectly flat, but I assume that has something to do with inflation and how rapidly the universe's initial expansion was.  That also resulted in its uniformity over vast distances (though the clumping of matter and voids over very large scales is also interesting, as are some of the patterns in the CMB- almost like tree rings lol.)


The idea of space expanding into itself also brings up the rather messy question of- what exactly is space?  According to Wheeler and some others, space consists of quantum foam, virtual particles and tiny wormholes that constantly blink into and wink out of existence.  It would be on the subplanck scale, since we can't quantize space on any scale that we can measure.  Quantizing does take care of messy singularity problems, if there is a certain minimum size for space it means that black holes (and also the big bang/bounce) could not have ever been smaller than that minimum value.  This minimum size value is also used by Loop Quantum Cosmology to show how a big bounce occurs once space reaches that minimum value.  Perhaps the same is the case on the interior of a black hole when a new universe is created within its own space.  There would be multiple event horizons in such a black hole.  The idea of space expanding into itself was also brought forward by Poplawski in his unifying conjecture of black hole cosmology, our universe existing inside a black hole inside a super-verse.  Smolin also mentioned the possibility of universes being born inside black holes inside larger universes and in a way similar to evolution, universes like ours (with life) are more likely to form.  When we have some cohesive theory of quantum gravity, maybe we'll figure all these things out.

though I dont understand why the source must be "destroyed"- can the source not be kept while a duplicate is generated?

No-cloning theorem.   This is something very fundamental to quantum mechanics, and it's easy to get lost in the weeds in terms of understanding rigorously why it must be true.  But there's a minutephysics video that does a pretty good job of explaining it more conceptually without too much of the math.


[youtube]owPC60Ue0BE[/youtube]

Thanks, Wat, so the 10^80 figure applies to the observable universe only?

That's right.  For all we know, the entire universe could be infinite (if the density is equal or less than the critical value so that space is flat or negatively curved).  There might be an infinite number of protons/electrons/anything in that case.

I wonder what a proton would decay into?

In order to decay there must be a violation of conservation of baryon number.  Since protons are the lightest known baryons, and we have so far no evidence of such a violation, this implies they don't decay at all.  
If they do decay, then the decay still must conserve quantities like charge and spin.  One way to do so is for it to decay to a pion and positron (and then the pion decays further into two photons).  Those photons could then be detected to provide evidence of the proton decay, depending on their energy and the angle between them.  That's what they'll be looking for in the next generation of detectors.

Though you could still use a triangle to prove what kind of curve would be the shortest path between two points.

  My favorite inversion of this idea is to find the shortest path between two points in spacetime.  It ends up behaving quite the opposite of intuition.  For example, the distance traced through spacetime by a photon is zero.  And for any two events separated more in time than in space, the straight path between them ends up being the longest path.  Spacetime physics is very weird and very fun.

The thing that I can think which violates the no cloning theorem is when near the event horizon of a black hole, Stephen Hawking figured out that a copy of whatever particle is there is made and one falls into the black hole while the other stats outside, though this may not really be a violation of it since we have no idea what happens to the one that goes into the black hole (it could be whisked away to another universe for all we know.)  This is part of the no information is ever lost idea.
Pions are really interesting- I think the pi meson is the one which is its own antiparticle!
What you said about space time makes a lot of sense- i wonder how that gets altered in the vicinity of black holes and especially spinning black holes with frame dragging and closed timelike curves (if they exist).  Being separated more in time than in space- is that because of intense gravitation like in the vicinity of black holes?  Warped space like that would be a reason why a straight line trip would be the longest path- basically when you go in a straight line in such a case you are almost going "against the (gravitational) current!"
The distance traversed by a photon in spacetime should be zero because traveling at the speed of light the photon doesn't experience the passage of time.  It makes me wonder how fast we need to travel to slowdown the passage of time enough so that we could make interstellar journeys without significant aging effects (though by then all our consciousness could be uploaded into mechanical bodies and we might never experience aging at all so the length of a space trip might be immaterial even without wormhole travel.)

The thing that I can think which violates the no cloning theorem is when near the event horizon of a black hole

It isn't making a copy because they don't both exist in the strict sense that you could bring them together at the same time.  It is impossible for any observer to measure both.  

Think of the question, "where is the thing that falls into the black hole located?  Is it inside the black hole or on its horizon?"   The answer is "who is measuring it?"  Which one you find depends on what you do.  If you hang around outside, then you say it's on the horizon.  If you dive in yourself to prove that, then you find it is not on the horizon -- it's all the way at the central singularity.

To get more insight on how this works I recommend reading Susskind.

Being separated more in time than in space- is that because of intense gravitation like in the vicinity of black holes?

No gravitation necessary!  Imagine yourself floating freely in an empty space-time.  You check your watch, it's 3pm.  You check your watch again some time later.  Now it's 4pm.  Each time you check your watch is another "event". 

In your frame of reference you haven't moved at all.  The distance in space between those events is zero.  But there is a distance between them in time: 1 hour.  So those events are separated more in time than in space.  In general you are always moving more in time than in space, because you are never moving at or faster than the speed of light.

Photons travel 1 light year in space for each year of time.  But the distance in space-time is defined as

(spacetime distance)² = (space distance)² - c²(time interval)² 

So if light goes 1 light year in space for each year in time, then the distance in spacetime is sqrt(1LY² - 1LY²) = 0.

 It makes me wonder how fast we need to travel to slowdown the passage of time enough so that we could make interstellar journeys without significant aging effects

You can use the special relativistic time dilation formula to compute it:



where γ how much slower your time is compared to someone not making the trip, and β is the fraction of the speed of light that you're going.  If you go 99% of the speed of light, then β=0.99, and γ=7.089, meaning your clocks tick 7 times slower.

Although the time dilation is real, getting to such speeds that it becomes significant is incredibly impractical. Even with the most efficient rocket possible (the photon rocket), the mass of propellant that you can convert to photons and shoot out the back must make up a huge fraction of the total mass of your rocket if you want to reach such highly relativistic speeds.

Is that the no hair theorem   I love reading that stuff!  Leonard Susskind!  That's right they can never be brought together in the same spacetime, at the singularity, might as well be in a different universe lol.  When you get past the event horizon, does space-time really become flipped, meaning movements which ordinarily would be considered movements in space, become movements in time instead?  Interesting thing I read about about supermassive black holes, when you enter their event horizon, you dont even know you're inside a black hole because they aren't dense and you can't even really feel gravitation- except for the fact that you can never get back out.  In that way it's actually a micro version of our own universe.

Wow, neat equation, Wat, that makes light the ultimate frame of reference for movement in the universe!  All positive mass particles must move slower than it so they all move less in space than they do in time. Anything that would (theoretically) move at the speed of light though would not experience the passage of time, but since all positive mass objects must move slower than it, we can only approach it.  I remember reading that for time dilation to become significant, we need to be moving at at least 0.9c and preferaby 0.95c.  The fastest that I read we can reasonably move within the next generation or so would be 0.2c using laser powered sails, but that would not be nearly enough for time dilation to be a factor (although it should be enough for generation ships to reach nearby stars in a reasonable amount of time.)

A-L-E-X wrote:
I heard there was a brown dwarf star found with a surface at room temperature close by- so could a brown dwarf actually be habitable?

From here.

Define habitable . I believe the brown dwarf you are referring to is WISE 1828+2650, once place-holder of the coldest brown dwarf on record with an average temperature of 250 to 400 K (−23 to 127 °C. The title has since been taken up by the ultra-cool WISE 0855−0714 brown dwarf that has a estimated 225 to 260 K (−48 to −13 °C) temperature.

Thanks are these both in SE?   haha, habitable hmmm, about as habitable as a gas giant in that temp range?  

Watsisname, in your professional opinion, what do you make of E₈ Theory?

[center][/center]


[left]I think it looks pretty, in about the same way that Doctor Strange's mystic shield looks pretty.  Granted, the former has a deep mathematical basis, while the latter probably did not.  But that doesn't mean it will be a productive path for extending our knowledge of fundamental physics.  I don't think anyone knows if it's the right path.  What matters is if it is consistent with observations and successfully predicts something new that other models did not.  There are many theoretical approaches these days, each with variations in motivation, utilizing different numbers of dimensions and so forth.  And there are a lot of very smart minds working on them, so hopefully sooner or later at least one of them will have a breakthrough.  I look forward to when they can connect with experimentalists and make practical testable predictions.  [/left]

[left](Moved from the cosmology thread because this question is more about theoretical approaches to extending the standard model of particle physics than about cosmology.)[/left]

A truly successful theory must incorporate all the previous successes of Relativity and Quantum Mechanics as well as making new predictions that are found to be successful that are beyond the scope of either of the other two (like removing singularities), which I remember you said before also.  I find Poplawski's idea somewhat more elegant- it gave me more of an AHA moment, although that doesn't sound very scientific lol.

[center]Cosmological Simulations[/center]

Introduction:

[left]As promised, I am now ready to share some results of cosmological simulations I have run on the computer, which model the evolution of universes filled with different amounts of matter, radiation, and dark energy.  Creating such simulations is something I have wanted to do for a long time, and An'shur's question earlier provided the motivation to finally do it. [/left]

[left]There is quite a bit to cover, so I'll break this up into spoilerized sections.  First, I'll review the equations and techniques for building a cosmological model.  Then I will present the standard Lambda-CDM model, which is currently the best model for observations.  We will see what the evolution of our universe looks like over time, and also encounter some surprises![/left]

[left]Then, to explore An'shur's question, I will flip things around and examine a model universe which started with a Big Bang, but contains much more matter -- enough to halt and reverse the expansion and eventually lead to a Big Crunch.  I will set the matter density so that after 13.8 billion years, this model universe is collapsing at negative 68km/s/Mpc -- the opposite of the expansion rate we observe in the real universe today.[/left]


[left]Building a Cosmological Model[/left]

► Show Spoiler

[left]On very large scales, the matter and energy in the universe is essentially uniformly distributed.  This is the cosmological principle, which is justified by observations on scales greater than ~100Mpc.  This is extremely convenient for our ability to model the universe, because it greatly simplifies the mathematics.  If we apply General Relativity to a universe with a uniform distribution of matter and energy, we obtain the Friedmann equations, which describe how the expansion rate of the universe changes over time.[/left]

[left]These equations may seem complicated and mysterious.  How can we be sure that they are accurate?  Well, for one thing they are exact solutions to GR, and for another, they are very well tested against observations.  Also, I think it is actually not that difficult to understand them, at least in a Newtonian context, by drawing an analogy to orbits.  If we launch something from a planet with too slow of a speed, then it will slow down and come back.  This is analogous to a "closed" universe, with too much gravitation such that its expansion slows down and then it collapses on itself in a finite amount of time.  An "open" universe, with too little gravitating matter, will continue to expand forever.  This is analogous to launching something from a planet on a hyperbolic orbit -- fast enough that it slows down, but never stops, and never comes back.  Finally there is a perfectly balanced "flat" universe, which has exactly the right amount of gravitation so that the expansion rate slows to zero, but after infinite time.  This is like a parabolic orbit, where the object was launched at exactly the escape speed.  [/left]

[left](Aside: "Flat" in the cosmological sense does not mean "like a pancake", but rather that the spatial geometry is 3D-Euclidean, the type of geometry we are all intimately familiar with.  It means two straight and initially parallel lines remain parallel, and the sum of angles in any triangle is 180 degrees.  A closed universe on the other hand will have those straight, parallel lines converge on each other, and the sum of angles is greater than 180°.  The 2D analogy is the surface of a sphere, where lines of longitude are straight, and parallel at the equator, yet converge at the poles.  This is also called "positive curvature".  An open universe will instead spread apart parallel lines, and the sum of angles in a triangle is less than 180 degrees.  That is "negative curvature", and a good 2D analogy is the region around a saddle point.)[/left]

[left]The Friedmann equations are the general relativistic version of this same logic.  In fact, if we use the Newtonian laws of gravity, apply it to an expanding, uniform sphere of particles that interact only by gravity, then we will come up with equations that look almost identical to the properly general relativistic Friedmann equations (save for a few numerical factors and constants, like the speed of light.)  Someday I might show the Newtonian derivation, which is quite a fun bit of physics, but for now let's take the equations as given.  How do we use them to simulate a universe?[/left]

[left]We'll start with the First Friedmann equation, written in a slightly different and more suggestive form:[/left]

[center][/center]

[left]Here "a" represents the size or "scale factor" of the universe.  It is set equal to 1 at the present time by convention.  If a grows to 2, then that means the distances between galaxies has doubled.  The moment of the Big Bang corresponds to a=0.  [/left]

"a" with a dot over it (which I'll write in text as a') represents the rate at which the scale factor is changing with time (like an expansion velocity).  In calculus terms, the dot stands for a time derivative.  Dividing a' by a means "expansion velocity per distance", which is the definition of the Hubble constant.  The Hubble constant today in the real universe is roughly 68km/s/Mpc.  This means a galaxy 1Mpc away is receding at 68km/s due to the expansion of space in between here and there.  A galaxy 2Mpc away recedes twice as fast (136km/s), because there is twice as much space in between.  For a wide range of cosmological distances, this linear relationship between distance and recession velocity holds, and this is known as Hubble's Law.

On the right hand side of the equation is H₀, which is the value of the Hubble constant at the present time, and then a bunch of funny symbols Ω (omega).  Each omega represents a density (for example Ω_m is the density of matter), but as a fraction of the "critical density".  The critical density is the exact density which would make the universe flat and slow down to zero expansion rate after infinite time.  As an example, if the matter density was exactly the critical density of the universe, then Ω_m would equal 1.

So, this first Friedmann equation says that the size of the universe changes at a rate which is governed by the densities of matter (Ω_m), radiation (Ω_r), and dark energy (Ω_Λ).  And there is an Ω_k, which is 1 minus the sum of the others.  That is, if the sum of the matter, radiation, and dark energy densities is exactly the critical density, then Ω_k=0.  The meaning of k is the spatial curvature.  Ω_k=0 is a flat universe, while Ω_k>0 is open (negatively curved), and Ω_k<0 is closed (positively curved).

One last thing to notice is how each Ω term is scaled by some power of a.  This has a very deep physical meaning.  Ω_m is divided by a³, because the density of matter decreases with the volume of the universe, or the size of the universe cubed.  Ω_r gets divided by a⁴ because radiation is not only dilluted just like matter, but each photon is also redshifted by expansion, which decreases its energy, and tacks on another factor of a.  So the energy density of radiation drops off as a⁴.  This is a weird way in which cosmological expansion appears to violate energy conservation -- the energy of radiation is not conserved in an expanding universe!

Ω_Λ is an interesting one.  It is divided by the scale factor with a weird exponent, containing another parameter, "w".  "w" is a knob which describes how the density of dark energy changes with expansion (in more precise terms it describes its equation of state).  The simplest model, and the one that best agrees with observations, is w=-1, which means dark energy does not dillute at all.  The interpretation is that dark energy is a property of space itself, rather than some substance like particles of matter.  The more the universe expands, the more space there is, and thus there more dark energy there is.  This is yet another example of how cosmic expansion seems to violate energy conservation.  (It doesn't really -- energy conservation just does not apply in the usual way to an expanding space-time!)


Now to turn this equation into something we can use -- implementing it into a computer code which evolves model universes.  To do so, we will convert it into an equation for acceleration.  This is a good calculus exercise, and I'll explain briefly the steps and then show the final result.

First, multiply both sides of the equation by a² to get a' by itself on the left side.  Then, take the time derivative of both sides.  Using the chain rule, this means the time derivative of a'² is 2a'a''. Finally, isolate a'' (the acceleration) by dividing both sides by 2a'.  The final result is:

[center][/center]

[left]This is a form of the 2nd Friedmann equation, also called the acceleration equation.  As you might guess it describes how the universe's expansion speeds up or slows down.  Notice that matter and radiation both act to slow it down (their coefficients are negative), while dark energy acts to speed it up.  This isn't immediately obvious since there is a minus sign in front of the dark energy term too, but remember that w=-1.[/left]

[left]At last, this acceleration equation can be implemented directly on the computer.  We begin by setting the values of the cosmological parameters at the present time (all the Ω's and Hubble constant), calculate the acceleration a'', and then iterate again for the expansion velocity a' and the change in scale factor a:[/left]

[center][/center]
[center][/center]
[center][/center]


[left]Those are the fundamental equations programmed into the model, and we will explore their consequences below.[/left]

[left](Aside:  The Friedmann equations may also (and often are) expressed and solved in integral form.  Here I have instead opted to work with them in differential form, solving for the evolution of the model universe by successive iteration through small steps of time from some starting condition.  In my case, starting at the present time with a value of H₀, and iterating both backwards and forwards.)[/left]

[left]Simulating the Lambda-CDM Universe[/left]

► Show Spoiler

[left]Let's look at the expansion history of the real universe.  Observations indicate the values of the cosmological parameters are:[/left]

[left]Ω_m ~ 0.31[/left]
[left]Ω_r ~ 9.2x10^-5  (the energy density of radiation is quite negligible in the universe today)[/left]
[left]Ω_Λ ~ 0.69[/left]
[left]H₀ ~ 67.5km/s/Mpc[/left]
[left]w = -1[/left]

[left]Here is the expansion history of this Lambda-CDM model universe:[/left]

[center][/center]

[left]The universe began expanding very quickly after the Big Bang, but slowed down due to the gravitational pull of all the matter and radiation present.  Radiation was then diluted away quite quickly (due to the 1/a⁴ dependence), leading to a matter dominated universe which still slowed down, but not as quickly.  But over time the matter was diluted away too, relative to the third, very mysterious component we call dark energy, or the cosmological constant.  This energy density stays constant as the universe expands, so while matter and radiation get diluted, it grows stronger by comparison, and eventually dominates.  We can see this by plotting their densities (in equivalent proton masses per cubic meter) over time:[/left]

[center][/center]

[left]At this plot scale we cannot see the radiation-dominated era, which was the first few 10,000 years.  But we can see both radiation and matter density dropping over time, radiation faster than matter, and we can see dark energy as the steady horizontal line, which began to dominate over matter when the universe was around 10 billion years old.  [/left]

[left]Next let's look at a space-time diagram of the universe:[/left]

[center][/center]

[left]There is a fair bit of information to digest in this figure![/left]

[left]I have time running along the horizontal axis in billions of years, and the "proper distance" running upward in billions of light years.  Proper distance is what you would measure if you could freeze the expansion everywhere and then stretch out a ruler to those locations.  I set the scale so that 1 year of time horizontally is the same as 1 light year of proper distance vertically.  The path of a light ray with this scale would be a 45° angle, if space was not expanding.  But it is expanding, which leads to some weird consequences.[/left]

[left]The present time is the vertical grey bar at 13.8 billion years.  Extending back from our location at the present is a yellow curve, which is the path taken by a photon just now reaching Earth, having been emitted at the Big Bang.  Notice how it initially moves away from us after the Big Bang.  For about 4 billion years this photon, which is always moving towards us, was pulled farther away by the expansion.  The path it took defines our "past light cone" -- the slice of spacetime that we observe.[/left]

[left]The thick dashed black curve shows the path of whatever source emitted that photon.  In truth there is no such source, as the early universe was opaque to light until about 380,000 years of age when electrons could finally join up with protons to form atoms, in the "recombination era".  But we can imagine a hypothetical source of some light-speed signal emitted from the Big Bang, just reaching us now.  Perhaps gravitational waves.  Anyway, it is interesting to ask "how far away is that source from us now"?  The source of the photon now reaching us from the Big Bang is now located at a proper distance of over 40 billion light years!  Much more than the speed of light times the age of the universe, because expansion has again pulled it further away.  This distance is what defines our "particle horizon" -- the most distant thing that we can in principle receive a signal from today.[/left]

[left]I show a number of thinner black dotted lines also radiating outward from the Big Bang.  These are the paths of galaxies.  I choose the specific ones that intersect the light cone at regular 2 billion year intervals.  This means each intersection is a galaxy that we observe when the universe was 2 billion years old, 4 billion years old, etc, up to 12 billion.  [/left]

[left]Finally, green dashed curves represent distances for which the expansion pulls things away at 1, 2, and 3 times the speed of light (from bottom-most to top-most curve, with the bottom 1c curve the thickest).  It might seem like the notion of things receding faster than the speed of light is contrary to relativity, but it isn't.  Special relativity says nothing can move through space faster than light, but here we are dealing with an expansion of space itself, which is a general relativistic effect, and there is no limit to how fast something can be pulled away from us by that expansion.[/left]


[left]Let's zoom in on the region closer to our past light cone:[/left]

[center][/center]

[left]Check out the galaxy represented by the second dotted curve from the top, intersecting the light cone right at its peak.  This is a galaxy we observe from when the universe was 4 billion years old, and it was at a proper distance of about 6 billion light years.  Notice it crosses both the light cone and the green "v=c" curve together.  At 4 billion years the distance where the recession velocity exceeds the speed of light overtook it, and for the next several billion years it hovered with a recession velocity a little slower than light.  Then, about a billion years ago, the dark-energy driven accelerated expansion brought that galaxy back into the region receding away from us faster than light, and it will continue to be receding faster than light for ever more.[/left]

[left]It gets even more extreme than that.  The galaxy represented by the first curve, which was at about 5 billion LY distance when the universe was 2 billion years old, 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.[/left]

[left]Referring back to the first space-time diagram, the most distant things we could see (in principle -- the particle horizon) now have recession velocities of about 3 times the speed of light.  [/left]

[left]There is some more still that we can explore with this standard Lambda-CDM model, particularly for the long-term evolution and questions of its eventual fate (heat death, big rip, etc?).  But I will save that discussion for another time.[/left]

A Collapsing Universe:

► Show Spoiler

Now we will explore your question, An'shur.  What if we lived in a universe which began with a Big Bang, but began contracting, and is now collapsing at H=-68km/s/Mpc?  (I know you asked for -72.5, but it does not make much of a difference.)

Let's set all the cosmological parameters to be the same as the Lambda-CDM model, except we'll make the matter density 9.9 times the critical density, which reverses the expansion rate to -67.5km/s/Mpc in 13.8 billion years.  Here's our plot for the universe scale factor over time:

[center][/center]

The densities over time:

[center][/center]


[left]And finally, the space-time diagram:[/left]

[center][/center]


[left]What can we tell from this?[/left]

[left]-The proper distance to the particle horizon will be about 18 billion light years, at least in this specific case.  Less than the real universe, because the expansion slowed down and reversed.
-We will observe galaxies nearby which are slightly blueshifted, while galaxies that we observe from when the universe is less than about 5 billion years old will still look redshifted, because the contraction has not yet "cancelled out" the same amount of expansion.
-Objects that we observe when the universe was less than about a billion years old will now be approaching us faster than light![/left]
[left]-Over time, more and more galaxies will move into the region that is approaching us faster than light.[/left]


[left]And, of course, we will be doomed to an sudden, horrifying fate in a Big Crunch.  (Actually I think that might rather be a lot more fun than a boring old heat death).   Speaking of which, [/left]


[left]

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?

[/left]
[left]I have not forgotten you, Gnargenox, but I am out of time for tonight and this post is already excessive.  So let's look at the Big Rip scenario next time, by playing with the parameter "w".   [/left]
[left]

[/left]

Wat, this is fantastic- is there a computer program you use for your simulations and can it be done graphically (meaning with a universe simulator like SE or Starry Night?)  And can physical constants be varied, etc.?

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]

You know I love this idea, and the possibility that the Big Rip might actually lead to a cyclic universe, as well as the universe as a cosmic quantum computer and you brought in the Holographic Principle also!  Loop Quantum Cosmology deals with a pre Big Bang universe, and quite a few sites favor that approach as the proper way of reconciling relativity with quantum mechanics, because it gets rid of the singularity at the big bang (as does Stephen Hawking's imaginary time), so the Big Bang becomes merely a point in space-time with a contracting universe before it and an expanding universe after it.

Hey everyone I came across this fairly recent paper : https://arxiv.org/pdf/1712.07962.pdf "A unifying theory of dark energy and dark matter: Negative masses and matter creation within a modified ΛCDM framework", that tries to explain dark energy and dark matter by unifying them in a "negative mass fluid". For this to work he obviously had to modify the ΛCDM cosmology (since ours only predicts positive masses) to make it compatible with the presence of negative mass. And his simultations are doing a pretty good job at modelling (not perfectly) the large-scale structures we have observed in the universe. 
It's interesting because we think of a lot of physical quantities as being able to be both positive and negative but never actually thought that way with mass (or I never came across it).

Thu Dec 06, 2018 2:37 pm

Hey everyone I came across this fairly recent paper:  "A unifying theory of dark energy and dark matter: Negative masses and matter creation within a modified ΛCDM framework"

I was thinking of making a post to discuss this paper.  In brief, the paper re-interprets dark matter and dark energy as a single substance with a constant and negative mass density.  In other words, it proposes there is no dark matter, but instead there is a negative matter that is being continuously generated.  So it is like Lambda-CDM cosmology, except changing the matter density to be equal to the baryon density, and changing the value and sign of the dark energy while keeping its equation of state the same at w=-1.

Proposing that there is negative matter being constantly created is a pretty crazy idea, but perhaps no more crazy than proposing additional invisible matter and dark energy were to begin with.  So the motivation itself isn't bad and I think is actually a rather interesting approach.  But does it agree with observations?

I don't think that it does.  It actually makes a lot of incorrect predictions that must then be explained away, which is what the author spends a big portion of the paper doing by either vigorous handwaving or re-interpreting observations.  I didn't find this very compelling.

• By making the total density of the universe less than the critical density, this model predicts a negative curvature, which contradicts observations showing the universe is very close to flat.  So the author argues the curvature must be very small.  This is problematic for the amount of negative matter needed, plus the removing of the positive-mass dark energy from Lambda-CDM.  The curvature should be very negative in this model, clearly observably so.
• By adding negative mass with equation of state w=-1, it predicts decelerating expansion.  What we observe is accelerating expansion.  So we have to re-interpret the high redshift supernova observations.
• It changes the age of the universe!  The more negative matter the model needs, the younger the universe must be, for the same current observed expansion rate.  With too much negative matter, this runs into problems with the measured ages of the oldest globular clusters.
• It predicts a different age-redshift relation.  The CMB and the first galaxies in this model must have formed much later, for the same observed redshifts.  So we would have to completely re-work our understanding of the early universe.  The paper did not attempt to do this.

I also could not find in the paper a clear, consistent choice of values for the cosmological parameters necessary for the model to work.  It requires very little negative matter to be consistent with the age of the universe and the curvature.  But at the same time, it does need a lot of negative matter to explain the galactic rotation curves.

Probably the simplest and most severe problem?  It doesn't predict the observation of the separation of dark matter from regular matter in the collision of the Bullet Cluster.  This is a natural prediction of proposing the existence of additional weakly-interacting positive matter within galaxies and clusters.  But if we instead remove that and replace it with negative matter, then no matter where the negative matter is located, we would not reproduce this observation.


In summary, I see a lot of outstanding issues with this model.  If I had been reviewing the paper for submission I would have suggested putting a clear explanation of the cosmological parameters used, either overall or in each analysis.  This would have made it much easier for other cosmologists to review and check their work.  I think an explanation of the Bullet Cluster in the context of this cosmology was also crucial. 

For future work, the model should be applied to predict the full CMB angular power spectrum (especially for the relative heights of the acoustic peaks, which are an evidence of dark matter!), and the implications for the early evolution of the universe.  A more thorough statistical analysis of the formation of structure would also make it more compelling.  Just because the model makes something that "looks like" the cosmic web doesn't mean much.  In Lambda-CDM cosmology the precise way the structure appears at different scales and redshifts is incredibly well studied.

Wat, this is fantastic- is there a computer program you use for your simulations and can it be done graphically (meaning with a universe simulator like SE or Starry Night?)  And can physical constants be varied, etc.?

Not exactly.  The simulations are run through a code I wrote in Python.  What it allows me to do is vary the cosmological parameters -- specifically, how much of each substance with a different equation of state there is.  So I can change the amount of matter, dark matter, radiation, and dark energy.  I can also change their equations of state, or introduce new substances that have different equations of state.

Then I hit a button, the machine churns through the code, and tells me "what happens".  It computes how the size of the universe changes through time, along with other quantities like the mass densities or the Hubble constant, which it then spits out graphically as plots.  It also traces the paths of photons through the universe, which can be plotted in a space-time diagram, and to show how much the light from different distances has been redshifted (or blueshifted, if the universe was collapsing).

So the simulations exist in a code rather than a GUI type of program, and the output is graphs rather than a visual rendition of what the universe were to look like if you were in it.  To turn it into something like SE or Starry Night that show what things look like in the sky is not possible -- it would require simulating the interactions of millions or billions of particles, as in the Millenium Simulations, and that would require a supercomputer.  

SE doesn't compute the evolution of the universe, either, for exactly that reason.  It's computationally infeasible.  Instead it emulates what space looks like today with procedural generation.  Universe Sandbox is more of a simulation, but only with gravity (and collision detection), rather than running through the Friedmann equations to simulate how the universe evolves.  And by the same limitations, it cannot show as many objects as SE does, or the real universe.


I haven't decided if I intend to release this code at any point, but honestly I'm leaning against it.  It might be fun for people to play with, but I also have made no effort to make it intuitive to use, or error-free if the user were to change things without being very careful.