One of the things cosmologists like to talk about is the fate of the universe. Some say it all depends on the density parameter omega: Ω. This started life as the average matter density of the universe divided by a “critical” matter density for the Friedmann universe:

*Based on the Friedmann universes public domain image by BenRG, see Wikipedia Commons and Wikipedia *

Nowadays when we talk about omega we don’t restrict ourselves to matter alone. That’s because energy doesn’t necessarily take the form of matter, and it’s the energy density that matters. But the same principle applies. So some cosmologists talk of closed universes, and others talk of open universes. Some talk of the Big Crunch, others talk of the Big Bounce. Others talk of the Big Chill or the Big Freeze. Or the Big Rip. So, what’s your poison? Let’s take it one step at a time, starting with the Friedmann universe.

*The Friedmann Universe*

The Friedmann universe was proposed by Alexander Friedmann in the early 1920s. He came up with a number of scenarios for a dynamical universe as opposed to Einstein’s static universe. In his 1922 paper Friedmann proposed three alternatives. He talked about a “monotonic world” of the first type, which expanded from a singularity and continued to expand. Then there was a monotonic world of the second type, which expanded from a non-zero initial size and continued to expand. Then there was a “periodic world” which expanded from a singularity then contracted. The latter is essentially the Big Crunch universe. See the 2012 Physics Today article Alexander Friedmann and the origins of modern cosmology. It was written by Ari Belenkiy, who wrote a related paper in 2013. He said this: *“The cosmos starts from the singularity r = 0, expands at a decelerating rate to maximum radius x _{1}, and then begins contracting back down to zero. The life of the cosmos is finite, ending in a Big Crunch”*. All three options feature positive curvature. In his 1924 paper Friedmann examined negative curvature too. He said the non-stationary negative curvature case was similar to the non-stationary positive curvature case, and the negative curvature might mean the universe is infinite. There’s a little issue here in that Newton’s universe was saved from gravitational collapse by being infinite. Hence one might question how an infinite universe can expand. However there are other issues too.

*The Friedmann equation*

Some people speak of the Friedmann equations as opposed to Friedmann equation. Some speak of the Friedmann acceleration equation. Some offer different versions of what are the same underlying expressions. But no matter, because what matters is what it all means. The Wikipedia Friedmann equations article gives what looks like a fair description. It says Friedmann derived his equations *“from Einstein’s field equations of gravitation for the Friedmann-Lemaître-Robertson-Walker metric and a perfect fluid with a given mass density ρ and pressure p”.* The density *ρ* is a rho. Friedmann came up with something that is said to describe *“the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity”*. That sounds reasonable enough. Especially since Friedmann is also said to have employed the *“simplifying assumption that the universe is spatially homogeneous and isotropic”*. We then read that the cosmological principle implies that the metric of the universe must be of the form ds² = a(t)² ds₃² – c²dt². The ds is a spacetime interval, a(t) is a scale factor, ds₃ is a 3D spatial metric with a positive, zero, or negative curvature, c is the speed of light, and dt is elapsed time. It’s perhaps easier to appreciate in the Friedman equation section of the Jodrell Bank Introduction to Cosmology course. This talks of a universe with a length scale R and an expansion speed Ṙ (R-dot), which is the rate of change of R. They can be combined as the Hubble parameter H = Ṙ / R. The Friedmann equation is then given as:

The G is Newton’s gravitational constant, ρ is the density, k is the curvature constant, and c is the speed of light. We can plot different curves for different initial densities and curvature constants. Then we can match those curves to observations to work out what sort of universe we live in. Of course, the expectation was that the expansion was slowing down. That’s because the consensus was that we lived in what you might call a cannonball universe.

*The cannonball universe*

That’s where the expanding universe is modelled like a cannonball fired aloft. See John Peacocks’s cosmological physics for an example: *“the dynamics of the entire universe are the same as those of a cannonball fired vertically against the Earth’s gravity. Just as the Earth’s gravity defines an escape velocity for projectiles, so a universe that expands sufficiently fast will continue to expand forever. Conversely, for a given rate of expansion there is a critical density that will bring the expansion asymptotically to a halt”. *Google on Friedmann escape velocity for other examples such as in The Road to Reality by Roger Penrose, the Teach Astronomy website, and the Hubblesite. The Friedman equation section of the Jodrell Bank *Introduction to Cosmology* course says *“we can more or less see where it comes from by using only Newton’s inverse-square law of gravity”*. It talks of Newton and his gravitational sphere and two galaxies, one in the middle of the sphere, one on the outer edge. The latter galaxy has a kinetic energy T = ½mv² and a potential energy V = -GMm/r. It says T + V is constant by virtue of conservation of energy, and the mass of the sphere M can be rewritten as average density ρ times volume 4/3πr³. With a few more steps it gives an expression which is almost the Friedman equation. Michael Redmond gives a similar description in the expanding universe, through Newtonian eyes. As does Karura in physics made easy:

*Image by **Karura**, see **physics made easy, cosmology II*

Again the expanding universe is treated as something fighting against gravity. Either the cannonball’s initial speed exceeds escape velocity and it departs the Earth forever. Or it doesn’t have enough initial speed and ends up falling back to Earth. Or it has exactly escape velocity and its speed asymptotes to zero. There’s just one little problem with this cannonball universe: it’s totally wrong.

*A critical clue*

The problem should have been telegraphed by the flatness problem. Density ρ is related to curvature constant k, and the idea is that if k is less than zero the universe expands forever. That’s because there isn’t enough gravitational attraction to stop the expansion. Alternatively if k is greater than zero, gravitational attraction will cause the universe to contract. If however k is zero, the universe has an Ω = 1 critical density where ρc = 3H²/8πG and just about manages to keep expanding slower and slower. That’s where we seemed to be. The standard Big Bang model is said to be balanced on a knife edge, such that density of the universe is incredibly close to the critical density. Alan Guth talked about it on page 176 of his 1997 book The Inflationary Universe. He referred to a pencil-point instability, and said one second after the Big Bang, omega must have been between 0.999999999999999 and 1.00000000000001. Guth claimed inflation is responsible for this because it drove omega to 1 with “exquisite” accuracy. He said it’s as obvious as blowing up a balloon, and that inflation makes the universe look flat for the same reason that the surface of the Earth looks flat. The trouble is, the surface of the Earth isn’t flat. You can see it isn’t flat from space. Or when you look out to sea and see a ship hull-down on the horizon. Inflation didn’t fix the flatness problem, it just swept it under the carpet. So here we are a hundred years after Einstein’s cosmological considerations, with the selfsame problem. We have a non-credible reheat of Einstein’s greatest blunder, wherein a slight increase in density would trigger contraction, and a slight decrease would trigger expansion. Even though it’s twenty years since the 1998 supernova observations. That’s when cosmologists tried to match the cannonball curves to observations to work out what sort of universe we live in. They expected to measure a slowdown in the expansion. Instead, what they measured, was an expansion that was speeding up.

*Omega gets split*

That didn’t fit with any of the cannonball curves. But instead of saying this killed the cannonball universe and kicked critical density into touch, we now have a density parameter Ω=1 split into a matter density of Ω_{M} = 0.3 and a dark energy density Ω_{Λ} = 0.7. The omega-m matter density is said to make the universe contract, whilst the omega-lambda dark energy density is said to make it expand. You can plot the evolution of the universe for the various values of Ω_{M} and Ω_{Λ} on Patrick Leahy’s PC3392 cosmology web page. Click on the left side of the chart where Ω_{M} = 1 and Ω_{Λ }= -2 and you get the k > 0 curve where your universe collapses. But click near the bottom of the chart where Ω_{M} = 0 and Ω_{Λ }= 1, and your universe expands forever:

*Image by **Patrick Leahy see his PC3392 cosmology** course *

It sounds plausible. Unfortunately there’s a whole bunch of problems. Starting with the c in the metric of the universe ds² = a(t)² ds₃² – c²dt², and in the Friedmann equation H² = 8πGρ/3 – kc²/R². Friedmann derived it from Einstein’s field equations of gravitation, but Einstein said a gravitational field is a place where the speed of light is spatially variable.

*One has to be wary of c in gravitational equations *

That’s why optical clocks go slower when they’re lower. The cannonball falls down *because* the speed of light is slower when it’s lower. There’s a similar issue for mass m in the Newtonian derivations. Kinetic energy is ½mv² and potential energy is GMm/r. But when you throw a cannonball up in the air you do work on it. You add kinetic energy to it. As the cannonball rises and slows down, gravity converts the kinetic energy into potential energy. This potential energy is in the cannonball, nowhere else. Hence there’s an increase in the cannonball mass m, which is the flip side of the mass deficit. Some might say mass m is E/c², but the mass m changed because c changed, so one has to be wary of c in gravitational equations. Especially when they describe the cannonball universe, which is totally wrong. It’s space that’s expanding, not the distribution of galaxies in space. A gravitational field might be a place where cannonballs fall down, but it isn’t some Chicken Little place where space is falling down. Hence cannonball gravity is not appropriate for the expanding universe.

*What curvature?*

If the only problem was a spatially variable c it wouldn’t be the end of the world, because you could say c is just a local speed or just a conversion factor between distance and time. But the problems don’t stop there. They get worse, particularly for curvature k. That’s because a gravitational field isn’t a place where space is curved. Matter doesn’t tell space how to curve. Matter tells *spacetime* how to curve. Only it’s not actually matter, it’s a non-uniform energy density that does this. There’s spacetime curvature around the Earth because the energy density of rock is much greater than the energy-density of space. As a result the surrounding space is inhomogeneous in a non-linear fashion, so we model it as curved spacetime. But this curved spacetime isn’t some combination of curved space and curved time. So Friedmann’s 1922 paper on the curvature of space, and his 1924 paper on the possibility of a world with constant negative curvature of space. were missing the trick: that spatial curvature is associated with electromagnetism, not cosmology. It’s clear he’d taken a lead from Einstein’s “outlandish” curvature of space which was *“not justified from our knowledge of gravitation”*, and that he didn’t know how gravity works. Sadly Friedmann died in 1925 and couldn’t fix the issues. But what’s now known as the Friedmann equation lives on. Even though light doesn’t curve towards the Earth because space is curved. Or because spacetime is curved. It curves because space is inhomogeneous. Spacetime curvatures relates to the tidal force, not the “force” of gravity. Light curves where there’s a gradient in potential, where the speed of light varies, where spacetime is “tilted” as opposed to curved. Yes, spacetime curvature is important because as per the rubber-sheet analogy, if you have no spacetime curvature your plot stays flat and level. But it’s the gradient that counts. To appreciate this take a stiff board, lift one side up, and roll a marble across it. The marble follows a curved path because the board is tilted, not because the board is curved:

*Marble from the **house of marbles**, board and arrow added by me*

It’s similar for the room you’re in. The force of gravity is 9.8 m/s² at the floor and at the ceiling, so there’s no detectable tidal force, and no detectable spacetime curvature. But your pencil still falls down. That’s detectable. However it doesn’t fall down because spacetime is curved. It isn’t the tidal force that makes your pencil fall down, it’s gravitational force. And that gravitational force isn’t there because space is curved. Spatial curvature is associated with electromagnetism, not gravity. See what Percy Hammond said in the 1999 Compumag: *“we conclude that the field describes the curvature that characterizes the electromagnetic interaction”. *It’s the electron, not the geon. Electrons do not drift towards some great attractor. Why would anybody think that space was somehow curved on a grand scale? The story goes that in days of old, people could not conceive of a world that was curved. They could only conceive of a world that was flat. Nowadays we have cosmologists who cannot conceive of a universe that’s flat. They still talk of curvature even though the 2003 Boomerang experiment indicated that the universe was flat. Even though the Wilkinson Microwave Anisotropy Probe confirmed it, as did the Planck mission, with a 0.4% margin of error.

*Images from James Schombert‘s UOregon 21st century science*

So when it comes to deriving an equation for the evolution of the universe from Einstein’s field equations of gravitation, one has to ask why spatial curvature is relevant at all. Or indeed any other curvature. So one has to ask: *what curvature?*

*What gravitation?*

The problems don’t stop there. Because Friedmann is said to have derived his equations to describe *“the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity”*. He is said to have derived his equations *“from Einstein’s field equations of gravitation” *employing the *“assumption that the universe is spatially homogeneous and isotropic”*. But in 1920 Einstein made it clear that a gravitational field is a place where space is neither homogeneous nor isotropic. A gravitational field is a place where space is *in*homogeneous. If your energy density is uniform, like it is inside the spherical shell, space is homogeneous. Then spacetime is not curved, light goes straight, and your pencil doesn’t fall down. It’s similar if you’re midway between two stars. Or if there are no stars at all. A homogeneous isotropic universe is a universe where there is no gravity. So one has to ask: *what gravitation?*

*What critical density?*

Hence when it comes to modelling the universe from Einstein’s field equations of gravitation, the cosmological principle doesn’t look like a good place to start, and the Friedmann equation doesn’t look like a good place to finish. The speed of light c is suspect because a gravitational field is a place where the speed of light varies. The curvature k is suspect because a gravitational field is not a place where space is curved. And the density ρ is suspect because when energy density is uniform there is no gravitational field. If he was still around Paul Dirac would be saying the gravitational constant G was suspect too. But whether it is or isn’t, right now on the largest scale, space is flat. As it was a billion years ago. As it will be a billion years from now. It’s flat because on that largest scale the energy density is uniform. So space is homogeneous. So light goes straight and your pencil doesn’t fall down, because there is no overall gravitational field for the universe. So what are we left with? A pressure term p, and since we talk of stress-energy, the density term ρ is perhaps just the other side of the coin. So I think it’s fair to say we’re still left with an equation of state. And I think it’s also fair to say spacetime is curved because space is expanding. But it isn’t expanding *because* spacetime is curved. Or because of some gravitational effect. You might say a gravitational field is a pressure-density gradient in space, but that doesn’t make space expand, or contract. It doesn’t make the sky fall down, or up. Space expands because that’s what space does. Because it has an innate stress-energy, stress being directional pressure. It’s expanding because it has that quality that Schrödinger called cosmic pressure. That means we’re left with a density parameter Ω = ρ / ρc which doesn’t much matter. And a flatness problem that never was a problem, because Ω doesn’t have to be the same as the critical density for the universe to be flat. There is no knife-edge. There is no pencil-point. The critical density isn’t critical at all. So one has to ask: *what critical density? *

*The Big Crunch*

All this means two out of three answers to the shape of the universe were always going to be wrong. And since the expansion of the universe is increasing, it isn’t just two out of three cannonball curves that were wrong. It’s *all of the above. *We do not live in some cannonball universe. Our universe is expanding, and that expansion is increasing. Which is bad news for the Big Crunch:

*Big Crunch image *© *copyright **howstuffworks.com*

See the 2009 Universe Today Big Crunch article by John Villanueva. He says scientists used to believe that only two factors were at work: gravitational attraction between all the galaxies, and their outward momentum due to the Big Bang. He says a logical prediction was that gravity would win in the end. But that prediction wasn’t logical at all. We talk of galaxies being gravitationally bound, in that space expands between the galaxies but not within. So it’s logical to say gravity can reduce the expansion of the universe. But it isn’t logical to say gravity can reverse it. Gravity alters the motion of light and matter through space, but it doesn’t suck space in. Your brakes will slow down your car. But once you’ve stopped your car, putting your foot on the brake pedal won’t make your car go backwards. Friedmann was a ballistics instructor on the Austrian front. But the universe is not some cannonball. If anything it’s more like a photon. And since optical clocks go slower when they’re lower because speed of light is spatially variable, the ascending photon speeds up. Which means it’s crunch time for the Big Crunch. It was never going to happen. The universe didn’t collapse due to its own gravity when it was small and dense, and it never ever will. As for what will happen, well that’s the $64,000 question, isn’t it? So what other options have we got?

*The Big Bounce*

The Big Bounce fell out of favour when cosmic inflation caught on. However inflation has come in for some stick recently, and the Big Bounce has enjoyed some renewed interest. Neil Turok advocates a Big Bounce universe. See the 2016 PhysOrg article The Big Bang might have been just a Big Bounce. It features Turok and Steffen Gielen. It quotes Gielen saying quantum mechanics could *“save the early universe from such violent beginnings and endings as the Big Bang and Big Crunch”*. That would mean the Big Bounce isn’t quite what some people say. Rather than a Big Crunch followed by a Big Bang, it’s neither, because it avoids singularities. That’s something I like. So when I read the associated paper Perfect Quantum Cosmological Bounce I feel sympathetic. Especially since they start by saying observations indicate a simple universe with a spatially flat geometry, nearly scale-invariant fluctuations, and no signals of primordial inflation. I feel happy enough with their conclusion too, wherein *“a valid semi-classical approximation to quantum cosmology with conformal matter can be obtained from complex classical paths which avoid the classical big bang singularity”*. The paper currently has 48 citations, so some people are thinking seriously about this sort of thing. One of the citers is Paul Steinhardt, a big critic of inflation. He and Turok wrote The Cyclic Model Simplified in 2004. Whilst I have no sympathy for *“the M-theoretic notion that our universe consists of two branes separated by a microscopic gap”*, I can’t point out any obvious flaw with the idea of a contraction prior to the Big Bang.

*Big bounce image from New Scientist article **From big bang to a big bounce** by Anil Ananthaswamy, also see **Baez*

I can however point out that there’s no evidence for it, and that a contracting universe prior to the Big Bang is a very different thing to our universe contracting in the future. Our universe is expanding. Due to dark energy, or cosmic pressure, or cosmic tension if you prefer. Since stress is directional pressure and stress and tension are associated with elastic things, one can conceive of a universe that expands and contracts repeatedly, like a Lagrangian mass on a spring. But I have to say that I think the increasing acceleration of the universe plus bubblegum makes it sound like wishful thinking.

*The Big Freeze*

The Big Freeze however sounds far more plausible. See what Luke Mastin says about it in physics of the universe: *“**Perhaps the most likely possibility, however, based on current knowledge, is a long, slow decline known as the “Big Freeze” (or the “Big Chill” or “Heat Death”). In this scenario, the universe continues expanding and gradually “runs down” to a state of zero thermodynamic free energy”. *Also see the Wikipedia future of an expanding universe article. It says much the same thing. But it also says stars are expected to form for 10^{12} to 10^{14} years. That’s a hundred *trillion* years. The article goes on to say *“as existing stars run out of fuel and cease to shine, the universe will slowly and inexorably grow darker, one star at a time”*. That sounds fair enough. Eventually, the universe would look like this:

*Heat death image by me** *

But the article also talks about proton decay and black holes disappearing as they emit Hawking radiation, which isn’t true. So when it talks about a universe with a temperature so uniform that no further work or life is possible, perhaps that isn’t true either. Particularly since the universe at large is rather cold already at 2.73 K, and most of it is dark. Look up at the stars on a clear night: most of the universe is darker than that. I think of it as a vast dark frozen plain. Here and there, separated by huge distances, are streetlamps. Each streetlamp shines both ultraviolet and infrared light down on the ground, warming a small patch of it, such that it is green with life. When the streetlamp goes out, the life is extinguished. But so much of the universe is dark frozen plain that the change isn’t major on the scale of things. And it doesn’t much matter to life if the temperature of the universe at large is 2.73 K or 0.73 K. The Big Freeze is here already, but here we are, and a hundred trillion years is plenty of time to work out how to do the GRB trick and make some stars of our own. Stars that are somewhat more efficient than those supplied by nature. That might buy enough time to convert space into energy or even answer the last question. Life finds a way. The Big Freeze is my favourite, because it is not the end.

*The Big Rip*

But is it the most likely? There are other scenarios, such as eternal inflation which smacks of turtles all the way down, as does conformal cyclic cosmology. There’s also the false vacuum, which has no actual scientific foundation. As per Adam Becker’s 2015 BBC article how will the universe end? the false vacuum *“sounds like something out of science fiction, and in a way it is”*. However it isn’t totally unlike the Big Rip. That’s where *“even spacetime itself, is progressively torn apart by the expansion of the universe”*.

*Big Rip image by Jeremy Teaford, Vanderbilt University, see New model of cosmic stickiness favors ‘Big Rip’ demise of universe*

I fear this could be the most likely scenario. I say that because of the balloon analogy. A balloon in vacuum is an analogy for the universe. The pressure of the air inside the balloon is counterbalanced by the tension in the skin, and you can make the balloon expand by blowing in more air. However since energy is pressure x volume, that would be in breach of conservation of energy. But you could also make the balloon bigger by reducing the tension in the skin. This sounds impossible until you think *bubblegum*. When you blow a bubblegum bubble, you are inflating a balloon of sorts. But as this balloon expands, the skin gets thinner and weaker, and less able to resist the expansion. So it expands further, so the skin gets weaker, and so on. This fits with what I know, as does the fact that bubblegum balloons sometimes end badly. See page 5 of New Physics at Low Accelerations (MOND): an Alternative to Dark Matter. Mordehai Milgrom mentions the strength of space. In a way, space has a tensile strength. It’s *harder than diamond and stronger than steel, because such things are made of it*. That’s why light waves propagate so fast. Space just doesn’t have an innate cosmic pressure, it has a tension too. If it didn’t have both, waves wouldn’t run through it. But everything has its limits. As for what might happen if space reaches its limits, there’s something in a blog article by Phil Plait that bothers me. It’s called Prince Rupert Drops. Exploding. He refers to it’s OK to be smart featuring Destin Sandlin from Smarter Every Day. The video is great. Take a fast-cooled long-tailed tadpole of glass and hit it on the head with a hammer. Nothing happens. But snip a little piece of tail off, and the whole thing explodes into fragments, with a fracture front moving at a mile a second. I rather think the universe could end like that. Like the way life always ends. Badly. In tears. Perhaps there’s a fracture front heading our way as I speak. As Adam Becker said in his BBC article, this is a bit of a bleak thought. If it upsets you, here is a picture of a cute kitten:

## Andy

14 Feb 2019On and off for a while I have been thinking of a website that talks about the things I am interested in. But what it seems like a large percentage of the rest of the world is interested in is cats. So I was thinking of combining physics and cats. Well done John. You beat me to the punchline. Oh, and congratulations on the blog. It is an impressive piece of work. I hope you get some recognition for it at some point.

## John Duffield

15 Feb 2019Many thanks Andy. We’ll see about the recognition, but I’m not holding my breath. I just thought I ought to “get it out there” and share what I know. As for the cat, I just creased myself when I read that BBC article by Adam Becker and saw that frightened kitten. LOL, I felt I just had to share that too!