The expansion is accelerating due to negative enough pressure

...no strings attached...

In recent years, I got used to the fact that Sean Carroll is confused about some very basic physics – the postulates of quantum mechanics as well as thermodynamics (and the very basic insight due to Boltzmann and others that its laws are microscopically explained by statistical physics and not, for example, by cosmology). And I won't even threaten your stomach by memories of the Boltzmann brains, doomsdays, and similar delusions.

But I thought he could rationally think at least about classical general relativity. His book was pretty good, I thought, although I have never read the whole volume. However, I don't think so anymore after I finished reading Carroll's insane tirade called

Why Does Dark Energy Make the Universe Accelerate?

in which he tries to assault an elementary fact that the reason behind the acceleration of the expansion of the Universe is the negative pressure (caused by dark energy).

His crusade is made even more paradoxical given the apparent fact that he knows the equation and other key pieces needed to understand why it's accurate to say that the acceleration is caused by negative pressure. But like a schoolkid who has just mindlessly memorized an equation but can't understand what it means, he just can't sort out what the basic implications of the equations are. So he wants to "ban" the fact that the negative pressure is the reason for the acceleration from expositions of cosmology. You may imagine that a progressive (i.e. Stalinist) like himself thinks that such a ban would be "a great step forward". Bans aren't a good step forward, especially not bans of key scientific insights.

(Brian Greene would be among those whose books would be banned; he wrote a crisp explanation of these matters in The Hidden Reality. Tony Zee's GR book would be on the black list, too. Zee mentions beginners' i.e. Carroll's confusion of the velocity and acceleration in 2nd paragraph on page 500 – and more generally, between pages 499 and 507.)

Since the late 1990s, we've known that the Universe was not only expanding but the rate of the expansion was increasing. It was a surprise for many because most people were expecting that the rate was slowing down. The substance driving this expansion is "dark energy" – the cosmological constant with \(p=-\rho\) is the simplest and most natural "subtype" or "more detailed explanation" of dark energy that is so far compatible with all statistically significant experimental results.




The property that allows dark matter or cosmological constant to accelerate the expansion is its negative pressure \(p\lt 0\). Why is that? Well, it is because of the so-called second Friedmann equation\[

\frac{\ddot a}{a} =-\frac{4\pi G}{3} (\rho + 3p)

\] The numerator on the left hand side contains the second derivative of the scale factor \(a\). You may literally imagine that in some units, \(a\) is nothing else than a distance between two particular galaxies (well, the proper length of a line that connects them through the \(t={\rm const}\) slice which is, let's admit, not a geodesic, but it is some coordinate distance, anyway).

The second derivative of this distance is fully analogous to the acceleration \(a_{\rm acc}=-\ddot h\) of a ball that you threw somewhere. Note that Earth's gravity implies \(a_{\rm acc}=-\ddot h=g\) which means that the ball will ultimately fall down (unless its speed exceeds the escape velocity: we would have to modify the equation if the ball could reach substantial distances from the surface) towards the Earth.

Note that we use the convention in which a positive acceleration \(a_{\rm acc}\gt 0\) means that the ball is attracted to the Earth i.e. the second derivative of its height is negative. That's why we had to insert the minus sign.




The second Friedmann equation is completely analogous. It's not just some vague popular analogy; it is a mathematical isomorphism. The distance between two galaxies is fully analogous to the distance between the ball and the Earth's surface. In both cases, they are attracted by the gravitational force (of a sort). In the second Friedmann case, the gravity follows somewhat more accurate laws imposed by the general theory of relativity – the Friedmann equations are what Einstein's equations of GR boil down to if we assume a uniform, isotropic Universe.

You may see that the role of the Earth's gravitational acceleration \(g\) is being played by\[

\frac{4\pi G}{3} (\rho + 3p)

\] The minus sign in front of the right hand side is there for the same reason as in the case of the ball: attraction (deceleration of the outward speed) is identified with a negative second derivative of "the" quantity (the height of the ball or the distance between two galaxies).

So the total force is "attractive" if\[

\rho + 3p \gt 0.

\] For example, if the Universe were filled with the dust only, and the dust has \(p=0\), this expression would surely be positive and we would get an attraction i.e. decelerated expansion. A positive energy density implies attraction for the same reason why the Earth's positive energy (and energy density) is able to attract the ball. Ordinary gravity is simply attractive. If the Universe were filled with radiation and nothing else, \(p=+\rho/3\) (with the plus sign) and the two terms would actually have the same sign and double: an even clearer deceleration.

The type of matter that has \(p=-\rho/3\) is actually "cosmic strings". If the Universe were filled with cosmic strings only (in chaotic directions), they would contribute nothing to the acceleration. Cosmic domain walls (membranes of a sort) would have \(p=-2\rho/3\) and the expression would already be negative. The domain walls would make the expansion accelerate.

Similarly, the cosmological constant – the most motivated type of dark energy – has \(p=-\rho\) so \(\rho+3p\) is negative. In general, you see that you get an accelerated expansion if \(p\) is not only negative but smaller than \(-\rho/3\),\[

p \lt -\frac{\rho}{3}.

\] This is the only refinement of the claim that "a negative pressure is the cause of the acceleration". In fact, we need a "sufficiently negative pressure", one obeying the inequality above. But otherwise the statement is 100% accurate – and not just at the level of popular presentations. It's also completely accurate to say that the total gravitational force operating in between the galaxies becomes repulsive – you may even call it "antigravity" – when the pressure in between is sufficiently negative.

Carroll tries to claim that there is something wrong with the proposition that "the acceleration is caused by a [sufficiently] negative pressure" but his argumentation seems utterly irrational. Well, the core of his would-be argument is probably the following:

But, while that’s a perfectly good equation — the “second Friedmann equation” — it’s not the one anyone actually uses to solve for the evolution of the universe. It’s much nicer to use the first Friedmann equation, which involves the first derivative of the scale factor rather than its second derivative (spatial curvature set to zero for convenience):\[

H^2 \equiv \zav{ \frac{\dot a}{a} }^2 = \frac{8\pi G}{3} \rho

\]

So Carroll told us that we should switch to this equation because "people use it more often" and it is "nicer". The problem with this would-be justification is that it is no justification at all. If an equation is used more often or looks "nicer" to someone (for other irrational reasons), it does not imply that this equation is the right equation to explain a pattern or to answer a question.

In this case, we want to explain why the acceleration is negative and the acceleration is simply related to the second derivative of the height or the second derivative of the scale factor \(a\). The last displayed equation above, the first Friedmann equation, doesn't include the second derivative \(\ddot a\) at all, so it can't possibly be the right equation that tells us whether the acceleration is positive or negative!

I am stunned that Carroll isn't capable of figuring this simple point out.

So the mathematical formalization of the reason why the expansion is accelerating is the second Friedmann equation and it doesn't matter a single bit whether this equation is used more often or less often to calculate other things or answer other questions.

What is the alternative proposition that Carroll proposed instead of the correct one? It isn't quite clear but it seems that it's the bold face sentence below:

Second, a constant energy density straightforwardly implies a constant expansion rate \(H\). So no problem at all: a persistent source of energy causes the universe to accelerate.

But this sentence is just incorrect. The energy density carried by dust or anything else is "persistent" in the sense that it remains nonzero forever but the dust implies a decelerating expansion (much like most other known types of energy density). If the word "persistent" were interpreted as "constant", the sentence above would be marginally correct but it would completely obscure the reason why the energy density is able to stay constant in an expanding Universe. The reason for this is the negative pressure, too; Sean has only offered a sleight-of-hand to mask the actual reason, the negative pressure. Even Sabine Hossenfelder knows that.

The relevant quantity for the sign of the acceleration is \(\rho+3 p\) and not just \(\rho\) (or by \(\dot\rho\)) as Carroll incorrectly suggests. This influence of the pressure on the curvature of the spacetime (in this particular case, the acceleration of its expansion) is one of the "refinements" that general relativity brought us relatively to Newton's gravity where only the total energy density mattered for all gravitational fields. In GR, the whole stress-energy tensor (not just the energy density but also the pressure and the density of momentum etc.) matters for various aspects of the spacetime curvature.

Carroll correctly states that the first Friedmann equation and the second Friedmann equation are consistent with one another because one may be derived from the other using the general relativistic form of the energy conservation law (which does depend on the pressure as well). All this stuff is OK but it changes nothing about the fact that he gave a completely incorrect answer to the key question which of the equations is the right one to calculate the sign of the acceleration of the expansion of the Universe.

These sentences of mine are no "popular presentations" and surely not "misleading popular presentations" and whoever understands them really understands what drives the acceleration etc. – it is not just an illusion of the understanding – while Carroll apparently does not understand these basic facts. He does not understand that the pressure has become relevant for some questions about the spacetime curvature – in particular, for the question whether the expansion is accelerating.

The relevant equation is unquestionably the second Friedmann equation, whether a pervert finds it nicer or not, and I urge all writers to keep on writing the absolutely valid claim that the negative pressure is the reason and notice that Sean Carroll is just [being?] an idiot.

And that's the memo.