Can Modifications to Gravity Explain the Accelerating Expansion of the Universe?

[inflector]

In Lee Smolin's book, The Trouble with Physics, pages 15 and 16, he states:

Or there is a new form of matter—or energy—that becomes relevant on these very large scales: That is, this new form of energy affects only the expansion of the universe. To do this, it cannot clump around galaxies or even clusters of galaxies, This strange new energy which we have postulated to fit the data, is called the dark energy.

Most kinds of matter are under pressure, but the dark energy is under tension—that is, it pulls things together rather than pushes them apart. For this reason, tension is sometimes called negative pressure. in spite of the fact that the dark energy is under tension, it causes the universe to expand faster. If you are confused by this, I sympathize. One would think that a gas with negative pressure would act like a rubber band connecting the galaxies and slow the expansion down. But it turns out that when the negative pressure is negative enough, in general relativity it has the opposite effect. It causes the expansion of the universe to accelerate.

Can someone explain how this works?

Does this mean that modifications to gravity causing tension at large intergalactic scales could account for the accelerating expansion of the universe as well?

 

[Chalnoth]

Can someone explain how this works?

Does this mean that modifications to gravity causing tension at large intergalactic scales could account for the accelerating expansion of the universe as well?

Well, in GR, it's that very tension that causes the accelerated expansion.

Perhaps the easiest way to see that such a type of matter causes an accelerated expansion is to examine the simplest case, where the density is a constant (a cosmological constant). In this case, the energy density is independent of both time and space, making our equations obscenely simple. Friedmann's equation can be written as:

$$H(a)^2 = \rho(a)$$

Here I've neglected the constants for clarity. $$H(a)$$ is the Hubble parameter, which parameterizes how fast the universe is expanding. I've written it as a function of the scale factor. $$\rho(a)$$ is the energy density of the universe.

So if we have a universe that is completely dominated by a cosmological constant, then we have a very simple situation:

$$H(a)^2 = \rho$$

Where $$\rho$$ is now a constant. To determine what this means, we have to make use of the definition of H:

$$H = \frac{1}{a}\frac{da}{dt}$$

So we have the simple differential equation:

$$\frac{1}{a}\frac{da}{dt} = const$$

I can call this constant $$H_0$$, and multiply both sides by a:

$$\frac{da}{dt} = H_0 a$$

If you know your differential equations, you know that this is the definition of an exponential function: a function whose growth is proportional to its size.

$$a(t) = a(0) e^{H_0 t}$$

Thus, if we have a universe dominated by a cosmological constant, then we have a universe with exponentially-accelerated expansion.

But what does all this have to do with pressure? That comes back to the property used to determine the expansion: how the energy density changes with time. The basic idea is that if we expand the universe by some amount, and have some stuff with no pressure, then the density drops: you have the same amount of energy, but now it's diluted. However, if you have pressure, and imagine an imaginary box around the stuff that's expanding, then the walls of the box end up doing work on the stuff as it expands.

For example, if we take photons, which have positive pressure, and cause them to expand, they have a force in the direction of expansion (due to the positive pressure), which means that the walls of the box do negative work on the photons (they want to expand), which means that the total energy of the photons drops.

But what if we want something to keep its energy density? To do that, it needs to not only keep its energy as it expands, it needs more energy: it needs the walls of the box to do positive work on it. Which means that its pressure must oppose the expansion. In that situation, stuff with negative pressure, as it expands, has work done on it by gravity which increases its total energy.

 

[inflector]

Thank you. That sure is counter-intuitive.

Consider a formula for gravity that took the form:

$$g = -\left(\frac{G_1}{r^2} + \frac{G_2}{r} + G_3r\right)$$

So at large distances, the $$G_3$$ term wound dominate and gravity would, in effect, act like a big spring, since the $$G_1$$ and $$G_2$$ terms go to zero as $${r}$$ goes to infinity. You'd be left with Hooke's law in effect:

$$g = -G_3r$$

So does that mean that a modification of gravity that took this form or any other form that at large distances looks like Hooke's Law would result in an expanding universe?

 

[Chalnoth]

Thank you. That sure is counter-intuitive.

Consider a formula for gravity that took the form:

$$g = -\left(\frac{G_1}{r^2} + \frac{G_2}{r} + G_3r\right)$$

So at large distances, the $$G_3$$ term wound dominate and gravity would, in effect, act like a big spring, since the $$G_1$$ and $$G_2$$ terms go to zero as $${r}$$ goes to infinity. You'd be left with Hooke's law in effect:

$$g = -G_3r$$

So does that mean that a modification of gravity that took this form or any other form that at large distances looks like Hooke's Law would result in an expanding universe?

No, I don't think so. In this case, if you take two test objects, and separate them by some large distance, they'll end up with some large positive gravitational potential energy.

By contrast, in the example I described above, the energy of the cosmological constant that grows with time can be thought of as being balanced by an increasingly negative gravitational potential energy. So the effect would be the exact opposite: such a universe, if it is dominated by normal matter, would tend to just collapse in on itself.

To get an accelerated expansion, you'd have to have $$G_3 < 0$$, which is adding a repulsive term to the potential.

 

[inflector]

So the effect would be the exact opposite: such a universe, if it is dominated by normal matter, would tend to just collapse in on itself.

To get an accelerated expansion, you'd have to have $$G_3 < 0$$, which is adding a repulsive term to the potential.

Hmm, now I'm really confused. What you just said matches my first intuition but not what I thought Smolin was saying.

If as Smolin says: "[dark energy] pulls things together," how is that different than a rubber-band like gravitational force?

How does your "repulsive term," as Smolin says, "pull things together?"

 

[Chalnoth]

Hmm, now I'm really confused. What you just said matches my first intuition but not what I thought Smolin was saying.

If as Smolin says: "[dark energy] pulls things together," how is that different than a rubber-band like gravitational force?

How does your "repulsive term," as Smolin says, "pull things together?"

The difference is gravity. In one situation, it's gravity doing the pulling. In the other, it's this form of matter, dark energy, doing the pulling. And the way that gravity acts on dark energy's attempt to pull causes it to actually repel.

As a side comment, the dark energy doesn't actually interact directly, so far as we know, with other forms of matter, so it only pulls itself. But the action of gravity is such that its attempt to pull itself supports an accelerated expansion.

 

[inflector]

I can tell it is going to take a while for me to understand this concept.

 

[Chalnoth]

I can tell it is going to take a while for me to understand this concept.

Well, one of the interesting things about gravity is that it doesn't just work on energy, but also pressure. A bunch of matter under positive pressure (that is, the pressure tries to blow apart the object) actually has an increased gravitational attraction due to that pressure. Some of the gravity of a neutron star, for instance, comes from the intense pressure that the nucleons feel (I'd have to look up exactly how much, as I can't remember off the top of my head).

One of the interesting things about gravity is that if you can think up a form of matter that feels extremely strong positive pressure, the gravitational attraction caused by that pressure will actually cause any matter of that sort to just collapse in on itself.

The way that gravity interacts with dark energy is just the reverse of this phenomenon: it has so much negative pressure, compared to its positive dark energy density, that the net effect of gravity is to cause a repulsion.

 

[inflector]

That's very interesting. I never thought about gravity acting on energy, but I guess that makes sense since it can curve photons.

I had another thought.

Is it not possible that if there were some kind of tension in gravity itself at very large distances (like the formula I listed above), that there could be acceleration on the inner part of the universe from the pull of the outer part of the universe (assuming it is of finite dimensions) left over from the energy of the Big Bang itself? And that if the entire universe is sufficiently larger than the visible universe and the visible universe is towards the center of the entire universe, we might only see the acceleration?

Consider what would happen if you had a giant ball of a 3D mesh of rubber-band like material, and you compressed it down to the smallest portion possible, and then quickly released it. Since a rubber-band has spring-like energy in compression and tension, the energy of compression would create an expansion force and resulting outward acceleration when the tension was released. At first, the outer-most layers would have their force vectors coming entirely from the center of the ball, so they'd experience the greatest acceleration, areas of the ball closer to the center would experience some combination of outward and inward force and therefore less net acceleration.

After a while, the force at the outer layer would diminish and the outer layers would stop accelerating as the mesh transitioned from compression to tension and the force turned from outward to inward at the outer layers. Since the outer layers had the most acceleration from the start of the "big bang", they would have the greatest kinetic energy, so this decelerating tension on the outer layer after it transitioned from compression to tension would result in an acceleration of the layers more towards the center, in a wave that moved towards the center over time.

Could something like this explain the acceleration of the visible universe given a large enough ratio between the entire universe and the visible portion?

 

[Chalnoth]

I'm not sure. The idea is a bit too vague for me to know precisely what you're talking about.

However, currently the question is entirely open as to whether the cause of the observed acceleration is some modification of gravity on very large distance scales, or some unknown form of matter/energy. We only use the term "dark energy" because it takes too long to say, "either some form of dark energy or modified gravity." Typically the modifications of gravity are proposed by considering a different action.

The action for gravity is:

$$\int \sqrt{-g} R d^4x = S_{matter}$$

(this is called the Einstein-Hilbert action, if you want to look into it in more detail)

On the left hand side of the equation we have the action due to gravity. The $${\sqrt{-g}}$$ is a normalization factor that makes it so that a change in coordinates doesn't change the physics. The meat of it, though, is the parameter $$R$$. This is a function of space and time that describes the amount of curvature in any particular place and time.

On the right hand side of the equation, we have the action due to matter, which we needn't go into, but suffice it to say when we evaluate the action to determine the equations of motion, the right hand side becomes a tensor that depends upon the energy, momentum, and stresses of matter.

When people want to come up with a modification to General Relativity, usually they look for ways to modified the left hand side of this equation. For example, what if the action isn't just R, but some other function of R? It could be, for instance:

$$R + \alpha R^2$$

...with $$\alpha$$ being some suitably small number that current experiments wouldn't have detected the difference. People have been looking at these possibilities for some time now, and it appears that it's actually pretty difficult to find a modification of gravity of this form that explains the acceleration we observe, but isn't already ruled out by measurements of gravity here on Earth and within our own solar system. This doesn't mean it's impossible, but it seems much easier to write down a theory with some sort of dark energy that isn't already ruled out.