I get email: Dark Energy and Universal Expansion

Some time ago, I wrote a 4-part tutorial on common misconceptions about the expanding universe: Part 1, Part 2, Part 3, Part 4. Following my latest cosmologically oriented io9 column, I got a an interesting query from a reader about the relationship between dark energy and the expanding universe. His question:

I read your article about dark energy a few days ago; and though I’m just an ignorant bystander in the world of science, I did a bit of Google research and read about how, within about two trillion years, dark energy should and/or might possibly — and I’m going to butcher the explanation — redshift all known methods of detection beyond our observation…

Why and how do we assume that dark energy, and whatever other factors are involved, will accelerate the expansion of the universe at our current estimated rate?

He also asks whether this is a possible doomsday scenario a subquestion which, I realize in retrospect, I completely failed to answer. (The answer to this latter part is yes, absolutely. The universe will get infinitely large and diffuse, and ultimately everything will become cold and evaporate and increasingly desolate. It’s the end of the universe in the most drawn out, depressing way possible).

However, the question is a good one, and so I wanted to do a “technical” post (see the tags) on the finer points of the content of the universe.

The universe will continue expanding exponentially at its current rate because of what a cosmological constant or dark energy is. I’ll get to that in a moment, but first I wanted to tell you a little about the rate of expansion of the universe. The rate of expansion at any given instant is governed by two things:

  1. The average density of all of the mass at that time. This includes everything: ordinary and dark matter, radiation, dark energy, and anything else you can think of. “But wait!” you might say. “Radiation, for instance, doesn’t have any mass at all. It’s made of photons, which are massless particles.” True enough, hypothetical reader, but Einstein showed that all forms of energy contribute to gravity, and to get the contribution of a particular energy source you simply invert his famous equation:
      m=frac{E}{c^2}
    Since c, the speed of light, is such a large number, this contribution is normally quite miniscule, and even though we’re surrounded by cosmic microwave background, the density is small enough that we really don’t care about it, at least today.

  2. The curvature of the universe. Our universe appears to be flat (and even if it’s not, it turns out not to affect this argument at all), so it means that the rate of expansion is simply governed by the total mass and energy out there.

Though I don’t put equations in my io9 column, I’ll give you one here to help make things more concrete:
  H^2=frac{8pi Grho}{3}
where I haven’t bothered including the curvature term at all, because, as I said, it seems to be zero. This equation is known as the “Friedman Equation,” and the terms that you might not be familiar with are:
H is the Hubble variable (a number describing how fast the universe is expanding), G is the gravitational constant, and rho is the density of the universe.

Imagine the universe made of dark matter. This is how people thought about things for a long time. As the universe expands, matter becomes more diffuse. Put mathematically:
  rho_{matter}propto frac{1}{a^3}
If the universe doubles in scale (that factor a), for example, the density could be expected to drop by a factor of 8. In a universe made of matter, you’d expect the Hubble constant to get lower and lower and lower.

But wait! Gravity cares about all contributions, not simply that of ordinary matter. Radiation also contributes. But here’s the thing, radiation also has a lot of pressure, and pressure plays an important role in Einstein’s model of gravity. Have you ever seen a radiometer?

width=200

Those are the little fan-like contraptions that spin from radiation pressure. Recall that the equivalent mass density of radiation is tiny, but nevertheless, ordinary light is able to spin a small top. How? Because the pressure is very high:
  P_{rad}=frac{1}{3}rho_{rad}
As the universe expands, different components of the universe lose different amounts of energy. Radiation not only becomes more diffuse, but it actually does work on the universe (losing net energy in the process):
  Delta E=-P Delta V
Because it has (relatively) lots of pressure, radiation does lots of work on the universe, and therefore loses lots of energy. As a result:
  rho_{rad}propto frac{1}{a^4}
As the universe doubles in scale factor, radiation density drops by a factor of 16, (as opposed to 8 for ordinary matter). Put another way, if you look far enough into the past, when the universe was much smaller than it is now, the radiation actually contributed to more of the energy density of the universe than mass did.

Dark Energy, on the other hand, is very, very weird. It has a negative pressure:
  P_{Lambda}=-rho_{Lambda}
This is a topic unto itself. Negative pressure is a tension, but that doesn’t make it any more intuitive. I wrote a previous post exploring some of the gravitational implications of dark energy, but that won’t necessarily make it more obvious why there should be a fluid in the universe with this bizarre relation.

I’m not going to derive the pressure of dark energy, but it’s something that we can measure based on the acceleration of the universe, and the pressure is the most “natural” value. The density of dark energy, is something like 10^{100} times smaller than the value that we’d naively expect from straightforward calculations of the vacuum energy density. As I’ve said on many occasions, I think this is the biggest unsolved problems in physics.

At any rate, it’s not the problem for today. The problem is why dark energy behaves like it does with the expansion of the universe. Since the pressure is negative, we find that:
  Delta E=-P Delta V > 0
As the universe expands, the total energy in dark energy increases! However, the dark energy also diffuses. These two effects exactly cancel, and the net energy density of the stuff stays constant as the universe expands. That’s why it’s a cosmological “constant.”

As the universe gets bigger and bigger, matter and radiation become smaller and smaller contributions, but dark energy remains the same, which means:
  H^2=frac{8pi Grho}{3}=constant
To get even a little more technical, this solves to:
  apropto e^{Ht}
We get exponential growth forever!

Sorry for the math-splosion, but hopefully some of you found this useful.

-Dave

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2 Responses to I get email: Dark Energy and Universal Expansion

  1. adam says:

    Well, I guess I found it useful for a reason you probably didn’t intend. Radiation pressure has already been brought into the standard model, so it is not a possible explanation for dark energy. Which was surprisingly difficult for me to figure out via search engine. So thanks! Even though I don’t understand almost anything else you’ve written…

  2. Graham says:

    Radiometers don’t actually spin due to radiation pressure. The momentum comes from a process called adsorption, when the residual gas atoms and molecules stick and are then ejected from the panes. If you look carefully the top actually spins in the opposite direction to that you would expect from radiation pressure.

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