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# Dark Matter

We determined in the last quiz that the fate of our expanding universe depends on a single parameter—its mass density.

If the density is larger than a critical value, gravity will prevail and the universe will collapse in on itself (a so-called closed universe). If the density is smaller than the critical value, the universe will expand forever (an open universe). If the density is exactly equal to the critical density, the universe is described as a flat.

In this quiz we will try to determine how dense our universe is, and how this compares with our critical value to determine the fate of our universe.

In our solar system, we can measure the total mass becaue we can see light emitted and reflected from it.

What is the dominant source of mass in our solar system?

If we assume that the rest of the universe is similar to our solar system, and most of the light is coming from stars, then we could estimate the mass of the universe by looking at the amount of light we see.

For example, if we assert that the Sun is a typical star, with about half of stars dimmer and the other half brighter, then we could measure the total luminosity from all stars in a region of space and deduce the mass density of that region using the measured mass and luminosity of the Sun.

Observations tell us that a cubic megaparsec ($$\si{\mega pc\cubed}$$) of space in the vicinity of our galaxy has a total luminosity of approximately $$\num{1.2e8}L_\odot$$. Estimate the mass density of this region of space in $$\si[per-mode=symbol]{\kilo\gram\per\meter\cubed}$$.

Details & Assumptions

• The mass of the Sun is $$\SI{1.99e30}{\kilo\gram}.$$
• $$\SI{1}{\mega pc} = \SI{3.09e22}{\meter}$$

In the last quiz we found that the critical density of the universe was $$\SI[per-mode=symbol]{8.3e-27}{\kilo\gram\per\meter\cubed}.$$ Based on the previous calculations, it would seem we live in a very open universe. If, indeed, most of the universe's mass is in stars, then the measured density is $$0.1\%$$ of the critical density. If this is correct, our universe will someday be emptier (and colder) than it already is.

The previous calculation hinges on the assumption that most matter in the universe emits visible light, which we based on the observation that most matter in our solar system is within the Sun. How can we tell is this assumption makes sense?

Other attempts to measure the density of matter have focused on light in the microwave and infrared bands of the spectrum, where both stars and colder matter emit light. Another way to estimate mass is to look at the effects it has on other mass—through gravity.

In the Gravity quiz, you estimated the mass of the Sun by looking at the speed and radius of the Earth's orbit. In a similar way we can estimate the mass of a galaxy by looking at the speed of objects orbiting its center.

Suppose, looking at a nearby elliptical galaxy (one that is spheroidal in shape), we see that the light from a star on the edge of it is blue-shifted relative to the rest of the galaxy with a z-value of $$0.001$$ (i.e., it is moving towards us with speed $$\Delta v=0.001c$$). Taking this as the velocity of the star's orbit of the galaxy, and measuring the galaxy's radius to be $$\SI{10}{kpc}$$, estimate the mass of the galaxy.

Details & Hint

• The speed of light is $$\SI[per-mode=symbol]{3e8}{\meter\per\second}.$$
• $$\SI{1}{kpc} = \SI{3.1e19}{\meter}.$$
• Equate the acceleration of the star due to gravity to the acceleration of a body in circular motion $$a = v^2/r.$$

Galaxies like the Milky way are shaped as a disc rather than a sphere, which complicates the analysis as they can no longer be modeled as if all of their mass were at the center.

However, images of spiral galaxies show that most of their light comes from near the center. If our assumption about most mass coming from stars is correct, we could expect most of the mass to be there too.

If this is the case, what should we expect to happen to the rotation speed as you get further into the outer reaches of the galaxy?

If our model were correct, we would expect to see the velocity of orbit decrease at greater distances, as the force of gravity decreases. This would correspond to line B on the graph below. The velocity increases up until the edge of the very massive central region, but then decreases.

In fact, we observe something more like line A. What does this suggest about the distribution of mass in our galaxy?

This observation of galaxy rotation curves was made in the 1930s, and formed the first evidence for dark matter—matter in the universe that does not emit light. Subsequent, precise measurements suggest that the mass of stars in the universe is dwarfed by matter that does not give off light.

Initial theories of how to explain this focused on objects known as MAssive Compact Halo Objects (MACHOs). This covers any objects consisting of cool matter that gives off little visible light, such as planets, dust, cool white dwarfs, etc.

Assuming these kinds of objects are too dim to see directly, in addition to looking at the effects of their gravity, how else could we observe them?

A rival theory is that dark matter is made of a hypothetical form of matter known as Weakly Interacting Massive Particles (WIMPs). Unlike MACHOs, WIMPs do not give off light, and they do not interact with light or any other kind of electromagnetic radiation, at all. This makes them very difficult to see!

However, one way we can detect the presence of either WIMPs or MACHOs is by a process called gravitational lensing. One of the consequences of Einstein's general theory of relativity is that gravity affects light as well as matter. This means that the light we receive from distant objects can be bent as it passes very massive objects. This technique has allowed us to estimate levels of dark matter in the universe. It also gives us a way of telling which type of dark matter we are seeing.

Considering the diagram below, how could we tell when making gravitational lensing observations whether the matter we are seeing could be best explained by WIMPs or MACHOs?

Gravitational lensing observations tell us that dark objects made of normal matter contribute more mass to the universe than stars, but still only come in at around $$5\%$$ of the universe's critical density. WIMPs, by contrast, appear to make up more like $$25\%$$, and this is what people mostly mean nowadays when they talk about dark matter. But despite having a name for them, we really have no idea what WIMPs are, and have never detected them on Earth. Since they only interact with the force of gravity (and possibly another force known as the "Weak Interaction") they are not going to be easy to study. Some scientists remain unconvinced, and pursue other explanations for these observations, such as an alternative theory of gravity.

In trying to find out the ultimate fate of our universe we have uncovered another great mystery - what is our universe actually made of? It seems that for every bit of normal matter that we know a bit about, there is five times as much dark matter, about which we know almost nothing. But at least we have answered our original question right? $$5\% + 25\% \lt 100\%$$ so our universe should be open and expand forever, right?

It turns out not to be that simple, as two important discoveries of the late 1990s and early 2000s proved. But first we need to explore in more detail what is meant by the terms "closed", "flat", and "open" universes by peering back into the early universe, and then skating over the surface of Einstein's master work—general relativity.

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