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Home , Scale factor (cosmology)

Today Supernovae revisited Type Ia Observations Physics 121 Galaxy Rotation Curves & December 4, 2009 Scaling Factor a(t) Course Evaluations fainter

brighter

Distant supernovae are fainter than nearby, and their light curves (observed at Earth) evolve more slowly, due to time dilation and other relativistic effects.

Supernova Results Supernova Results:The Context

Hubble’s observations found that Galaxies at small The results of the supernova (Z~0.003) are streaming away from us, brightness measurements surprised with velocities proportional to distances. The fainter is expanding. everyone. They show that supernova in high galaxies are fainter Observations of supernovae in more distant galaxies brighter than expected. Crudely speaking, this (Z~1.00) found that they are fainter than expected. means that they are more distant than The expansion of the universe is accelerating, that is, the speeds with which galaxies are moving away predicted by the Hubble law. from one another are increasing. This was a surprise, because gravitational forces This implies that the expansion of the between the galaxies should cause them to decelerate. We conclude that there must be some universe is accelerating. The fainter sort of anti-gravity force, which we call “dark expansion causes the supernova light energy” or “.” brighter to be spread more thinly over space, so that less of it is captured by the What is this dark energy? Let’s put that question aside for the moment and look at yet another kind of telescope. mysterious stuff.

1 Galaxy Rotation Curves Galaxy Rotation Curves, continued

Measurement of Doppler shifts of stars in our measured Galaxy (and therefore their speeds) give a surprising result: speeds do not get smaller as distance from the center gets larger. Stars in the outer reaches of our Galaxy are expected moving faster around their orbits than expected. R v Something must be wrong with our model of the galaxy: the net force on a given star is larger than that expected from the gravitational attraction of other stars in the Galaxy. Conclusion: there must be additional matter in our galaxy, pulling on stars and causing them to Stars in spiral galaxies such as our own Milky Way In reality, the net force on a star results from follow nearly circular orbits around the centers of the gravitational interactions with all the other stars in the move more quickly in their orbits. We call this galaxies. Consider a star at distance R from the galaxy, so it is a complicated thing to calculate. r v stuff “dark matter:” center of the galaxy, moving at speed v. We expect v Nevertheless, because most of the stars are relatively to be large for small R and small for large R. close to the center, if motion of stars is due only to • “dark:” we can’t see it gravity from other stars, then we expect velocity, v, to This is easy to prove if the force on each star is • “matter:” it interacts via the gravitational be smaller for stars at larger distance, R. dominated by gravity from a single massive lump in force, just like normal matter. the center of the galaxy. Milky Way rotation curve: Sofue et al 1999, Astrophysical Journal 523: 136

Galaxy Rotation Curves, continued

Measurements in other galaxies give similar results: velocities are more-or- less constant as distance from the center of the galaxy is increased.

In a typical spiral galaxy, stars can be found as far as 10-15 kiloparsecs from the center. Beyond that, there are no stars, but there are clouds of gas which allow us to measure rotation curves to Left: Expected drop-off of speed with distance from center of galaxy larger distances. The rotation curves Right: Constant speed at all distances from center of galaxy remain flat to remarkably large distances

(The animation might work better in a browser than in PowerPoint.)

Animation from: http://find.uchicago.edu/~pryke/dasi/documents/200309_plancourse/mgp00034.html

Rotation curves: Bosma 1978 PhD Thesis via Faber & Gallagher 1979 Annual Review of Astronomy and Astrophysics

2 What is Dark Matter? The story so far

MACHOS: MAssive Compact Halo Objects? Too much gravity on small scales (<~0.1 Megaparsec) In and around galaxies, things are moving too fast. The attractive forces “Machos” is basically a fancy name for “dead stars, which you can’t see any more:” “Massive” because they are heavy, “compact” because dead stars are very dense, “Halo” because they between masses are larger than we expect. We infer that there must be dark would form a halo around the galaxy. matter: massive, gravitationally interacting stuff which we cannot otherwise see. There are, in fact, dead stars in the galaxy, but are there enough to make up the dark matter? The answer turns out to be no. Dead stars will occasionally come close to the line-of-sight between us and ordinary stars. Their gravity of these dead stars will distort the space through Anti-gravity on large scales (~1000 Megaparsec) which light travels from the ordinary star to us. This is called a “microlensing event.” Such Galaxies are rushing away from each other, and this process is accelerating. events have been seen, but not enough to account for the dark matter. Our usual explanation is that there must be a force or phenomenon (such as Einstein’s cosmological constant) which is not part of our standard physical WIMPS: Weakly Interacting Massive Particles? laws. This is sometimes called dark energy. We can think of dark energy as a These would be tiny particles (tiny in the sense that protons and electrons are tiny). They could peculiar sort of stuff which has mass (so it does gravitationally interact) but not interact through the electromagnetic force; otherwise we would see them. Hence they must which also interacts through a new, not-yet-understood, repulsive (anti-gravity) interact through the weak force (and gravity). Physicists are actively searching for WIMPs, but force. none have been found. Something else entirely? Perhaps the most exciting possibility is that neither of these is right, and that there are as-yet-discovered physical processes, different from Newton/Einstein gravity, contributing to motions around galaxies.

So What’s the Universe Made of Today? So What’s the Universe Made of Today?

Dark Energy 72% Only 5% of the content of the universe (measured by energy content) is “normal matter,” stuff we can see.

The other 95% is dark energy and dark matter.

Radiation (Light) <0.01% In this jar, 95% of the jellybeans are black. Normal Matter 4.5% Imagine how different this would look if we could only see the colored jellybeans.

Dark Matter 23%

Image Credit: Fermilab. To see this jar constructed, search on “jelly bean universe” on YouTube.

3 So What’s the Universe Made of Today? The Metric for an Evolving Universe

The table shows the energy content of the universe, tabulated using all available evidence. Refence: M. Fukugita and P. J. E. Peebles, 2004, “The Cosmic Energy Inventory”, Astrophysical Journal, 616: 643. For a copy of the full paper, do a web search on “astro-ph/0406095”. We can describe the expansion of the universe by defining a “scale function”, a(t). By definition, a=1 right now. At earlier times t, when the universe was smaller, a(t) was smaller. As the universe expands, a(t) is getting larger.

We can take the flat-space metric,

Δs2 = Δt2 – (Δx2 + Δy2 + Δz2)

And modify it to allow for an expanding (or contracting) universe:

Δs2 = Δt2 – a2(t)(Δx2 + Δy2 + Δz2)

We model the universe as homogenous, so a(t) is the same everywhere. The universe is not in steady state, so a(t) changes with time.

[Note: in an earlier handout, I called the scale factor R(t). Starting today, I am changing notation and calling it a(t) instead.]

normal matter or dark matter w = 0 The scale factor a(t) changes over time. a˙˙ 4#G 2 = " $(1+ 3w) light w = 1/3 da d a a 2 Write its first and time derivatives as: a˙ = a˙˙ = 3c cosmological constant (dark energy) w = –1 dt dt2

General relativity gives a formula for the acceleration of the scale factor. It depends on ! two characteristics of the stuff which! causes! the acceleration, which we label ε and w: a˙˙ 4#G = " $(1+ 3w) a 3c2

Here G is Newton’s gravitation constant, G=6.67×10-11 m3/kg s2.

Typically in physics we write! “ρ” (greek letter rho) to denote density, that is, mass divided by volume. However, when we talk about stuff that doesn’t have mass in the usual sense (dark energy) we need to do something different. So we use “ε” to represent energy density. Thus ε is the amount of energy in a given volume of space. For normal matter at rest, E=mc2, so ε = energy/volume=mc2/V. The variable “w” is the “equation of state” parameter. It is related to how the pressure of something changes with density. Here are the rules: normal matter or dark matter w = 0 light () w = 1/3 dark energy w = –1

4 ˙a˙ 4#G normal matter or dark matter w = 0 = " $(1+ 3w) light w = 1/3 a 2 3c cosmological constant (dark energy) w = –1

How does this fit our picture of the evolution of the Universe? ! In the early Universe (but after ...more about that later), the Universe was dense, and was dominated by normal matter. Plugging w=0 into the above equation, we see that ä is negative, so the scale factor, a(t), decelerates over time. The Universe was rapidly expanding, but the expansion was slowing over time. Later on, as the Universe expanded, the density of ordinary matter got smaller. However, the density of dark energy did not. (If Einstein’s cosmological constant is the correct model, the density of dark energy never changes.) If most of the energy is in dark energy, with w= –1 in the above equation, then ä becomes positive and the scale factor a(t) accelerates. This is the picture: • The Universe has been expanding since the . • Early on, its expansion slowed down over time. (It continued to expand, but more and more slowly). • Later, as dark energy came to dominate the Universe, the expansion started speeding up.

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