Physics 121 December 2, 2009

relativistic Doppler shift

c v

1 1 1

λ λλZ

c v c v

e

er ≈− −

+ =

− =

approximation for small velocities

v = c (Z +1)2 −1 (Z +1)2 +1 � ��

� ��

Redshift and Velocity

400 nm 500 nm 600 nm 700 nm

400 nm 500 nm 600 nm 700 nm

• What range of redshifts did Edwin Hubble observe?

Example

We are not special! More specifically: There is nothing special about our particular place in the universe.

This idea has been influential throughout the history of physics:

Earth as center of the solar system? Sun as center of the solar system. The Earth-centered model of the solar This works much better: orbits are simple system requires complicated models to ellipses. We give up the idea of the Earth explain the motion of the planets: as a “special place” in the solar system.

The idea that we are not in a special place (or a special time) is sometimes called the Copernican principle, after Nicolaus Copernicus, who proposed the Sun-centered solar system model.

No special places in the universe: Homogeneity On large scales, the universe is homogeneous: every point is like every other. There are no special places.

Homogenous Not homogenous Inhomogeneous on small scales, but homogenous on large scales

On small scales, our universe is inhomogeneous: some places are relatively dense (galaxies, say), other places are nearly empty.

If we step back and look over very large scales, 100s of Megaparsecs, say, every place looks pretty much like every other place: for example, the number of galaxies per unit volume doesn’t vary by much.

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No special directions in the universe: Isotropy On large scales, the universe is isotropic: every direction is like every other.

Not isotropic (vertical, horizontal directions are different). But it is homogenous on large scales.

Isotropic, from the point of view of the center of the region. But not homogenous: the center is a special place.

Homogenous and isotropic.

This is an idealized picture of an ideal universe on large scales: very boring!

On small scales, our universe is non-isotropic: look in one direction and you will see more nearby galaxies than in another direction.

On large scales, it is isotropic: you see the same number of distant galaxies no matter which direction you look in. (You will also see nearly the same intensity of microwave background radiation from all directions. We will discuss this relic of the big bang on in another class.)

A related (incorrect) idea: Steady State Universe The idea of a steady state universe is that no point in time is special.

This contradicts the observed fact of the big bang, since the moment at which the big bang happened is surely a special point in time. So the “steady state” idea is wrong.

Nevertheless, the idea of a steady state universe played an important role in defining how we talk about the universe today.

Einstein developed general relativity before Hubble’s discovery of the expanding Universe. Einstein thought that the universe was both homogenous (no special points in space) and steady state (no special points in time).

There was a problem with this picture, which we can understand in Newtonian terms. Every bit of mass is gravitationally attracted to every other bit of mass. If the universe is absolutely perfectly homogenous (it has uniform mass density), then each bit of mass will be pulled equally in all directions, and will not move. (Which is not a problem.) If however, there are areas of the universe which are just slightly denser (more massive) than their surroundings, they will tend to pull their surroundings in toward them. This is a problem: as more mass gets pulled in, the attractive force is larger, and the process snowballs. For an infinitely old universe, this is a very big problem, because it creates large mass inhomogeneity.

Homogeneity and isotropy are related to one of the core principles of physics:

The laws of physics are the same at all times, in all places, in all reference frames, and in all directions. If you a perform an experiment, and then you repeat the exact same experiment in a different place or a different time, you will get the same result.

We are now going a step farther. We are saying that not only that the laws of physics the same everywhere, but also that the “stuff” that the universe is made of is the same in all places and in all directions. On the other hand, the "stuff" is not the same at all times (the universe was more dense in the past) and is not the same in all reference frames.

Steady State Universe, continued Einstein solved the problem of the buildup of density inhomogenities by introducing the cosmological constant. This is an extra term in the equations of general relativity which acts like a repulse force.

Here’s his idea:

• Because of the cosmological constant, every point in space repels every other point in space.

• At the same, time because of mass distributed throughout the universe, every point attracts every other point by the gravitational force.

• These effects exactly balance, so that the universe remains in steady state, with mass (such as galaxies) neither rushing towards each other nor rushing away.

The rationale for the cosmological constant was wrong. The universe is not in a steady state: as Hubble showed, it is expanding. Upon learning of Hubble’s discovery, Einstein called the cosmological constant “the biggest blunder of my life.”

Recent astronomical observations have shown that, in fact, a cosmological constant seems to be needed after all. (We’ll see why soon.)

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The Fate of the Universe All the Galaxies are rushing away from each other. Yet, they are attracted to each other by gravity. We expect the gravitational force to decelerate the galaxies’ motions. (If the direction of an object’s acceleration is opposite to the direction of its motion, the object slows down.) In this deceleration scenario, there are two possibilities:

• If the deceleration is strong enough, the galaxies eventually reverse course and start hurling at one another. The universe ends in a “big crunch” billions of years into the future. Watch out!

• If the deceleration is not too strong, the Universe keeps expanding forever, but it goes more and more slowly over time.

Various astronomical observational programs have been undertaken in order to determine which of these possibilities is the fate of the Universe. To the surprise of everyone, neither of these possibilities turned out to be correct.

p + p → 2H + e+ + νe 2H + p → 3He + γ

2H + 3He → 4He + 2p + γ

3He + 4He → 7Be + γ

7Be + e– → 7Li + νe 7Li + p → 2 4He

7Be + e– → 8B + νe 8B → 8Be + e+ + νe

8Be → 2 4He

86% 14%

14% 0.02%

Gas Pressure

Gravitational Pressure

In a star like our sun, outward pressure of hot gas (heated by thermonuclear reactions) balances inward pressure from gravity.

Eventually the thermonuclear reactions cease—the nuclear fuel is “used up”—in which case the star collapses.

Back to Hubble’s Strategy Hubble’s strategy was to measure distances and redshifts (velocities) of distant galaxies. This remains a powerful way of studying cosmology. The goal is to measure galaxies at larger and larger distances.

The basic idea remains the same: observe something of known luminosity (power), measure the intensity of its light at the Earth, and use the formula

PI = 4πr 2

to find the distance, r.

The challenge is to find something of “known luminosity.” Sometimes the term “standard candle” is used to describe such objects. A problem arises:

• Some types of stars, such as Cepheid variables, are good standard candles. However, these stars are not bright enough to be seen in very distant galaxies, so they won’t work.

• Galaxies themselves are bright enough to be seen, but there is nothing “standard” about them. Looking at a galaxy, you can’t tell how bright it is supposed to be, so you have no “P” to plug into the I= P/4πr^2 formula.

• We need something else! What we need are ....... supernovae!

Supernovae In the 1930s, Walter Baade and Fritz Zwicky realized that the collapses of some types of stars would trigger explosions, blowing most or all of the star outward. We call these events supernovae. They happen during the death throes of very massive stars (stars that initially have, say, 8 times more mass than our own Sun).

In 1054, Chinese astronomers noted that an extremely bright new star had appeared in the sky: we now know this to have been a supernova. Over weeks and months it faded away. Even now, the remnant of the supernova can be seen. In fact, it is still expanding outward into space!

Supernova are very rare: there is only one every hundred years or so in our own Galaxy. (The most recent nearby supernova was in 1987, in the Large Magellanic cloud, a satellite galaxy of our own Milky Way galaxy.)

Supernova are very bright and hence can be seen at great distances.

Crab Supernova remnant

Fritz Zwicky

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Supernovae as Cosmological Probes

Because supernovae can be seen very far, they are potentially useful to help us measure distances.

But: we need to find “standard candles,” categories of objects all of which have the same intrinsic luminosity. Are supernovae standard candles? Most are not. The luminosity of a supernova explosion depend on the composition of the star and the circumstances of the explosion.

Hubble Space Telescope observations of supernova SN 1997cj as it fades out (Garnavich et al. 1998).

Example of a supernova “light curve” showing intensity versus time measured in four different colors.

(Magnitudes are crazy units used by optical astronomers to measure brightness. A difference of 2.5 magnitudes corresponds to a difference in intensity of a factor of 10.)

Riess et al. 1999

Type Ia Supernovae In the mid-to-late 1980s, astronomers working on categorizing supernovae realized that one particular subgroup, “type Ia supernovae”, all had very similar light curves, so they would make good standard candles.

These supernovae arise under a very specific circumstance: a white dwarf star orbits another star, and matter flows from that other star onto the white dwarf. The white dwarf slowly gains mass until, when it is just heavy enough, it explodes.

(Type Ia supernovae are distinguished by spectral lines; they show lines from the element silicon but not from the element hydrogen.)

Finding Type Ia Supernovae in Distant Galaxies

A problem with Type Ia supernovae is that they are very rare events.

In a typical galaxy (such as our own), there is a supernova explosion every 100 years. Around 10% of these are Type Ia supernova, so we would have to wait around 1000 years for a Type Ia.

Supernovae are unpredictable: you don’t know ahead of time when one will occur.

How can supernovae be found? More important, how can supernovae be found in such a way that their discovery can be followed up efficiently by telescopes, for example, so that their light curves (intensity measured over time) can be measured for several days or weeks in a row?

The solution is monitor thousands upon thousands of galaxies, looking for supernovae in each of them. Two research groups began following this strategy in the mid-1990s. The programs are called the “Supernova Cosmology Project,” and the “High-Z Supernova Search,” and they follow the same basic strategy.

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Perlmutter, Physics Today, 2003.

How to Analyze Supernova Observations

1. Discover a supernova by scanning many thousands of galaxies on a regular basis (monthly).

2. Observe the supernova over many weeks and months to measure its light curve (intensity versus time).

3. Recall that the intensity of radiation detected from something emitting power in all directions is proportional to 1/r2, where r is its distance from us.

Compare the light curve of the newly discovered supernova to the light curves of similar supernova whose distances are known. Use the relative intensities to deduce the distance to the newly discovered supernova.

4. Observe the galaxy in which the supernova is located to measure spectral lines of stars and hence to measure the redshift of the galaxy.

5. Make a plot showing the distances and redshifts of all the supernovae.

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fainter

brighter

The results of the supernova brightness measurements surprised everyone. They show that supernova in the highest redshift galaxies are fainter than expected. This means that they are more distant than predicted by the Hubble law.

This implies that the expansion of the universe is accelerating. The expansion causes the supernova light to be spread more thinly over space, so that less of it is captured by the telescope.

Supernova Results

fainter

brighter

Why is the Universe accelerating?

If we put Einstein’s cosmological constant back into the equations of relativity, then they are able to describe this acceleration of the Universe.

But the question remains: why is there a cosmological constant? Is there some physical mechanism to which we can attribute the acceleration?

The term we use is “dark energy,” mysterious stuff whose dominant force is repulsion rather than attraction. There seems to be more “dark energy” in the universe than there is normal matter.

Next class we will talk more about dark energy and we will take a survey of it and the other types of stuff in the Universe.

The Accelerating Universe

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