Universe Expansion

Hubble's Law and the Expanding Universe

In the early 1900s (1910s), V. M. Slipher made the first measurements of the redshifts of nearby galaxies. Slipher found that Andromeda (M31) was approaching the Milky Way but that most galaxies were running away from us. Edwin Hubble followed up on this week and greatly extended it. Hubble found that the light from all distant galaxies showed increasing redshifts with distance from the Earth. Hubble presented his results in plots (often times as also done today) as speed vs distance rather than their measured redshift vs distance. Hubble (as hinted at by Slipher) got the interesting result that all distant galaxies are running away from us. This suggests the natural interpretation that we are the center of the Universe and that the other galaxies are simply streaming away from us, like in an explosion. The modern interpretation of Hubble's Law, that is, this streaming will be presented later.
Question: How did Hubble and Slipher Accomplish Their Feat?

Spectroscopes and Rainbows
Hubble and Slipher acquired spectra of distant galaxies (see the image below of MAssive Cluster Survey item 416 [MACS 0416], a cluster of galaxies located a distance 4.3 billion light years from the Earth).

Hubble and Slipher passed the light from each galaxy through a dispersive device (a device which separates blends of light into their constituent colors), a spectroscope. In the picture to the left, the prism is the dispersive device that separates the light into its constituent colors.
The bottom left picture of a rainbow shows a naturally occurring dispersive device, namely, droplets of water which spread the blended light from the Sun into its constituent colors. Using a telescope we find

Spectra and Absorption Lines
To the left are shown spectra of different types of stars. The dark absorption lines are the fingerprints of individual elements. Each pattern is unique to the element. absorption lines of hydrogen, helium, carbon, and other elements are shown. Because distant galaxies are made of stars, one expects that the light from a galaxy will appear as the combination of hundreds of billions of stars! Hubble and Slipher acquired spectra of many galaxies and noticed something quite interesting. They found that the lines for all of the elements present in the spectrum were shifted toward the red end of the spectrum, the light was redshifted.

Redshift, z
A redshift is a shift in the measured wavelength of some spectral feature to a value greater than its value as measured in a laboratory on the Earth. The redshift, z, is defined as

Here the Greek letter lambda represents the wavelength of the light. The redshift is the relative change in the wavelength of the spectral line (feature).
A redshift may arise if the source of the waves and the observer are in relative motion, If there is no relative motion then there is no shift and z = 0. This type of redshift is the Doppler effect. The Doppler effect is shown by waves of all kinds, for example, it is shown by sound waves (train whistle), and light waves. The frequency of a wave (for example, the pitch of a sound wave) is higher when the source of the waves is approaching; the frequency of a wave is lower when the source of the wave is running away from the observer.

Hubble's Law
Hubble demonstrated that the larger the redshift, z, the greater the distance to the object. This can be easily seen in the left image where the sources with the smaller redshifts present larger images on the sky. Since we know that angles (apparent sizes of objects on the sky) decrease as you move an object farther away roughly as angle ~ size/distance, smaller appearing objects must be farther away,

Hubble found what is referred to as Hubble's Law.

Hubble's Law → redshift z is proportional to distance. In their plot, they actually used cz, where c is the speed of light, 300,000 km per second. In this way, they expressed redshifts as speeds.

We have cz = speed = H_oD

Hubble Law and Hubble Constant
The Hubble diagram produced by the Hubble Space Telescope is shown to the left. For this particular version of Hubble's Law, speeds are measured in kilometers per second and distances from Earth in millions of parsecs. The expansion rate of the Universe is measured by the slope of Hubble's Law as determined by galaxies near the Earth (distances close to 0, see lower left). The slope is called Hubble's Constant, H_o. The best current estimate for H_o determined using the method of Hubble is H_o = 73 km per second per million parsecs.
The slope of the Hubble diagram changes at large distances, What might this be telling us?

Interpretation of Hubble Law

The Hubble Law is an empirical fact. The challenge is to interpret what the Hubble Law means. Here, we will give two possible interpretations.
Before we begin, let's make an observation

The light from the galaxies observed by Hubble and Slipher left their sources at different times depending upon how far they were from the Earth because light travels at a finite speed, 300,000 km per second.
Light from the galaxies close to the Earth left their galaxies after the light from the more distant galaxies.
This means that light from nearby galaxies shows what the Universe is like now (when the Universe is old); light from distant galaxies shows what the Universe was like in the past (when the Universe was young).

We are looking back in time when we look at distant objects

Hubble Law and Expansion of Universe I
Imagine a universe where the galaxies are initially confined to one region. At some point in the past an enormous explosion takes place that throws out galaxies in all directions at all speeds but leaves us stationary at the center.
What do we observe?
After say 1 billion years, the slower moving galaxies will have traveled less far and so will be closer to us. Galaxies that were traveling twice as fast as the slow ones will be twice as far away. Galaxies that were traveling three times as fast would be three times as far away

→ we would find Hubble's Law

This scenario reproduces Hubble's Law, but is it reasonable?

Hubble Law and Expansion of Universe II
Imagine a very large universe in the form of an uncooked loaf of raisin bread. Here, the universe could be very large but fixed in shape with the raisins embedded in the loaf. Now cook the raisin bread so that it doubles in size from 20 cm to 40 cm. As the raisin bread doubles in size, the distance between every raisin also doubles in size. That is, the raisin that was initially 5 cm away is now 10 cm away and the raisin that was 10 cm away is now 20 cm away. This means that if a wave stretched between each of these raisins, the wavelength of each will have doubled because of the expansion. Alternatively, it also means that it would appear that the more distant the raisin initially, the faster it would appear to move away from us!
→ This again is Hubble's Law

Note that if the universe is sufficiently large and we change our reference raisin, this same result would also hold, that is, the new raisin would also observe Hubble's Law. In fact, any arbitrary raisin in the raisin bread universe would see every other galaxy running away from them as well thus making each look as if they were at the center of an expanding universe, as long as the universe was sufficiently large.

Homework 1

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