Space and Time

394-409

In this section we will look at the basic underlying assumptions made in modern cosmology as well as the basic concepts of space, time and spacetime.

The Basic Assumptions - The Cosmological Principle

There are three fundamental ideas upon which modern cosmology is built and are summarized in table 15.2

For both homogeneity and isotropy the he idea of "sameness" is intended in the sense of large scale structure. We see clusters of galaxies and superclusters etc. So to will an astronomer 1000 Mpc away. Not the same clusters and superclusters but clusters and superclusters none the less.

 These assumptions, taken together, make up the Cosmological Principle which is the fundamental idea of cosmology. By invoking this principle we are claiming that our statements about the universe are "global".

The Cosmological Redshift

At several points earlier you were "cautioned" not to think to the redshift of galaxies and especially the redshift measured for very distant galaxies are due the Doppler effect (ie motion induced change in wavelength). We will now be much more explicit and identify 3 major ways in which redshift can occur:

1. Motion: When a wave source moves the waves that are emitted will be Doppler shifted. This is an idea that we introduced early on and have used in many different ways. For example, when looking at galactic rotation we are measuring light from different parts of a galaxy moving at different rates. Motion can produce both red and blue shifts - it all depends on whether the motion is away from you or toward you.
2. Gravitation: Gravitational fields can create a redshift. This is a relativistic effect due to the effect that gravitational fields have on the basic geometry of spacetime.
3. Universal Expansion: The expansion of the universe literally "stretches" space in such a way that photons are "stretched" as they travel through space. The net effect is that as a photon travels its wavelength lengthens. This is what an astronomer is measuring for very distant galaxies. To help understand this consider the applet Cosmological Redshift given as Figure 15.5

Figure 15.5 Cosmological Redshift illustrates how universal expansion and photon travel time produce the cosmological redshift.

Example 15.4 You receive photons from two sources. Each source emitted 500 nm wavelength photons but photons from source "A" traveled for 1 billion years while photons from source "B" traveled for 2.5 billion years. How will their wavelengths compare?

Solution: A little experimentation with the applet Cosmological Redshift will help emphasize the key point - the more time a photon spends in flight the more its wavelength is stretched by the expanding scale of the universe. For example, after traveling for 1 billion years the photons from "A" will be redshifted from 500 nm to 531 nm. After traveling for 2.5 billion years the photons from "B" will be redshifted from 500 nm to 593.5 nm.

The cosmological redshift is a consequence of the expansion of space. In Example 15.4 a 500 nm photon is stretched as it travels across the universe and the amount of stretching depends on both the rate of expansion of the universe (Hubble's Constant) and the time of flight for the photon.

Example 15.5 Determine the redshift "z-value" for the photons from sources "A" and "B" in the previous example.

Solution: Use the redshift formula:.

For photons from "A" the z-value is

For photons from "B" the z-value is

Einstein's Theory of Gravity, Curved Space and Model Universes

On the largest scale of the universe gravity (we think) is the dominant shaping force. In order to erect a mathematics of cosmology we need as complete a theory of gravity as possible. Einstein extended Newton's theory of gravitation during the first part of the 20th century. In simplest terms Einstein's theory can be summarized as:

So What's "spacetime"?

Perhaps the most profound implication of Einstein's Theory of Relativity is the union of the ideas of space and time. If you move a clock through space, for example, you alter its rate of measuring time. Since we can no longer treat these as independent ideas physicists now use the term spacetime to discuss both space and time.

Some Features of Einstein's Universe

• spacetime is curved in the vicinity of matter
• black holes are a possibility
• time slows down in the presence of gravitational fields
• light is deflected in gravitational fields

Einstein's equations governing the structure of space are very complex. In 1917 Einstein was vexed by a problem that these equations implied. Left on its own the universe would expand! This is particularly puzzling since, if anything you might suspect that it would contract. To "fix" this problem Einstein introduced into his equations the now famous cosmological constant. By the 1920's a number of mathematicians had published solutions of Einstein's equations that made universal expansion a key feature of the universe and in 1929 Hubble provided the empirical evidence that our universe is expanding. Einstein rued that introducing the cosmological constant was "the greatest blunder" of his career.

The Critical Density and The Curvature of Space

One of the key questions of cosmology concerns the "shape" or geometry of space. According to Einstein's theory matter distorts space. The amount of distortion or curvature will depend on the amount of matter - more accurately the density of matter - in the universe. It is fairly easy to relate the shape of space to the density of matter in the universe. If the density of the universe is

Space - In 3 Flavours!

	Positive Curvature: This is sometimes called the closed universe and is analogous to a spherical shape. Imagine that you are an ant living on a spherical surface. Your space is closed, finite and without bound. Some consequences of positive space would be density > critical density If the universe has a positive curvature it will expand but the rate of expansion will slow to zero and then begin to contract back to a point.
Figure 15.6a 2-dimensional analog of positively curved space
	Zero Curvature: This is the "normal" space that we experience in every day life. All the laws of geometry are Euclidean. density = critical density space would expand forever but the expansion rate would slow to zero (in the limit)
Figure 15.6b 2-dimensional analog of flat space
	Negative Curvature: This is sometimes called the open universe and is analogous to a saddle shape. Negative space is open,infinite and without bound. Some consequences of negative space would be density< critical density the universe would expand forever in this case.
Figure 15.6c 2-dimensional analog of negatively curved space
Table 15.3 The 3 possible geometries of space

So where are we right now? Our best estimate of the universal density - based on what we see - is about 5 x 10^-27 kg/m³ or about 10% of the critical density. This implies that the universe is either flat or open. In the next section we will consider the reasons why current evidence favours a flat universe.

Dark Matter

There is good reason to believe that we live in a flat universe. This immediately poses two significant puzzles:

1. the Hubble Age of a flat universe is given by the formula which would imply an age of 9.1 billion years which is younger than many globular clusters!
2. The best estimate for the density of observable matter in the universe is at most 1/10th of the critical density to produce a flat universe.

In the next section we will address the first puzzle. The second puzzle is, of course the same one we saw with anomalous rotation rates in galaxies, clusters of galaxies moving too fast and gravitational lensing. All of these point to the necessity of dark matter. But what is dark matter?

WIMPS and MACHOS!

Two competing views on dark matter can be summarized as:

WIMPS (Weakly Interacting Massive Particles): are an exotic form of matter quite different than protons and neutrons which are from a general class of matter called baryons or baryonic matter. Most of what you and I see and touch in the universe around us consists of baryonic matter. WIMPS would be examples of nonbaryonic matter.

MACHOS (Massive Compact Halo Objects) are made of normal matter that is just to faint to be seen. Brown dwarf stars, neutron stars or even planets from dead stars could be such examples.

Hot and Cold Dark Matter

Even though we don't really know much about dark matter we can make some general statements about how dark matter can be related to the formation of the earliest galaxies. If the dark consisted of particles moving at very high velocity at the time of the transition from a radiation dominated universe to a matter dominated universe (the time when the cosmic background radiation emerged)

then these particles would have had little influence on the subsequent development of galaxies. We call this form hot dark matter. On the other hand, if the particles were moving slowly at this time then they would be significant participants in the formation of galaxies. This form is called cold dark matter. Figure 15.7 is a Hubble Ultra Deep Field image that shows the most distant and hence earliest galaxies. We now know that galaxies formed much earlier than astronomers had anticipated. Given the density of normal matter this argues strongly that dark matter played a significant role in galaxy formation and then suggest that dark matter is likely in the cold dark matter form.

Figure 15.7 Hubble Ultra Deep Field image showing earliest galaxies. (image courtesy of NASA/ESA/S. Beckwith(STScI) and The HUDF Team )

At this point in time the true nature of dark matter and whether it is WIMPS, MACHOS, a combination of the two, whether it is "hot" or "cold" or something quite different is still a very open question. What is becoming clear however is that "normal baryonic matter" (the stuff you and I are made from) can only make up a tiny fraction of the mass of the universe. This is very clearly shown by recent measurement of the amount of Lithium-7 and Deuterium found in distant clouds of gas. The amount of either of these isotopes produced during the earliest phase of the Big Bang is related critically to the density of matter in the universe. Evidence suggests that normal matter can only make up about 4% of the universe's mass. This leaves the startling fact that almost 96% of the universe is in a form we cannot see directly! What is even stranger is the growing understanding that dark matter can only account for part of this "missing mass"!

Practice

1. Explain why it is possible for a nearby galaxy to have a blue shift while all very distant galaxies show red shifts.
2. Why do we associate the cosmic background radiation ("primeval fireball") with the time at which the universe switched from being radiation dominated to matter dominated?
3. A 3000 K black body emits most of its energy at a wavelength of 1000 nm. In Example 15.3 you saw that the observed background microwaves radiation emits most of its radiation at a wavelength of 1 mm. By how much has the light emitted at the the transition from the radiation dominated to matter dominated eras been redshifted?
4. Why do we think most dark matter is of the cold dark matter variety?
5. How do the abundances of Deuterium and Lithium-7 constrain the amount of normal matter in the universe?