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Introduction to Astronomy

35.2 Galactic Recession An early indication that many "nebula" were in fact galaxies is that their spectra show a substantial redshift, i.e. they are receding from us at great speed (~10,000 km/s!).   As Hubble studied numerous galaxies in the 1920s, he measured both their distance and their Doppler shifts, and calculated their speed. Interestingly, Hubble found that all distant galaxies are receding from us. Also, Hubble found that the more distant the galaxy, the greater its speed of recession. More specifically, when Hubble plotted their recessional speed v vs. their distance r, he found a linear relationship, called the Hubble Law: v = H0 r   The slope of this line, H0, is known as the Hubble Constant, and it has a value H0 = 21 km/s/Mly (± 2 km/s/Mly) .   For example, the Virgo cluster is 50 Mly away, which means that it is receding at a speed of v = 21 km/s/Mly x 50 Mly = 1050 km/sec

35.3 The Expansion of the Universe How can we explain the fact that every distant galaxy is receding away from us? Certainly we are not at the center of the Universe; there must be some other explanation! Einstein's Theory of General Relativity provides the answer.   The Newtonian idea of space is that it is static and endless. It is "nothing"; objects exist in it but are not connected to it.   On the other hand, General Relativity requires that space be "elastic". It is capable of being compressed, stretched, and even twisted by the presence of the mass and energy it contains. Question: what examples of this have we already seen?   Hubble's observation of galactic redshifts demonstrated that the Universe itself must be expanding. The space between any pair of galaxies increases with time; the galaxies separate without actual motion. The further apart they are, the faster they separate. An important consequence of this is that no galaxy (including our own) is at the "center" of the Universe.   As a photon from one galaxy travels to another, it will have its wavelength "stretched", producing a cosmological redshift. The farther the photon travels, the longer it takes, so the more stretching occurs.   Galaxies do not expand: because of their large mass and compact size, their gravitational force counteracts the expansion. So, only intergalactic (and primarily intercluster) space expands. Nevertheless, because of their large mass, superclusters of galaxies will still have a significant contracting effect on the Universe.   Before Hubble announced his observations, Einstein had realized that this contracting effect must exist. However, Einstein had philosophical difficulty with the notion of a contracting Universe. Einstein therefore introduced an expansive force into his theory, called the cosmological constant, which would just balance the effect of gravity and create a static Universe.   As we have seen, however, the Universe is neither contracting nor static, but instead expanding. The simplest explanation of the observed expansion is that the Universe received an initial large "push", easily overcoming gravity, which became known as the Big Bang. This prompted Einstein to remark that the cosmological constant was the "biggest blunder" of his life (even he is subject to the prick of Occam's Razor!).

35.4 The Geometry of the Universe General Relativity also predicts that the Universe must have a distinct "shape", which depends directly on how much mass is in it and on how fast it is expanding (i.e. the Hubble Constant).   The relevant parameter here is the critical density of mass and energy, which, for H0 = 23 km/s/Mly, is calculated to be dc = 2.4 x 10-26 kg/m3 . This is roughly fourteen hydrogen atoms per cubic meter, or about one Milky Way-sized galaxy for every 4 Mly in each direction.   Because of the cosmological principle, the actual density of the Universe must be the same everywhere (at least over scales larger than about 700 Mly). There are therefore only three possibilities for the geometry of the universe: flat, hyperbolic, and spherical.   If the Universe has exactly the critical density, it is said to be a critical or flat Universe. Imagine that instead of being three dimensional, the Universe was two-dimensional. Imagine also that you could stand "outside" of it and look at it (although in actuality this is not possible -- everything is inside the Universe!). A flat Universe would then appear to be a plane (it would have zero curvature). Picture Information From within a flat Universe, we could determine its geometry by the fact that two parallel beams of light will always remain parallel to each other. A flat Universe must have no edges and therefore must extend indefinitely in every direction (or else it would violate the cosmological principle).   One way for a flat Universe to extend indefinitely is if the Universe is infinite in size. Such a Universe is also said to be open. In an open Universe, a light beam will never return to its starting point.   Another way for a flat Universe to extend indefinitely is if the Universe connects back on itself after a certain distance, in which case it is finite in size. Such a Universe is also said to be closed. In the picture below, you can see how a finite Universe appears to repeat itself over and over again (but it is actually only the size of the colored square). Picture Information In each picture, the red edge connects to the red edge and the yellow edge connect to the yellow edge, etc. In the left-hand picture the connection is direct, while in the right-hand picture the connection is to the opposite end, making the space "twisted" (like a Möbius strip). With a three-dimensional space, there are even more ways for a closed Universe to be connected together. In a closed Universe, a light beam will eventually return to its starting point, and then retraverse the same distance, over and over again. Question: For these two Universes, how do they differ with respect to the distance travelled by a light beam before it returns to its starting point?   If the Universe is closed, we should be able to "see ourselves" by looking far enough away in the distance. The only problem is that we are also looking back in time, since light takes a finite time to traverse the Universe and return to us. This distorts our view since we would be seeing our galaxy when it was much younger, and we might not recognize it!   If the Universe has less than the critical density, gravity is relatively weak, and the expansion causes space to curve away from itself. The Universe is then said to be hyperbolic. To an imaginary outside observer, a hyperbolic Universe would appear to have a saddle shape (it would have negative curvature). Picture Information Within a hyperbolic Universe, two parallel beams of light will diverge from each other. A hyperbolic Universe, like a flat universe, must extend indefinitely far in every direction, and again could be either open or closed.   If the Universe has more than the critical density, gravity is relatively strong, and causes space to fold over towards itself. The Universe is then said to be spherical. To an imaginary outside observer, a spherical Universe would appear like the surface of a sphere (it would have positive curvature). Note that the spherical Universe does not include the interior of the sphere (and therefore has no center, in compliance with the cosmological principle). Picture Information Within a spherical Universe, two parallel beams of light will converge together and cross each other. Unlike a flat or hyperbolic Universe, the spherical Universe is necessarily finite in size and closed. The spherical Universe still extends indefinitely, just like here on the surface of the Earth, where you can circle the globe forever if you chose to do so.

35.5 The Cosmic Dilemma As we have seen, it is very difficult to determine the amount of mass and energy in the Universe, because so much of it is in the form of dark matter. However, by observing the motions of galaxies, clusters, and superclusters, we can still estimate the total amount. Recent results suggest that there is about ten times as much dark matter as luminous matter, and that the total density of matter is d ~ 5 x 10-27 kg/m3 ~ dc /10 . In other words, it would seem that we live in a hyperbolic Universe.   On the other hand, observations of the most distant galaxies fail to reveal any evidence of curvature, suggesting instead that we live in a flat Universe!   Resolving the contradiction between these two sets of observations will require a better understanding of how the Universe is evolving.