To begin, let us introduce a commonly used unit of distance for very large scales. One parsec (derived from "parallax of one arc-second") is defined from the relation tan(0) =

1 parsec where 0 is an angle of one arc-second, i.e. 0 = 1'' = 2n/(360° x 602) (Figure 6.1).

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Figure 6.1 A triangle illustrating the definition of the parsec unit.

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Figure 6.1 A triangle illustrating the definition of the parsec unit.

Table 6.1 clarifies the various measures of distance that we have encountered, and their relationships to the meter.

There is now excellent evidence for Hubble's law, which states that the recessional velocity v of a galaxy is proportional to its distance d from us, that is, where H0 is called Hubble's constant, although since it varies in time, it fails to meet the criteria for being called "constant." The reason for the terminology "constant" for H0 is that Hubble was originally surprised that its value now should be a constant, that is, that v and d should be linearly related in this fashion (Figures 6.2 and 6.3). The Hubble constant frequently also appears in di-mensionless form, h = H0/100 km s-1 Mpc-1.

The exact value of the Hubble "constant" now is still somewhat uncertain, but H0 is generally believed to be around 70 kilometers per second for every megaparsec, so it has units of km/sec/Mpc. This means that h ~ 0.7 and a galaxy at a distance of 1 megaparsec from us will move away from us at a speed of about 70 km/sec, while a second galaxy 100 megaparsecs away recedes at 100 times this speed.

TABLE 6.1 Measures of Distance and Their Relationship to the Meter | |

MEASURE |
RELATIONSHIP TO THE METER |

meter (m) |
The fundamental unit of length in the metric system. Equal to about 3.3 feet or 1.1 yard. |

kilometer (km) |
Equal to 1,000 meters. |

Astronomical Unit (AU) |
The commonly used unit of distance in the solar system; it is equal to the average Earth-Sun distance, or 149,000,000 km. |

light year (ly) |
A commonly used unit of distance on galactic scales, defined to be the distance traveled by light in one year, or 9,460,000,000,000 km. |

parsec (pc) |
The preferred unit of distance in astronomy (outside the solar system). Defined as the distance at which 1 Astronomical Unit subtends an angle of 1 second of arc (1/3600 of a degree). Equal to 3.26 light years or 30,800,000,000,000 km. |

kiloparsec (kpc) |
1,000 parsecs. |

megaparsec (Mpc) |
1 million parsecs. |

Astronomers have studied the extrapolations of galactic trajectories backwards in time. The observation was that they converge, and this would seem to imply a high-density initial state. One is tempted to say that this initial state would have been exciting to see; however, it wouldn't have been visible at all before the time of last scattering! This is something that we will explain shortly, and the time of last scattering is defined following Equation 6.2.

The cosmological principle states that the universe appears the same in every direction from every point in space. It amounts

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2 x 106 parsecs 106 parsecs Distance 2 x 106 parsecs © Infobase Publishing Figure 6.2 An early Hubble diagram, illustrating the approximate linear relationship between distance (in parsecs) and recessional velocity, measured with observations of galaxies. to asserting that our position in the universe—when viewed on the very largest distance scales—is nothing special. One sometimes sees the word "Copernican" associated with this idea, since Copernicus was the first to carefully show that our position in the solar system is not central. Of course, Copernicus would have known nothing of galaxies, or of the large-scale structure of spacetime, but this sort of assertion is in the nature of Copernicus' defiance of the previously prevailing doctrine that we are in the center of anything. There is considerable observational evidence for the cosmo-logical principle, including the measured distributions of galaxies and faint radio sources, though the best evidence comes from the near-perfect uniformity of the cosmic microwave background (CMB) radiation, a type of energy which has its origins in the
early universe and yet is still passing through the Earth today and can be measured by satellites. The cosmological principle means that any observer anywhere will enjoy much the same view that we have of the large-scale structure of the universe, including the observation that the other galaxies appear to be receding. The expansion of the universe can be difficult to visualize. One analogy that is often used is the following one. Imagine that you are a two-dimensional creature. Like a thin piece of rubber, you may bend or curve, but regardless of the curvature, you have a certain length and width, but negligible height. Now, imagine that your "universe" is the surface of a perfectly spherical balloon, which is slowly being blown up. (Real balloons are more oval, and have a "neck" where the air goes in—for the purpose of this illustration, ignore both of these features.) As the balloon is blown up, the distance between all neighboring points grows; the two-dimensional universe expands but there is no preferred center. One may also imagine that someone has taken a marker and drawn uniformly distributed points all over the surface of the balloon. You (the two-dimensional version) are standing on one of the dots, but you can see a few of the others by means of light rays that travel geodesic paths along the curved surface to reach you. The other points that you can see appear to be receding from you. Note that, in this analogy, you should most certainly not conclude that you are in the center of the universe! Remember that the surface of a sphere has no center. The interior of the sphere does, of course, have a center, but we have assumed that the universe is the boundary. Now, we are not flat creatures, so it is perhaps more useful to think of the "space" part of our universe as also being the (three-dimensional) boundary of some four-dimensional thing that is being blown up. As strange as it may sound, this is the expansion that Edwin Hubble observed! This "balloon model" is also very consistent with general relativity. The Friedmann-Robertson-Walker solution of Einstein's equations with a perfect fluid matter-energy distribution (note that we are now interpreting the individual galaxies as particles in a viscous fluid!) gives a curved geometry that is well described by the balloon analogy. A simple form of this geometry has the schematic form ds2 - dt2 - a(t)(dx2 + dy2 + dz2) (6.1) where a(t) is an increasing function of t, such as a(t) = eHt, and dx2 + dy2 + dz2 represents the length in the usual flat three-dimensional geometry. Here, ds2 is a generalization of the spacetime interval that we already encountered in the case of special relativity. The time-dependent function a (t) is called the scale factor. The definition of the Hubble parameter is H = a'(t)/a (t), with H0 defined to be its value today and a'(t) is the derivative of a (t). Note that the ds2 in Equation 6.1 has the property, consistent with special relativity, that the squared length of a relativistic interval can be negative if the space part of the interval is longer than the time part. This is fine; in practice, one almost never needs to consider the square root of this negative quantity! The stretching of the wavelengths of photons implied by the expansion of the universe, and the associated growing of the scale factor, accounts for the redshift from distant galaxies: the wavelength of the radiation we see today is larger by the factor a (now)/ a (then). Astronomers denote this factor by 1 + z, which means that an object at redshift z emitted the light seen today when the universe was smaller by a factor of 1 + z. Normalizing the scale factor to unity today gives a (emission) = 1/(1 + z). |

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