Chapter 1 8/25/05 68 Cosmology: A Cosmic Perspective Chapter 1: Evolutionary Cosmology Even if we ignore for a moment the discovery of cosmic expansion, Huble?s use of various distance indicators to demonstrate the extree remotenes of some galaxies has mind-bogling implications. In the first place, light travels at a finite sped (~10 13 km in a year) so that the light, which we receive now from a galaxy a bilion light years away, had to leave there a bilion years ago. We se the galaxy now by light emited then, so we se it as it was a bilion years ago. The finite sped of light means that loking out in space is loking back in time. Under everyday circumstances in our tiny world, light sems to travel instantaneously fro one point to another, but the sped of light is an important isue when considering cosmological distances. Figure 1a below shows how we can represent motion graphicaly whether a particles or a burst of radiation moving through space. The dashed line describes the motion of a burst of radiation. In time t?, it travels a distance D light , which is the largest distance that can be traversed in that time since no particle with mas nor any signal (no form of information) can travel through space faster than light does. A particle traveling at v = 0.5c (half the sped of light) is also shown. It travels at half the distance of the light signal in the same time, t?. This is a two-dimensional spacetime diagram, so only one space dimension can be shown. Supose the ?distance? diension coresponds to the altitude above the surface of the earth of a rifle bulet fired verticaly. The third curve represents the trajectory of that bulet rising from the surface, reaching a maximum altitude, then faling back to its starting point at time t?. Clearly the slope of a trajectory shown on this diagram coresponds to the sped of motion: the steper the slope the slower the sped since a smaler distance is covered in a fixed time increment. FIG. 1a Also, if the line slopes from lower left to uper right, motion is away from the observer because distance is then sen to increase with time. Conversely, if the line slopes from lower right to uper left, the motion is toward the observer. The position of the observer defines the point in space from which distance is measured; thus, by Chapter 1 8/25/05 69 definition, she ?stands stil? in space, but moves only (and inexorably) upward along the time axis in the diagram. No slope is shalower than that coresponding to the sped of light; so al trajectories that intersect the origin of the graph (time = 0, distance = 0) must always lie above the dashed line. Using this same kind of diagram, we represent in Fig. 1b the history of motion of galaxies relative to us as observers, taking into acount both the cosmic expansion and the finite sped of light, as in Fig. 1b. FIG. 1b In this diagram, only two galaxies are shown, one at 10 Mpc receding at 500 km/sec. The other is twice as far away moving twice as fast (a la Huble). If we se these galaxies at the present moment they must lie along that single line sloping to the uper left which coresponds to the path of any light that reaches our position ?now? (where ?now? is the point in spacetime shown). One may also construct the lines that give the past trajectories of the galaxies in their outward motion. These lines slope to the uper right, the slower twice as step since it moves half as fast. If the light lines were realisticaly portrayed on the same scale in this diagram, they would be almost indistinguishable from horizontal lines. From simple geometry, since the distances of these galaxies are in a ratio of two and their speds in a ratio of ½, their past trajectories converge at the position of the observer Chapter 1 8/25/05 70 in the remote past (?then?) as shown. In fact, the Huble law implies that any galaxy considered would be found at ?our position? back at a time labeled t = 0 in the diagram. Perhaps there is a simpler, les formal way to visualize this phenomenon. Supose you are watching a ovie of cosmological expansion in which galaxies and clusters of galaxies are sen to be retreating acording to the Huble Law. The projectionist sudenly decides to reverse the machine and we se the universe closing in on us (galaxies apear blueshifted now). The longer we view, the closer those once remote galaxies aproach us. Obviously this claustrophobic situation culminates in a singular state. Galaxies eventualy merge and the density of mater becomes arbitrarily large as we aproach the ?initial? moment, t = 0. Clearly a universe so constructed is quite diferent from the static, eternal cosmos of Newton. An expanding universe sems inescapably to embody the concept of age. This is a universe that ?begins? at some point in tie. One might say it is semi-eternal in the sense that it never ceases to be, but it apears necesary to suspend the notion that it has always existed (at least in a form we can presently understand). How old, then, is the Universe? This question sems to have a simple answer. From the top of a tal tower we watch the trafic on a straight freway leading from the tower?s base to the next city. We can judge the distance to a receding car by its aparent size (we know its true size) and we might use a Dopler radar system to measure its sped (perhaps a befed-up version of the kind used by police). In any case, supose we determine that the distance is 5 miles, and increasing at the rate of 50 miles per hour. Asuing the car has maintained this sped, we would have found it at the base of our tower about: Dnc tvel ped of = epsd ti, or 5 ml 50 mi/hour =0.1 ago That is, knowing the distance and recesion sped, one may easily estimate the elapsed travel time of a car from our location to any reote position at which we observe it. Using this same reasoning, because we can observe the distance and recesion sped of a remote galaxy, we may estimate the time taken to move from our location to its presently observed position. By the Huble law, if the distance is D, the recesion sped is the Huble constant, H, times D. Thus if D is 10 megaparsecs (10Mpc), and if H were 50km/s/Mpc: lapsd ti= dinc pe! ="0.02 pc km # $ % & ? ( s"20 bilion yers One Mpc is about 3 x 10 19 km, so: elpsd ti0.02310 19 sec = 610 17 sec There are about 3 x 10 7 seconds in a year, thus: la!20" 9 ye (20 bilion year). In examining this calculation more closely, we discover that the numerical value of ?elapsed time? for galaxy motion is independent of the particular galaxy chosen, regardles of distance! This can be sen imediately by noting that once the distance D of a galaxy is measured, the recesion sped folows acording to the Huble law: sped =H. Chapter 1 8/25/05 71 By substitution we find that the distance D is not relevant: elapsd tm= disnc peH! = 1 "0.02 Mpc km # $ % & ? ( sc"20 bilion years. But in the context of an expanding universe, this makes sense. Initialy, the galaxies we now se as remote would have ben arbitrarily close to us. As time pased, the Huble expansion would place them at their curently observed distances, thus the time elapsed for this motion is the sae for any galaxy and equivalent to the age of the universe. Recent observations indicate H ? 72km/s/Mpc, so the age is more like 14 bilion years. Of course this estimate asumes the recesion sped of a given galaxy has always ben the same. This necesarily means the Huble ?constant? is not constant, that distant galaxies were moving at the same high sped, even when closer. If we acount for the fact that the gravitational atraction betwen distant mases has gradualy decelerated that motion, then the speds were actualy greater in the past and the age estimate must be corespondingly smaler. This posibility is shown in the spacetime diagram Fig. 1b as the curved trajectories for the galaxies, indicating higher sped (shalower slope) in the past and a consequent reduction in the time interval betwen their intersection with our position in the past, and our curent sighting of them. The efect shown in the diagram is exagerated. For a low-density universe (as evidence sugests) this efect is sal and the corected age is perhaps a litle les than 14 bilion years. This age does not disagre with other estimates based on the ages of the oldest stars, and on radioactive decay measurements. Recal that this expansion is completely consistent with the Cosmological Principle. The Universe does not apear to expand from our position only, but from any (and consequently every) observer?s position. In the past the Universe did not begin as a super dense blob of mater at our position?the blob was everywhere, it filed the universe by definition. Furthermore, if the Universe is infinite now, it was also infinite then. The entire Universe was simply denser in the past and is continualy becoming more and more rarefied (a proces completely unrelated to its finitude or infinitude); but there is a subtle question dealing with the Cosmological Principle. In this interpretation of the expansion picture?and it sems a rather reasonable interpretation?the Universe is clearly evolving, changing at least in the sense of becoming les dense. So if we say that the Universe loks the same to al observers, we ust specify when it loks the same. The only satisfactory answer is: ?when the density is the same.? It turns out that because the sped of light is the same for al observers, Newton?s concept of absolute time must be discarded. Consider Fig. 1c which shows thre observers, A and B are at rest relative to each other and C is moving past A toward B. Two events, E 1 and E 2 , ocur at the positions in spacetime shown. * When in time do they hapen? Observers A, B and C would al disagre. * Each of these events could be thought of as simply the emision of a burst of light by a flashbulb. Chapter 1 8/25/05 72 Fig. 1c From the light trajectories shown, it is clear that A would claim the events to be siultaneous and B definitely ses E 2 before E 1 . The situation is more complicated for C. The light trajectories shown do not corespond to c = 3 x 10 5 km/sec from his moving frame of reference. The details can be worked out, and even though C is precisely at A?s position when A observes the events to be simultaneous, C wil swear that he ses E 1 before E 2 . So there is no absolute time that can be established for each and every observer. Measurements of time are necesarily related to position in space and motion through space. As you might gues the reverse is true also: measureent of space intervals depends on the motion of the observer and over what tie interval the measurements are made. This inter-relationship betwen space and time is fundamental to the relativity theory of Einstein. This problem does not imply esential dificulties for the Cosmological Principle. The Universe is asumed hoogeneous if parts are compared at the same ?age?: when the Huble ?constants? are the same; or alternatively, when the density of galaxies is the same (on the large scale) for two observers. It is interesting to examine the history of estimates of the age of the Universe begining with the biblical genealogies of the 17 th Century up through curent estimates based on the Huble constant. As shown in the diagram below, not only has the Universe as comprehended by man ben geting ?larger? over the past few centuries, but ?older? as wel?at least for those stories that involve a creation event. Chapter 1 8/25/05 73 FIG. 1d Is it posible to conceive of a model for the Universe that is consistent with the observational facts but does not evolve and thus does not have a begining. In such a model the density of galaxies must always be the same in spite of expansion. Also, the Huble constant must be truly constant so that galaxies must acelerate to higher and higher speds as they move farther away. A model eting these requirements can be constructed as shown in the diagra below. As time progreses (as one oves up the diagram), nothing changes. The patern of trajectories loks the same along any line. This was the Steady State Theory of Bondi, Gold, and Hoyle. It embodied the Perfect Cosmological Principle: The Universe loks the same for any observer, any time. It had two ad hoc ingredients: a mechanism to apropriately acelerate galaxies, and another to continualy create mater throughout space, forming stars and galaxies at precisely the rate necesary to maintain a constant density despite the expansion. The rate is only a secondary stumbling block because we do not even have a physical theory that describes the spontaneous, universal creation of mater in the sense required. However, we canot rule out the posibility of such a proces on the basis of laboratory investigations. The required creation rate, les than one atom per century in a volume 10 meters on a side, is quite unobservable now and with any foreseable technology. Chapter 1 8/25/05 74 FIG. 1e The Steady State Theory paints a world-picture almost as bleak as the static, eternal, infinite cosmos of the previous century?continual otion and no change, a Sisyphean nightmare. But that objection is hardly scientific, and since we live in an age that clais to have sucesfuly decoupled science and value judgments, we must have scientific reasons for rejecting a theory that may not be apealing. * Does there exist independent evidence in favor of an evolving cosmology? There are at least thre observational clues: Quasars, radio source counts, and the so-caled Cosmic Background Radiation (CBR). Interpreting their high redshifts cosmologicaly, Quasars lie only in the remote past and apear to represent an early phase in the evolution of galaxies. In any case, they do imply change. An interesting check on the nature of the large scale properties of the Universe is to use a radio telescope to count the number, N, of radio sources (objects, generaly galaxies or galaxy-like objects, which emit radio waves) brighter than a certain aparent brightnes, B, for a large range of B. If d is the distance to which one is searching for uniformly distributed sources, N is proportional to d 3 ?that is, the number is simply proportional to the volume being examined. Secondly, because of the inverse square * Read this sentence carefuly; it expreses a viewpoint worth discusing. Chapter 1 8/25/05 75 law, B is proportional to 1/d 2 if we asume al sources are of the same intrinsic brightnes. * Thus ! - ,hencB -1/2 , ndN! -3/2 . If the number of sources brighter than B is ploted against B -3/2 a straight line should result. As sen in Fig. 1f, maters are not so simple. The number and/or brightnes of faint (distant) sources are to large, relative to what one expects from a theory based on a uniform distribution of sources and source properties. Thus radio galaxies as wel as optical quasars sem to imply an evolving universe whose remote past was quite diferent from the conditions we observe ?nearby?. But neither of these phenomena is necesarily fatal to the Steady State hypothesis. The real chalenge came fro the Cosmic Background Radiation of Gamow, Dicke, Penzias and Wilson. FIG. 1f * The argument holds clas by clas, and thus for the total, even if there are many clases of intrinsic brightnes. Gregory Putzel Microsoft Word - Chapter11.doc
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