Class 5
Class 5
Class 4
In spite of the fact that one needs General Relativity to accurately describe the evolution of the universe as a whole, classical mechanics at a high school level can give a perfectly accurate picture of the key issues in determining what factors will determine the future of the expansion. In fact, we can derive Einstein's equations of general relativity for an expanding universe using purely Newtonian arguments.
The global geometry of our universe can be classified by its final state of evolution. Either the expansion will continue unabated forever, or it will stop in some finite time, and the universe will recollapse. In the former case the universe is said to be open; in the latter it is closed. At the boundary between these two cases is a situation where the expansion of the universe will continue forever, but it will slow down, asymptotically approaching zero, but never quite stopping in any finite amount of time. This very special case is called a flat universe, and many cosmologists currently think this describes the universe in which we live.
Although these terms derive from geometric concepts associated with the curvature of spacetime, their operational meaning, in terms of the eventual fate of the universe, is perhaps more significant. We can understand under exactly which circumstances each type of universe will result, using only the concepts of classical mechanics which appear in any high school physics course. In so doing, we will also begin to appreciate how the uncertainty in the Hubble constant permeates throughout cosmology.
If the Universe is isotropic and homogeneous on large scales, then a spherical region which is small compared to the size of the universe, but large compared to the average distance between galaxies, can be considered in isolation. Namely, if the universe as a whole will continue expanding, so will this region, whereas if the universe recollapses this region will as well. As long as the expansion velocities of the galaxies at the edge of this region compared to the center are small compared to the speed of light, then Newton's Laws should hold.
Recall that if we write the total energy of an object as: E = Kinetic + Potential, then if an object has positive total energy E, it will be able to escape from the earth, while if it has negative E, it will fall back. Since the kinetic energy depends on velocity, this implies that the velocity must be larger than some minimum value, called the escape velocity, in order for an object launched from the earth's surface to escape. Newton showed that the field on the surface of a spherical mass is the same as if all the mass were at the center, and that the field inside a spherical hole in a uniform mass distribution is zero. We can then calculate whether a spherical region will expand or contract using the same principles we use to calculate whether an object will escape from the earth's surface. The outside galaxies have no net effect on motions inside this region.
We begin by considering the total energy associated with a galaxy of mass m on the edge of a spherical region of radius r which contains a total mass M:
E = Kinetic + Potential = (1/2)mv2 - GmM/r (1)
Now, using the fact that the uniform Hubble expansion relates the velocity of a galaxy at the edge of this region to the distance from the center by the relation v = Hr, where H is the Hubble constant, we can rewrite (1) as:
E = (1/2)mH2r2 - GmM/r (2)
Dividing both sides by mr2/2, we obtain:
2E/(mr2) = H2 - 2GM/r3 (3)
If the mass density inside the sphere is rho, the total mass in the sphere is the volume times the density, or (4p/3)r3r. Using this expression in (3), and recognizing that E/m is constant in time and does not depend on m, we can introduce the constant k = - 2E/m and rewrite (3) in its final form
k/r2 = 8prG/3 - H2, (4)
This equation is precisely Einstein's equation for an expanding universe, with all the 2's and pi's correct. In general relativity the constant k is called the curvature constant. If k is negative, the universe will go on expanding forever and is said to be open, while if k is positive, the universe will recollapse, and is said to be closed. We can see, based on this derivation and the relation k = -2E/m, that, in the Newtonian sense, k is precisely related to the total energy of an average galaxy trying to escape from the gravitational pull of its neighbors. If k is negative, the energy of the galaxy is positive, and it can escape, while if k is positive the energy is negative, and it will remain bound.
The isotropy and homogeneity of the Universe tells us that if an average region will recollapse, the universe will as well. Thus, equation (4) tells us that the future evolution of the Universe comes down to comparing the average density of matter with the expansion rate as determined by the Hubble constant. If we measure both of these quantities, and then subtract them in the relation shown in (4), if the answer is positive the universe is open, and if it is negative, the universe is closed.
As simple as this sounds in principle, in practice we still do not know the answer to this question. We have already seen how our uncertainty in the Hubble constant is large (about a factor of 2). It turns out that measuring the Hubble constant is the easy part. Determining the total average density of material in the Universe is much harder, because our primary tool is the telescope, which measures light, not mass. To turn an observation of light into an estimate of mass, we have to make educated guesses about how much mass corresponds to a given amount of light. If we assume that the light from distant galaxies comes from stars and gas much like our own galaxy, we find that the corresponding mass associated with light is only about 1/100 of the amount which would be needed to cause the universe to recollapse, or to "close the universe" in modern parlance.
Of course, just adding up the visible material merely gives a lower limit on the total mass of the universe. Many objects like planets, rocks, snowballs, cosmologists, do not shine, and therefore would be invisible to telescopes. To get a better idea of the total mass, we must weigh the Universe. Much work has been done on this problem, but the hunt for elusive forms of "dark matter" is still on.
Exercise:
Assuming the Hubble constant is equal to 100 h kilometers/sec/Megaparsec, where h is a constant between 1/2 and 1, calculate in grams/cubic meter the minimum average density of the universe in order for it to be closed. Remember that a parsec is approximately 3.16 light years, and that the speed of light is 3×108 meters/sec.
The Hubble constant directly gives an upper limit on the age of the universe, as follows. First, using the fact that gravity is an attractive force, and therefore must be slowing the expansion rate of the universe (assuming that the matter which dominates the universe today experiences gravity as normal matter does---a reasonable assumption), then at earlier times galaxies were moving apart faster than they are today. Thus, if we observe a galaxy at a distance r from us, and know its velocity v away from us, we can extrapolate back to determine the time when it was very close to us (the time of the Big Bang). Assuming that the galaxy moved at a constant velocity throughout, the time since the big bang is t = r/v. This is clearly an overestimate of t if the galaxy was moving at a faster speed in the past. Combining this estimate with the Hubble relation, v = Hr, we see that t = 1/H, or H = 1/t. Thus, if you think about it, the Hubble constant should be expressible in units of 1/sec. This is indeed the case. The normal way of expressing the Hubble constant is in units of km/sec/Megaparsec. Recalling that a Megaparsec is a unit of length, we see that it will cancel with km, so that we can re-express the Hubble constant in the units of 1/sec.
Exercise
Find an upper limit on the age of the universe, in years, if H = 100 km/sec/Megaparsec and if H = 50 km/sec/Megaparsec. The true value of H is probably between these two extremes.
In fact, because the expansion has been slowing down, if the universe is dominated by normal matter, the actual expansion age of the universe is between 2/3 and 1 times 1/H, depending on the amount of matter in the universe. If the universe were empty, there would be no gravitational attraction slowing down the expansion, so t = 1/H would be exact. According to current estimates, there is, at most, just enough matter to "close the universe" and make it "flat" in which case t = 2/3 times 1/H, as we show below.
The law of expansion
We will work out only the case where E =0, or k =0, corresponding to a flat universe. The other cases lead to more complicated formulas. Since the actual universe is very close to being "flat" and we are very far from "recollapsing", the simple formulas are good to describe what has happened, but not what will ultimately happen.
Taking E = 0 in Eq. (1) and canceling out m, we get the "escape velocity" relation
(1/2)v2 = GM/r (5)
We see from this relation that in the past, when the distance between galaxies was small, their relative velocity was large, becoming apparently infinite at r = 0. In fact, one can show from Eq. (5) (with a little bit of calculus) that v is proportional to the inverse cube root of time, while r is proportional to t2/3. If we know very little calculus, we can make the inspired guess that
r = Atb (6)
where A and b are constants to be determined. It follows that
v = dr/dt = A bt(b-1) (7)
and by substituting back into Eq. (5), we find that b = 2/3, as anticipated, and also that A3 = 9GM/2.
Even without computing A, we see from (6) and (7) that
v/r = 2/3t. (8)
But v/r is the inverse of the Hubble constant H. So, we found that t = (2/3)(1/H), or in words, that the "true time" since the Big Bang is 2/3 of the "Hubble time" 1/H.
Adapted, in part, from Teaching about Cosmology, © 1995 AAPT, L.M. Krauss and G. Starkman