1 00:00:02,560 --> 00:00:05,515 Now let us take a look at the Friedmann equation. 2 00:00:05,515 --> 00:00:10,563 This is the basic equation of cosmology, and it's solution tells how universe will 3 00:00:10,563 --> 00:00:15,248 evolve on large scales. We can derive it, actually, in a fairly 4 00:00:15,248 --> 00:00:19,083 simple Newtonian approximation, if we accept a couple results. 5 00:00:19,083 --> 00:00:21,913 First, there is, so called, Birkhoff's theorem, 6 00:00:21,913 --> 00:00:26,188 which says that, for a spherically symmetric system, the force due to 7 00:00:26,188 --> 00:00:31,091 gravity at some radius is determined only by the mass interior to this radius, 8 00:00:31,091 --> 00:00:34,738 which makes sense. And then also, the energy contributes to 9 00:00:34,738 --> 00:00:38,573 gravitational mass density. So you really need to add them up 10 00:00:38,573 --> 00:00:41,590 together. So, total energy or mass density can be 11 00:00:41,590 --> 00:00:45,740 expressed as one or the other modules, square of speed of light. 12 00:00:47,540 --> 00:00:53,210 So how do we go about this. Imagine a sphere with a mean density of 13 00:00:53,210 --> 00:00:57,781 row and radius of R. And acceleration to a particle in 14 00:00:57,781 --> 00:01:04,382 surface, which is the second derivative of the radius, is given by the familiar 15 00:01:04,382 --> 00:01:08,776 formula. Now, if, if the density changes, it 16 00:01:08,776 --> 00:01:12,709 changes according to the inverse cube of the radius. 17 00:01:12,709 --> 00:01:16,717 So if we, again, use subscript zero for present moment, 18 00:01:16,717 --> 00:01:22,844 then you can derive the density at any given time through the cube of the radius 19 00:01:22,844 --> 00:01:27,987 divided the present one, which we can assume to be unity for 20 00:01:27,987 --> 00:01:32,407 simplicity. So now, plug that in the, in the first 21 00:01:32,407 --> 00:01:36,075 equation. And we have the following equation for 22 00:01:36,075 --> 00:01:40,431 the second derivative. Well, right away, this is telling us 23 00:01:40,431 --> 00:01:45,168 something interesting. If the density is not zero, the universe 24 00:01:45,168 --> 00:01:51,282 must be either expanding or contracting, because the second derivative is finite. 25 00:01:51,282 --> 00:01:55,561 And this sign depends, obviously, on the first derivative. 26 00:01:55,561 --> 00:02:02,060 But right away, this tells us that relativistic universe cannot be static. 27 00:02:02,060 --> 00:02:06,222 Well, we can integrate that in the following fashion. 28 00:02:06,222 --> 00:02:11,106 We multiply the whole thing with first derivative of r, r dot. 29 00:02:11,106 --> 00:02:17,030 And now, we remember that, from your calculus, that derivative of the square 30 00:02:17,030 --> 00:02:21,193 of the first derivative is given by this expression. 31 00:02:21,193 --> 00:02:26,157 So we can plug it in, and the equation now looks a little more 32 00:02:26,157 --> 00:02:29,599 complicated. But there is a reason for this. 33 00:02:29,599 --> 00:02:33,876 And we also remember that, the following will apply. 34 00:02:33,876 --> 00:02:39,712 That's the derivative of one over radius, divided by time as an independent 35 00:02:39,712 --> 00:02:43,880 coordinate, is given by default in formula. 36 00:02:43,880 --> 00:02:49,046 So now we plug that in, and we have an interesting, simple differential 37 00:02:49,046 --> 00:02:52,071 equation, which says that derivative time 38 00:02:52,071 --> 00:02:56,057 derivative of this thing in brackets is actually zero. 39 00:02:56,057 --> 00:03:01,223 And therefore, the thing in brackets must be constant, so it's called K. 40 00:03:01,223 --> 00:03:07,817 Turns out that's curvature constant. Now, if we replace present day density 41 00:03:07,817 --> 00:03:11,630 row zero with general density at any given time, 42 00:03:11,630 --> 00:03:16,157 rho, scaling by, appropriate scale factor, and then divide by r2. 43 00:03:16,157 --> 00:03:20,207 squared, we get the following equation. That looks simpler. 44 00:03:20,207 --> 00:03:26,005 That, in fact, is the Friedmann equation in the absence of the cosmological 45 00:03:26,005 --> 00:03:31,777 constant. So, finally we get this result, 46 00:03:31,777 --> 00:03:36,530 and let's take a, just, inspection of this. 47 00:03:36,530 --> 00:03:41,513 If K0 as it could be, then R dot must be always positive, 48 00:03:41,513 --> 00:03:48,070 because you move the gravity turn to the right side and the expansion goes 49 00:03:48,070 --> 00:03:51,893 forever. But it slows down, because the density 50 00:03:51,893 --> 00:03:55,592 goes down. This is so called critical or flat 51 00:03:55,592 --> 00:03:59,456 universe. If curvature constant is greater than 52 00:03:59,456 --> 00:04:04,414 zero, then initially R dot will be positive, and that means the universe 53 00:04:04,414 --> 00:04:09,420 will be expanding, but then at some point, we'll flip the sign and start 54 00:04:09,420 --> 00:04:15,132 contracting, so that closed the universe. And if curvature constant is less than 55 00:04:15,132 --> 00:04:20,561 zero then r dot must be always positive. It's never zero and universe expands 56 00:04:20,561 --> 00:04:25,850 forever, so that's the open universe. Alright, so this is the Friedmann 57 00:04:25,850 --> 00:04:28,862 equation, and we've neglected details including 58 00:04:28,862 --> 00:04:34,053 what happens with cosmological constant. But if cosmological constant is included, 59 00:04:34,053 --> 00:04:38,474 that is just yet another term. You can think of it as another density 60 00:04:38,474 --> 00:04:43,120 term, although it behaves differently from other kinds of density. 61 00:04:43,120 --> 00:04:48,193 So Friedmann equation is the equation of motion for a universe, for homogeneous, 62 00:04:48,193 --> 00:04:55,252 isotropic, relativistic universe. And solving this equation for a given 63 00:04:55,252 --> 00:04:58,340 parameters gives good cosmological models. 64 00:04:58,340 --> 00:05:04,296 But in order to do this we need one more equation, which is the equation of state, 65 00:05:04,296 --> 00:05:09,663 which tells us how is the density of whatever is filling up the universe 66 00:05:09,663 --> 00:05:14,062 changing as the universe expands. So next time we'll talk about 67 00:05:14,062 --> 00:05:18,600 cosmological parameters, and express Friedman equation in those terms.