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Problem 47 Hard Difficulty

In Section 13.6, we discussed dark matter along with one proposal for the origin of dark matter: WIMPs, or
weakly interacting massive paricles. Another proposal is that dark matter consists of large planet-sized objects, called MACHOs, or massive astrophysical compact halo objects, that drift through interstellar space and are not bound to a solar system. Whether WIMPs or MACHOs, suppose astronomers perform theoretical calculations and determine the average density of the observable Universe to be 1.20$\rho_{c}$ . If this value were correct, how many times larger will the Universe become before it begins to collapse? That is, by what factor will the distance between remote galaxies increase in the future?




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Video Transcript

Okay. First stop. Critical density of the universe. Um, our density of the universe has given us row equals 1.2 times two critical density. And we know that critical density is three times Hubble constant squared over eight pi g. This being the gravitational constant are now consider galaxy That is at a distance, R okay. And so for this galaxy are the mass is equal to density times volume on. And so we're gonna use the average density of the universe. And so this will be 1.2 times three h squared over eight pi chili and volume. The sphere as ah four pi over three times are cute. That's the distance to their to the galaxy. Ah, and so this works out to be, and so you can see a few things cancel out here and there, and this works out to be 0.6 times h square times are cubed over a key. That's the mess. OK, so now, um, both now and in the future, when the galaxy has slowed to rest from V equals H times are two vehicle zero. Um, we're going to compute the energetic six moving the initial energy will equal to the final energy. Yeah, because the f um And so initially, you have kinetic energy Won't happen to square building that, um minus the gravitational potential in gene G m small am over our and finally you have kinetic energy zero. But its gravitational potential energy in GMM over big are, uh this, um s o the energy of this up of the galaxy in the sphere system will be constant when the galaxy moves to its distance. Are so this are is what? The final distance. And so that will tell us, uh, want the expansion factor. It's right. So you saw for our, um And, of course, all those small arms will cancel here. Uh, and, um and so this is 1/2 times, uh, h squared times are screened for V squared. Ah, minus gm over, uh, car equal to minus gm over big are, uh, and so now we plug in would be found for em. Begin, uh, which is, um, put six, uh, eight squared are cute over. And she, um And so, um And so, actually, let's let's pull out this term here. Some of it's that, uh and so you have you do have cheesy, but you have times 1/1 of her are plus, whenever big are all right. Cheese, cancel. Um and so that is equal to zero. Good. Um, and so and so eventually, you'll be left toe. Once you work out the algebra, you'll be off to it. Minus plate one is equal to 6.6 times are over. Big are, uh and, um And that's because, of course, the H squares cancel. You have 0.5 r squared, Uh, minus 0.6 are cube tons. Whatever. Small arms, wherever big art. And in essence, this is where you end up on dso are, will be equal to six times are and so the universe expanded six times.

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