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Calculating the Variation of R(t) and the Age of the Universe

Our goal is to calculate how the radius of the universe, R(t), varies with time and derive the age of the universe, t0. We will use physics laws, such as gravity, to determine the relationship between R(t) and time. By plugging it back into the equation cdt^2 = R(t)^2 dr^2 / (1 - kr^2), we can find t versus R(t). Through simple estimates, we can estimate t0 = 1/H0, where H0 is the expansion rate of the universe today. This expansion rate is related to the critical density, rc, and the parameter W0. We will also explore the effect of the cosmological constant, L, on the universe's acceleration. This analysis will help us predict the future of the universe based on its past.

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Calculating the Variation of R(t) and the Age of the Universe

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  1. Our Goal: take R(t) and physics (gravity) to calculate how R(t) varies with time. Then plug back into (cdt)2 = R(t)2dr2/(1-kr2) Get t versus R(t) and derive age of universe (t0) versus W0 and H0 Simple estimate of t0 = 1/H0 H0 = 50-70 km/sec-Mpc => 1/H0 has units of time = 19-14 billion years Mpc = megaparsec = 3 million lt-years = 3 x 1024 cm

  2. Want to show where the following come from: • H0 = expansion rate for universe today • rc = critical density = 3H0/8pG • W0 = r0/rc <=> k relation • q0 = de-acceleration parameter • L = cosmological constant <=> pressure and why positive L (and WL causes an accelerating universe) 2

  3. The distance light travels on the surface is greatly affected by the value of k. k = -1 open k = 0 flat k = 1 closed And, R(to)r for the observed object translates into a distance to the object today, and our goal is to figure out how to calculate R and r

  4. For the related figures, see page 217 (shows geometry) , 283 (shows R changing in different ways), and 299 (shows R for k = -1, 0, +1)

  5. For the math we will do, assume that there is no dark energy (cosmological constant) until further notice

  6. Predicting the Future from the past: A primary goal of the cosmologist is to tell us what will happen to R as function of time, based on fitting models to the data

  7. Predicting the Future from the past: • Measure R(t) by looking back in time • Measure how the geometry of the universe affects our measure of distance or apparent size. R(t0)/R(t) = 1+ z t = the age of the universe when light left the object t0 = age of the universe today by definition cf. pages 374-376

  8. Predicting the Future from the past: Also, R(t0)/R(t) = lob/lem =lambda(observed)/lambda(emitted). the universe is expanding R(t0) is always greater than R(t) (for us today) lambda(observed) must always be > lambda (emitted) longer lambda (now this means wavelength of light) means redder, we call this a redshift!

  9. How to get R(t) We need to relate R(t) to some “force” The Universe affects itself. It has self gravity Self-gravity will slow down expansion

  10. Equate potential energy (GMm/R) with kinetic energy [(1/2) mv2] M is the self-gravitating mass of the universe R is the scale factor of the universe. M = density(r) x volume[(4/3) x p R3)]

  11. How to get R(t), part 1, cont. density = r ; volume = 4pR3 => M = r4pR3 . v = R Aside: A subscript 0 means “today” (R(t0) = R0 ) to keep from writing R(t) or R(t0). . => (1/2)mv2 = (1/2)mR2 and GMm/R = Gr4pR3m/3R = Gr4pmR2/3

  12. How to get R(t), part 1, cont. KE > PE, we get “escape” KE < PE, the universe will collapse on itself. (1/2)mv2 = GMm/R, KE = PE The little m’s cancel out. Put an energy term on the KE side to allow us to describe “to escape or not to escape”

  13. . R2 + kc2 = G8prR2/3 R2 = G8prR2/3, now adding in the extra term . Yes! The k we used for our geometry and c is the speed of light. . R02 + kc2 = G8pr0R02 /3; today

  14. The KE, kc2, and PE connection . 2 2 R0 = G8pr0R0/3- kc2 So, k = -1 means the KE is more than the PE, and we get escape, and vice versa

  15. Critical density = when pull of gravity (PE) just balances the BB push (KE), i.e. the density when k = 0 !

  16. How to get R(t), part 1, cont. So, rc as it is called is when k = 0 and we have rc = 3R0/(8pGR0), but R0/R0 = H0 ! (another old friend) = the expansion rate of the universe today . . 2 2 2 2 Or, rc = 3H0/8pG 2 2 Or, 1 +kc2/(H0 R0 )= r0/rc = ? W0 2 2 Or kc2/(H0R0) = W0 -1 We see the relationship between k and W0 and the fate of the universe!

  17. Aside on H0: • How to use to get distances (good to 1+z of about 1.2) • D = v/H0 where v = velocity of recession • use km/sec along with H0 = 50 km/sec-Mpc for example • D = v/H0 is the “Hubble Relation” • Observation of this told us Universe is expanding • For z << 1, z = v/c (approximately) z <=> v

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