Entropy, Heat Death, and the like The answer is we don’t know. The book good point probably won’t “bounce” because we will have gained entropy (disorder) therefore be so “disordered”, the universe won’t start “afresh.”
But maybe at high temperatures and density, our “normal” definition of entropy fails! So really, we don’t know!
Difference between energy density and entropy Suppose start with 2 separate gases: Hydrogen and He. Low entropy, highly ordered box box H He H He Wall removed begins to mix wall
contract box (like contracting universe) energy per unit volume goes up But arrow of time goes forward => H and He continue to mix! System proceeds to higher entropy!
In a closed universe, can we see the back of our heads? For our standard, L = 0 universe, the universe will re-collapse by the time the light from the back of our heads reaches our eyes.
How does q vary for the case where k = 0? Answer: not at all!= 1/2! Comes from 1 + kc2/R2 = W = 2q
Before the “L,” life was simple: k and W0 and q0 and the fate of the universe were all uniquely linked page 305, table 11.1: k W0 q0 fate -1 0 < < 1 0< < 1/2 expand forever 0 1 1/2 almost forever +1 1< 1/2 < big crunch _ _
. Look at (R/R)2 + kc2/R2 = (G8p/3) x r As prologue to “inflation” (why “we” like it):
Write r as r0(R0/R)3 , for matter (fixed number of particles, change volume) density term goes up faster (as must the R2/R2 term) as 1/R3 increases faster than the kc2/R2 term! => k becomes negligible => becomes effectively 0 => . . (R/R)2 + kc2/R2 = (G8p/3) x r0(R0/R)3 W tends to 1 as we go back Also, stays 1 if it started as 1.
Einstein’s “biggest blunder” • Einstein didn’t know the universe was expanding. • Static (and no L) means • r = (3kc2/8pGR2); set R = 0 and solve escape eq • rc2 = -3p (from dU = -pdV and previous eq )* • For ordinary matter and light (radiation), r = < 0 or p < 0 is unphysical => • Einstein added a “fudge factor.” . * math given as an “appendix”
Einstein called this the cosmological constant • (sometimes written l and sometimes L; • we’ll stickto using L) • To avoid p negative first • r => r + L/8pG = r • p => p - c2L/8pG = p • use new r and p in our equations => • Then using rc2 = -3p = 3kc4/(8pGR2) • for p = 0 (matter dominated) then kc2/R2 = L and • r = L/4pG <=> L > 0, r > 0 , what we want => • means k > 0 or k = +1 , p < 0 ~ ~ ~ ~ ~ ~ ~
Summary • Einstein wanted static universe and ordinary matter density and pressure positive • Couldn’t do this without adding in a fudge factor (cosmological constant, L) • For matter dominated era this (r ,ordinary > 0) drives • L > 0 and then k = +1 • and p < 0 ! ~
~ ~ But then dU = -pdV with p < 0 and the universe gains energy by expanding (if not static) => The cosmological constant gives us in modern models, and accelerating universe! (more later)
. R2/R2 = (G8p/3)r0(R0/R)3 + L/3- c2/R2 KE PE (attractive) Repulsive for L/3 > c2/R2 R decreases, then density (PE) term takes over and “rules” over L/3- c2/R2 collapse occurs R increases, then density term drops, no longer “rules”, L/3- c2/R2 rules, since L/3- c2/R2 > 0 is “repulsive”, expansion rules as L/3 will rule over c2/R2 as R increases => This situation is not stable.
This is OK, because we observe the universe expanding today and even accelerating
.. . q0 = -R0R0/R02 • We have two Omegas • Wm, WL and, WL= • q = Wm/2 -WL ; Wm+ WL = 1, k =0 => • q = Wm/2 -WL = 3Wm/2 -1= 1/2 - (3/2)WL • k = + 1 can still expand forever • See page 312, table 11.2 • Also from can see that q0 < 0 • means R0 > 0, accelerating universe • From Lecture 11 slide 2, see that WL will overtake Wm and we’ll always progress to an accelerating universe for k = 0 L/8pGrc ..
Games people play with L ^ • Lemaitre universe • Inflation in an “empty” universe (father of current day inflation models) • Current day models of “accelerating universe”
=> Lamaitre Universe ^ => Allows universe to be (much) older than implied by “normal expansion” equations. This allows for “seeing” in the backs of our heads (or complementary images of distant objects 180 degrees away.) But this effect is not seen in nature=> we reject these models.
^ Lemaitre see book (page 311) for good plot R L takes over Cooking phase r causes slow down t
De Sitter Universe, father of Modern Inflation • Assume q = -1 and flat (k = 0) , => Wm = 0 ! = “empty universe” => From escape equation => • R = Rx((sqrt(L/3)) or R = Riexp(tx[sqrt(L/3)]) • Key points are: R grows exponentially with time AND at t = 0 R has a finite size = no BB “singularity” = R does not go to an actual “point” as t goes to 0 .
. R can gets so large it exceeds the speed of light (by a lot), more on this later.
Appendix: U =rc2R3 ,dU = rc23R2dR + dr(c2R3) =-pdR3 = -p3R2dR For our static universe, r = 3c2/8pGR2 and drc2R3 = -(2dR/R3)(3/8pG)c2R3 x(R2/R2) = -2c2rR2dR => -2c2rR2dR+3R2rc2dR = R2rc2dR = -p3R2dR Or rc2 = -3p