8.286 Leture 9 Otober 10, 2018 DYNAMICS OF HOMOGENEOUS - - PowerPoint PPT Presentation

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8.286 Leture 9 Otober 10, 2018 DYNAMICS OF HOMOGENEOUS - - PowerPoint PPT Presentation

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8.286 Leture 9 Otober 10, 2018 DYNAMICS OF HOMOGENEOUS EXPANSION, PART IV Summary of Leture 8 Age of a Flat Matter-Dominated Universe: 2 2 = 3 1 a ( t ) / = ) = t t H 3 1 1 F or = 67 : 7 0 : 5

slide-1
SLIDE 1 8.286 Le ture 9 O tober 10, 2018 DYNAMICS OF HOMOGENEOUS EXPANSION, PART IV
slide-2
SLIDE 2 Summary
  • f
Le ture 8 Age
  • f
a Flat Matter-Dominated Universe: a(t) / t 2=3 = ) t = 2 3 H 1 F
  • r
H = 67:7
  • 0:5
km-s 1
  • Mp
1 , age = 9.56 { 9.70 billion y ears | but stars are
  • lder.
Con lusion:
  • ur
univ erse is nearly at, but not matter-dominated. The Big Bang Singularity: a(0) = 0, with innite densit y , is a feature
  • f
  • ur
mo del, but not ne essarily the real univ erse. Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {1{
slide-3
SLIDE 3 Summary
  • f
Le ture 8 Age
  • f
a Flat Matter-Dominated Universe: a(t) / t 2=3 = ) t = 2 3 H 1 F
  • r
H = 67:7
  • 0:5
km-s 1
  • Mp
1 , age = 9.56 { 9.70 billion y ears | but stars are
  • lder.
Con lusion:
  • ur
univ erse is nearly at, but not matter-dominated. The Big Bang Singularity: a(0) = 0, with innite densit y , is a feature
  • f
  • ur
mo del, but not ne essarily the real univ erse. Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {1{
slide-4
SLIDE 4 Horizon Distan e: the presen t distan e
  • f
the furthest parti les from whi h ligh t has had time to rea h us. ` ph ys;horizon (t) = a(t) Z t a(t ) dt : a(t) / t 2=3 = ) ` ph ys;horizon = 3 t = 2 H 1 : Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {2{
slide-5
SLIDE 5 Equations for a Matter-Dominated Universe (\Matter-dominated" = dominated b y nonrelativist i matter.) F riedmann equations: 8 > : _ a a 9 > ; 2 = 8 3 G
  • k
2 a 2 ;
  • a
=
  • 4
3 G(t)a : Matter
  • nserv
ation: (t) / 1 a 3 (t) ;
  • r
(t) =
  • a(t
1 ) a(t)
  • 3
(t 1 ) for an y t 1 . An y t w
  • f
the ab
  • v
e equations an allo w us to nd the third. Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {3{
slide-6
SLIDE 6 Evolution
  • f
a Closed Universe
  • _
a a
  • 2
= 8 3 G
  • k
2 a 2 ; (t)a 3 (t) =
  • nstan
t ; k > : Re all [a(t)℄ = meter/not h, [k ℄ = 1/not h 2 . Dene new v ariables: ~ a(t)
  • a(t)
p k ; ~ t
  • t
(b
  • th
with units
  • f
distan e) Multiplyi ng F riedmann eq b y a 2 =(k 2 ): 1 k 2
  • da
dt
  • 2
= 8 3 Ga 2 k 2
  • 1
: Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {4{
slide-7
SLIDE 7 1 k 2
  • da
dt
  • 2
= 8 3 Ga 2 k 2
  • 1
= 8 3 Ga 3 k 3=2 2 p k a
  • 1
: (4:15) Rewrite as
  • d
~ a d ~ t
  • 2
= 2 ~ a
  • 1
; where
  • 4
3 G ~ a 3 2 : [℄ = meter.
  • is
  • nstan
t, sin e a 2 is
  • nstan
t. Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {5{
slide-8
SLIDE 8
  • d
~ a d ~ t
  • 2
= 2 ~ a
  • 1
= ) d ~ t = ~ a d ~ a p 2 ~ a
  • ~
a 2 : Then ~ t f = Z ~ t f d ~ t = Z ~ a f ~ a d ~ a p 2 ~ a
  • ~
a 2 ; where ~ t f is an arbitrary hoi e for a \nal time" for the al ulation, and ~ a f is the v alue
  • f
~ a at time ~ t f . Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {6{
slide-9
SLIDE 9 Evolution
  • f
a Closed Universe t = (
  • sin
  • )
; a p k = (1
  • s
  • )
: t =
  • 2jH
j(
  • 1)
3=2
  • ar sin
  • 2
p
  • 1
  • 2
p
  • 1
  • :
Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {7{
slide-10
SLIDE 10 {8{
slide-11
SLIDE 11 Evolution
  • f
a Closed Universe t = (
  • sin
  • )
; a p k = (1
  • s
  • )
: t =
  • 2jH
j(
  • 1)
3=2
  • ar sin
  • 2
p
  • 1
  • 2
p
  • 1
  • :
Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {9{
slide-12
SLIDE 12 Evolution
  • f
a Closed Universe t = (
  • sin
  • )
; a p k = (1
  • s
  • )
: t =
  • 2jH
j(
  • 1)
3=2
  • ar sin
  • 2
p
  • 1
  • 2
p
  • 1
  • :
Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {9{
slide-13
SLIDE 13 t =
  • 2jH
j(
  • 1)
3=2
  • ar sin
  • 2
p
  • 1
  • 2
p
  • 1
  • :
Quadran t Phase
  • Sign
Choi e sin 1 () 1 Expanding 1 to 2 Upp er to
  • 2
2 Expanding 2 to 1 Upp er
  • 2
to
  • 3
Con tra ting 1 to 2 Lo w er
  • to
3 2 4 Con tra ting 2 to 1 Lo w er 3 2 to 2 Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {10{
slide-14
SLIDE 14 Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {11{
slide-15
SLIDE 15 Alan Guth Massa husetts Institute
  • f
T e hnology 8.286 Le ture 9, O tober 10, 2018 {12{