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

8.286 Le ture 9 O tober 10, 2018 DYNAMICS OF HOMOGENEOUS EXPANSION, PART IV

Summary of Le ture 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 km-s -Mp , age = 9.56 { 9.70 billion H y ears | but stars are older. Con lusion: our univ erse is nearly �at, but not matter-dominated. a (0) = 0, with in�nite densit y , is a The Big Bang Singularity: feature of our mo del, but not ne essarily the real univ erse. Alan Guth Massa husetts Institute of T e hnology {1{ 8.286 Le ture 9, O tober 10, 2018

Summary of Le ture 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 km-s -Mp , age = 9.56 { 9.70 billion H y ears | but stars are older. Con lusion: our univ erse is nearly �at, but not matter-dominated. a (0) = 0, with in�nite densit y , is a The Big Bang Singularity: feature of our mo del, but not ne essarily the real univ erse. Alan Guth Massa husetts Institute of T e hnology {1{ 8.286 Le ture 9, O tober 10, 2018

the presen t distan e of the furthest parti les Horizon Distan e: from whi h ligh t has had time to rea h us. t Z 0 ( t ) = a ( t ) d t ` : ys ; horizon ph a ( t ) 0 0 2 = 3 � 1 a ( t ) / = ) = 3 t = 2 H t ` : ys ; horizon ph Alan Guth Massa husetts Institute of T e hnology {2{ 8.286 Le ture 9, O tober 10, 2018

Equations for a Matter-Dominated Universe (\Matter-dominated" = dominated b y nonrelativist i matter.) F riedmann equations: 2 2 a _ 8 � k 8 9 = G� ; � > > : ; a 3 a 2 4 � � a = G� ( t ) a : � 3 Matter onserv ation: 3 1 � a ( t ) � 1 � ( t ) ; or � ( t ) = � ( t ) for an y t . / 1 1 a 3 ( t ) a ( t ) An y t w o of the ab o v e equations an allo w us to �nd the third. Alan Guth Massa husetts Institute of T e hnology {3{ 8.286 Le ture 9, O tober 10, 2018

Evolution of a Closed Universe 2 2 � a _ � 8 � k 3 = G� ; � ( t ) a ( t ) = onstan t ; k > 0 : � a 3 a 2 2 Re all [ a ( t )℄ = meter/not h, [ k ℄ = 1/not h . De�ne new v ariables: a ( t ) ~ a ( t ) ~ ; t t (b oth with units of distan e) p � � k 2 2 Multiplyi ng F riedmann eq b y a = ( k ): 2 2 1 � da � 8 � G�a = 1 : � k dt 3 k 2 2 Alan Guth Massa husetts Institute of T e hnology {4{ 8.286 Le ture 9, O tober 10, 2018

2 2 1 � da � 8 � G�a = 1 � k 2 dt 3 k 2 (4 : 15) p 3 8 � G�a k = 1 : � 3 = 2 3 k 2 a Rewrite as 2 � d ~ a � 2 � = 1 ; � ~ a ~ d t where 3 4 � G� ~ a � : � 3 2 2 [ � ℄ = meter. � is onstan t, sin e �a is onstan t. Alan Guth Massa husetts Institute of T e hnology {5{ 8.286 Le ture 9, O tober 10, 2018

2 � � ~ 2 � ~ ~ d a a d a ~ p = � 1 = ) = d t : ~ ~ a d t 2 2 � ~ � ~ a a Then ~ t a ~ ~ ~ Z Z a d a f f ~ ~ p = = t d t ; f 2 2 � ~ � ~ a a 0 0 ~ where is an arbitrary hoi e for a \�nal time" for the al ulation, t f ~ and ~ is the v alue of ~ at time . a a t f f Alan Guth Massa husetts Institute of T e hnology {6{ 8.286 Le ture 9, O tober 10, 2018

Evolution of a Closed Universe = � ( � � sin ) t � ; a p = � (1 � os ) � : k p p � � � � � 2 � � 1 2 � � 1 = ar sin � � t : 3 = 2 2 j H j (� � 1) � � Alan Guth Massa husetts Institute of T e hnology {7{ 8.286 Le ture 9, O tober 10, 2018

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Evolution of a Closed Universe = � ( � � sin ) t � ; a p = � (1 � os ) � : k p p � � � � � 2 � � 1 2 � � 1 = ar sin � � t : 3 = 2 2 j H j (� � 1) � � Alan Guth Massa husetts Institute of T e hnology {9{ 8.286 Le ture 9, O tober 10, 2018

Evolution of a Closed Universe = � ( � � sin ) t � ; a p = � (1 � os ) � : k p p � � � � � 2 � � 1 2 � � 1 = ar sin � � t : 3 = 2 2 j H j (� � 1) � � Alan Guth Massa husetts Institute of T e hnology {9{ 8.286 Le ture 9, O tober 10, 2018

p p � � � � � 2 � � 1 2 � � 1 = ar sin � � t : 3 = 2 2 j H j (� � 1) � � � 1 Quadran t Phase � Sign Choi e sin () � 1 Expanding 1 to 2 Upp er 0 to 2 � 2 Expanding 2 to 1 Upp er to � 2 3 � 3 Con tra ting 1 to 2 Lo w er to � 2 3 � 4 Con tra ting 2 to 1 Lo w er to 2 � 2 Alan Guth Massa husetts Institute of T e hnology {10{ 8.286 Le ture 9, O tober 10, 2018

Alan Guth Massa husetts Institute of T e hnology {11{ 8.286 Le ture 9, O tober 10, 2018

Alan Guth Massa husetts Institute of T e hnology {12{ 8.286 Le ture 9, O tober 10, 2018