Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~brian/A3002/lect4.pdf Дата изменения: Tue Oct 3 10:28:27 2006 Дата индексирования: Tue Oct 2 01:07:29 2012 Кодировка: Поисковые слова: eugene cernan

Key Results from last lecture
a 2 + kc 2 =
2 8"G 2 1 % da ( 8"G #a $ 2 ' * = 2 #a 2 + k 3 c & dt ) 3c

Friedman Eq.
1/ 2

% da ( 2 8"G 4 dt a0 a #a + ka 2 , = ' *= 3c 2 d, c a0 & d, )
4 - d % a (0 % a( k30 # % a ( / ' *2 = ' * + k' * & a0 ) . d, & a0 )1 30 + 1 # 0 & a0 ) 2 2

a0 H 0 "0 # 1 c

=k

Boundary Condition G.R. Eq. II Robertson-Walker Metric Equation of State of Normal Matter

a=+

!

! % dr 2 ( ds2 = (cdt ) 2 " a( t ) 2 ' + r 2 ( d# 2 + sin 2 #d$ 2 )* 2 &1 " kr ) redefine as conformal time dt +=c, d- = d# 2 + sin 2 #d$ 2 a(t) % . dr 2 1( ds2 = ( a(+)) 2 '( d+) 2 " 0 + r 2 ( d-)3* 2 / 1 " kr 2) &

4 "G (# + 3 p)a 3

!

Model Content of Universe by the Equation of State of the different forms of Matter/Energy

P %(1+ w i ) wi " i # i $ (Volume) #i e.g., w=0 for normal matter -4/3 " ! V01 w=1/3 for photons w=!1 for Cosmological Constant

Figuring Out the Equation of State
w" p #c
2

"!V

#i a

3wi + 3

= constant
3w + 3

$ # '$ a ' & )& ) % # 0 (% a0 (

=1

&M $ & ) # $M $ !=% 0 $) ! &V % 0" $ $V %0 V &a# =$ ! V0 $ a0 ! %"
3

# ! ! " # ! ! "

!
demo
!

M E &a# = =$ ! M 0 E 0 $ a0 ! %"
?

?

H o w d o e s M /E g o e s a s th e s c a le fa c to r ?

3w + 3 = 3 * ? w = * ?/ 3

&a# $! 3( ? & ) # $ a0 ! & ) #& a # $ ! % "3 ' $ !$ ! = 1 $ !$ ! $) != % 0" & a # % ) 0 "% a0 " $! $a ! % 0" ? = 0 for normal matter ? = -1 for photons ? = 3 for Cosmological Constant

Flat Universe ­Matter Dominated
1 " da $ c 2 # dt a y+ a0 % 2 8(G 2 )a * k '= & 3c 2 dy dy da 1 da , = = , k =0 dt da dt a0 dt

1

2 2 " dy % 2 8(G) 0 2 " ) %" a % 1 2 " ) %" a % H 0 $ '$ ' = H 0 $ '$ ' Friedman Equation for a flat Universe $ '= 2 # dt & 3H 0 ,0 # ) 0 &# a0 & # ) 0 &# a0 &

" ) %" a % $ '$ ' = 1 for matter dominated universe # ) 0 &# a0 & " %*1 " dy % 2 2a 2 $ ' = H0 $ ' = H0 y # dt & # a0 & y dy = H 0 dt 2 y 3
3/2 *1

3

'&a# $ ! =1 ' 0 $ a0 ! %"

3

dy = H 0 t
2/3

" 3H t % y =$ 0 ' #2&

!


F r. E q f ro m la s t page O n ly w a y to r e c o n c ile th e s e o u r s u b s titu tio n w ith o u r e q u a tio n is f o r

& dy # $ $ d' ! ! % "

2

# dy & 2 # )2 )0 & 0 * %y* % (= ( 4()0 * 1)2 $ 2()0 * 1) ' $ d" '

2

# dy & 2 # )0 )2 )0 & 0 *% (1 * cos+ ) * % (= ( 4()0 * 1)2 $ 2()0 * 1) 2()0 * 1) ' $ d" '
1& '# sin 2 ' = $1 + cos ! 2% 2"

2

!

Flat Universe ­ Radiation Dominated
& dy # 2 & ( #& a # $ ! = H 0 $ !$ ! $ ( !$ a ! % dt " % 0 "% 0 "
4 2 2

& ( #& a # $ !$ ! = 1 for radiation dominated universe $ ( !$ a ! % 0 "% 0 " & dy # 2& a # $ ! = H0 $ ! $a ! % dt " % 0" ydy = H 0 dt y2 = H0t 2 y = 2H 0 t
2 '2

=

H y

2 0 2

(

)

1/ 2

Flat Universe ­Cosmological Constant Dominated
·
& dy # 2 & ' #& a # $ ! = H 0 $ !$ ! $ ' !$ a ! % dt " % 0 "% 0 "
0 2 2

Domination of the Universe
A s U n iv e rs e E x p a n d s
­ Photon density decays as a4 ­ Matter density decays as a3 ­ Cosmological Constant density decays as a0

& ' #& a # $ !$ ! = 1 for cosmological constant dominated universe $ ' !$ a ! % 0 "% 0 " & dy # 2& a # 2 $! $ ! = H0 $ ! = H0 y % dt " % a0 " 1 dy = H 0 dt y ln( y ) = H 0 t y=e
H0t 2 2 2

) rad a = = (1 + z ) )M a0 )" ( a =& ) M & a0 ' % # = (1 + z ) # $
!3 !3

· · ·

N o te th a t e x a c tly fla t U n iv e r s e r e m a in s fla t ­ i.e . " # i = 1 C o s m o lo g ic a l C o n s ta n t M o d e ls te n d to w a r d s fla tn e s s o v e r tim e O th e r m o d e ls te n d a w a y fr o m fla tn e s s o v e r tim e .


Log(a)

Cosmological Constant

matter radiation Log(t)