Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~brian/A3002/lect5.pdf Дата изменения: Wed Oct 4 05:32:51 2006 Дата индексирования: Tue Oct 2 01:07:46 2012 Кодировка: Поисковые слова: universe

O b s e rv a b le s
· So far we have talked about the Universe's dynamic evolution with two Observables
­ a ­the scale factor (tracked by redshift) ­ t ­ time So a cosmological test would be to compare age with redshift. But cosmic clocks are in short supply.

Proper Distance
· Defined as what a ruler would measure at an instantaneous time. Use R-W Metric. For a fixed t, angle, going from r=0 to r=r.
!

0
2

% dr 2 ds2 = (cdt ) 2 " a 2 ( t )' + r 2 ( d# 2 + sin 2 #d$ 2 & 1 " kr dt = 0, d# = 0, d$ = 0 , arcsin(r) r% . a( t ) dr ( d proper = + ds = + ' r * = a( t )& 1 " kr 2 ) . 0 /arcsinh( r) a0 d a

( * ) k = +10 . k = 0 1 = a( t ) S ( r) . k = "12

d

proper

( t 0 , r) = a0 S ( r) =

proper

( t, r)

d

comoving

3

a d a0

proper

= aS ( r)

F irs t G o a t H u b b le L a w
& d d a & v = ( proper (t , r ) )= (aS (r ) ) = aS ( R) = d d dt dt a v = H (t )d proper ! H 0 d if d is small.
proper

Origin of Redshift
Massless particles (e.g. photons) travel on geodesics in GR ­ e.g.
ds 2 = 0 for all massless particles / dr 2 2 2 2 2, ds 2 = 0 = (cdt ) 2 ) a 2 (t )- 1 ) kr 2 + r (d2 + sin 2d1 * * . +

r,t 0,t

0

Objects which have an apparent velocity proportional to their distance. The Hubble Law. Somewhat tricky use of v here, which is not really A velocity, but rather a redshift.

0
t
0

0
1/ 2

0
t t
0

r / dr 2 , cdt =+ r 2 (d2 2 + sin 2 2d1 2 ) * * a 0 - 1 ) kr 2 + 0.

0
t

r / dr cdt =a 0 - 1 ) kr 0.

2

( arcsin(r ) k = +1% ," " *=' r k = 0 $ ! S (r ) * + "arcsinh (r ) k = )1" & #

S(r) represents a comoving coordinate ­ it is invariant as the Universe expands
t + "t

#
t

cdt = a

t 0 + "t

0

t

#
0

cdt = S (r ) a

"

"=c!t #=1/!t

In the limit that !t (and hence !t0) are small, integral becomes
!0 " !e Definition of the Redshift !e !0 Observed wavelength of light (Observer's frame) !e Emitted wavelength of light (rest frame)
z#

c"t c"t = a a0

0

t

t+!t

"t0 a0 = "t a "t0 $0 / c $0 a0 = = = "t $e / c $e a

When light is emitted from a object, its wavelength is represents the time interval between the wavelength peaks.

$0 a !1 = 0 !1 $e a $0 ! $e a = z = 0 !1 $e a

As the Universe Expands, Light is redshifted!


The Expansion Stretches Light Waves
Close, Recent Far, Ancient

T h e w a v e l e n g th o f l i g h t i n c r e a s e s a s i t tr a v e l s th r o u g h th e e x p a n d i n g U n i v e r s e "Redshift" T h e l o n g e r th e l i g h t h a s b e e n tr a v e l l i n g , th e more it's Redshifted ­ Not a Doppler Shift!

Luminosity Distance
· We observe the Universe via luminous objects. How bright they appear as a function of redshift is a fundamental observable.
f= L inverse square law for light in Euclidean Universe 4(d 2
1/ 2

&L# $ ! dL ' $ ! % 4(f "

Imagine we have a monochromatic bit of light Emitted in all direction. How bright will it appear As a function of distance

f=

L &a# $! 2 4(d p $ a0 ! %"

2

&L# $ ! & dL ' $ ! % 4(f "

1/ 2

· Observed flux is Luminosity spread out over the surface area of a sphere with radius of the proper distance. 2 L &a# $! f= 2 4'd p $ a0 ! %"
But! There is time dilation ­ photons arrive more slowly (same idea of redshift) as scale factor changes But! There is energy dimunition.Wavelength increases...So E per photon diminished. Add two factors of a/a0

&a# d L = d p $ ! = d p (1 + z ) $a ! % 0"


Angular Size Distance...
How big does a rod (of length l) look as a function of distance?
"# = l for Euclidean d l dt,dr,d$ = 0 dA $ "# ' dr 2 * ds2 = (cdt ) 2 % a 2 ( t )) + r 2 ( d# 2 + sin 2 #d& 2 , 2 ( 1 % kr + l=

Move to observable space...
, a (t ) = a0 *1 + * + d 2a (t 2 q 0 . - dt da (t dt da (t0 ) dt H0 . a0 ) (t - t0 ) 1 d 2 a (t - t0 ) 2 da (t0 ) (t0 ) + ... ' 2 ' dt a0 2 dt 2 a0 (
0

Taylor Expansion

)a )
2

0

Deceleration Parameter.

0

Hubble Constant 1 # 2 t0 ) - q0 H 0 (t - t0 ) 2 + ...! 2 " q0 ) 2 2 ' H 0 (t0 - t ) + ... 2( # q) + 0 ' z 2 + ...! 2( "

-

ds =

-

a( t ) rd# = ar"# ( k = 0) = a d a0
proper

dA = ar = d

comoving

=

dp dL = (1 + z) (1 + z)

2

& a (t ) = a0 $1 + H 0 (t % , z = H 0 (t0 - t ) + *1 + + 1& , (t0 - t ) = z - *1 H0 $ + %

Substitute in deceleration and Hubble Constant Non-trivial algebra...

!

True as derived here for flat Universe But always true!

q0 * 2 2 ( H 0 (t0 ' t ) + ... 2) # q* + 0 ( z 2 + ...! 2) " for a light ray, using R - W metric and ds 2 = 0 z = H 0 (t0 ' t ) + +1 + , 1& (t0 ' t ) = z ' +1 H0 $ , %
t
0

d d

proper

r - a (t )dr = 5 ds = 5 + + 2 0 , 1 ' kr

4 arcsin(r ) k = +11 * . ( = a (t ). r k = 0 0 = a (t ) S (r ) 3 ( ) .arcsinh (r ) k = '1. 2 /

proper

= ar

.
t

cdt = a
t

r

.
0

dr 1 ' kr
2

= S (r ) = r for the flat case
t

# c0 c 0& - q* 2 (1 + z )dt = . $1 + H 0 (t ' t0 ) + +1 + 0 ( H 0 (t ' t0 ) 2 + ...! dt = r a0 . a0 t % 2) , " t c& $(t a0 % c r= a0 H 0 r=
0

Substitute in For z

# H 0 (t0 ' t ) + ...! 2 " &1 # z ' (1 + q0 ) z 2 + ...! $2 % " ' t) +

2

2

Integrate Substitute in for t0-t

1 & # 2 a (t ) = a0 $1 + H 0 (t ' t0 ) ' q0 H 0 (t ' t0 ) 2 + ...! 2 % " # 1 & - q0 * 2 (t0 ' t ) = z ' +1 + ( z + ...! Substitute in for t0 H0 $ , 2) % " c& 1 # 2 r= z ' (1 + q0 ) z + ...! Multiply a(t) and r a0 H 0 $ 2 % " c &1 # expansions to get 2 d proper = z + (1 ' q0 ) z + ...! H 0 (1 + z ) $ 2 % " d L = (1 + z )d
proper

-t in a(t)

= (1 + z ) 2 d

A