Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mrao.cam.ac.uk/~anthony/aac_lecture15_overlays_2006.pdf Дата изменения: Thu Nov 9 17:13:52 2006 Дата индексирования: Mon Oct 1 22:52:00 2012 Кодировка: ISO8859-5 Поисковые слова: п п п п п п п п п п п п п п п п

Lecture 15
Plan for Lecture Will complete derivation of Angular Diameter Distance and then go through a specific example-- Einstein de Sitter universe Then look at what it becomes in a `flat ' universe (currently favoured) Then discuss velocities and the Hubble Law (actually derive this) And then specific examples of what the data on Luminosity distance seems to be saying about the dynamics of the universe Finally, a new topic -- horizons, in par ticular the concept of par ticle horizons, which have some surprising features!



Angular Diameter Distance in a EdS Universe
Consider the following problem: Show that, in an Einstein-de-Sitter universe, the angle subtended by an object has a minimum value for z = 1.25. What is this minimum angle for a typical galaxy? Here is a worked answer for this. = D /d where D = physical size of object, d = angular diameter distance = R0 S ()/(1 + z ) in general. So just have to find S () for an EdS universe. Now (t ) =
t t
0

c dt R (t ) dt -dz = . R R0 H (z )

and dz = -(1 + z )H (z )dt


Thus (z ) =

z 0

c /(R0 H (z )) dz so we just need H (z ).
2/3

Now in EdS, we know that R t more general results later). Thus have H

(will come back to this and

R t -1/3 1 1 2/3 3/2 (1 + z )3/ R t t R

2

Thus in EdS we have H (z ) = H0 (1 + z )3/2 . This gives (z ) = 2c R0 H0 1- 1 (1 + z )1/
2

.

Also, k = 0 (for EdS) S () . So = D (1 + z ) DH0 (1 + z ) = R0 2c [1 - (1 + z )-
1/2

]

.


8 7 /(DH0 /c ) 6 5 4 3 2 1 0 1/z Actual function

d ()/dz = 0 z = 5/4 at which point min = 27DH0 /(8c ). Taking D = 20 kpc as a galactic size, then yields min = 3.2 arcsec (for H0 = 70 km s-1 Mpc-1 ).

1 2 3 Redshift z

4


Angular Diameter Distance in a flat Universe

Above we looked at the specific example of the angular diameter distance relation in an Einstein de Sitter universe, and found that angular diameters, for fixed proper length, star ted getting bigger again after z = 1.25. However, we now know (from e.g. the supernovae data, and the combined CMB and large scale structure results) that while our universe is close to flat, it is far from EdS, and is in fact dominated by `dark energy', most likely a cosmological constant. What would the results look like in this case?


When we get back to discussing dynamics, below, we will show that for a flat universe with cosmological constant , then defining the `density of dark energy' = one can show H (z ) = H where
0 0

3H

2

(1 - 0 )(1 + z )3 +

1/ 2 0

is the value of today.

Tracing through the same steps as we went through for the EdS universe, we find that in this (still flat) case, we get = DH0 c
0 z

1+z dz (1 - 0 )(1 + z )3 +
1/ 2 0


8 7 /(DH0 /c ) 6 5 4 3 2 1 0 1 2 3 Redshift z 4
0

Function has to be evaluated numerically (or one can use elliptic functions) Get results as shown for 0 = 0 (EdS again) and 0 = 0.7. It is clear that as 0 is increased, there is still a minimum angular diameter, but that the point of minimum moves fur ther out in z and the curve is generally flatter at higher z .

=0 = 0.7

0





F I N D I N G T H E VA L U E O F еМ
Suppose Ъ и Using Ъ
еи

Л

ДЕЛ
ви

ив , for some в.

в Н and Ъ
ЪЪ ЪО

ДЕ ДЕ

Л вДв НЕив О we see that вДв НЕиОв О Н З Н Двив Н ЕО в
corresponds to a matter dominated

Thus е и

еМ

М

spatially flat, i.e. Einstein-de-Sitter universe. The idea is thus to use obser vations of what we hope are standard candles and plot their flux versus redshift, and compare this what what would be predicted for e.g. еМ

М

.

By this means could deter mine the geometr y of the universe. Prediction can be either using the exact Р о relation for a model, or we could do an expansion for low о , and thus get a generic result for how we expect flux to go with redshift, for low redshifts (which is where most of our obser vations are made).

ДЕ




/H0





Par ticle Horizons I

Next major topic will be the dynamics of the universe, and solutions for general density There introduce a new quantity -- the conformal time -- which is useful for parameterising these solutions. However, we'll first introduce it in an interesting and surprising context, that of par ticle horizons in cosmology. A `par ticle horizon' is the coordinate of the most distant par ticle/object that can be seen by a given observer at a given time. We will write it as p , and clearly in any model with a big bang origin we have