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Lecture 17

Plan for Lecture Will continue with `par ticle horizons' Then look at dynamics of the universe, and get out solutions for general density in Friedmann case ( = 0). Then begin discussion of the age of the universe in these models, and how it compares with ages measured in real objects


Par ticle Horizons II
t1

p =
0

c dt R (t )

where t1 is the epoch of observation. The link with conformal time is that we shall see later that the conformal time at some given cosmic time t1 is given by precisely this expression, i.e.
t
1

(t1 ) =
0

c dt R (t )

To see that p is not necessarily infinity, i.e. that par ts of the universe are not accessible to observation, consider the early stages of the universe. We showed in Section 3 of Handout PtII-21 that R t 2/(3+ ) for any Friedmann model, for sufficiently small t.


Par ticle Horizons III
Here 3p/(c 2 ) encodes the `equation of state': for `dust' and = 1 for radiation. =0

Suppose now that t1 is some small time > 0, and consider an observer at that time trying to see the most distant object he or she can. Then photon reaching the observer must have been emitted at the big bang itself, i.e. t = 0. We thus have
t1

p =
0

c dt R (t )

t1 0

dt t
2/(3+ )

-1 = lim t 0 1 - 2 3+

t -
2 3+

1

+1

,

which tends to a finite quantity if 2/(3 + ) < 1, which it is for all the values of we have considered so far.






Dynamics of the Universe I
First let's remind ourselves of the basic dynamical equations, in the most general form in which we met them in Par t II Astrophysics (see e.g. Handout PtII-21). These are Ё R 4 G + (1 + ) - = 0, R 3 3 R R
2

(A )

-

8 G kc 2 - =- 2. 3 3 R

(B )

tells us how impor tant the pressure is. We define = 3p , c 2 = 1, 0.

and thus have radiation dominated equation of state: matter dominated equation of state:


Dynamics of the Universe II
Here = ordinar y density for matter, and energy density/c 2 for radiation. A useful initial aim is to transform these equations into equations involving some parameters we have already met, namely H= R , R = , 3H 2 q=- Ё RR , R2 m = 3H /8 G
2

3H 2 /8 G is of course the critical density if there is no . Translating equation (A ) into these parameters we have -qH 2 + H and thus q= The other equation is
2

m (1 + ) - H 2 = 0 2 m (1 + ) - 2




Dynamics of the Universe III
H 2 - m H 2 - H 2 = - leading to 1 - m - = - The universe is thus closed flat open (+ve curvature) (zero curvature) (-ve curvature) if m + > 1 if m + = 1 if m + < 1. kc 2 R2H
2

kc 2 R2

The q equation is interesting in telling us that the universe is accelerating if m (1 + ) < . The current best guesses for 2 these parameters are
m0

0.3,

0

0,



0

0.7

leading to q0 -0.55, a substantial acceleration!


Dynamics of the Universe IV
Analytical solutions to the equations are difficult to achieve unless either We restrict attention to = 0 (Friedmann models), or We allow = 0, but restrict ourselves to flat models. We will look at both these possibilities. For the former, while the universes involved may not be currently favoured, they were the object of intense study for many decades, and have quite pretty mathematical forms, and thus form par t of the knowledge one requires to understand a lot of (still existing) literature and even popular expositions. For the latter (flat models with ), the details of these models are not too hard to work out, but can still lead to difficult integrals, so we will content ourselves with just discussing `ages' in these models, which will be considered next time.


Analytic solution of Friedmann models I
These models still constitute a large class, since we can consider open closed or flat models, and radiation or dust as the dominant constituent in each. Flat models are simple -- we did these in Par t II. We found (e.g. Handout PtII-21, Section 3) R t 2/3 for matter dominated flat, and R t 1/2 for radiation dominated flat. The approach for more general models is to work equation (B ) into a simpler form involving dimensionless variables, and achieve a solution parameterised not by cosmic time t , but by the conformal time
t
1

(t1 ) =
0

c dt R (t )

introduced in the last Handout.


Analytic solution of Friedmann models II
Let us change the independent variable to this. The R derivative then becomes dR d c dR dR = = R= dt d dt R d Also, we will replace the variable by an expression in R , using the `energy conservation' relation we found by considering the work done when a sphere containing cosmic fluid expands. This yielded (Handout PtII-21, Section 1) R
-(3+ )

Making these changes in the (B ) equation then gives us 1 R2 c dR R d
2 3 8 G0 R0 + kc 2 - =- 2 3R 3+ R


Analytic solution of Friedmann models III
We introduce a dimensionless variable to replace R , defining a = R /R0 , and multiply both sides by R 2 /c 2 , obtaining 1 2R a
2 0

da R0 d

2

-

2 8 G0 R0 = -k 3c 2 a1+

If we denote the dimensionless combination
2 8 GR0 0 1 = am+ 3c 2

then finally we can write as a `master equation' for all the cases 1 da a d
2

=

am a

1+

-k

Now star t with closed matter-dominated, so k = 1 and = 0, and our master equation becomes


Analytic solution of Friedmann models IV
1 da a d
2

=

am -1 a

which is now getting simple enough to solve easily! Thiking initially of the par t of the solution coming out of the big bang, so a increases with , we have da = d
a

a(am - a)

=
0

d =
0

dx x (am - x )

(Note a = 0 at the big bang ( = 0).) We do the integral via the substitution x = am sin2 y , so dx = 2am sin y cos y dy and =
0 y (a )

2am sin y cos y am sin y am cos y
2 2

dy = 2y (a)


Analytic solution of Friedmann models V 2
where y (a) is such that am sin y (a) = a. Thus = 2y (a) = am sin2 ( /2) = a = R = R0 Then


am (1 - cos ) 2

t=
0

R ( ) d c

=

t=

R0 am ( - sin ) c2

We thus get a cycloid in the t - R plane! As can be seen from the figure, am is well-named, since (when multiplied by the current scale factor, which brings in the dimensions) it gives the scale factor of the universe at maximum expansion, before the recollapse begins. The image of the universe's evolution being given by a circle rolling along a line is of course very suggestive -- could there be fur ther revolutions after the `big crunch', or before the `big bang'??


Analytic solution of Friedmann models VI
Scale factor R

R0 am R0
am 2



BIG BANG
Cosmic time t

BIG CRUNCH
t = 2
R0 am c2

Figure: Plot showing the development of the scale factor with cosmic time for a closed matter-dominated universe. The scale factor develops as a `cycloid', i.e. the locus of a point on a circle as it rolls along a line.


Analytic solution of Friedmann models VII
We can tie the above quantities through to m0 and H0 as follows. We know from the general relation above that 1 - m = - kc 2 H 2R
2

(if = 0)

Thus evaluating at the present R0 = c | H0
m0

- 1|-

1/ 2

and this is true for k = ±1. Also, perhaps surprisingly, am =
m0
1 1+

m0

-1

(We use mod signs again so that the result can be applied in the k = -1 case as well).


Analytic solution of Friedmann models VIII

Why does (1) follow? Our asser tion (for the k = +1 case), is that 2 8 GR0 0 m 0 = 2 m 0 - 1 3c i.e., using (1) for R0 8 Gc 2 0 = m 2 3c 2 H0
0

which is indeed the definition of m0 . A similar analysis for the k = +1 radiation case ( = 1) gives 1 da a d
2

=

am a

2

-1

which leads to the simple solution R = R0 am sin , t= R0 am (1 - cos ) c

This is actually a semicircle in the t - R plane!


Analytic solution of Friedmann models IX
Scale factor R R0 am

BIG BANG

BIG CRUNCH
Cosmic time t

Figure: Plot showing the development of the scale factor with cosmic time for a closed radiation-dominated universe. In this case the scale factor develops as a semicircle.

Note only gets as far as here.


Open models I
Here we are dealing with k = -1. These exactly parallel the closed case, but with sin sinh and cos cosh, and some changes of sign. The results are Dust case: R = R0 am (cosh - 1) , 2 t= R0 am (sinh - ) c2

Radiation case: R = R0 am sinh , t= R0 am (cosh - 1) c

Both of these expand indefinitely (more rapidly than Einstein de Sitter). Thus it is only if m0 > 1 that the universe re-collapses.


Open models II

R

open, dust open, radiation

t


Ages in Friedmann models I
We will now show that it is possible to calculate ages just from purely local observations, i.e. using q0 and H0 . For example, consider the k = +1 matter-dominated model, and let us evaluate R at the current time. R0 = R0 But am = and thus 0 = cos
-1

am (1 - cos 0 ) 2
m0

=

cos 0 = 1 -

2 am

m0

-1

( = 0)

1-

m 0 - 1 m0 /2

= cos

-1

1 -1 q0

using q0 = m0 /2 if 0 = 0. Plugging in the expressions above for t and R0 , then yields


Ages in Friedmann models II

t0 =

1 q0 H0 (2q0 - 1)3

/2

cos-

1

1 1 - 1 - (2q0 - 1)1 q0 q0

/2

So can indeed deduce the age of a model universe just from observations which we can in principle make locally. A very similar expression can be found in the open case. Again working just with a matter-dominated universe, one finds t0 = 1 q0 H0 (1 - 2q0 )3/ 1 (1 - 2q0 )1/2 - cosh q0
-1

2

1 -1 q0


Ages of real objects in the Universe
Having obtained formulae for the ages in general models, we should make a comparison with the ages of actual objects in the universe. In the mid results on Telescope (80 km s-1 1990's, this star ted causing real difficulties, as the Hubble constant from the Hubble Space star ted coming in. These favoured `high' Mpc-1 and above) values of H0 .

Meanwhile, the theoretical prejudice was still that the universe was Einstein de Sitter, with an m0 close to 1. The problems this led to, in comparison with data on real ages from globular clusters, are shown in the Figure showing `Universe age' versus m0 . We now discuss how constraints on the real ages of objects can be found.