Cosmology Freedman Equations

PHYM432 Relativity and Cosmology 18. Cosmology Freedman Equations Proper Distance - From the RW metric, we know the coordinate time t represents the cosmic time, which is the proper time for all the fundamental observers, moving with the Hubble flow. But what is the relationship between co-moving coordinates and the proper distance at some arbitrary time t, which a fundamental observer would measure?  ds = c dt 2

2

2

dr2 R (t) + r2 d 2 (1 kr ) 2

2

+ r2 sin2 d⇥2

proper distance is measured simultaneously so dt=0 

2 dr 2 ds2 = d⇥ 2 = R2 (t) + r d (1 kr2 )

2

+ r2 sin2 d⇤2

The proper distance is still a function of the proper time, as the scale factor will expand and contract in an expanding/contracting universe. The co-moving coordinates, remember, do not change their values.

To find the proper distance, we need to integrate d The problem is simpler if we use the isotropic/homogeneous properties to position our coordinates such that we only integrate along a radial direction from the origin to the distance desired.

d = d⇥ = 0

d = R(t) =

k=+1 k=0 k=-1

Z

(1

d = R(t)

= R(t) = R(t)

= R(t)

Z

Z

0

Z0

0

Z

dr kr2 )1/2 (1

dr kr2 )1/2

r=0 r=

dr = R(t) sin 2 1/2 (1 r ) dr = R(t)⇥ 1/2 (1) dr = R(t) sinh 2 1/2 (1 + r )

1

⇥

1

⇥

From these relations, we can work out how quickly the proper distance between fundamental observers changes with respect to the expansion or contraction of the universe. d dt

k=+1

d dR = sin dt dt

k=0

d dR = ⇥ dt dt

k=-1

d dR = sinh dt dt

1

⇥

1

⇥

k=+1

We can relate R

= sin

k=0 1

⇥

R

=⇥

k=-1 R

= sinh

1

⇥

So in all three cases

d 1 dR = dt R dt

vp = H(t)dp

H(t) =

1 dR R dt

H(t) is the Hubble parameter

The radial velocity of two fundamental observers (galaxies) relates to the proper distance between them by, vp = H(t)dp This equation tells us that at any time t, every fundamental observer (galaxy) is moving rapidly relative to every other galaxy, with a proper speed which is related by the proper distance between them. In an expanding universe, the velocity is positive, and galaxies are reshifted away. In a contracting universe, the velocity is negative, and galaxies are blueshifted, moving toward each other.

Freedman equations We have already worked out the Robertson-Walker metric should look like, based upon a few basic assumptions, backed up by observations. What we need now is the full solution to Einstein’s field equation, which will relate the energy and mass of the universe to the curvature, thus telling us how the dynamics of the universe and cosmological expansion depend upon the universe’s content, how much radiation or matter for example. The more massive a universe is, presumably the slower expansion would be (for instance). The Freedman equations are the solution to Einstein’s Field equation, and relate the dynamics of expansion (contraction) to a Universe’s cosmic composition. Of course we could solve this formally, using the field equation.

R↵

1 g↵ R = 2

8 G T↵ 4 c

R↵

1 g↵ R = 2

8 G T↵ 4 c

We would then have to solve the Ricci tensor

↵

1 = g↵ 2

✓

g g + X X

ds = c dt 2

g00 g11 = g22 = g33 =

2

2

g X 

◆

dr2 2 R (t) + r d (1 kr2 ) 2

2

+ r2 sin2 d⇥2

= c2 R2 (t)/(1 kr2 ) R2 (t)r2 R2 (t)r2 sin2

and assume an ideal fluid The derivation is quite straightforward, though a bit long. Along with conservation of energy, you end up with two independent equations.

r↵ T ↵ = 0

A first equation is the energy equation, which is the fundamental equation governing the expansion of the universe.

⇥1

⇤ dR 2 R dt

=

kc2 R2

8 G 3 ⇢

The energy equation

While conservation of energy gives the fluid equation. d dt

=

3 dR R dt

⇢+

p c2

The fluid equation

A third equation, the acceleration equation is also used 1 d2 R R dt2

=

4 G 3

⇢+

3p c2

The acceleration equation

Each equation can be derived from the other two.

In the derivation, we assume the matter, radiation, and energy is described by an ideal fluid

T↵

2

2

c 6 0 =6 4 0 0

0 p 0 0

0 0 p 0

3

0 0 7 7 0 5 p

ideal fluid

energy density

Tµ

2

00

T 6 T 10 =6 4 T 20 T 30

01

T T 11 T 21 T 31 shear stress

02

T T 12 T 22 T 32

03

3

T T 13 7 7 23 5 T T 33 pressure

Energy-Momentum tensor

R↵

1 g↵ R = 2

8 G T↵ 4 c

Rather than going through the derivation with Einstein’s field equation. It is (perhaps) more enlightening and simpler to use basic physical arguments to derive the Freedman equations. First law of Thermodynamics for FRW models (fluid equation) The entire evolution of a homogeneous isotropic universe is contained within the scale factor R(t). The evolution of matter and radiation (for example) only depend on time t, since the Universe is homogeneous. The first law of thermodynamics is an expression of energy conservation, and connects the scale factor to the matter and radiation densities. The first law states that for any change d( V ) in a volume V containing a fixed number of particles, the total energy of the volume is the work done, minus the heat that flows out. With isotropic conditions, there can be no heat flow. In other words, homogeneous conditions dictate that the temperature must be the same everywhere at a given time.

d( E) =

pd( V )

From the stress-energy tensor, the total energy density is

c2

c2 V is energy density in the element of volume, use spherical geometry

4 3 V = R (t) 3

E= c

2

V 4 3 2 E = c ⇥(t) R (t) 3 d 2 4 3 d 4 3 (c ⇥(t) R (t)) = p ( R (t)) dt 3 dt 3 d 3 2 d 3 c ( (t)R (t)) = p (R (t)) dt dt 2d 3 2 2 dR 2 dR c R (t) + c (t)3R = 3pR dt dt dt d dt

=

3 dR R dt

⇢+

p c2

The fluid equation

Energy equation for FRW models For the case of non-relativistic matter, a second Friedman equation can be derived from conservation of energy and Newtonian mechanics. Consider a particle in a gravitational field due to a continuous density of matter ⇢

The kinetic energy of a co-moving particle with unity mass at a distance R(t) is = R(t)r = R(t)(r = 1) = R(t)

T = 12 R˙ 2 The mass interior to the position of the particle is

M=

4 3

⇥R3

So the potential energy is

V =

GM R

=

G 43

R3 ⇥R

The total energy is

T + V = constant We must identify the constant from the results of GR to be

T +V =

1 ˙2 2R

4 3

⇥GR2 = constant = E

1 ˙2 2R

4 3

⇥GR2 =

kc2 /2

R˙ 2

8 3

⇥GR2 =

kc2

⇥1

⇤ dR 2 R dt

=

8 G 3 ⇢

kc2 R2

The energy equation

kc2 2

Particles will be able to escape to infinity if and only if the Energy is positive E 0 which means k=0 or -1 For k=+1 they will have less than escape velocity, so the expansion will eventually stop and particles will fall back toward each other. Cosmological Composition There are three sources of gravitation that are dealt with in cosmology Matter, Radiation, and Vacuum energy Matter Matter in galaxies is well approximated by a pressure-less gas. Since there is no internal energy, the energy density ⇢m is just the rest energy density of the galaxies d( E) dt 2 3 d( m c R ) dt

=0 =0

d(

2 3 mc R )

dt

=0

essentially expresses conservation of mass. Equivalently m (t) =

m (t0 )

h

R(t0 ) R(t)

i3

where t0 is the present instant of time. The overall time dependance is thus dependant upon the scale factor. Radiation For gas of blackbody radiation at temperature T, energy density ⇢r and pressure pr

pr =

2 rc

3

d( (t)c2 R3 (t)) dt

r (t) =

=

d(R3 (t)) p dt

r (t0 )

h

R(t0 ) R(t)

i4

doubling R(t) halves the energy of each particle, reduces the energy density, and the effective mass density by a total factor of 16

doubling R(t) halves the energy of each particle (double wavelength), reduces the energy density, and the effective mass density by a total factor of 16. Thus in an expanding universe, the density of radiation will decline more rapidly than that for matter. Vacuum energy The final case is the (still mysterious) energy density of the vacuum. Currently, there is no fundamental theory that fixes the value of the vacuum energy. Not even the sign is known. We restrict our attention to the case that the vacuum energy is constant in space and time. We also take the case of a positive energy density as found from observations. The first law of thermodynamics then implies

c2 =

p

a pressure which is constant in space and time but negative. A negative pressure is like a tension in a rubber band. It takes work to expand the volume rather than work to compress it.

A homogeneous negative pressure associated with dark energy has the same effect has a cosmological constant. Thus the vacuum energy density is often written for historical purposes as

p =

c = 2

c4 8 G

If there is a nonzero vacuum energy, it remains constant while the energy densities in matter and radiation decay away as the universe expands. The long-term future of a universe that expands indefinitely is dominated by vacuum energy.

energy density in our expanding universe as a function of cosmological time