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Home , Friedmann equations

Some History

TEXTBOOKS FOR FURTHER REFERENCE

1) Physical Foundations of , Viatcheslav Mukhanov

2) Cosmology, Michael Rowan-Robinson

3) A short course in , J. Foster and J.D. Nightingale

DERIVATION OF FRIEDMANN EQUATIONS IN A NEWTONIAN COSMOLOGY

THE COSMOLOGICAL PRINCIPAL

Viewed on a sufficiently large scale, the properties of the are the same for all observers.

This amounts to the strongly philosophical statement that the part of the Universe which we can see is a fair sample, and that the same physical laws apply throughout.

=> the Hubble expansion is a natural property of an expanding universe which obeys the

Distribution of galaxies on the sky

Distribution of 2.7 K microwave radiation on the sky

vA = H0. rA

vB = H0. rB

v and r are position and velocity vectors

VBA = VB – VA = H0.rB – H0.rA = H0 (rB - rA)

The observer on galaxy A sees all other galaxies in the universe receding with velocities described by the same Hubble law as on . In fact, the Hubble law is the unique expansion law compatible with and .

Co-moving coordinates: express the distance r as a product of the co-moving distance x and a term a(t) which is a function of time only:

rBA = a(t) . x BA

The original r coordinate system is known as physical coordinates.

Deriving an equation for the universal expansion thus reduces to determining a function which describes a(t)

Newton's Shell Theorem

The force acting on A, B, C, D— which are particles located on the surface of a sphere of radius r—is the gravitational attraction from the internal to r only, acting as a point mass at O.

Write the total energy of a particle of mass m at A, B, C, D as the usual sum of kinetic and gravitational potential energy:

Switching to co-moving coordinates:

Re-arrange to obtain the Friedmann equation:

What is k?

The first equation implies that k is independent of x. (this then implies that U scales as x2)

The second equation implies that k is independent of t, since U (total energy) is conserved and dx/dt=0 by definition.

So k is a constant, independent of space and time.

• A positive k implies negative U, so that V > T —the expansion will at some time t halt and reverse itself

• A negative k implies positive U, so that V < T —the expansion will continue forever

• If k = 0, the total energy is also U = 0 and the expansion of the universe will slow down, but only halt at t = infinity

From v= H0.r and r= a(t).x , we write:

H- the Hubble constant

Substitute into Friedmann equation, to get time evolution Critical for re-collapse: of Hubble parameter:

To describe the time evolution of the of the universe, a(t), we need an additional equation describing the time evolution of the density ρ of material in the universe. The first law of thermodynamics states that the change in energy of an expanding system equals the work done by the :

dE= -pdV Using E=mc2= 4π/3 a3 ρ c2 , and taking the time derivative:

also

===>

This is the fluid equation , which tells us that there are two terms contributing to the change in density as the universe expands. The first isthe dilution in the density because the volume has increased.The second term corresponds to the loss in energy because the pressure of the material has done work as the universe’s volume increased. In cosmology the assumption is usually made that there is a unique pressure associated with each density, so that p = p(ρ). Such a relationship is known as the equation of state. The form of the equation of state depends on the nature of the constituent of the universe we are considering:

• Non-relativistic matter has negligible pressure, p = 0. Examples are galaxies (their only interaction is through ), , and atoms in general once the universe has expanded and cooled. Cosmologists sometimes refer to this component as ‘dust’, precisely to indicate matter with negligible pressure.

• Light, or more generally any highly relativistic particle, has an associated pressure (radiation pressure) p = ρ c2 /3

We derive an expression for the acceleration of the Universe by differentiating the Friedmann equation:

. Substituting the fluid equation for ρ gives:

. Substituting the Friedman equation for (a/a)2 gives the acceleration equation:

Pressure here acts to increase the gravitational force, and so further decelerates the expansion. Note that the constant k has cancelled out of this equation.

The case of Dust. ( k=0 case is called Einstein-)

With p=0, the fluid equation gives:

This can be rewritten: i.e.

This has solution:

(the density of matter falls off in proportion to the volume of the expanding universe)

Substituting into Friedmann equation with k=0:

For H0 = 70 km s-1 Mpc-1 ; t0= 9.5 Gyr (too small) For the case of radiation pressure, the fluid equation gives:

following same steps, yields the solution:

TIME EVOLUTION: In the early Universe, matter is relativistic and radiation pressure dominates:

The density of radiation then falls more quickly than that of dust. When dust dominates:

so dust domination increases. General Relativity generalizes and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time. In particular, the of space-time is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the

Rμν -1/2 gμν R = 8πG/c4 Tμν

Rμν is the tensor, R is the , gμν the metric tensor and Tμν the stress-energy tensor.

The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Such a solution consists of giving the metric in specific coordinates). Matter and geometry must satisfy Einstein's equations. The metric tensor may be thought of as the generalization of the Newtonian gravitational field. The metric captures all the geometric and causal structure of space-time, being used to define notions such as distance, volume, curvature, angle etc

Mathematically, space-time is represented by a 4-dimensional differentiable manifold M. Physicists usually work in local coordinates xμ (i.e. coordinates defined on some local patch of M). With the quantity dxμ being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant line element ds2 , which is written

ds2= gμν dxμ dxν

The metric components obviously depend on the chosen local coordinate system. Flat space time: ds2 = -c2dt2 + dx2 +dy2 +dz2 In spherical coordinates: 2 2 2 2 2 2 2 2 2 2 ds = -c dt + r +r dΩ ; dΩ = dθ +sin θ dφ The Friedmann–Lemaître–Robertson–Walker (FLRW) metric describes a homogeneous, isotropic expanding or contracting universe. The generic metric which meets conditions of homogeneity and isotropy may be written:

ds2= -c2 dt2 + a(t)2 dΣ2 where Σ ranges over a 3-dimensional space of uniform curvature. In polar coordinates:

k is a constant representing the curvature of the space.

Two-dimensional analogue of the 3-dimensional curvature in General Relativity

Solution to the Einstein Equations (see GR text book, e.g. Foster & Nightingale for full derivation)

The components of the Robertson-Walker metric are:

The components of the stress-energy tensor are:

A lengthy derivation leads to:

So, the Friedmann equations are the same, whether derived under Newtonian dynamics or General Relativity. It is only the interpretation of the constant k which is different: in the Newtonian universe k is related to the mechanical energy of an expanding mass shell. In Einstein’s universe it is the present value of the curvature of the universe.

The Einstein’s . Before the discovery of the cosmic expansion by Hubble in 1929, the universe was thought to be static. This then implies that the scale factor is a constant a0, so that a˙ = a¨ = 0 (and the is infinite). The Friedmann equations are then:

Since ρ0 must be a positive number, k must be +1. Note also that this leads to the surprising result that the pressure of matter p0 is negative! Einstein corrected for this in 1917 by introducing a constant Lorentz invariant Λgμν into his field equations. Friedmann's equations then take the form:

Λc2/8πG corresponds to an energy density, the vacuum energy density, and Λc4/8πG corresponds to a pressure term. A positive value of Λ corresponds to a repulsive force counteracting the conventional attractive gravitation. A negative value corresponds to an additional attractive force.

After Hubble discovered the expansion, Einstein declared this to be his “biggest blunder”. As we will show in the next lecture, such a term is needed to describe the observed Universe! Our Universe, spatially flat with non-zero cosmological constant....