Section 3 Flashcards
Consider a spherical shell of radius r within a spherical region of the universe filled with pressureless matter of uniform density
The universe’s expansion proceeds in the same way for all shells
Mass conservation in a matter-dominated universe
a^3(t)ρ(t) = a^3(t₀)ρ(t₀) = ρ₀
A simple pressureless model of the Universe
Assume a homogeneous and isotropic universe filled with ‘dust’ of uniform density ρ(t)
Let universe expand:
Conservation of total energy E
E = K(t) + U(t)
where K(t) = 1/2mv^2(t) and U(t) = -G Mrm/r(t)
Mr = 4/3 π r^3(t)ρ(t)
Friedmann’s equation derivation
K = 1/2 mṙ^2 = 1/2 mȧ^2 s^2
U = -GMm/r = - 4/3π a^2 s^2 Gρm
E total = 1/2 ms^2 [ ȧ^2 - (8πGρa^2)/3 ] = const
giving
ȧ^2/a^2 - 8πGρ/3 = -kc^2/a^2
where k is a constant
What does Friedmann’s equation describe
it describes how gravity slows the rate of the expansion of the Universe
the constant k determines the curvature of the universe describe the three classes of solutions
k > 0 closed Universe, positive curvature
k < 0 open Universe, negative curvature
k = 0 flat Universe, zero curvature
show that a(t) = Kt^(2/3)
(da/dt)^2 = A/a
ȧa^(1/2) = A^(1/2)
a^(1/2) da = A^(1/2) dt
∫ a^(1/2) da = ∫ A^(1/2) dt
2/3 a^(3/2) = A^(1/2) t
a^(3/2) = 3/2 A^(1/2) t
a(t) + Kt^(2/3)
Solution of Friedmann’s equation for k = 0
(da/dt)^2 = (8πGρa^2)/3
assuming the universe is matter dominated
(da/dt)^2 = A/a
Redshift in a flat Universe
a = 1/1+z
ρ(z) = ρ₀(1+z)^3
at t₀ , a/a₀ = (t/t₀)^(2/3)
1 + z = (t/t₀)^(-2/3)
Critical density of the Universe
(ȧ/a)^2 = 8πGρ/3
H = ȧ/a
ρ = 3H^2 / 8πG
ρ is the critical density denoted by ρc
present day value is
ρc₀ = 3H₀^2/8πG
If ρ > ρc
the universe recollapses
if ρ < ρc
the universe expands indefinitely
Density Parameter
Ω(t) = ρ(t)/ρc(t) = 8πGρ/3H^2
Ω > 1
Universe is closed
Ω < 1
Universe is open
Ω = 1
Universe is flat
Although Ω(t) can change with time, it can be shown that its state of being closed, open or flat
cannot change
Derivation of fluid equations
we have a first fluid equation governing the flow and continuity of mass
ρ(dot) + ∇.(ρ→v) = -(P/c^2)∇.→v
ρ is the denstiy and includes the internal energy
ρ = ρ₀ + U/c^2
P is the pressure
→v is the velocity field
Derivation of Hubble-Lemaitre Law
∇.(ρ→v) = ∇ρ.→v + ρ∇.→v = ρ∇.→v
therefore the mass continuity equation becomes
ρ(dot) + (ρ + P/c^2) ∇.→v = 0
by isotropy
∂xvx = ∂yvy = ∂zvz = H(t)
So 𝛿→v = H(t)𝛿→r
Continuity of mass derivation
∇.→v = 3H(t)
this gives us the equation for continuity of mass in its final form
ρ(dot) + 3(ρ + P/c^2) H = 0
Equation of state
typical equation of state
P(ρ) = {Kρ^(5/3) adiabatic case
{ Kρ isothermal case
The momentum equation derivation
The equation of motion four our shell is then
ȧ^2 = -GM/a^2 = -G/a^2 [ 4π/3 a^3 (ρ+3P/c^2)]
therefore the momentum equation can be written as
ä = -4πGa/3 (ρ+3P/c^2)
aρ(dot)/ȧ + 3(ρ + P/c^2) = 0
P/c^2 = - aρ(dot)/3ȧ - ρ
insert into the momentum equation
ä = -4π/3 Ga (-aρ(dot)/ȧ - 2ρ)
äȧ = 4π/3 G(a^2ρ(dot) + 2ρȧa)
is a perfect differential equation
d/dt(1/2ȧ^2) = d/dt (4π/3G(a^2ρ)
ȧ^2 - 8π/3 Ga^2ρ + kc^2 = 0
The momentum equation depends on the constant k the solutions tell us that
k < 0 ȧ →c √-k - open universe
k > ȧ = 0 at some critical radius followed by a callapse phase - closed universe
k = 0 ȧ→0 at t →∞ - flat universe
The momentum equation can be written as
H^2(t) = 8π/3 Gρ(t) - k c^2/a^2
Einstein-De Sitter Universe
is a special case of a homogeneous, isotropic universe of zero curvature described by Newtonian gravity
ρ = ρ₀(a₀/a)^η
ȧ^2 - 8π/3 Ga^2ρ + kc^2 = 0
ȧ^2 = 8π/3 Ga^2ρ ρ₀a₀^η a^(-η)
a^(η/2 -1) ȧ = (8π/3 Gρ₀a₀^η)^1/2
2/η ∂/∂t a^(η/2) = (8π/3 Gρ₀a₀^η)^1/2
a^(η/2) = η/2 (8π/3 Gρ₀a₀^η)^1/2 t + const
The temporal behaviour f the scale factor is given by
a(t) ∝ t^(2/η)
a(t) ∝ { t^2/3 matter-dominated
{ t^1/2 radiated -dominated
The deceleration paramter
q(t) = -aä/ȧ^2
for a matter dominated flat universe
p~a^-3 and pa^2 = p0/a
substitute into fried for k =0
a(dot) = (8πGp/3)^1/2 a^(-1/2)
a(dot) a^1/2= (8πGp/3)^1/2
integrate
a(t) ~ t^3/2
for a radited dominated flat universe
p~a^-4 and pa^2=p0/a^2
substitute into fried for k =0
a(dot) = (8πGp/3)^1/2 a^-1
a(dot) a= (8πGp/3)^1/2
integrate
a(t) ~ t^3/2
a(dot)^2 =
H^2 a^2
H0^2/H^2 =
Ω/Ω0 (1+z)^(-3)