Derivations Flashcards
Derive luminosity distance
d^2(lum) = L/S
S = L/(a^2(0)r^2(0))
photons lose energy ∝ (1+z) so have less energy on arrival, and
photons arrive less frequently, also ∝ (1+z)
S = L/[a^2(0)r^2(0)(1+z)^2]
giving formula on crib sheet
Derive the angular diameter distance
d(diam) = l/sinθ ~ l/θ
l = r(0)a(t(e))θ
θ = l/r(0)a(t(e)) = l(1+z)/a(0)r(0)
giving formula on crib sheet
derive the galaxy number counts
N = n(0) 4/3 πr^3 Ω/4π
F = L/4πr^2
N(>F) = const n(0) L^(3/2) F^(-3/2) Ω
N(>F) = const n(0)F^(-3/2) Ω(∞ ∫ 0) f(L) L^(3/2) dL
giving formula on crib sheet
Derive the Friedmann equation
K = 1/2mr^2(dot) = 1/2 ma^2(dot) s^2
U = -GMm/r = -4/3 π a^2s^2Gpm
total energy = constant
giving the Friedmann equation COS2 equation sheet
Solution of Friedmann’s equation for k = 0
(da/dt)^2 = 8πGpa^2/3
Mass is proportional to pa^3
(da/dt)^2 = A/a
Redshift in flat Universe
a = 1/1+z
1+z = λ(obs)/λ(em)
p(z) = p(0)(1+z)^3
a/a(0) = (t/t(0))^(2/3)
1 + z = (t/t(0))^(-2/3)
the critical density of the universe
when k = 0
(a(dot)/a)^2 = 8πGp/3
H = (a(dot)/a)
p = 3H^2/8πG
p(c,0) = 3H(0)^2/8πG
Derivation of Hubble-lamaitre law
start from the fluid equation
p = p(0) + U/c^2
p(dot) + (p+P/c^2)∇ . v(v) = 0
∇ . v(v) = ∂xvx+∂yvy+∂zvz
∂xvx = ∂yvy = ∂zvz = H(t)
δv = H(t) δr
Derive the Friedmann equation starting from the momentum equation
starting from the momentum/acceleration equation equation
from fluid equation rearrange for P and substitute in
multiply by a(dot)
perfect differential
leads to fried equation
Derive the Einstein De-Sitter solutions
k = 0 in the friedmann equation
p = p(0) (a(0)/a)^η
sqrt to find a(dot)
get a on LHS and rest on RHS
derivative in terms of t
Derive q(t) = 1/2Ω(t)
substitute a(d dot) and a(dot) into q(t)
identify Ω
Derive Ω = Ω(0)/[Ω(0) + (1-Ω(0))(1+z)^(2-η)]
equate present Friedmann equations
replace a/a(0) for 1/1+z
replace H^2/H(0)^2
replacing p/p(0) = (a(0)/a)^η
Derive the Jeans Length
PV = NkT
P = Gp^2L^2
Gp^2L^2 ≥ pkT/m
gives Jeans length
and thus jeans mass as M = pL^3 or M(J) = p(0)λ^3(J)
Derive a single equation for the evolution of the density perturbation.
start from the continuity equation
p(dot) + p(0) ∇.(v) = 0
take derivative
substitute the momentum equation
substitute Poisson’s equation
substitute for sound speed
∇ P(1) = c^2(s)∇ p(1)
Derive Ω - 1 = [Ω(0) - 1]/[1+Ω(0)z]
Friedman equation = Friedmann equation present time
substitute in the density parameter
substitute redshift (1+z) factor
(H(0)/H)^2 = p(0)/p Ω/Ω(0) = Ω/Ω(0) (1+z)^(-3)
from the single equation for the evolution of the density perturbation derive the critical mass M(J)
assume plane wave with p(1) = exp{i(kr(v).ωt)}
substitute into the single equation for the evolution of the density perturbation
get ω on one side
for stability RHS > 0
gives λ(J)
M(J) = λ^3(J)p(0)
Demonstrate that M(J) ∝ p(0)^(3γ-4)/2
M(J) = π^3/2 c^3(s) p(0)^(-1/2) G^(-3/2)
P ∝ p^γ
M(J) ~ p^(-1/2) p^(3(γ-1)/2)
~ p^1/2(3γ-4)
M(J) ∝ p(0)^(3γ-4)/2
starting from the Friedmann equation demonstrate that the expansion of a flat universe under the influence of gravity aloe proceeds more rapidly if the universe is matter dominated than if radiated-dominated.
Friedmann equation for flat k = 0
matter-dominated p ~ 1/a^3
radiated-dominated p ~ 1/a^4
a on LHS and constant on RHS
integrate
a(t) ~ t^2/3
and a(t) ~ t^1/2
linearisation of continuity equation
continuity equation
∇.(Hr) = 3H
p0(dot) + 3Hp(0) = 0
s(dot) + Hr.∇s + ∇.u = 0
a(d dot) = -4πGp(0)/3a^2
gives
a^2(dot) - 8πGp(0)/3a = -H^2(0)(Ω-1)
take integral
rearrange and substitute
derive
M = 3σ(vr)^2Re/G
<K> = 3/2Mσ(vr)^2
<u> = -GM^2/Re
0 = <K>+1/2<u>
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