F_{x} = −\frac{∂U}{∂x}

\begin{aligned}M_{int} & = \frac{4π}{3} ρ(t)r(t)^{3} \\ & = \frac{4π}{3} ρ_{0}r_{0}^{3} \end{aligned}

KE + GPE = \frac{1}{2}mv^{2} − \frac{GM_{int}m}{r} = −\frac{1}{2}mkc^{2}ω̃^{2}

r(t) ≡ ω̃a(t)

ω̃ comoving coordinate meters
a(t) scale factor  no units

ω̃^{2} (\frac{da}{dt})^{2} − \frac{8π}{3} Gρω̃^{2}a^{2} = −kc^{2}ω̃^{2}

(\frac{da}{dt})^{2} − \frac{8π}{3} Gρa^{2} = −kc^{2}

H(t) = \frac{1}{a(t)} \frac{da}{dt}

[H(t)^{2} − \frac{8π}{3} Gρ(t)] a(t)^{2} = −kc^{2}

[H_{0}^{2} − \frac{8π}{3} Gρ_{0}] a_{0}^{2} = −kc^{2}

Ω_{M} ≡ \frac{ρ_{0}}{ρ_{crit}}

(\frac{da}{dt})^{2} − \frac{8π}{3} Gρ(t)a(t)^{2} = −kc^{2}

\begin{aligned}ρ_{0}a_{0}^{3} & = ρ(t)a(t)^{3} \\ ρ_{0}1^{3} & = ρ(t)a(t)^{3} \\ \frac{ρ_{0}}{a(t)^{2}} & = ρ(t)a(t)^{2} \end{aligned}

(\frac{da}{dt})^{2} − (\frac{8π}{3} Gρ_{0}) \frac{1}{a(t)} = −kc^{2}

a(x) = \frac{1}{2} \frac{Ω_{M}}{Ω_{M} − 1} (1 − cos(x))

t(x) = \frac{1}{2H_{0}} \frac{Ω_{M}}{(Ω_{M} − 1)^{3/2}} (x − sin(x))

a(t) = (6πGρ_{0})^{1/3} t^{2/3}

a(x) = \frac{1}{2} \frac{Ω_{M}}{1 − Ω_{M}} (cosh(x) − 1)

t(x) = \frac{1}{2H_{0}} \frac{Ω_{M}}{(1 − Ω_{M})^{3/2}} (sinh(x) − x)

z = \frac{λ_{obs} − λ_{emit}}{λ_{emit}} = \frac{a_{obs} − a_{emit}}{a_{emit}}

q(t) ≡ \frac{a(t) (d^{2}a/dt^{2})}{(da/dt)^{2}}

q(t) = \frac{1}{2} Ω_{M}(t)

(ds)^{2} = (c dt)^{2} − a(t)^{2} (\frac{dω̃^{2}}{1 − kω̃^{2}} )

c^{2} dt^{2} = a(t)^{2} (\frac{dω̃^{2}}{1 − kω̃^{2}} )

c \frac{dt}{a(t)} = \frac{dω̃}{\sqrt{1 − kω̃^{2}}}

c ∫_{0}^{t_{obs}} \frac{dt}{a(t)} = ∫ ω̃_{e} ; 0 \frac{dω̃}{\sqrt{1 − kω̃^{2}}}

ω̃(z) = \frac{2c}{H_{0}} (\frac{1}{1 + z} ) \frac{1}{Ω_{M}^{2}} (Ω_{M}z − (2 − Ω_{M}) [\sqrt{1 + Ω_{M}z} − 1])

f =^{?} \sqrt{\frac{L}{4πω̃^{2}}}

f_{photon} = \sqrt{\frac{L}{4πω̃^{2}(1 + z)}}

D_{L photon} = \sqrt{\frac{L}{4πf_{photon}}} = ω̃\sqrt{1 + z}

f_{energy} = \sqrt{\frac{L}{4πω̃^{2}(1 + z)(1 + z)}}

D_{L energy} = \sqrt{\frac{L}{4πf_{energy}}} = ω̃(1 + z)

total energy = \frac{1}{2}mv^{2} − \frac{GM_{int}m}{r} − \frac{1}{6}Λmc^{2}r^{2} = −\frac{1}{2}mkc^{2}ω̃^{2}

q_{0} = \frac{1}{2}Ω_{M} − \frac{Λc^{2}}{3H_{0}^{2}}

ρ = density (\frac{kg}{m^{3}} ) Ω_{M} = \frac{ρ}{ρ_{c}} (no units)

Λ = lambda (\frac{1}{m^{2}} ) Ω_{Λ} = \frac{Λc^{2}}{3H_{0}^{2}} (no units)

Ω_{tot} = Ω_{M} + Ω_{Λ}

Ω_{K} ≡ 1 − Ω_{tot} = 1 − (Ω_{M} + Ω_{Λ})

D_{M} = \frac{c}{H_{0}\sqrt{|Ω_{K}|}} sinn [\sqrt{|Ω_{K}|} ∫_{0}^{z1} [(1 + z)^{2}(1 + Ω_{M}z) − z(2 + z)Ω_{Λ}]^{−1/2}dz]

D_{A} = D_{M} (\frac{1}{1 + z} )
photon − based D_{L} = D_{M} \sqrt{1 + z}
energy − based D_{L} = D_{M}(1 + z)