Components (submodels)
Spacetimes
SymBoltz.metric — Function
metric(; name = :g, kwargs...)Create a symbolic component for the perturbed FLRW spacetime metric in the conformal Newtonian gauge with sign signature diag(-1, +1, +1, +1).
g = SymBoltz.metric()Equations
equations(g)\[ \begin{align*} z\left( \tau \right) &= -1 + \frac{1}{a\left( \tau \right)} \\ \dot{z}\left( \tau \right) &= \frac{\mathrm{d} ~ z\left( \tau \right)}{\mathrm{d}\tau} \\ \mathscr{H}\left( \tau \right) &= \frac{\frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau}}{a\left( \tau \right)} \\ H\left( \tau \right) &= \frac{\mathscr{H}\left( \tau \right)}{a\left( \tau \right)} \end{align*} \]
Gravitational theories
SymBoltz.general_relativity — Function
general_relativity(g; name = :G, kwargs...)Create a symbolic component for the general relativistic (GR) theory of gravity in the spacetime with the metric g.
G = SymBoltz.general_relativity(g)Equations
equations(G)\[ \begin{align*} \frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau} &= \left( a\left( \tau \right) \right)^{2} ~ \sqrt{\frac{8}{3} ~ \rho\left( \tau \right) ~ \pi} \\ \mathtt{F_1}\left( \tau \right) &= 0 \\ \mathtt{F_2}\left( \tau \right) &= \frac{\mathrm{d}^{2} ~ a\left( \tau \right)}{\mathrm{d}\tau^{2}} + \frac{ - \left( \frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau} \right)^{2} ~ \left( 1 + \frac{ - 3 ~ P\left( \tau \right)}{\rho\left( \tau \right)} \right)}{2 ~ a\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) &= \frac{ - k^{2} ~ \Phi\left( \tau, k \right)}{3 ~ \mathscr{H}\left( \tau \right)} + \frac{ - \frac{4}{3} ~ \left( a\left( \tau \right) \right)^{2} ~ \mathtt{\delta\rho}\left( \tau, k \right) ~ \pi}{\mathscr{H}\left( \tau \right)} - \Psi\left( \tau, k \right) ~ \mathscr{H}\left( \tau \right) \\ k^{2} ~ \left( - \Psi\left( \tau, k \right) + \Phi\left( \tau, k \right) \right) &= 12 ~ \left( a\left( \tau \right) \right)^{2} ~ \Pi\left( \tau, k \right) ~ \pi \\ \mathtt{\dot{\Psi}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Psi\left( \tau, k \right) \\ \mathtt{\dot{\Phi}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \end{align*} \]
Initialization equations
initialization_equations(G)\[ \begin{align*} \end{align*} \]
SymBoltz.brans_dicke — Function
brans_dicke(g; name = :G, kwargs...)Create a symbolic component for the Brans-Dicke (BD) theory of gravity in the spacetime with the metric g.
G = SymBoltz.brans_dicke(g)Equations
equations(G)\[ \begin{align*} \frac{\mathrm{d}^{2} ~ \phi\left( \tau \right)}{\mathrm{d}\tau^{2}} &= \frac{8 ~ \left( a\left( \tau \right) \right)^{2} ~ \left( - 3 ~ P\left( \tau \right) + \rho\left( \tau \right) \right) ~ \pi}{3 + 2 ~ \omega} + \frac{ - 2 ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} ~ \frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau}}{a\left( \tau \right)} \\ \frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau} &= \sqrt{\frac{\frac{8}{3} ~ \left( a\left( \tau \right) \right)^{4} ~ \rho\left( \tau \right) ~ \pi}{\phi\left( \tau \right)} + \left( \frac{\frac{1}{2} ~ a\left( \tau \right) ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau}}{\phi\left( \tau \right)} \right)^{2} + \frac{1}{6} ~ \left( \frac{a\left( \tau \right) ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau}}{\phi\left( \tau \right)} \right)^{2} ~ \omega} + \frac{ - \frac{1}{2} ~ a\left( \tau \right) ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau}}{\phi\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{\delta\phi}\left( \tau, k \right) + \left( k^{2} + 3 ~ \left( \mathscr{H}\left( \tau \right) \right)^{2} \right) ~ \mathtt{\delta\phi}\left( \tau, k \right) - 2 ~ k^{2} ~ \Phi\left( \tau, k \right) ~ \phi\left( \tau \right) + \frac{1}{2} ~ \left( - 2 ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{\delta\phi}\left( \tau, k \right) + \left( \frac{\frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau}}{\phi\left( \tau \right)} \right)^{2} ~ \mathtt{\delta\phi}\left( \tau, k \right) \right) ~ \omega - 8 ~ \left( a\left( \tau \right) \right)^{2} ~ \left( \mathtt{\delta\rho}\left( \tau, k \right) + 2 ~ \Psi\left( \tau, k \right) ~ \rho\left( \tau \right) \right) ~ \pi}{3 ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} + 6 ~ \mathscr{H}\left( \tau \right) ~ \phi\left( \tau \right)} \\ \frac{\mathrm{d}^{2}}{\mathrm{d}\tau^{2}} ~ \mathtt{\delta\phi}\left( \tau, k \right) &= \frac{ - 8 ~ \left( a\left( \tau \right) \right)^{2} ~ \left( 3 ~ \mathtt{{\delta}P}\left( \tau, k \right) - \mathtt{\delta\rho}\left( \tau, k \right) \right) ~ \pi}{3 + 2 ~ \omega} + 2 ~ \Psi\left( \tau, k \right) ~ \frac{\mathrm{d}^{2} ~ \phi\left( \tau \right)}{\mathrm{d}\tau^{2}} - 2 ~ \mathscr{H}\left( \tau \right) ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{\delta\phi}\left( \tau, k \right) + \left( 3 ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} + \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Psi\left( \tau, k \right) \right) ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} - k^{2} ~ \mathtt{\delta\phi}\left( \tau, k \right) + 4 ~ \Psi\left( \tau, k \right) ~ \mathscr{H}\left( \tau \right) ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} \\ G\left( \tau \right) &= \frac{4 + 2 ~ \omega}{\phi\left( \tau \right) ~ \left( 3 + 2 ~ \omega \right)} \\ \mathtt{F_1}\left( \tau \right) &= 0 \\ \mathtt{F_2}\left( \tau \right) &= - \frac{ - 4 ~ \left( a\left( \tau \right) \right)^{3} ~ P\left( \tau \right) ~ \pi}{\phi\left( \tau \right)} - \frac{ - \frac{1}{2} ~ a\left( \tau \right) ~ \frac{\mathrm{d}^{2} ~ \phi\left( \tau \right)}{\mathrm{d}\tau^{2}}}{\phi\left( \tau \right)} + \frac{\mathrm{d}^{2} ~ a\left( \tau \right)}{\mathrm{d}\tau^{2}} - \frac{ - \frac{1}{2} ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} ~ \frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau}}{\phi\left( \tau \right)} - \frac{\left( \frac{\mathrm{d} ~ a\left( \tau \right)}{\mathrm{d}\tau} \right)^{2}}{2 ~ a\left( \tau \right)} + \frac{1}{4} ~ \left( \frac{\frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau}}{\phi\left( \tau \right)} \right)^{2} ~ a\left( \tau \right) ~ \omega \\ \Psi\left( \tau, k \right) &= \Phi\left( \tau, k \right) + \frac{ - 12 ~ \left( a\left( \tau \right) \right)^{2} ~ \Pi\left( \tau, k \right) ~ \pi}{k^{2} ~ \phi\left( \tau \right)} + \frac{ - \mathtt{\delta\phi}\left( \tau, k \right)}{\phi\left( \tau \right)} \\ \mathtt{\dot{\Psi}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Psi\left( \tau, k \right) \\ \mathtt{\dot{\Phi}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \end{align*} \]
Initialization equations
initialization_equations(G)\[ \begin{align*} \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{\delta\phi}\left( \tau, k \right) &= 0 \end{align*} \]
Species
Generic
SymBoltz.species_constant_eos — Function
species_constant_eos(g, _w, ẇ = 0, _σ = 0; analytical = true, θinteract = false, adiabatic = false, name = :s, kwargs...)Create a symbolic component for a particle species with equation of state w ~ P/ρ in the spacetime with the metric g.
using SymBoltz: τ
@variables w(τ)
s = SymBoltz.species_constant_eos(g, w)Equations
equations(s)\[ \begin{align*} \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{3 ~ \left( 1 + w\left( \tau \right) \right)}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= w\left( \tau \right) \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \delta\left( \tau, k \right) &= \left( -1 - w\left( \tau \right) \right) ~ \left( \theta\left( \tau, k \right) - 3 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \right) - 3 ~ \mathscr{H}\left( \tau \right) ~ \left( - w\left( \tau \right) + \mathtt{c_s^2}\left( \tau \right) \right) ~ \delta\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \theta\left( \tau, k \right) &= \mathtt{{\theta}interaction}\left( \tau, k \right) + \frac{k^{2} ~ \mathtt{c_s^2}\left( \tau \right) ~ \delta\left( \tau, k \right)}{1 + w\left( \tau \right)} + k^{2} ~ \Psi\left( \tau, k \right) - k^{2} ~ \sigma\left( \tau, k \right) - \mathscr{H}\left( \tau \right) ~ \left( 1 - 3 ~ w\left( \tau \right) \right) ~ \theta\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ \mathtt{\dot{u}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ u\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= 0 \\ \mathtt{{\theta}interaction}\left( \tau, k \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(s)\[ \begin{align*} \delta\left( \tau, k \right) &= - \frac{3}{2} ~ \Psi\left( \tau, k \right) ~ \left( 1 + w\left( \tau \right) \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \end{align*} \]
Relativistic
SymBoltz.radiation — Function
radiation(g; name = :r, kwargs...)Create a particle species for radiation (with equation of state w ~ 1/3) in the spacetime with metric g.
r = SymBoltz.radiation(g)Equations
equations(r)\[ \begin{align*} T\left( \tau \right) &= \frac{\mathtt{T{_0}}}{a\left( \tau \right)} \\ \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{4}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= \frac{1}{3} \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \delta\left( \tau, k \right) &= \left( -1 - w\left( \tau \right) \right) ~ \left( \theta\left( \tau, k \right) - 3 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \right) - 3 ~ \mathscr{H}\left( \tau \right) ~ \left( - w\left( \tau \right) + \mathtt{c_s^2}\left( \tau \right) \right) ~ \delta\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \theta\left( \tau, k \right) &= \mathtt{{\theta}interaction}\left( \tau, k \right) + \frac{k^{2} ~ \mathtt{c_s^2}\left( \tau \right) ~ \delta\left( \tau, k \right)}{1 + w\left( \tau \right)} + k^{2} ~ \Psi\left( \tau, k \right) - k^{2} ~ \sigma\left( \tau, k \right) - \mathscr{H}\left( \tau \right) ~ \left( 1 - 3 ~ w\left( \tau \right) \right) ~ \theta\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ \mathtt{\dot{u}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ u\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= 0 \\ \mathtt{{\theta}interaction}\left( \tau, k \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(r)\[ \begin{align*} \delta\left( \tau, k \right) &= - \frac{3}{2} ~ \Psi\left( \tau, k \right) ~ \left( 1 + w\left( \tau \right) \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \end{align*} \]
SymBoltz.photons — Function
photons(g; polarization = true, lmax = 6, name = :γ, kwargs...)Create a particle species for photons in the spacetime with metric g.
γ = SymBoltz.photons(g; lmax = 6)Equations
equations(γ)\[ \begin{align*} \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{F0}\left( \tau, k \right) &= 4 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) - k ~ F\_{1}\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{1}\left( \tau, k \right) &= \frac{ - \frac{4}{3} ~ \mathtt{\dot{\kappa}}\left( \tau \right) ~ \left( - \theta\left( \tau, k \right) + \mathtt{{\theta}b}\left( \tau, k \right) \right)}{k} + \frac{1}{3} ~ k ~ \left( 4 ~ \Psi\left( \tau, k \right) + \mathtt{F0}\left( \tau, k \right) - 2 ~ F\_{2}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{2}\left( \tau, k \right) &= \frac{1}{5} ~ k ~ \left( 2 ~ F\_{1}\left( \tau, k \right) - 3 ~ F\_{3}\left( \tau, k \right) \right) + \mathtt{\dot{\kappa}}\left( \tau \right) ~ \left( - \frac{1}{10} ~ \Pi\left( \tau, k \right) + F\_{2}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{3}\left( \tau, k \right) &= \frac{1}{7} ~ k ~ \left( 3 ~ F\_{2}\left( \tau, k \right) - 4 ~ F\_{4}\left( \tau, k \right) \right) + \mathtt{\dot{\kappa}}\left( \tau \right) ~ F\_{3}\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{4}\left( \tau, k \right) &= \frac{1}{9} ~ k ~ \left( 4 ~ F\_{3}\left( \tau, k \right) - 5 ~ F\_{5}\left( \tau, k \right) \right) + \mathtt{\dot{\kappa}}\left( \tau \right) ~ F\_{4}\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{5}\left( \tau, k \right) &= \frac{1}{11} ~ k ~ \left( 5 ~ F\_{4}\left( \tau, k \right) - 6 ~ F\_{6}\left( \tau, k \right) \right) + \mathtt{\dot{\kappa}}\left( \tau \right) ~ F\_{5}\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{6}\left( \tau, k \right) &= \frac{ - 7 ~ F\_{6}\left( \tau, k \right)}{\tau} + k ~ F\_{5}\left( \tau, k \right) + \mathtt{\dot{\kappa}}\left( \tau \right) ~ F\_{6}\left( \tau, k \right) \\ \delta\left( \tau, k \right) &= \mathtt{F0}\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ \theta\left( \tau, k \right) &= \frac{3}{4} ~ k ~ F\_{1}\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= \frac{F\_{2}\left( \tau, k \right)}{2} \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ \Pi\left( \tau, k \right) &= G\_{2}\left( \tau, k \right) + \mathtt{G0}\left( \tau, k \right) + F\_{2}\left( \tau, k \right) \\ \mathtt{\dot{\Pi}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Pi\left( \tau, k \right) \\ \mathtt{{\Theta}0}\left( \tau, k \right) &= \frac{\mathtt{F0}\left( \tau, k \right)}{4} \\ \Theta\_{1}\left( \tau, k \right) &= \frac{F\_{1}\left( \tau, k \right)}{4} \\ \Theta\_{2}\left( \tau, k \right) &= \frac{F\_{2}\left( \tau, k \right)}{4} \\ \Theta\_{3}\left( \tau, k \right) &= \frac{F\_{3}\left( \tau, k \right)}{4} \\ \Theta\_{4}\left( \tau, k \right) &= \frac{F\_{4}\left( \tau, k \right)}{4} \\ \Theta\_{5}\left( \tau, k \right) &= \frac{F\_{5}\left( \tau, k \right)}{4} \\ \Theta\_{6}\left( \tau, k \right) &= \frac{F\_{6}\left( \tau, k \right)}{4} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{G0}\left( \tau, k \right) &= - k ~ G\_{1}\left( \tau, k \right) + \mathtt{\dot{\kappa}}\left( \tau \right) ~ \left( \mathtt{G0}\left( \tau, k \right) - \frac{1}{2} ~ \Pi\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ G\_{1}\left( \tau, k \right) &= \frac{1}{3} ~ k ~ \left( - 2 ~ G\_{2}\left( \tau, k \right) + \mathtt{G0}\left( \tau, k \right) \right) + G\_{1}\left( \tau, k \right) ~ \mathtt{\dot{\kappa}}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ G\_{2}\left( \tau, k \right) &= \frac{1}{5} ~ k ~ \left( 2 ~ G\_{1}\left( \tau, k \right) - 3 ~ G\_{3}\left( \tau, k \right) \right) + \left( G\_{2}\left( \tau, k \right) - \frac{1}{10} ~ \Pi\left( \tau, k \right) \right) ~ \mathtt{\dot{\kappa}}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ G\_{3}\left( \tau, k \right) &= \frac{1}{7} ~ k ~ \left( 3 ~ G\_{2}\left( \tau, k \right) - 4 ~ G\_{4}\left( \tau, k \right) \right) + G\_{3}\left( \tau, k \right) ~ \mathtt{\dot{\kappa}}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ G\_{4}\left( \tau, k \right) &= \frac{1}{9} ~ k ~ \left( 4 ~ G\_{3}\left( \tau, k \right) - 5 ~ G\_{5}\left( \tau, k \right) \right) + G\_{4}\left( \tau, k \right) ~ \mathtt{\dot{\kappa}}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ G\_{5}\left( \tau, k \right) &= \frac{1}{11} ~ k ~ \left( 5 ~ G\_{4}\left( \tau, k \right) - 6 ~ G\_{6}\left( \tau, k \right) \right) + G\_{5}\left( \tau, k \right) ~ \mathtt{\dot{\kappa}}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ G\_{6}\left( \tau, k \right) &= \frac{ - 7 ~ G\_{6}\left( \tau, k \right)}{\tau} + k ~ G\_{5}\left( \tau, k \right) + G\_{6}\left( \tau, k \right) ~ \mathtt{\dot{\kappa}}\left( \tau \right) \\ T\left( \tau \right) &= \frac{\mathtt{T{_0}}}{a\left( \tau \right)} \\ \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{4}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= \frac{1}{3} \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \mathtt{c_s^2}\left( \tau \right) &= w\left( \tau \right) \end{align*} \]
Initialization equations
initialization_equations(γ)\[ \begin{align*} \mathtt{F0}\left( \tau, k \right) &= - 2 ~ \Psi\left( \tau, k \right) \\ F\_{1}\left( \tau, k \right) &= \frac{2}{3} ~ k ~ \Psi\left( \tau, k \right) ~ \tau \\ F\_{2}\left( \tau, k \right) &= \frac{ - \frac{8}{15} ~ k ~ F\_{1}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ F\_{3}\left( \tau, k \right) &= \frac{ - \frac{3}{7} ~ k ~ F\_{2}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ F\_{4}\left( \tau, k \right) &= \frac{ - \frac{4}{9} ~ k ~ F\_{3}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ F\_{5}\left( \tau, k \right) &= \frac{ - \frac{5}{11} ~ k ~ F\_{4}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ F\_{6}\left( \tau, k \right) &= \frac{ - \frac{6}{13} ~ k ~ F\_{5}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ \mathtt{G0}\left( \tau, k \right) &= \frac{5}{16} ~ F\_{2}\left( \tau, k \right) \\ G\_{1}\left( \tau, k \right) &= \frac{ - \frac{1}{16} ~ k ~ F\_{2}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ G\_{2}\left( \tau, k \right) &= \frac{1}{16} ~ F\_{2}\left( \tau, k \right) \\ G\_{3}\left( \tau, k \right) &= \frac{ - \frac{3}{7} ~ k ~ G\_{2}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ G\_{4}\left( \tau, k \right) &= \frac{ - \frac{4}{9} ~ k ~ G\_{3}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ G\_{5}\left( \tau, k \right) &= \frac{ - \frac{5}{11} ~ k ~ G\_{4}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \\ G\_{6}\left( \tau, k \right) &= \frac{ - \frac{6}{13} ~ k ~ G\_{5}\left( \tau, k \right)}{\mathtt{\dot{\kappa}}\left( \tau \right)} \end{align*} \]
Non-relativistic
SymBoltz.matter — Function
matter(g; name = :m, kwargs...)Create a particle species for matter (with equation of state w ~ 0) in the spacetime with metric g.
m = SymBoltz.matter(g)Equations
equations(m)\[ \begin{align*} \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{3}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= 0 \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \delta\left( \tau, k \right) &= \left( -1 - w\left( \tau \right) \right) ~ \left( \theta\left( \tau, k \right) - 3 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \right) - 3 ~ \mathscr{H}\left( \tau \right) ~ \left( - w\left( \tau \right) + \mathtt{c_s^2}\left( \tau \right) \right) ~ \delta\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \theta\left( \tau, k \right) &= \mathtt{{\theta}interaction}\left( \tau, k \right) + \frac{k^{2} ~ \mathtt{c_s^2}\left( \tau \right) ~ \delta\left( \tau, k \right)}{1 + w\left( \tau \right)} + k^{2} ~ \Psi\left( \tau, k \right) - k^{2} ~ \sigma\left( \tau, k \right) - \mathscr{H}\left( \tau \right) ~ \left( 1 - 3 ~ w\left( \tau \right) \right) ~ \theta\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ \mathtt{\dot{u}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ u\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= 0 \\ \mathtt{c_s^2}\left( \tau \right) &= w\left( \tau \right) \\ \mathtt{{\theta}interaction}\left( \tau, k \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(m)\[ \begin{align*} \delta\left( \tau, k \right) &= - \frac{3}{2} ~ \Psi\left( \tau, k \right) ~ \left( 1 + w\left( \tau \right) \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \end{align*} \]
SymBoltz.cold_dark_matter — Function
cold_dark_matter(g; name = :c, kwargs...)Create a particle species for cold dark matter in the spacetime with metric g.
c = SymBoltz.cold_dark_matter(g)Equations
equations(c)\[ \begin{align*} \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{3}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= 0 \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \delta\left( \tau, k \right) &= \left( -1 - w\left( \tau \right) \right) ~ \left( \theta\left( \tau, k \right) - 3 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \right) - 3 ~ \mathscr{H}\left( \tau \right) ~ \left( - w\left( \tau \right) + \mathtt{c_s^2}\left( \tau \right) \right) ~ \delta\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \theta\left( \tau, k \right) &= \mathtt{{\theta}interaction}\left( \tau, k \right) + \frac{k^{2} ~ \mathtt{c_s^2}\left( \tau \right) ~ \delta\left( \tau, k \right)}{1 + w\left( \tau \right)} + k^{2} ~ \Psi\left( \tau, k \right) - k^{2} ~ \sigma\left( \tau, k \right) - \mathscr{H}\left( \tau \right) ~ \left( 1 - 3 ~ w\left( \tau \right) \right) ~ \theta\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ \mathtt{\dot{u}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ u\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= 0 \\ \mathtt{c_s^2}\left( \tau \right) &= w\left( \tau \right) \\ \mathtt{{\theta}interaction}\left( \tau, k \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(c)\[ \begin{align*} \delta\left( \tau, k \right) &= - \frac{3}{2} ~ \Psi\left( \tau, k \right) ~ \left( 1 + w\left( \tau \right) \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \end{align*} \]
SymBoltz.baryons — Function
baryons(g; recombination = true, reionization = true, Hswitch = 1, Heswitch = 6, name = :b, kwargs...)Create a particle species for baryons in the spacetime with metric g.
b = SymBoltz.baryons(g)Equations
equations(b)\[ \begin{align*} \frac{\mathrm{d} ~ \mathtt{\_\kappa}\left( \tau \right)}{\mathrm{d}\tau} &= \frac{ - 1.9944 \cdot 10^{-20} ~ a\left( \tau \right) ~ \mathtt{ne}\left( \tau \right)}{3.2408 \cdot 10^{-18} ~ h} \\ \mathtt{\dot{\kappa}}\left( \tau \right) &= \frac{\mathrm{d} ~ \mathtt{\_\kappa}\left( \tau \right)}{\mathrm{d}\tau} \\ \kappa\left( \tau \right) &= \mathtt{\_\kappa}\left( \tau \right) - \mathtt{{\kappa}0} \\ I\left( \tau \right) &= e^{ - \kappa\left( \tau \right)} \\ v\left( \tau \right) &= - e^{ - \kappa\left( \tau \right)} ~ \frac{\mathrm{d} ~ \kappa\left( \tau \right)}{\mathrm{d}\tau} \\ \mathtt{\dot{v}}\left( \tau \right) &= \frac{\mathrm{d} ~ v\left( \tau \right)}{\mathrm{d}\tau} \\ \mathtt{c_s^2}\left( \tau \right) &= \frac{1.3806 \cdot 10^{-23} ~ \left( T\left( \tau \right) + \frac{ - \frac{\mathrm{d} ~ T\left( \tau \right)}{\mathrm{d}\tau}}{3 ~ \mathscr{H}\left( \tau \right)} \right)}{\mathtt{{\mu}c^2}\left( \tau \right)} \\ \mathtt{{\mu}c^2}\left( \tau \right) &= \frac{1.5044 \cdot 10^{-10}}{1 - 0.74816 ~ \mathtt{YHe} + \left( 1 - \mathtt{YHe} \right) ~ \mathtt{Xe}\left( \tau \right)} \\ \mathtt{DT}\left( \tau \right) &= \frac{ - 4.0265 \cdot 10^{-43} ~ \left( \mathtt{T\gamma}\left( \tau \right) \right)^{4} ~ a\left( \tau \right) ~ \mathtt{Xe}\left( \tau \right) ~ \mathtt{{\Delta}T}\left( \tau \right)}{2.6551 \cdot 10^{-39} ~ \left( 1 + \mathtt{fHe} + \mathtt{Xe}\left( \tau \right) \right) ~ h} - 2 ~ \mathscr{H}\left( \tau \right) ~ T\left( \tau \right) \\ \mathtt{DT\gamma}\left( \tau \right) &= \frac{\mathrm{d} ~ \mathtt{T\gamma}\left( \tau \right)}{\mathrm{d}\tau} \\ \frac{\mathrm{d} ~ \mathtt{{\Delta}T}\left( \tau \right)}{\mathrm{d}\tau} &= - \mathtt{DT\gamma}\left( \tau \right) + \mathtt{DT}\left( \tau \right) \\ T\left( \tau \right) &= \mathtt{T\gamma}\left( \tau \right) + \mathtt{{\Delta}T}\left( \tau \right) \\ \mathtt{nH}\left( \tau \right) &= 94.012 ~ h^{2} ~ \left( 1 - \mathtt{YHe} \right) ~ \rho\left( \tau \right) \\ \mathtt{nHe}\left( \tau \right) &= \mathtt{fHe} ~ \mathtt{nH}\left( \tau \right) \\ \mathtt{ne}\left( \tau \right) &= \mathtt{Xe}\left( \tau \right) ~ \mathtt{nH}\left( \tau \right) \\ \mathtt{rec.nH}\left( \tau \right) &= \mathtt{nH}\left( \tau \right) \\ \mathtt{rec.nHe}\left( \tau \right) &= \mathtt{nHe}\left( \tau \right) \\ \mathtt{rec.ne}\left( \tau \right) &= \mathtt{ne}\left( \tau \right) \\ \mathtt{rec.T}\left( \tau \right) &= T\left( \tau \right) \\ \mathtt{Xe}\left( \tau \right) &= \mathtt{rei1.Xe}\left( \tau \right) + \mathtt{rei2.Xe}\left( \tau \right) + \mathtt{rec.Xe}\left( \tau \right) \\ \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{3}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= 0 \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \delta\left( \tau, k \right) &= \left( -1 - w\left( \tau \right) \right) ~ \left( \theta\left( \tau, k \right) - 3 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \right) - 3 ~ \mathscr{H}\left( \tau \right) ~ \left( - w\left( \tau \right) + \mathtt{c_s^2}\left( \tau \right) \right) ~ \delta\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \theta\left( \tau, k \right) &= \mathtt{{\theta}interaction}\left( \tau, k \right) + \frac{k^{2} ~ \mathtt{c_s^2}\left( \tau \right) ~ \delta\left( \tau, k \right)}{1 + w\left( \tau \right)} + k^{2} ~ \Psi\left( \tau, k \right) - k^{2} ~ \sigma\left( \tau, k \right) - \mathscr{H}\left( \tau \right) ~ \left( 1 - 3 ~ w\left( \tau \right) \right) ~ \theta\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ \mathtt{\dot{u}}\left( \tau, k \right) &= \frac{\mathrm{d}}{\mathrm{d}\tau} ~ u\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= 0 \\ \mathtt{rec.\beta}\left( \tau \right) &= \frac{1}{1.3806 \cdot 10^{-23} ~ \mathtt{rec.T}\left( \tau \right)} \\ \mathtt{rec.{\lambda}e}\left( \tau \right) &= \frac{6.6261 \cdot 10^{-34}}{\sqrt{\frac{5.7236 \cdot 10^{-30}}{\mathtt{rec.\beta}\left( \tau \right)}}} \\ \mathtt{rec.H}\left( \tau \right) &= 3.2408 \cdot 10^{-18} ~ h ~ H\left( \tau \right) \\ \mathtt{rec.{\alpha}H}\left( \tau \right) &= \frac{4.309 \cdot 10^{-19} ~ \mathtt{rec.FH}}{\left( \frac{\mathtt{rec.T}\left( \tau \right)}{10000} \right)^{0.6166} ~ \left( 1 + 0.6703 ~ \left( \frac{\mathtt{rec.T}\left( \tau \right)}{10000} \right)^{0.53} \right)} \\ \mathtt{rec.{\beta}H}\left( \tau \right) &= \frac{e^{ - 5.4468 \cdot 10^{-19} ~ \mathtt{rec.\beta}\left( \tau \right)} ~ \mathtt{rec.{\alpha}H}\left( \tau \right)}{\left( \mathtt{rec.{\lambda}e}\left( \tau \right) \right)^{3}} \\ \mathtt{rec.KH}\left( \tau \right) &= \frac{7.1484 \cdot 10^{-23} ~ \mathtt{rec.KHfitfactor}\left( \tau \right)}{\mathtt{rec.H}\left( \tau \right)} \\ \mathtt{rec.CH}\left( \tau \right) &= 0.5 ~ \left( 1 + \frac{1 + 8.2246 ~ \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right) ~ \mathtt{rec.nH}\left( \tau \right) ~ \mathtt{rec.KH}\left( \tau \right)}{1 + \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right) ~ \left( 8.2246 + \mathtt{rec.{\beta}H}\left( \tau \right) \right) ~ \mathtt{rec.nH}\left( \tau \right) ~ \mathtt{rec.KH}\left( \tau \right)} + \left( 1 + \frac{-1 - 8.2246 ~ \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right) ~ \mathtt{rec.nH}\left( \tau \right) ~ \mathtt{rec.KH}\left( \tau \right)}{1 + \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right) ~ \left( 8.2246 + \mathtt{rec.{\beta}H}\left( \tau \right) \right) ~ \mathtt{rec.nH}\left( \tau \right) ~ \mathtt{rec.KH}\left( \tau \right)} \right) ~ \tanh\left( 1000 ~ \left( - \mathtt{rec.XlimC} + \mathtt{rec.XH^+}\left( \tau \right) \right) \right) \right) \\ \frac{\mathrm{d} ~ \mathtt{rec.XH^+}\left( \tau \right)}{\mathrm{d}\tau} &= \frac{ - a\left( \tau \right) ~ \left( \mathtt{rec.ne}\left( \tau \right) ~ \mathtt{rec.XH^+}\left( \tau \right) ~ \mathtt{rec.{\alpha}H}\left( \tau \right) - \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right) ~ \mathtt{rec.{\beta}H}\left( \tau \right) ~ e^{ - 1.634 \cdot 10^{-18} ~ \mathtt{rec.\beta}\left( \tau \right)} \right) ~ \mathtt{rec.CH}\left( \tau \right)}{3.2408 \cdot 10^{-18} ~ h} \\ \mathtt{rec.{\alpha}He}\left( \tau \right) &= \frac{1.803 \cdot 10^{-17}}{\left( 1 + \sqrt{\frac{\mathtt{rec.T}\left( \tau \right)}{3}} \right)^{0.289} ~ \left( 1 + \sqrt{\frac{\mathtt{rec.T}\left( \tau \right)}{1.3002 \cdot 10^{5}}} \right)^{1.711} ~ \sqrt{\frac{\mathtt{rec.T}\left( \tau \right)}{3}}} \\ \mathtt{rec.{\beta}He}\left( \tau \right) &= \frac{4 ~ e^{ - 6.3633 \cdot 10^{-19} ~ \mathtt{rec.\beta}\left( \tau \right)} ~ \mathtt{rec.{\alpha}He}\left( \tau \right)}{\left( \mathtt{rec.{\lambda}e}\left( \tau \right) \right)^{3}} \\ \mathtt{rec.KHe}\left( \tau \right) &= \frac{1}{\mathtt{rec.invKHe1}\left( \tau \right) + \mathtt{rec.invKHe0}\left( \tau \right) + \mathtt{rec.invKHe2}\left( \tau \right)} \\ \mathtt{rec.invKHe0}\left( \tau \right) &= 1.2597 \cdot 10^{23} ~ \mathtt{rec.H}\left( \tau \right) \\ \mathtt{rec.CHe}\left( \tau \right) &= 0.5 ~ \left( 1 + \frac{e^{ - 9.6491 \cdot 10^{-20} ~ \mathtt{rec.\beta}\left( \tau \right)} + 51.3 ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right) ~ \mathtt{rec.KHe}\left( \tau \right)}{e^{ - 9.6491 \cdot 10^{-20} ~ \mathtt{rec.\beta}\left( \tau \right)} + \left( 51.3 + \mathtt{rec.{\beta}He}\left( \tau \right) \right) ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right) ~ \mathtt{rec.KHe}\left( \tau \right)} + \left( 1 + \frac{ - e^{ - 9.6491 \cdot 10^{-20} ~ \mathtt{rec.\beta}\left( \tau \right)} - 51.3 ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right) ~ \mathtt{rec.KHe}\left( \tau \right)}{e^{ - 9.6491 \cdot 10^{-20} ~ \mathtt{rec.\beta}\left( \tau \right)} + \left( 51.3 + \mathtt{rec.{\beta}He}\left( \tau \right) \right) ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right) ~ \mathtt{rec.KHe}\left( \tau \right)} \right) ~ \tanh\left( 1000 ~ \left( - \mathtt{rec.XlimC} + \mathtt{rec.XHe^+}\left( \tau \right) \right) \right) \right) \\ \mathtt{rec.DXHe^+}\left( \tau \right) &= \frac{ - a\left( \tau \right) ~ \mathtt{rec.CHe}\left( \tau \right) ~ \left( \mathtt{rec.ne}\left( \tau \right) ~ \mathtt{rec.{\alpha}He}\left( \tau \right) ~ \mathtt{rec.XHe^+}\left( \tau \right) - e^{ - 3.303 \cdot 10^{-18} ~ \mathtt{rec.\beta}\left( \tau \right)} ~ \mathtt{rec.{\beta}He}\left( \tau \right) ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) \right)}{3.2408 \cdot 10^{-18} ~ h} \\ \frac{\mathrm{d} ~ \mathtt{rec.XHe^+}\left( \tau \right)}{\mathrm{d}\tau} &= \mathtt{rec.DXHe^+}\left( \tau \right) + \mathtt{rec.DXHet^+}\left( \tau \right) \\ \mathtt{rec.RHe^+}\left( \tau \right) &= \frac{e^{ - 8.7187 \cdot 10^{-18} ~ \mathtt{rec.\beta}\left( \tau \right)}}{\left( \mathtt{rec.{\lambda}e}\left( \tau \right) \right)^{3} ~ \mathtt{rec.nH}\left( \tau \right)} \\ \mathtt{rec.XHe^{+ +}}\left( \tau \right) &= \frac{2 ~ \mathtt{fHe} ~ \mathtt{rec.RHe^+}\left( \tau \right)}{\left( 1 + \mathtt{fHe} + \mathtt{rec.RHe^+}\left( \tau \right) \right) ~ \left( 1 + \sqrt{1 + \frac{4 ~ \mathtt{fHe} ~ \mathtt{rec.RHe^+}\left( \tau \right)}{\left( 1 + \mathtt{fHe} + \mathtt{rec.RHe^+}\left( \tau \right) \right)^{2}}} \right)} \\ \mathtt{rec.Xe}\left( \tau \right) &= \mathtt{rec.XH^+}\left( \tau \right) + \mathtt{rec.XHe^{+ +}}\left( \tau \right) + \mathtt{fHe} ~ \mathtt{rec.XHe^+}\left( \tau \right) \\ \mathtt{rec.KHfitfactor}\left( \tau \right) &= 1 - 0.14 ~ e^{ - \left( \frac{7.28 + \log\left( a\left( \tau \right) \right)}{0.18} \right)^{2}} + 0.079 ~ e^{ - \left( \frac{6.73 + \log\left( a\left( \tau \right) \right)}{0.33} \right)^{2}} \\ \mathtt{rec.{\tau}He}\left( \tau \right) &= \frac{5.3949 \cdot 10^{9} ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right)}{\mathtt{rec.invKHe0}\left( \tau \right)} \\ \mathtt{rec.invKHe1}\left( \tau \right) &= - \mathtt{rec.invKHe0}\left( \tau \right) ~ e^{ - \mathtt{rec.{\tau}He}\left( \tau \right)} \\ \mathtt{rec.{\gamma}2ps}\left( \tau \right) &= \frac{4.8487 \cdot 10^{26} ~ \mathtt{fHe} ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right)}{4.8748 \cdot 10^{26} ~ \sqrt{\frac{6.2832}{5.9736 \cdot 10^{-10} ~ \mathtt{rec.\beta}\left( \tau \right)}} ~ \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right)} \\ \mathtt{rec.invKHe2}\left( \tau \right) &= \frac{5.3949 \cdot 10^{9} ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right)}{1 + 0.36 ~ \left( \mathtt{rec.{\gamma}2ps}\left( \tau \right) \right)^{0.86}} \\ \mathtt{rec.{\alpha}Het}\left( \tau \right) &= \frac{4.9431 \cdot 10^{-17}}{\left( 1 + \sqrt{\frac{\mathtt{rec.T}\left( \tau \right)}{3}} \right)^{0.239} ~ \left( 1 + \sqrt{\frac{\mathtt{rec.T}\left( \tau \right)}{1.3002 \cdot 10^{5}}} \right)^{1.761} ~ \sqrt{\frac{\mathtt{rec.T}\left( \tau \right)}{3}}} \\ \mathtt{rec.{\beta}Het}\left( \tau \right) &= \frac{1.3333 ~ \mathtt{rec.{\alpha}Het}\left( \tau \right) ~ e^{ - 7.6388 \cdot 10^{-19} ~ \mathtt{rec.\beta}\left( \tau \right)}}{\left( \mathtt{rec.{\lambda}e}\left( \tau \right) \right)^{3}} \\ \mathtt{rec.{\tau}Het}\left( \tau \right) &= \frac{1.102 \cdot 10^{-19} ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) ~ \mathtt{rec.nHe}\left( \tau \right)}{25.133 ~ \mathtt{rec.H}\left( \tau \right)} \\ \mathtt{rec.pHet}\left( \tau \right) &= \frac{1 - e^{ - \mathtt{rec.{\tau}Het}\left( \tau \right)}}{\mathtt{rec.{\tau}Het}\left( \tau \right)} \\ \mathtt{rec.{\gamma}2pt}\left( \tau \right) &= \frac{4.788 \cdot 10^{19} ~ \mathtt{fHe} ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right)}{4.861 \cdot 10^{26} ~ \sqrt{\frac{6.2832}{5.9736 \cdot 10^{-10} ~ \mathtt{rec.\beta}\left( \tau \right)}} ~ \left( 1 - \mathtt{rec.XH^+}\left( \tau \right) \right)} \\ \mathtt{rec.CHetnum}\left( \tau \right) &= 177.58 ~ \left( \frac{1}{3 ~ \left( 1 + 0.66 ~ \left( \mathtt{rec.{\gamma}2pt}\left( \tau \right) \right)^{0.9} \right)} + \mathtt{rec.pHet}\left( \tau \right) \right) ~ e^{ - 1.8337 \cdot 10^{-19} ~ \mathtt{rec.\beta}\left( \tau \right)} \\ \mathtt{rec.CHet}\left( \tau \right) &= \frac{1 \cdot 10^{-9} + \mathtt{rec.CHetnum}\left( \tau \right)}{1 \cdot 10^{-9} + \mathtt{rec.{\beta}Het}\left( \tau \right) + \mathtt{rec.CHetnum}\left( \tau \right)} \\ \mathtt{rec.DXHet^+}\left( \tau \right) &= \frac{ - \left( \mathtt{rec.{\alpha}Het}\left( \tau \right) ~ \mathtt{rec.ne}\left( \tau \right) ~ \mathtt{rec.XHe^+}\left( \tau \right) - 3 ~ e^{ - 3.1755 \cdot 10^{-18} ~ \mathtt{rec.\beta}\left( \tau \right)} ~ \mathtt{rec.{\beta}Het}\left( \tau \right) ~ \left( 1 - \mathtt{rec.XHe^+}\left( \tau \right) \right) \right) ~ a\left( \tau \right) ~ \mathtt{rec.CHet}\left( \tau \right)}{3.2408 \cdot 10^{-18} ~ h} \\ \mathtt{rei1.Xe}\left( \tau \right) &= 0.5 ~ \left( 1 + \mathtt{fHe} + \left( 1 + \mathtt{fHe} \right) ~ \tanh\left( 0.4528 ~ \left( 25.534 - \sqrt{1 + z\left( \tau \right)} ~ \left( 1 + z\left( \tau \right) \right) \right) \right) \right) \\ \mathtt{rei2.Xe}\left( \tau \right) &= 0.5 ~ \left( \mathtt{fHe} + \mathtt{fHe} ~ \tanh\left( 2 ~ \left( 3.5 - z\left( \tau \right) \right) \right) \right) \end{align*} \]
Initialization equations
initialization_equations(b)\[ \begin{align*} \delta\left( \tau, k \right) &= - \frac{3}{2} ~ \Psi\left( \tau, k \right) ~ \left( 1 + w\left( \tau \right) \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \end{align*} \]
Neutrinos
SymBoltz.massless_neutrinos — Function
massless_neutrinos(g; lmax = 6, name = :ν, kwargs...)Create a particle species for massless neutrinos in the spacetime with metric g.
ν = SymBoltz.massless_neutrinos(g; lmax = 6)Equations
equations(ν)\[ \begin{align*} \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{F0}\left( \tau, k \right) &= 4 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) - k ~ F\_{1}\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{1}\left( \tau, k \right) &= \frac{1}{3} ~ k ~ \left( 4 ~ \Psi\left( \tau, k \right) + \mathtt{F0}\left( \tau, k \right) - 2 ~ F\_{2}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{2}\left( \tau, k \right) &= \frac{1}{5} ~ k ~ \left( 2 ~ F\_{1}\left( \tau, k \right) - 3 ~ F\_{3}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{3}\left( \tau, k \right) &= \frac{1}{7} ~ k ~ \left( 3 ~ F\_{2}\left( \tau, k \right) - 4 ~ F\_{4}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{4}\left( \tau, k \right) &= \frac{1}{9} ~ k ~ \left( 4 ~ F\_{3}\left( \tau, k \right) - 5 ~ F\_{5}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{5}\left( \tau, k \right) &= \frac{1}{11} ~ k ~ \left( 5 ~ F\_{4}\left( \tau, k \right) - 6 ~ F\_{6}\left( \tau, k \right) \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ F\_{6}\left( \tau, k \right) &= \frac{ - 7 ~ F\_{6}\left( \tau, k \right)}{\tau} + k ~ F\_{5}\left( \tau, k \right) \\ \delta\left( \tau, k \right) &= \mathtt{F0}\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ \theta\left( \tau, k \right) &= \frac{3}{4} ~ k ~ F\_{1}\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= \frac{F\_{2}\left( \tau, k \right)}{2} \\ u\left( \tau, k \right) &= \frac{\theta\left( \tau, k \right)}{k} \\ T\left( \tau \right) &= \frac{\mathtt{T{_0}}}{a\left( \tau \right)} \\ \Omega\left( \tau \right) &= \frac{\mathtt{\Omega{_0}}}{\left( a\left( \tau \right) \right)^{4}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= \frac{1}{3} \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \mathtt{c_s^2}\left( \tau \right) &= w\left( \tau \right) \end{align*} \]
Initialization equations
initialization_equations(ν)\[ \begin{align*} \delta\left( \tau, k \right) &= - 2 ~ \Psi\left( \tau, k \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \\ \sigma\left( \tau, k \right) &= \frac{1}{15} ~ \tau^{2} ~ k^{2} ~ \Psi\left( \tau, k \right) \\ F\_{3}\left( \tau, k \right) &= \frac{3}{7} ~ k ~ F\_{2}\left( \tau, k \right) ~ \tau \\ F\_{4}\left( \tau, k \right) &= \frac{4}{9} ~ k ~ F\_{3}\left( \tau, k \right) ~ \tau \\ F\_{5}\left( \tau, k \right) &= \frac{5}{11} ~ k ~ F\_{4}\left( \tau, k \right) ~ \tau \\ F\_{6}\left( \tau, k \right) &= \frac{6}{13} ~ k ~ F\_{5}\left( \tau, k \right) ~ \tau \end{align*} \]
SymBoltz.momentum_quadrature — Function
momentum_quadrature(f, N; u = x -> 1/(1+x/100), x = u -> 100*(1-u)/u, dx_du = u -> -100/u^2, x1 = 0.0, x2 = Inf)Compute $N$ dimensionless momentum bins $xᵢ$ and integral weights $Wᵢ$ for integrating $∫dx x²f(x)g(x)$ from $0$ to $∞$ against arbitrary weight functions $g(x)$ with $N$-point Gaussian quadrature using QuadGK.jl. The returned weights Ws approximates the integral for arbitrary functions g(x) with the sum $sum(Ws .* g.(xs))$.
The keyword arguments specifies an integral substitution $x(u)$ with derivative $\mathrm{d}x/\mathrm{d}u$ and inverse $u(x)$ to apply. The default transformation first maps $x$ on the infinite domain $(0, ∞)$ to $x/L$ with $L = 100$, which is an approximate decay length of the massive neutrino distribution function. It then performs a rational transformation of $x/L$ into $u$ on the finite domain $(0, 1)$ to make the numerical integral well-defined.
SymBoltz.massive_neutrinos — Function
massive_neutrinos(g; nx = 4, lmax = 4, name = :h, kwargs...)Create a particle species for massive neutrinos in the spacetime with metric g.
h = SymBoltz.massive_neutrinos(g; nx=4, lmax = 5)Equations
equations(h)\[ \begin{align*} T\left( \tau \right) &= \frac{\mathtt{T{_0}}}{a\left( \tau \right)} \\ y\left( \tau \right) &= \mathtt{y{_0}} ~ a\left( \tau \right) \\ \mathtt{In}\left( \tau \right) &= 1.8031 \\ \mathtt{I\rho}\left( \tau \right) &= 0.55272 ~ E\_{1}\left( \tau \right) + 0.99943 ~ E\_{2}\left( \tau \right) + 0.24384 ~ E\_{3}\left( \tau \right) + 0.0070934 ~ E\_{4}\left( \tau \right) \\ \mathtt{IP}\left( \tau \right) &= \frac{0.85619}{E\_{1}\left( \tau \right)} + \frac{10.944}{E\_{2}\left( \tau \right)} + \frac{10.543}{E\_{3}\left( \tau \right)} + \frac{0.98827}{E\_{4}\left( \tau \right)} \\ \rho\left( \tau \right) &= \frac{1.165 \cdot 10^{-16} ~ \left( T\left( \tau \right) \right)^{4} ~ N ~ \mathtt{I\rho}\left( \tau \right)}{1.4143 \cdot 10^{-8} ~ h^{2}} \\ P\left( \tau \right) &= \frac{3.8835 \cdot 10^{-17} ~ \left( T\left( \tau \right) \right)^{4} ~ N ~ \mathtt{IP}\left( \tau \right)}{1.4143 \cdot 10^{-8} ~ h^{2}} \\ w\left( \tau \right) &= \frac{P\left( \tau \right)}{\rho\left( \tau \right)} \\ \Omega\left( \tau \right) &= \frac{8}{3} ~ \rho\left( \tau \right) ~ \pi \\ \mathtt{I\delta\rho}\left( \tau, k \right) &= 0.55272 ~ \mathtt{{\psi}0}\_{1}\left( \tau, k \right) ~ E\_{1}\left( \tau \right) + 0.99943 ~ \mathtt{{\psi}0}\_{2}\left( \tau, k \right) ~ E\_{2}\left( \tau \right) + 0.24384 ~ \mathtt{{\psi}0}\_{3}\left( \tau, k \right) ~ E\_{3}\left( \tau \right) + 0.0070934 ~ \mathtt{{\psi}0}\_{4}\left( \tau, k \right) ~ E\_{4}\left( \tau \right) \\ \delta\left( \tau, k \right) &= \frac{\mathtt{I\delta\rho}\left( \tau, k \right)}{\mathtt{I\rho}\left( \tau \right)} \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ u\left( \tau, k \right) &= \frac{0.68792 ~ \psi\_{1,1}\left( \tau, k \right) + 3.3072 ~ \psi\_{2,1}\left( \tau, k \right) + 1.6033 ~ \psi\_{3,1}\left( \tau, k \right) + 0.083727 ~ \psi\_{4,1}\left( \tau, k \right)}{\frac{\mathtt{IP}\left( \tau \right)}{3} + \mathtt{I\rho}\left( \tau \right)} \\ \theta\left( \tau, k \right) &= k ~ u\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= \frac{\frac{2}{3} ~ \left( \frac{10.944 ~ \psi\_{2,2}\left( \tau, k \right)}{E\_{2}\left( \tau \right)} + \frac{10.543 ~ \psi\_{3,2}\left( \tau, k \right)}{E\_{3}\left( \tau \right)} + \frac{0.85619 ~ \psi\_{1,2}\left( \tau, k \right)}{E\_{1}\left( \tau \right)} + \frac{0.98827 ~ \psi\_{4,2}\left( \tau, k \right)}{E\_{4}\left( \tau \right)} \right)}{\frac{\mathtt{IP}\left( \tau \right)}{3} + \mathtt{I\rho}\left( \tau \right)} \\ \mathtt{c_s^2}\left( \tau, k \right) &= \frac{\frac{10.543 ~ \mathtt{{\psi}0}\_{3}\left( \tau, k \right)}{E\_{3}\left( \tau \right)} + \frac{0.85619 ~ \mathtt{{\psi}0}\_{1}\left( \tau, k \right)}{E\_{1}\left( \tau \right)} + \frac{10.944 ~ \mathtt{{\psi}0}\_{2}\left( \tau, k \right)}{E\_{2}\left( \tau \right)} + \frac{0.98827 ~ \mathtt{{\psi}0}\_{4}\left( \tau, k \right)}{E\_{4}\left( \tau \right)}}{\mathtt{I\delta\rho}\left( \tau, k \right)} \\ E\_{1}\left( \tau \right) &= \sqrt{1.549 + \left( y\left( \tau \right) \right)^{2}} \\ E\_{2}\left( \tau \right) &= \sqrt{10.95 + \left( y\left( \tau \right) \right)^{2}} \\ E\_{3}\left( \tau \right) &= \sqrt{43.235 + \left( y\left( \tau \right) \right)^{2}} \\ E\_{4}\left( \tau \right) &= \sqrt{139.32 + \left( y\left( \tau \right) \right)^{2}} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{{\psi}0}\_{1}\left( \tau, k \right) &= 0.96627 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) + \frac{ - 1.2446 ~ k ~ \psi\_{1,1}\left( \tau, k \right)}{E\_{1}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{{\psi}0}\_{2}\left( \tau, k \right) &= \frac{ - 3.3091 ~ k ~ \psi\_{2,1}\left( \tau, k \right)}{E\_{2}\left( \tau \right)} + 3.1924 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{{\psi}0}\_{3}\left( \tau, k \right) &= \frac{ - 6.5754 ~ k ~ \psi\_{3,1}\left( \tau, k \right)}{E\_{3}\left( \tau \right)} + 6.5662 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \mathtt{{\psi}0}\_{4}\left( \tau, k \right) &= \frac{ - 11.804 ~ k ~ \psi\_{4,1}\left( \tau, k \right)}{E\_{4}\left( \tau \right)} + 11.803 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{1,1}\left( \tau, k \right) &= \frac{0.41487 ~ k ~ \left( - 2 ~ \psi\_{1,2}\left( \tau, k \right) + \mathtt{{\psi}0}\_{1}\left( \tau, k \right) \right)}{E\_{1}\left( \tau \right)} + 0.25879 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{1}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{2,1}\left( \tau, k \right) &= \frac{1.103 ~ k ~ \left( - 2 ~ \psi\_{2,2}\left( \tau, k \right) + \mathtt{{\psi}0}\_{2}\left( \tau, k \right) \right)}{E\_{2}\left( \tau \right)} + 0.32158 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{2}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{3,1}\left( \tau, k \right) &= \frac{2.1918 ~ k ~ \left( - 2 ~ \psi\_{3,2}\left( \tau, k \right) + \mathtt{{\psi}0}\_{3}\left( \tau, k \right) \right)}{E\_{3}\left( \tau \right)} + 0.33287 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{3}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{4,1}\left( \tau, k \right) &= \frac{3.9345 ~ k ~ \left( - 2 ~ \psi\_{4,2}\left( \tau, k \right) + \mathtt{{\psi}0}\_{4}\left( \tau, k \right) \right)}{E\_{4}\left( \tau \right)} + 0.33333 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{4}\left( \tau \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{1,2}\left( \tau, k \right) &= \frac{0.24892 ~ k ~ \left( 2 ~ \psi\_{1,1}\left( \tau, k \right) - 3 ~ \psi\_{1,3}\left( \tau, k \right) \right)}{E\_{1}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{2,2}\left( \tau, k \right) &= \frac{0.66182 ~ k ~ \left( 2 ~ \psi\_{2,1}\left( \tau, k \right) - 3 ~ \psi\_{2,3}\left( \tau, k \right) \right)}{E\_{2}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{3,2}\left( \tau, k \right) &= \frac{1.3151 ~ k ~ \left( 2 ~ \psi\_{3,1}\left( \tau, k \right) - 3 ~ \psi\_{3,3}\left( \tau, k \right) \right)}{E\_{3}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{4,2}\left( \tau, k \right) &= \frac{2.3607 ~ k ~ \left( 2 ~ \psi\_{4,1}\left( \tau, k \right) - 3 ~ \psi\_{4,3}\left( \tau, k \right) \right)}{E\_{4}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{1,3}\left( \tau, k \right) &= \frac{0.1778 ~ k ~ \left( 3 ~ \psi\_{1,2}\left( \tau, k \right) - 4 ~ \psi\_{1,4}\left( \tau, k \right) \right)}{E\_{1}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{2,3}\left( \tau, k \right) &= \frac{0.47273 ~ k ~ \left( 3 ~ \psi\_{2,2}\left( \tau, k \right) - 4 ~ \psi\_{2,4}\left( \tau, k \right) \right)}{E\_{2}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{3,3}\left( \tau, k \right) &= \frac{0.93934 ~ k ~ \left( 3 ~ \psi\_{3,2}\left( \tau, k \right) - 4 ~ \psi\_{3,4}\left( \tau, k \right) \right)}{E\_{3}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{4,3}\left( \tau, k \right) &= \frac{1.6862 ~ k ~ \left( 3 ~ \psi\_{4,2}\left( \tau, k \right) - 4 ~ \psi\_{4,4}\left( \tau, k \right) \right)}{E\_{4}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{1,4}\left( \tau, k \right) &= \frac{0.13829 ~ k ~ \left( 4 ~ \psi\_{1,3}\left( \tau, k \right) - 5 ~ \psi\_{1,5}\left( \tau, k \right) \right)}{E\_{1}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{2,4}\left( \tau, k \right) &= \frac{0.36768 ~ k ~ \left( 4 ~ \psi\_{2,3}\left( \tau, k \right) - 5 ~ \psi\_{2,5}\left( \tau, k \right) \right)}{E\_{2}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{3,4}\left( \tau, k \right) &= \frac{0.7306 ~ k ~ \left( 4 ~ \psi\_{3,3}\left( \tau, k \right) - 5 ~ \psi\_{3,5}\left( \tau, k \right) \right)}{E\_{3}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{4,4}\left( \tau, k \right) &= \frac{1.3115 ~ k ~ \left( 4 ~ \psi\_{4,3}\left( \tau, k \right) - 5 ~ \psi\_{4,5}\left( \tau, k \right) \right)}{E\_{4}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{1,5}\left( \tau, k \right) &= \frac{0.11315 ~ k ~ \left( - 6 ~ \left( \frac{8.8381 ~ \psi\_{1,5}\left( \tau, k \right) ~ E\_{1}\left( \tau \right)}{k ~ \tau} - \psi\_{1,4}\left( \tau, k \right) \right) + 5 ~ \psi\_{1,4}\left( \tau, k \right) \right)}{E\_{1}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{2,5}\left( \tau, k \right) &= \frac{0.30083 ~ k ~ \left( - 6 ~ \left( \frac{3.3242 ~ \psi\_{2,5}\left( \tau, k \right) ~ E\_{2}\left( \tau \right)}{k ~ \tau} - \psi\_{2,4}\left( \tau, k \right) \right) + 5 ~ \psi\_{2,4}\left( \tau, k \right) \right)}{E\_{2}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{3,5}\left( \tau, k \right) &= \frac{0.59776 ~ k ~ \left( 5 ~ \psi\_{3,4}\left( \tau, k \right) - 6 ~ \left( - \psi\_{3,4}\left( \tau, k \right) + \frac{1.6729 ~ \psi\_{3,5}\left( \tau, k \right) ~ E\_{3}\left( \tau \right)}{k ~ \tau} \right) \right)}{E\_{3}\left( \tau \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \psi\_{4,5}\left( \tau, k \right) &= \frac{1.073 ~ k ~ \left( 5 ~ \psi\_{4,4}\left( \tau, k \right) - 6 ~ \left( - \psi\_{4,4}\left( \tau, k \right) + \frac{0.93193 ~ \psi\_{4,5}\left( \tau, k \right) ~ E\_{4}\left( \tau \right)}{k ~ \tau} \right) \right)}{E\_{4}\left( \tau \right)} \end{align*} \]
Initialization equations
initialization_equations(h)\[ \begin{align*} \mathtt{{\psi}0}\_{1}\left( \tau, k \right) &= - 0.48313 ~ \Psi\left( \tau, k \right) \\ \mathtt{{\psi}0}\_{2}\left( \tau, k \right) &= - 1.5962 ~ \Psi\left( \tau, k \right) \\ \mathtt{{\psi}0}\_{3}\left( \tau, k \right) &= - 3.2831 ~ \Psi\left( \tau, k \right) \\ \mathtt{{\psi}0}\_{4}\left( \tau, k \right) &= - 5.9017 ~ \Psi\left( \tau, k \right) \\ \psi\_{1,1}\left( \tau, k \right) &= 0.12939 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{1}\left( \tau \right) ~ \tau \\ \psi\_{2,1}\left( \tau, k \right) &= 0.16079 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{2}\left( \tau \right) ~ \tau \\ \psi\_{3,1}\left( \tau, k \right) &= 0.16643 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{3}\left( \tau \right) ~ \tau \\ \psi\_{4,1}\left( \tau, k \right) &= 0.16667 ~ k ~ \Psi\left( \tau, k \right) ~ E\_{4}\left( \tau \right) ~ \tau \\ \psi\_{1,2}\left( \tau, k \right) &= 0.032209 ~ \tau^{2} ~ k^{2} ~ \Psi\left( \tau, k \right) \\ \psi\_{2,2}\left( \tau, k \right) &= 0.10641 ~ \tau^{2} ~ k^{2} ~ \Psi\left( \tau, k \right) \\ \psi\_{3,2}\left( \tau, k \right) &= 0.21887 ~ \tau^{2} ~ k^{2} ~ \Psi\left( \tau, k \right) \\ \psi\_{4,2}\left( \tau, k \right) &= 0.39345 ~ \tau^{2} ~ k^{2} ~ \Psi\left( \tau, k \right) \\ \psi\_{1,3}\left( \tau, k \right) &= 0 \\ \psi\_{2,3}\left( \tau, k \right) &= 0 \\ \psi\_{3,3}\left( \tau, k \right) &= 0 \\ \psi\_{4,3}\left( \tau, k \right) &= 0 \\ \psi\_{1,4}\left( \tau, k \right) &= 0 \\ \psi\_{2,4}\left( \tau, k \right) &= 0 \\ \psi\_{3,4}\left( \tau, k \right) &= 0 \\ \psi\_{4,4}\left( \tau, k \right) &= 0 \\ \psi\_{1,5}\left( \tau, k \right) &= 0 \\ \psi\_{2,5}\left( \tau, k \right) &= 0 \\ \psi\_{3,5}\left( \tau, k \right) &= 0 \\ \psi\_{4,5}\left( \tau, k \right) &= 0 \end{align*} \]
Dark energy
SymBoltz.cosmological_constant — Function
cosmological_constant(g; name = :Λ, kwargs...)Create a particle species for the cosmological constant (with equation of state w ~ -1) in the spacetime with metric g.
Λ = SymBoltz.cosmological_constant(g)Equations
equations(Λ)\[ \begin{align*} \delta\left( \tau, k \right) &= 0 \\ \theta\left( \tau, k \right) &= 0 \\ \sigma\left( \tau, k \right) &= 0 \\ \Omega\left( \tau \right) &= \mathtt{\Omega{_0}} \\ \rho\left( \tau \right) &= \frac{3 ~ \Omega\left( \tau \right)}{8 ~ \pi} \\ w\left( \tau \right) &= -1 \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \mathtt{c_s^2}\left( \tau \right) &= w\left( \tau \right) \end{align*} \]
Initialization equations
initialization_equations(Λ)\[ \begin{align*} \end{align*} \]
SymBoltz.w0wa — Function
w0wa(g; kwargs...)Create a particle species for the w₀-wₐ dark energy (CPL) parametrization in the spacetime with metric g.
X = SymBoltz.w0wa(g)Equations
equations(X)\[ \begin{align*} w\left( \tau \right) &= \mathtt{w0} + \mathtt{wa} ~ \left( 1 - a\left( \tau \right) \right) \\ \dot{w}\left( \tau \right) &= \frac{\mathrm{d} ~ w\left( \tau \right)}{\mathrm{d}\tau} \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \frac{\mathrm{d} ~ \rho\left( \tau \right)}{\mathrm{d}\tau} &= - 3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \rho\left( \tau \right) \\ \mathtt{c_a^2}\left( \tau \right) &= w\left( \tau \right) + \frac{ - \dot{w}\left( \tau \right)}{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right)} \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \delta\left( \tau, k \right) &= \left( -1 - w\left( \tau \right) \right) ~ \left( \theta\left( \tau, k \right) - 3 ~ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \Phi\left( \tau, k \right) \right) - 3 ~ \left( \mathtt{c_s^2} - w\left( \tau \right) \right) ~ \mathscr{H}\left( \tau \right) ~ \delta\left( \tau, k \right) - 9 ~ \left( \frac{\mathscr{H}\left( \tau \right)}{k} \right)^{2} ~ \left( \mathtt{c_s^2} - \mathtt{c_a^2}\left( \tau \right) \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right) \\ \frac{\mathrm{d}}{\mathrm{d}\tau} ~ \theta\left( \tau, k \right) &= \frac{k^{2} ~ \mathtt{c_s^2} ~ \delta\left( \tau, k \right)}{1 + w\left( \tau \right)} - \left( 1 - 3 ~ \mathtt{c_s^2} \right) ~ \mathscr{H}\left( \tau \right) ~ \theta\left( \tau, k \right) + k^{2} ~ \Psi\left( \tau, k \right) - k^{2} ~ \sigma\left( \tau, k \right) \\ \Delta\left( \tau, k \right) &= \frac{3 ~ \mathscr{H}\left( \tau \right) ~ \left( 1 + w\left( \tau \right) \right) ~ \theta\left( \tau, k \right)}{k^{2}} + \delta\left( \tau, k \right) \\ \sigma\left( \tau, k \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(X)\[ \begin{align*} \delta\left( \tau, k \right) &= - \frac{3}{2} ~ \Psi\left( \tau, k \right) ~ \left( 1 + w\left( \tau \right) \right) \\ \theta\left( \tau, k \right) &= \frac{1}{2} ~ k^{2} ~ \Psi\left( \tau, k \right) ~ \tau \end{align*} \]
SymBoltz.quintessence — Function
quintessence(g[, v]; name = :ϕ, kwargs...)Create a species with a quintessence scalar field in the potential v in the spacetime with metric g.
Q = SymBoltz.quintessence(g)Equations
equations(Q)\[ \begin{align*} \frac{\mathrm{d}^{2} ~ \phi\left( \tau \right)}{\mathrm{d}\tau^{2}} &= - 2 ~ \mathscr{H}\left( \tau \right) ~ \frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau} - \left( a\left( \tau \right) \right)^{2} ~ \mathtt{V\prime}\left( \tau \right) \\ K\left( \tau \right) &= \frac{\left( \frac{\frac{\mathrm{d} ~ \phi\left( \tau \right)}{\mathrm{d}\tau}}{a\left( \tau \right)} \right)^{2}}{2} \\ \rho\left( \tau \right) &= V\left( \tau \right) + K\left( \tau \right) \\ P\left( \tau \right) &= - V\left( \tau \right) + K\left( \tau \right) \\ w\left( \tau \right) &= \frac{P\left( \tau \right)}{\rho\left( \tau \right)} \\ \mathtt{m^2}\left( \tau \right) &= \mathtt{V\prime\prime}\left( \tau \right) \\ \mathtt{{\epsilon}s}\left( \tau \right) &= \frac{\left( \frac{\mathtt{V\prime}\left( \tau \right)}{V\left( \tau \right)} \right)^{2}}{16 ~ \pi} \\ \mathtt{{\eta}s}\left( \tau \right) &= \frac{\mathtt{V\prime\prime}\left( \tau \right)}{8 ~ V\left( \tau \right) ~ \pi} \\ \delta\left( \tau, k \right) &= 0 \\ \sigma\left( \tau, k \right) &= 0 \\ \mathtt{c_s^2}\left( \tau \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(Q)\[ \begin{align*} \end{align*} \]
Curvature (effective)
SymBoltz.curvature — Function
curvature(g; name = :K, kwargs...)Create a species that effectively accounts for curvature in the spacetime with metric g.
K = SymBoltz.curvature(g)Equations
equations(K)\[ \begin{align*} w\left( \tau \right) &= \frac{-1}{3} \\ \rho\left( \tau \right) &= \frac{3 ~ \mathtt{\Omega_0}}{8 ~ \left( a\left( \tau \right) \right)^{2} ~ \pi} \\ P\left( \tau \right) &= w\left( \tau \right) ~ \rho\left( \tau \right) \\ \delta\left( \tau, k \right) &= 0 \\ \theta\left( \tau, k \right) &= 0 \\ \mathtt{c_s^2}\left( \tau \right) &= 0 \\ \sigma\left( \tau, k \right) &= 0 \end{align*} \]
Initialization equations
initialization_equations(K)\[ \begin{align*} \end{align*} \]
Effective
SymBoltz.effective_species — Function
effective_species(g, species; effective_name = "", kwargs...)Create an effective "read-only" species for several given species with metric g. Additive properties (like $ρ$, $P$ and $δρ$) are summed, and used to express non-additive properties (like $w$ and $δ$).
@named m = SymBoltz.effective_species(g, [c, b, h]; effective_name = "Late-time matter")Equations
equations(m)\[ \begin{align*} \rho\left( \tau \right) &= \mathtt{c.\rho}\left( \tau \right) + \mathtt{b.\rho}\left( \tau \right) + \mathtt{h.\rho}\left( \tau \right) \\ P\left( \tau \right) &= \mathtt{h.P}\left( \tau \right) + \mathtt{c.P}\left( \tau \right) + \mathtt{b.P}\left( \tau \right) \\ w\left( \tau \right) &= \frac{P\left( \tau \right)}{\rho\left( \tau \right)} \\ \delta\left( \tau, k \right) &= \frac{\mathtt{h.\delta}\left( \tau, k \right) ~ \mathtt{h.\rho}\left( \tau \right) + \mathtt{c.\rho}\left( \tau \right) ~ \mathtt{c.\delta}\left( \tau, k \right) + \mathtt{b.\rho}\left( \tau \right) ~ \mathtt{b.\delta}\left( \tau, k \right)}{\rho\left( \tau \right)} \\ \theta\left( \tau, k \right) &= \frac{\mathtt{h.\theta}\left( \tau, k \right) ~ \left( 1 + \mathtt{h.w}\left( \tau \right) \right) ~ \mathtt{h.\rho}\left( \tau \right) + \mathtt{c.\rho}\left( \tau \right) ~ \mathtt{c.\theta}\left( \tau, k \right) ~ \left( 1 + \mathtt{c.w}\left( \tau \right) \right) + \mathtt{b.\rho}\left( \tau \right) ~ \left( 1 + \mathtt{b.w}\left( \tau \right) \right) ~ \mathtt{b.\theta}\left( \tau, k \right)}{P\left( \tau \right) + \rho\left( \tau \right)} \\ \Delta\left( \tau, k \right) &= \frac{\mathtt{h.\Delta}\left( \tau, k \right) ~ \mathtt{h.\rho}\left( \tau \right) + \mathtt{c.\rho}\left( \tau \right) ~ \mathtt{c.\Delta}\left( \tau, k \right) + \mathtt{b.\rho}\left( \tau \right) ~ \mathtt{b.\Delta}\left( \tau, k \right)}{\rho\left( \tau \right)} \end{align*} \]
Initialization equations
initialization_equations(m)\[ \begin{align*} \end{align*} \]