aurel.solutions

Here are a couple solutions to Einstein’s field equations. Functions in these modules generate numerical data based on analytical expressions.

solutions.Collins_Stewart

This is a Collins and Stewart 1971 solution that describes a Bianchi II γ-law perfect fluid homogeneous solution. See section 3.3 of 2211.08133

aurel.solutions.Collins_Stewart.Kdown3(t, x, y, z)[source]: Extrinsic curvature

aurel.solutions.Collins_Stewart.data(t, x, y, z)[source]: Returns dictionary of Collins Stewart data

aurel.solutions.Collins_Stewart.gammadown3(t, x, y, z, analytical=False)[source]: Spatial metric

aurel.solutions.Collins_Stewart.gdown4(t, x, y, z, analytical=False)[source]: Spacetime metric

aurel.solutions.Collins_Stewart.press(t, x, y, z)[source]: Pressure

aurel.solutions.Collins_Stewart.rho(t, x, y, z)[source]: Energy density

solutions.Conformally_flat

aurel.solutions.Conformally_flat.Kdown3(t, x, y, z)[source]: Extrinsic curvature

aurel.solutions.Conformally_flat.Omega(x)[source]: Returns the conformal factor

aurel.solutions.Conformally_flat.Tdown4(t, x, y, z)[source]: Stress-energy tensor, from Einstein’s field equations

aurel.solutions.Conformally_flat.alpha(t, x, y, z)[source]: Lapse

aurel.solutions.Conformally_flat.data(t, x, y, z)[source]: Returns dictionary of Collins Stewart data

aurel.solutions.Conformally_flat.dxOmega(x)[source]: Returns the 1st derivative of the conformal factor

aurel.solutions.Conformally_flat.dxdxOmega(x)[source]: Returns the 2nd derivatire of the conformal factor

aurel.solutions.Conformally_flat.gammadown3(t, x, y, z, analytical=False)[source]: Spatial metric

aurel.solutions.Conformally_flat.gdown4(t, x, y, z, analytical=False)[source]: Spacetime metric

aurel.solutions.Conformally_flat.st_RicciS(x)[source]: Spacetime Ricci scalar

solutions.EdS

FLRW spacetime with an Einstein-de Sitter model

aurel.solutions.EdS.Hconf(t)[source]: Conformal Hubble function

aurel.solutions.EdS.Hprop(t)[source]: Proper Hubble function

aurel.solutions.EdS.Kdown3(t, x, y, z)[source]: Extrinsic curvature

aurel.solutions.EdS.Omega_m(t)[source]: Matter density parameter

aurel.solutions.EdS.a(t)[source]: Scale factor

aurel.solutions.EdS.a_func_z(z)[source]: Scale factor from redshift

aurel.solutions.EdS.alpha(t, x, y, z)[source]: Lapse

aurel.solutions.EdS.an_today(t)[source]: Scale factor normalised by a(z=0)

aurel.solutions.EdS.betaup3(t, x, y, z)[source]: Shift

aurel.solutions.EdS.fL(t)[source]: Growth index = d ln (delta) / d ln (a)

aurel.solutions.EdS.gammadown3(t, x, y, z)[source]: Spatial metric

aurel.solutions.EdS.press(t)[source]: Pressure

aurel.solutions.EdS.redshift(t)[source]: Redshift

aurel.solutions.EdS.rho(t)[source]: Energy density

aurel.solutions.EdS.t_func_Hprop(Hprop)[source]: Proper time from Hubble

aurel.solutions.EdS.t_func_a(a)[source]: Proper time from scale factor

aurel.solutions.EdS.t_func_z(z)[source]: Proper time from redshift

solutions.Harvey_Tsoubelis

This is a A.Harvey and T.Tsoubelis solution that describes a vacuum Bianchi IV plane wave homogeneous spacetime page 191 of ‘Dynamical Systems in Cosmology’ by J.Wainwright and G.F.R.Ellis See section 3.5 of 2211.08133

aurel.solutions.Harvey_Tsoubelis.Kdown3(t, x, y, z)[source]: Returns the extrinsic curvature

aurel.solutions.Harvey_Tsoubelis.Tdown4(t, x, y, z)[source]: Energy stress tensor

aurel.solutions.Harvey_Tsoubelis.alpha(t, x, y, z)[source]: Returns the lapse function

aurel.solutions.Harvey_Tsoubelis.betaup3(t, x, y, z)[source]: Returns the shift vector

aurel.solutions.Harvey_Tsoubelis.data(t, x, y, z)[source]: Returns dictionary of Harvey Tsoubelis data

aurel.solutions.Harvey_Tsoubelis.gammadown3(t, x, y, z, analytical=False)[source]: Returns the spatial metric

aurel.solutions.Harvey_Tsoubelis.gdown4(t, x, y, z, analytical=False)[source]: Returns the spacetime metric

aurel.solutions.Harvey_Tsoubelis.press(t, x, y, z)[source]: Returns the pressure

aurel.solutions.Harvey_Tsoubelis.rho(t, x, y, z)[source]: Returns the energy density

aurel.solutions.Harvey_Tsoubelis.uup4(t, x, y, z)[source]: Fluid 4 velocity

solutions.ICPertFLRW

See: https://arxiv.org/pdf/2302.09033 and: https://arxiv.org/pdf/1307.1478

aurel.solutions.ICPertFLRW.Kdown3(sol, fd, t, Rc)[source]: Extrinsic curvature, nonlinear from gammmadown3

aurel.solutions.ICPertFLRW.Rc_func(x, y, z, amp, lamb)[source]: Comoving curvature perturbation

aurel.solutions.ICPertFLRW.delta1(sol, fd, t, Rc)[source]: Linear density contrast

aurel.solutions.ICPertFLRW.gammadown3(sol, fd, t, Rc)[source]: Spatial metric with 1st order perturbations

solutions.LCDM

FLRW spacetime with a \(\Lambda\)CDM model

aurel.solutions.LCDM.Hconf(t)[source]: Conformal Hubble function

aurel.solutions.LCDM.Hprop(t)[source]: Proper Hubble function

aurel.solutions.LCDM.Kdown3(t, x, y, z)[source]: Extrinsic curvature

aurel.solutions.LCDM.Omega_m(t)[source]: Matter density parameter

aurel.solutions.LCDM.a(t, analytical=False)[source]: Scale factor

aurel.solutions.LCDM.alpha(t, x, y, z)[source]: Lapse

aurel.solutions.LCDM.an_today(t)[source]: Scale factor normalised by a(z=0)

aurel.solutions.LCDM.betaup3(t, x, y, z)[source]: Shift

aurel.solutions.LCDM.fL(t)[source]: Growth index = d ln (delta) / d ln (a)

aurel.solutions.LCDM.gammadown3(t, x, y, z, analytical=False)[source]: Spatial metric

aurel.solutions.LCDM.redshift(t)[source]: Redshift

aurel.solutions.LCDM.rho(t)[source]: Energy density

solutions.Non_diagonal

See section 3.2 of 2211.08133

aurel.solutions.Non_diagonal.A(z, analytical=False)[source]: Conformal factor

aurel.solutions.Non_diagonal.Kdown3(t, x, y, z)[source]: Extrinsic curvature

aurel.solutions.Non_diagonal.Tdown4(t, x, y, z)[source]: Stress-energy tensor, from Einstein’s field equations

aurel.solutions.Non_diagonal.data(t, x, y, z)[source]: Returns dictionary of Non-diagonal data

aurel.solutions.Non_diagonal.dzA(z)[source]: Conformal factor 1st derivative

aurel.solutions.Non_diagonal.dzdzA(z)[source]: Conformal factor 2nd derivative

aurel.solutions.Non_diagonal.gammadown3(t, x, y, z, analytical=False)[source]: Spatial metric

aurel.solutions.Non_diagonal.gdown4(t, x, y, z, analytical=False)[source]: Spacetime metric

solutions.Rosquist

This is a Rosquist and Jantzen solution that describes a Bianchi VI tilted γ ̃-law perfect fluid homogeneous solution with vorticity See section 3.4 of 2211.08133

aurel.solutions.Rosquist_Jantzen.Kdown3(t, x, y, z)[source]: Returns the extrinsic curvature

aurel.solutions.Rosquist_Jantzen.Tdown4(t, x, y, z)[source]: Returns the energy-stress tensor

aurel.solutions.Rosquist_Jantzen.data(t, x, y, z)[source]: Returns dictionary of Roquist Jantzen data

aurel.solutions.Rosquist_Jantzen.gammadown3(t, x, y, z, analytical=False)[source]: Returns the spatial metric

aurel.solutions.Rosquist_Jantzen.gdown4(t, x, y, z, analytical=False)[source]: Returns the spacetime metric

solutions.Schwarzschild_isotropic

This is the Schwarzschild solution in in isotropic coordinates with maximal slicing. See https://arxiv.org/pdf/0904.4184

aurel.solutions.Schwarzschild_isotropic.Kdown3(t, x, y, z)[source]: Returns the extrinsic curvature

aurel.solutions.Schwarzschild_isotropic.Kretschmann(t, x, y, z)[source]: Kretschmann scalar

aurel.solutions.Schwarzschild_isotropic.Tdown4(t, x, y, z)[source]: Returns the energy-stress tensor

aurel.solutions.Schwarzschild_isotropic.alpha(t, x, y, z, analytical=False)[source]: Returns the lapse function

aurel.solutions.Schwarzschild_isotropic.betaup3(t, x, y, z)[source]: Returns the shift vector

aurel.solutions.Schwarzschild_isotropic.data(t, x, y, z)[source]: Returns dictionary of Schwarzschild data

aurel.solutions.Schwarzschild_isotropic.gammadown3(t, x, y, z, analytical=False)[source]: Returns the spatial metric

aurel.solutions.Schwarzschild_isotropic.gdown4(t, x, y, z, analytical=False)[source]: Returns the spacetime metric

aurel.solutions.Schwarzschild_isotropic.null_ray_exp_out(t, x, y, z)[source]: Outward null ray expansion

solutions.Szekeres

This is a \(\Lambda\) - Szekeres solution which is perturbed solution of the flat dust FLRW + LCDM spacetime. See section 3.1 of 2211.08133

aurel.solutions.Szekeres.Kdown3(t, x, y, z)[source]: Returns the extrinsic curvature

aurel.solutions.Szekeres.Z_terms(t, x, y, z, analytical=False)[source]: Returns F, Z, dtZ functions

aurel.solutions.Szekeres.alpha(t, x, y, z)[source]: Returns the lapse function

aurel.solutions.Szekeres.betaup3(t, x, y, z)[source]: Returns the shift vector

aurel.solutions.Szekeres.data(t, x, y, z)[source]: Returns dictionary of Szekeres data

aurel.solutions.Szekeres.gammadown3(t, x, y, z, analytical=False)[source]: Returns the spatial metric

aurel.solutions.Szekeres.gdown4(t, x, y, z, analytical=False)[source]: Returns the spacetime metric

aurel.solutions.Szekeres.press(t, x, y, z)[source]: Returns the pressure

aurel.solutions.Szekeres.rho(t, x, y, z)[source]: Returns the energy density