primordial: inflationary equation solver¶

primordial:	inflationary equation solver
Author:	Will Handley
Version:	0.0.14
Homepage:	https://github.com/williamjameshandley/primordial
Documentation:	http://primordial.readthedocs.io/

Description¶

primordial is a python package for solving cosmological inflationary equations.

It is very much in beta stage, and currently being built for research purposes.

Example Usage¶

Plot Background evolution¶

import numpy
import matplotlib.pyplot as plt
from primordial.solver import solve
from primordial.equations.inflation_potentials import ChaoticPotential
from primordial.equations.t.inflation import Equations, KD_initial_conditions
from primordial.equations.events import Inflation, Collapse

fig, ax = plt.subplots(3,sharex=True)
for K in [-1, 0, +1]:
     m = 1
     V = ChaoticPotential(m)
     equations = Equations(K, V)

     events= [Inflation(equations),                    # Record inflation entry and exit
              Inflation(equations, -1, terminal=True), # Stop on inflation exit
              Collapse(equations, terminal=True)]      # Stop if universe stops expanding

     N_p = -1.5
     phi_p = 23
     t_p = 1e-5
     ic = KD_initial_conditions(t_p, N_p, phi_p)
     t = numpy.logspace(-5,10,1e6)

     sol = solve(equations, ic, t_eval=t, events=events)

     ax[0].plot(sol.N(t),sol.phi(t))
     ax[0].set_ylabel(r'$\phi$')

     ax[1].plot(sol.N(t),sol.H(t))
     ax[1].set_yscale('log')
     ax[1].set_ylabel(r'$H$')

     ax[2].plot(sol.N(t),1/(sol.H(t)*numpy.exp(sol.N(t))))
     ax[2].set_yscale('log')
     ax[2].set_ylabel(r'$1/aH$')

ax[-1].set_xlabel('$N$')

Plot mode function evolution¶

import numpy
import matplotlib.pyplot as plt
from primordial.solver import solve
from primordial.equations.inflation_potentials import ChaoticPotential
from primordial.equations.t.mukhanov_sasaki import Equations, KD_initial_conditions
from primordial.equations.events import Inflation, Collapse, ModeExit

fig, axes = plt.subplots(3,sharex=True)
for ax, K in zip(axes, [-1, 0, +1]):
    ax2 = ax.twinx()
    m = 1
    V = ChaoticPotential(m)
    k = 100
    equations = Equations(K, V, k)

    events= [
            Inflation(equations),                    # Record inflation entry and exit
            Collapse(equations, terminal=True),      # Stop if universe stops expanding
            ModeExit(equations, +1, terminal=True, value=1e1*k)   # Stop on mode exit
            ]


    N_p = -1.5
    phi_p = 23
    t_p = 1e-5
    ic = KD_initial_conditions(t_p, N_p, phi_p)
    t = numpy.logspace(-5,10,1e6)

    sol = solve(equations, ic, t_eval=t, events=events)

    N = sol.N(t)
    ax.plot(N,sol.R1(t), 'k-')
    ax2.plot(N,-numpy.log(sol.H(t))-N, 'b-')

    ax.set_ylabel('$\mathcal{R}$')
    ax2.set_ylabel('$-\log aH$')

    ax.text(0.9, 0.9, r'$K=%i$' % K, transform=ax.transAxes)

axes[-1].set_xlabel('$N$')

To do list¶

Eventually would like to submit this to JOSS. Here are things to do before then:

Cosmology¶

Slow roll initial conditions
add \(\eta\) as independent variable
add \(\phi\) as independent variable

Code¶

Documentation
Tests
- 100% coverage
- interpolation

primordial package¶

Subpackages¶

primordial.equations package¶

Subpackages¶

primordial.equations.N package¶

Submodules¶

primordial.equations.N.cosmology module¶

class primordial.equations.N.cosmology.Equations(H0, Omega_r, Omega_m, Omega_k, Omega_l)[source]¶

Bases: primordial.equations.cosmology.Equations

Cosmology equations in time

Solves background variables in cosmic time for curved and flat universes using the Friedmann equation.

Independent variable:: N: efolds
Variables:: t: cosmic time

Methods

`H`(t, y)	Hubble parameter
`H2`(t, y)	The square of the Hubble parameter, computed using the Friedmann equation
`__call__`(N, y)	The derivative function for underlying variables, computed using the Klein-Gordon equation
`add_variable`(*args)	Add dependent variables to the equations
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

class primordial.equations.N.cosmology.initial_conditions(Ni)[source]¶

Bases: object

Methods

__call__

primordial.equations.N.inflation module¶

class primordial.equations.N.inflation.Equations(K, potential)[source]¶

Bases: primordial.equations.inflation.Equations

Background equations in time

Solves background variables in cosmic time for curved and flat universes using the Klein-Gordon and Friedmann equations.

Independent variable:: N: e-folds (log a)
Variables:: phi: inflaton field dphi: d/dN (phi) t: cosmic time

Methods

`H`(t, y)	Hubble parameter
`H2`(N, y)	The square of the Hubble parameter, computed using the Friedmann equation
`V`(t, y)	Potential
`__call__`(N, y)	The derivative function for underlying variables, computed using the Klein-Gordon equation
`add_variable`(*args)	Add dependent variables to the equations
`d2Vdphi2`(t, y)	Potential second derivative
`dVdphi`(t, y)	Potential derivative
`dlogH`(N, y)	d/dN log H
`inflating`(N, y)	Inflation diagnostic
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

H2(N, y)[source]¶: The square of the Hubble parameter, computed using the Friedmann equation

dlogH(N, y)[source]¶: d/dN log H

inflating(N, y)[source]¶: Inflation diagnostic

class primordial.equations.N.inflation.Inflation_start_initial_conditions(N_e, phi_e)[source]¶

Bases: object

Methods

__call__

primordial.equations.N.mukhanov_sasaki module¶

class primordial.equations.N.mukhanov_sasaki.Equations(K, potential, k)[source]¶

Bases: primordial.equations.N.inflation.Equations

Methods

`H`(t, y)	Hubble parameter
`H2`(N, y)	The square of the Hubble parameter, computed using the Friedmann equation
`V`(t, y)	Potential
`__call__`(N, y)	The derivative function for underlying variables, computed using the Mukhanov-Sasaki equation equation
`add_variable`(*args)	Add dependent variables to the equations
`d2Vdphi2`(t, y)	Potential second derivative
`dVdphi`(t, y)	Potential derivative
`dlogH`(N, y)	d/dN log H
`inflating`(N, y)	Inflation diagnostic
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

class primordial.equations.N.mukhanov_sasaki.Inflation_start_initial_conditions(N_e, phi_e)[source]¶

Bases: primordial.equations.N.inflation.Inflation_start_initial_conditions

Methods

__call__

Module contents¶

primordial.equations.t package¶

Submodules¶

primordial.equations.t.cosmology module¶

class primordial.equations.t.cosmology.Equations(H0, Omega_r, Omega_m, Omega_k, Omega_l)[source]¶

Bases: primordial.equations.cosmology.Equations

Cosmology equations in time

Solves background variables in cosmic time for curved and flat universes using the Friedmann equation.

Independent variable:: t: cosmic time
Variables:: N: efolds

Methods

`H`(t, y)	Hubble parameter
`H2`(t, y)	The square of the Hubble parameter, computed using the Friedmann equation
`__call__`(t, y)	The derivative function for underlying variables, computed using the Klein-Gordon equation
`add_variable`(*args)	Add dependent variables to the equations
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

class primordial.equations.t.cosmology.initial_conditions(Ni)[source]¶

Bases: object

Methods

__call__

primordial.equations.t.inflation module¶

class primordial.equations.t.inflation.Equations(K, potential)[source]¶

Bases: primordial.equations.inflation.Equations

Background equations in time

Solves bacgkround variables in cosmic time for curved and flat universes using the Klein-Gordon and Friedmann equations.

Independent variable:: t: cosmic time
Variables:: N: efolds phi: inflaton field dphi: d (phi) / dt

Methods

`H`(t, y)	Hubble parameter
`H2`(t, y)	The square of the Hubble parameter, computed using the Friedmann equation
`V`(t, y)	Potential
`__call__`(t, y)	The derivative function for underlying variables, computed using the Klein-Gordon equation
`add_variable`(*args)	Add dependent variables to the equations
`d2Vdphi2`(t, y)	Potential second derivative
`dVdphi`(t, y)	Potential derivative
`inflating`(t, y)	Inflation diagnostic
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

H2(t, y)[source]¶: The square of the Hubble parameter, computed using the Friedmann equation

inflating(t, y)[source]¶: Inflation diagnostic

class primordial.equations.t.inflation.Inflation_start_initial_conditions(N_e, phi_e)[source]¶

Bases: object

Methods

__call__

class primordial.equations.t.inflation.KD_initial_conditions(t0, N_p, phi_p)[source]¶

Bases: object

Methods

__call__

primordial.equations.t.mukhanov_sasaki module¶

class primordial.equations.t.mukhanov_sasaki.Equations(K, potential, k)[source]¶

Bases: primordial.equations.t.inflation.Equations

Methods

`H`(t, y)	Hubble parameter
`H2`(t, y)	The square of the Hubble parameter, computed using the Friedmann equation
`V`(t, y)	Potential
`__call__`(t, y)	The derivative function for underlying variables, computed using the Mukhanov-Sasaki equation equation
`add_variable`(*args)	Add dependent variables to the equations
`d2Vdphi2`(t, y)	Potential second derivative
`dVdphi`(t, y)	Potential derivative
`inflating`(t, y)	Inflation diagnostic
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

class primordial.equations.t.mukhanov_sasaki.Inflation_start_initial_conditions(N_e, phi_e)[source]¶

Bases: primordial.equations.t.inflation.Inflation_start_initial_conditions

Methods

__call__

class primordial.equations.t.mukhanov_sasaki.KD_initial_conditions(t0, N_p, phi_p)[source]¶

Bases: primordial.equations.t.inflation.KD_initial_conditions

Methods

__call__

Module contents¶

Submodules¶

primordial.equations.cosmology module¶

class primordial.equations.cosmology.Equations(H0, Omega_r, Omega_m, Omega_k, Omega_l)[source]¶

Bases: primordial.equations.equations.Equations

Cosmology equations

Solves background variables in cosmic time for curved and flat universes using the Friedmann equation.

Independent variable:: N: efolds
Variables:: t: cosmic time

Methods

`H`(t, y)	Hubble parameter
`H2`(t, y)	The square of the Hubble parameter, computed using the Friedmann equation
`__call__`(t, y)	Vector of derivatives
`add_variable`(*args)	Add dependent variables to the equations
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

H(t, y)[source]¶: Hubble parameter

H2(t, y)[source]¶: The square of the Hubble parameter, computed using the Friedmann equation

sol(sol, **kwargs)[source]¶: Post-process solution of solve_ivp

primordial.equations.equations module¶

class primordial.equations.equations.Equations[source]¶

Bases: object

Base class for equations.

Allows one to compute derivatives and derived variables. Most of the other classes take ‘equations’ as an object.

Attributes:	i : dict dictionary mapping variable names to indices in the solution vector independent_variable : string name of independent variable

Methods

`__call__`(t, y)	Vector of derivatives
`add_variable`(*args)	Add dependent variables to the equations
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Amend solution from from solve_ivp

add_variable(*args)[source]¶

Add dependent variables to the equations

creates an index for the location of variable in y
creates a class method of the same name with signature name(self, t, y) that should be used to extract the variable value in an index-independent manner.

Parameters:	*args : str Name of the dependent variables

set_independent_variable(name)[source]¶

Set name of the independent variable

Parameters:	name : str Name of the independent variable

sol(sol, **kwargs)[source]¶: Amend solution from from solve_ivp

primordial.equations.events module¶

class primordial.equations.events.Collapse(equations, direction=0, terminal=False, value=0)[source]¶

Bases: primordial.equations.events.Event

Tests if H^2 is positive

Methods

__call__

class primordial.equations.events.Event(equations, direction=0, terminal=False, value=0)[source]¶

Bases: object

Base class for events.

Gives a more usable wrapper to callable event to be passed to scipy.integrate.solve_ivp

Parameters:	equations: Equations The equations for computing derived variables. direction: int [-1, 0, +1], optional, default 0 The direction of the root finding (if any) terminal: bool, optional, default False Whether to stop at this root value: float, optional, default 0 Offset to root

Methods

__call__(t, y) Vector of derivatives

class primordial.equations.events.Inflation(equations, direction=0, terminal=False, value=0)[source]¶

Bases: primordial.equations.events.Event

Inflation entry/exit

Methods

__call__

class primordial.equations.events.ModeExit(equations, direction=0, terminal=False, value=0)[source]¶

Bases: primordial.equations.events.Event

When mode exits the horizon aH

Methods

__call__

class primordial.equations.events.UntilN(equations, direction=0, terminal=False, value=0)[source]¶

Bases: primordial.equations.events.Event

Stop at N

Methods

__call__

primordial.equations.inflation module¶

class primordial.equations.inflation.Equations(K, potential)[source]¶

Bases: primordial.equations.equations.Equations

Base class for inflation equations

Methods

`H`(t, y)	Hubble parameter
`H2`(t, y)	Hubble parameter squared
`V`(t, y)	Potential
`__call__`(t, y)	Vector of derivatives
`add_variable`(*args)	Add dependent variables to the equations
`d2Vdphi2`(t, y)	Potential second derivative
`dVdphi`(t, y)	Potential derivative
`set_independent_variable`(name)	Set name of the independent variable
`sol`(sol, **kwargs)	Post-process solution of solve_ivp

H(t, y)[source]¶: Hubble parameter

H2(t, y)[source]¶: Hubble parameter squared

V(t, y)[source]¶: Potential

d2Vdphi2(t, y)[source]¶: Potential second derivative

dVdphi(t, y)[source]¶: Potential derivative

sol(sol, **kwargs)[source]¶: Post-process solution of solve_ivp

primordial.equations.inflation_potentials module¶

class primordial.equations.inflation_potentials.ChaoticPotential(m=1)[source]¶

Bases: primordial.equations.inflation_potentials.Potential

Simple potential

Methods

__call__
d
dd

d(phi)[source]¶

dd(phi)[source]¶

class primordial.equations.inflation_potentials.Potential[source]¶: Bases: object

Module contents¶

primordial.test package¶

Submodules¶

primordial.test.test_cosmology module¶

primordial.test.test_cosmology.test_cosmology()[source]¶

primordial.test.test_inflation module¶

primordial.test.test_inflation.test_inflation()[source]¶

primordial.test.test_mukhanov_sasaki module¶

Module contents¶

Submodules¶

primordial.solver module¶

primordial.solver.solve(equations, ic, interp1d_kwargs={}, *args, **kwargs)[source]¶

Solve differential equations

This is a wrapper around scipy.integrate.solve_ivp, with easier reusable objects for the equations and initial conditions.

Parameters:	equations: primordial.equations.Equations callable to compute the equations ic: initial conditions callable to set the initial conditions interp1d_kwargs: dict kwargs to pass to the interpolation functions All other arguments are identical to ``scipy.integrate.solve_ivp``
Returns:	solution Monkey-patched version of the Bunch type usually returned by solve_ivp

primordial.units module¶

Useful cosmological units

Module contents¶

primordial: inflationary equation solver¶

primordial:	inflationary equation solver
Author:	Will Handley
Version:	0.0.14
Homepage:	https://github.com/williamjameshandley/primordial
Documentation:	http://primordial.readthedocs.io/

Description¶

primordial is a python package for solving cosmological inflationary equations.

It is very much in beta stage, and currently being built for research purposes.

Example Usage¶

Plot Background evolution¶

import numpy
import matplotlib.pyplot as plt
from primordial.solver import solve
from primordial.equations.inflation_potentials import ChaoticPotential
from primordial.equations.t.inflation import Equations, KD_initial_conditions
from primordial.equations.events import Inflation, Collapse

fig, ax = plt.subplots(3,sharex=True)
for K in [-1, 0, +1]:
     m = 1
     V = ChaoticPotential(m)
     equations = Equations(K, V)

     events= [Inflation(equations),                    # Record inflation entry and exit
              Inflation(equations, -1, terminal=True), # Stop on inflation exit
              Collapse(equations, terminal=True)]      # Stop if universe stops expanding

     N_p = -1.5
     phi_p = 23
     t_p = 1e-5
     ic = KD_initial_conditions(t_p, N_p, phi_p)
     t = numpy.logspace(-5,10,1e6)

     sol = solve(equations, ic, t_eval=t, events=events)

     ax[0].plot(sol.N(t),sol.phi(t))
     ax[0].set_ylabel(r'$\phi$')

     ax[1].plot(sol.N(t),sol.H(t))
     ax[1].set_yscale('log')
     ax[1].set_ylabel(r'$H$')

     ax[2].plot(sol.N(t),1/(sol.H(t)*numpy.exp(sol.N(t))))
     ax[2].set_yscale('log')
     ax[2].set_ylabel(r'$1/aH$')

ax[-1].set_xlabel('$N$')

Plot mode function evolution¶

import numpy
import matplotlib.pyplot as plt
from primordial.solver import solve
from primordial.equations.inflation_potentials import ChaoticPotential
from primordial.equations.t.mukhanov_sasaki import Equations, KD_initial_conditions
from primordial.equations.events import Inflation, Collapse, ModeExit

fig, axes = plt.subplots(3,sharex=True)
for ax, K in zip(axes, [-1, 0, +1]):
    ax2 = ax.twinx()
    m = 1
    V = ChaoticPotential(m)
    k = 100
    equations = Equations(K, V, k)

    events= [
            Inflation(equations),                    # Record inflation entry and exit
            Collapse(equations, terminal=True),      # Stop if universe stops expanding
            ModeExit(equations, +1, terminal=True, value=1e1*k)   # Stop on mode exit
            ]


    N_p = -1.5
    phi_p = 23
    t_p = 1e-5
    ic = KD_initial_conditions(t_p, N_p, phi_p)
    t = numpy.logspace(-5,10,1e6)

    sol = solve(equations, ic, t_eval=t, events=events)

    N = sol.N(t)
    ax.plot(N,sol.R1(t), 'k-')
    ax2.plot(N,-numpy.log(sol.H(t))-N, 'b-')

    ax.set_ylabel('$\mathcal{R}$')
    ax2.set_ylabel('$-\log aH$')

    ax.text(0.9, 0.9, r'$K=%i$' % K, transform=ax.transAxes)

axes[-1].set_xlabel('$N$')

To do list¶

Eventually would like to submit this to JOSS. Here are things to do before then:

Cosmology¶

Slow roll initial conditions
add \(\eta\) as independent variable
add \(\phi\) as independent variable

Code¶

Documentation
Tests
- 100% coverage
- interpolation