primpy.efolds package

Calculations for the primordial Universe w.r.t. e-folds N.

primpy.efolds.inflation module

Differential equations for inflation w.r.t. e-folds N.

class primpy.efolds.inflation.InflationEquationsN(K, potential, track_time=False, track_eta=False, verbose=False)[source]

Bases: InflationEquations

Background equations during inflation w.r.t. e-folds N.

Solves background variables with e-folds N of the scale factor as independent variable for curved and flat universes using the Klein-Gordon and Friedmann equations.

Independent variable:

_N: e-folds of the scale-factor (the underscore here means that this is the as of yet uncalibrated scale factor)

Dependent variables:

phi: inflaton field
dphidN: d(phi)/dN
t: time (optional)
eta: conformal time (optional)

Methods

`H2`(x, y)	Hubble parameter squared.
`__call__`(x, y)	System of coupled ODEs for underlying variables.
`get_H2`(N, dphi, V, K)	Get the Hubble parameter squared from the background equations.
`get_d2H`(N, H, dH, dphi, d2phi, K)	Get the 2nd time derivative of the Hubble parameter from the background equations.
`get_d2phi`(H2, dH_H, dphi, dV)	Get the 2nd time derivative of the inflaton field from the background equations.
`get_d3H`(N, H, dH, d2H, dphi, d2phi, d3phi, K)	Get the 3rd time derivative of the Hubble parameter from the background equations.
`get_d3phi`(H, dH, d2H, dphi, d2phi, dV, d2V)	Get the 3rd time derivative of the inflaton field from the background equations.
`get_d4phi`(H, dH, d2H, d3H, dphi, d2phi, ...)	Get the 4th time derivative of the inflaton field from the background equations.
`get_dH`(N, H, dphi, K)	Get the 1st time derivative of the Hubble parameter from the background equations.
`get_dH_H`(N, H2, dphi, K)	Get the 1st time derivative of the Hubble parameter normalised by the Hubble parameter..
`get_delta_1`(H, dH, dphi, d2phi)	Get the 2nd slow-roll parameter for n=1.
`get_delta_2`(H, dH, d2H, dphi, d2phi, d3phi)	Get the 2nd slow-roll parameter for n=2.
`get_delta_3`(H, dH, d2H, d3H, dphi, d2phi, ...)	Get the 2nd slow-roll parameter for n=3.
`get_epsilon_1H`(H, dH)	Get the 1st Hubble flow parameter.
`get_epsilon_2H`(H, dH, d2H[, kind])	Get the 2nd Hubble flow parameter.
`get_epsilon_3H`(H, dH, d2H, d3H[, kind])	Get the 3rd Hubble flow parameter.
`inflating`(x, y)	Inflation diagnostic for event tracking.
`sol`(sol, **kwargs)	Post-processing of `scipy.integrate.solve_ivp()` solution.
`w`(x, y)	Equation of state parameter.

__call__(x, y)[source]: System of coupled ODEs for underlying variables.

static get_H2(N, dphi, V, K)[source]

Get the Hubble parameter squared from the background equations.

Parameters:

Nfloat or array_like: Number of e-folds N=ln(a).
dphifloat or array_like: 1st derivative of inflaton field.
Vfloat or array_like: Inflation potential at phi.
Kint: Curvature parameter.

Returns:

H2float or array_like: Hubble parameter squared.

static get_dH(N, H, dphi, K)[source]

Get the 1st time derivative of the Hubble parameter from the background equations.

Parameters:

Nfloat or array_like: Number of e-folds N=ln(a).
Hfloat or array_like: Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
Kint: Curvature parameter.

Returns:

dHfloat or array_like: 1st derivative of Hubble parameter.

static get_dH_H(N, H2, dphi, K)[source]

Get the 1st time derivative of the Hubble parameter normalised by the Hubble parameter..

Parameters:

Nfloat or array_like: Number of e-folds N=ln(a).
H2float or array_like: Hubble parameter squared.
dphifloat or array_like: 1st derivative of inflaton field.
Kint: Curvature parameter.

Returns:

dH_Hfloat or array_like: 1st derivative of Hubble parameter normalised by Hubble parameter: dH/H.

static get_d2H(N, H, dH, dphi, d2phi, K)[source]

Get the 2nd time derivative of the Hubble parameter from the background equations.

Parameters:

Nfloat or array_like: Number of e-folds N=ln(a).
Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.
Kint: Curvature parameter.

Returns:

d2Hfloat or array_like: 2nd derivative of Hubble parameter.

static get_d3H(N, H, dH, d2H, dphi, d2phi, d3phi, K)[source]

Get the 3rd time derivative of the Hubble parameter from the background equations.

Parameters:

Nfloat or array_like: Number of e-folds N=ln(a).
Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.
d3phifloat or array_like: 3rd derivative of inflaton field.
Kint: Curvature parameter.

Returns:

d3Hfloat or array_like: 3rd derivative of Hubble parameter.

static get_d2phi(H2, dH_H, dphi, dV)[source]

Get the 2nd time derivative of the inflaton field from the background equations.

Parameters:

H2float or array_like: Hubble parameter squared.
dH_Hfloat or array_like: 1st derivative of Hubble parameter normalised by Hubble parameter dH/H.
dphifloat or array_like: 1st derivative of inflaton field.
dVfloat or array_like: 1st derivative of inflation potential at phi.

Returns:

d2phifloat or array_like: 2nd derivative of inflaton field.

static get_d3phi(H, dH, d2H, dphi, d2phi, dV, d2V)[source]

Get the 3rd time derivative of the inflaton field from the background equations.

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.
dVfloat or array_like: 1st derivative of inflation potential at phi.
d2Vfloat or array_like: 2nd derivative of inflation potential at phi.

Returns:

d3phifloat or array_like: 3rd derivative of inflaton field.

static get_d4phi(H, dH, d2H, d3H, dphi, d2phi, d3phi, dV, d2V, d3V)[source]

Get the 4th time derivative of the inflaton field from the background equations.

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
d3Hfloat or array_like: 3rd derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.
d3phifloat or array_like: 3rd derivative of inflaton field.
dVfloat or array_like: 1st derivative of inflation potential at phi.
d2Vfloat or array_like: 2nd derivative of inflation potential at phi.
d3Vfloat or array_like: 3rd derivative of inflation potential at phi.

Returns:

d4phifloat or array_like: 4th derivative of inflaton field.

static get_epsilon_1H(H, dH)[source]

Get the 1st Hubble flow parameter.

This definition of the 1st Hubble flow parameter was suggested e.g. by Stewart & Lyth (1993) in eq. (23). https://arxiv.org/abs/gr-qc/9302019

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.

Returns:

epsilon_1Hfloat or array_like: 1st Hubble flow parameter.

static get_epsilon_2H(H, dH, d2H, kind=None)[source]

Get the 2nd Hubble flow parameter.

This definition of the 2nd Hubble flow parameter was suggested e.g. by Leach, Liddle, Martin & Schwarz (2003) in eq. (15). https://arxiv.org/abs/astro-ph/0202094

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
kindstr or None, default=None: Switch to alternative formulation given by Gong (2004).

Returns:

epsilon_2Hfloat or array_like: 2nd Hubble flow parameter.

static get_epsilon_3H(H, dH, d2H, d3H, kind=None)[source]

Get the 3rd Hubble flow parameter.

This definition of the 3rd Hubble flow parameter was suggested e.g. by Leach, Liddle, Martin & Schwarz (2003) in eq. (15). https://arxiv.org/abs/astro-ph/0202094

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
d3Hfloat or array_like: 3rd derivative of Hubble parameter.
kindstr or None, default=None: Switch to alternative formulation given by Gong (2004).

Returns:

epsilon_3Hfloat or array_like: 3rd Hubble flow parameter.

static get_delta_1(H, dH, dphi, d2phi)[source]

Get the 2nd slow-roll parameter for n=1.

This definition of the 2nd slow roll parameter was suggested e.g. by Stewart & Lyth (1993) in eq. (23). https://arxiv.org/abs/gr-qc/9302019

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.

Returns:

delta_1float or array_like

static get_delta_2(H, dH, d2H, dphi, d2phi, d3phi)[source]

Get the 2nd slow-roll parameter for n=2.

This definition of higher orders of the 2nd slow roll parameter was suggested e.g. by Stewart & Gong (2001) in eq. (22). https://arxiv.org/abs/astro-ph/0101225

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.
d3phifloat or array_like: 3rd derivative of inflaton field.

Returns:

delta_2float or array_like

static get_delta_3(H, dH, d2H, d3H, dphi, d2phi, d3phi, d4phi)[source]

Get the 2nd slow-roll parameter for n=3.

This definition of higher orders of the 2nd slow roll parameter was suggested e.g. by Stewart & Gong (2001) in eq. (22). https://arxiv.org/abs/astro-ph/0101225

Parameters:

Hfloat or array_like: Hubble parameter.
dHfloat or array_like: 1st derivative of Hubble parameter.
d2Hfloat or array_like: 2nd derivative of Hubble parameter.
d3Hfloat or array_like: 3rd derivative of Hubble parameter.
dphifloat or array_like: 1st derivative of inflaton field.
d2phifloat or array_like: 2nd derivative of inflaton field.
d3phifloat or array_like: 3rd derivative of inflaton field.
d4phifloat or array_like: 4th derivative of inflaton field.

Returns:

delta_3float or array_like

H2(x, y)[source]: Hubble parameter squared.

w(x, y)[source]: Equation of state parameter.

inflating(x, y)[source]: Inflation diagnostic for event tracking.

sol(sol, **kwargs)[source]: Post-processing of scipy.integrate.solve_ivp() solution.

primpy.efolds.perturbations module

Curvature perturbations with respect to e-folds _N.

class primpy.efolds.perturbations.PerturbationN(background, k, **kwargs)[source]

Bases: Perturbation

Curvature perturbation for wavenumber k with respect to e-folds _N=ln(a).

Solves the Mukhanov–Sasaki equations w.r.t. number of e-folds _N of the scale factor a for curved universes.

Parameters:

backgroundBunch object same as returned by scipy.integrate.solve_ivp(): Background solution as returned by primpy.efolds.inflation.InflationEquationsN.sol().
kfloat: wavenumber

class primpy.efolds.perturbations.ScalarModeN(background, k, **kwargs)[source]

Bases: ScalarMode

Template for scalar modes.

Methods

`__call__`(x, y)	Vector of derivatives.
`get_vacuum_ic_HD`()	Get initial conditions for scalar modes for HD vacuum w.r.t.
`get_vacuum_ic_RST`()	Get initial conditions for scalar modes for RST vacuum w.r.t.
`get_vacuum_ic_k`()	Get initial conditions for scalar modes for HD approximation w.r.t.
`mukhanov_sasaki_frequency_damping`()	Frequency and damping term of the Mukhanov-Sasaki equations for scalar modes.

__call__(x, y)[source]: Vector of derivatives.

mukhanov_sasaki_frequency_damping()[source]

Frequency and damping term of the Mukhanov-Sasaki equations for scalar modes.

Frequency and damping term of the Mukhanov-Sasaki equations for the comoving curvature perturbations R w.r.t. e-folds _N, where the e.o.m. is written as ddR + 2 * damping * dR + frequency**2 * R = 0.

get_vacuum_ic_k()[source]: Get initial conditions for scalar modes for HD approximation w.r.t. e-folds _N.

get_vacuum_ic_HD()[source]: Get initial conditions for scalar modes for HD vacuum w.r.t. e-folds _N.

get_vacuum_ic_RST()[source]: Get initial conditions for scalar modes for RST vacuum w.r.t. e-folds _N.

class primpy.efolds.perturbations.TensorModeN(background, k, **kwargs)[source]

Bases: TensorMode

Template for tensor modes.

Methods

`__call__`(x, y)	Vector of derivatives.
`get_vacuum_ic_HD`()	Get initial conditions for tensor modes for HD vacuum w.r.t.
`get_vacuum_ic_RST`()	Get initial conditions for tensor modes for RST vacuum w.r.t.
`get_vacuum_ic_k`()	Get initial conditions for tensor modes for HD approximation w.r.t.
`mukhanov_sasaki_frequency_damping`()	Frequency and damping term of the Mukhanov-Sasaki equations for tensor modes.

__call__(x, y)[source]: Vector of derivatives.

mukhanov_sasaki_frequency_damping()[source]

Frequency and damping term of the Mukhanov-Sasaki equations for tensor modes.

Frequency and damping term of the Mukhanov-Sasaki equations for the tensor perturbations h w.r.t. e-folds _N, where the e.o.m. is written as ddh + 2 * damping * dh + frequency**2 * h = 0.

get_vacuum_ic_k()[source]: Get initial conditions for tensor modes for HD approximation w.r.t. e-folds _N.

get_vacuum_ic_HD()[source]: Get initial conditions for tensor modes for HD vacuum w.r.t. e-folds _N.

get_vacuum_ic_RST()[source]: Get initial conditions for tensor modes for RST vacuum w.r.t. e-folds _N.