"""Differential equations for inflation w.r.t. time `t`."""
import numpy as np
from primpy.inflation import InflationEquations
[docs]
class InflationEquationsT(InflationEquations):
"""Background equations during inflation w.r.t. time `t`.
Solves background variables in cosmic time for curved and flat universes
using the Klein-Gordon and Friedmann equations.
Independent variable:
``t``: cosmic time
Dependent variables:
* ``_N``: number of e-folds
* ``phi``: inflaton field
* ``dphidt``: `d(phi)/dt`
* ``eta``: conformal time (optional)
"""
def __init__(self, K, potential, track_eta=False, verbose=False):
super(InflationEquationsT, self).__init__(K=K, potential=potential, verbose=verbose)
self._set_independent_variable('t')
self.add_variable('phi', 'dphidt', '_N')
self.track_eta = track_eta
if track_eta:
self.add_variable('eta')
[docs]
def __call__(self, x, y):
"""System of coupled ODEs for underlying variables."""
N = self._N(x, y)
H2 = self.H2(x, y)
H = np.sqrt(H2)
dphidt = self.dphidt(x, y)
dVdphi = self.dVdphi(x, y)
dy = np.zeros_like(y)
dy[self.idx['phi']] = dphidt
dy[self.idx['dphidt']] = self.get_d2phi(H2=H2, dH_H=None, dphi=dphidt, dV=dVdphi)
dy[self.idx['_N']] = H
if self.track_eta:
dy[self.idx['eta']] = np.exp(-N)
return dy
[docs]
@staticmethod
def get_H2(N, dphi, V, K): # noqa: D102
return (dphi**2 / 2 + V) / 3 - K * np.exp(-2 * N)
[docs]
@staticmethod
def get_dH(N, H, dphi, K): # noqa: D102
# here: dH/dt
return -dphi**2 / 2 + K * np.exp(-2 * N)
[docs]
@staticmethod
def get_dH_H(N, H2, dphi, K): # noqa: D102
# here: dH/dt
H = np.sqrt(H2)
return -dphi**2 / (2 * H) + K * np.exp(-2 * N) / H
[docs]
@staticmethod
def get_d2H(N, H, dH, dphi, d2phi, K): # noqa: D102
# here: d2H/dt2
return -d2phi * dphi - 2 * K * H * np.exp(-2 * N)
[docs]
@staticmethod
def get_d3H(N, H, dH, d2H, dphi, d2phi, d3phi, K): # noqa: D102
# here: d3H/dt3
return - d3phi * dphi - d2phi**2 + 2 * K * (2 * H**2 - dH) * np.exp(-2 * N)
[docs]
@staticmethod
def get_d2phi(H2, dH_H, dphi, dV): # noqa: D102
# here: d2phi/dt2
H = np.sqrt(H2)
return -3 * H * dphi - dV
[docs]
@staticmethod
def get_d3phi(H, dH, d2H, dphi, d2phi, dV, d2V): # noqa: D102
return -3 * H * d2phi - d2V * dphi + 3 * dphi**3 / 2
[docs]
@staticmethod
def get_d4phi(H, dH, d2H, d3H, dphi, d2phi, d3phi, dV, d2V, d3V): # noqa: D102
return -3 * H * d3phi - d2V * d2phi - d3V * dphi**2 - 3 * d2H * dphi - 6 * d2phi * dH
[docs]
def H2(self, x, y): # noqa: D102
return self.get_H2(N=self._N(x, y), dphi=self.dphidt(x, y), V=self.V(x, y), K=self.K)
[docs]
def w(self, x, y): # noqa: D102
V = self.V(x, y)
dphidt_2 = self.dphidt(x, y)**2
p = dphidt_2 / 2 - V
rho = dphidt_2 / 2 + V
return p / rho
[docs]
def inflating(self, x, y): # noqa: D102
return self.V(x, y) - self.dphidt(x, y)**2
[docs]
def sol(self, sol, **kwargs): # noqa: D102
sol = super(InflationEquationsT, self).sol(sol, **kwargs)
return sol