integrate_backwards=True in EMRIInspiral returns inverted physical phases

[perturber]

From FEWv2.0.0 tag:

When generating a backward trajectory using the integrate_backwards=True flag, the initial phase signs are flipped to use the same integrator module (see few.trajectory.inspiral.EMRIInspiral.get_inspiral Line 242).

However, when returning the trajectory in Line 269, the phases are not flipped back, which results in $-1 \Phi_{\alpha}(t)$ as the output. Here, $\Phi_{\alpha}(t)$ are the time evolutions of the true initial physical phases with $\alpha = \phi, \theta, r$.

Reproduce the bug:

import numpy as np
from few.trajectory.inspiral import EMRIInspiral
from few.trajectory.ode import KerrEccEqFlux
from few.utils.utility import get_p_at_t

kerr_traj = EMRIInspiral(func=KerrEccEqFlux)
T = 1.0
m1, m2, a, e0, xI0 = 1e6, 1e1, 0.9, 0.2, 1.0
Phi_phi0, Phi_theta0, Phi_r0 = 1.0, 0.0, 1.0
p0 = 12.0

# 1. Forward integration
traj = kerr_traj(m1, m2, a, p0, e0, xI0, 
                 Phi_phi0=Phi_phi0, Phi_theta0=Phi_theta0, Phi_r0=Phi_r0, 
                 T=T, err_tol=1e-11)

# 2. Backward integration from the final state
traj_back = kerr_traj(m1, m2, a, traj[1][-1], traj[2][-1], traj[3][-1], 
                      Phi_phi0=-traj[4][-1], # <= user must supply the negative of the final phases
                      Phi_theta0=-traj[5][-1], 
                      Phi_r0=-traj[6][-1], 
                      T=T, upsample=False, integrate_backwards=True, err_tol=1e-11)

# The raw returned phase is the negative of the true physical phase
print(f"Input Initial Phase: {Phi_phi0}")
print(f"Raw Returned t=0 Phase: {traj_back[4][0]}")
print(f"Negated Returned t=0 Phase: {-traj_back[4][0]}")

Output:

Input Initial Phase: 1.0
Raw Returned t=0 Phase: -0.9999967547482811
Negated Returned t=0 Phase: 0.9999967547482811

Proposed fix:
Add a condition to negate the phase columns of the out array right before return:

if temp_kwargs["integrate_backwards"]:
    out[:, 4] *= -1
    out[:, 5] *= -1
    out[:, 6] *= -1

t, p, e, x, Phi_phi, Phi_theta, Phi_r = out.T.copy()
return t, p, e, x, Phi_phi, Phi_theta, Phi_r

[cchapmanbird]

There is definitely some ambiguity about this. Phases are returned negative as this is "correct" (as the inspiral is going backwards in time). However we assume the final phases provided have not had this convention applied, leading to this issue.

The correct solution is probably just to not invert the initial phases (as really, the user should be responsible for the initial conditions). The phases should be returned negative though I would say.