How to Plot the Light Path in a Space-Time Diagram Using Python?

[AdrianD]

TL;DR Summary

Space-time diagram with python

I am trying to replicate the space time plot (the 2nd plot with Proper distance vs Time) as in this thread: space-time
I wrote everything in python using the astropy cosmology package.
Everything went smooth, but I am stuck at plotting the light path on the 'purple path', as per the above example. I don't know how to plot the light path for the light emitted by the galaxy at t0 = 0 (now), with the proper distance of aprox 13.8 Glyr, that will reach us a t=49 Gyr.
Here is the code in python:

from astropy.cosmology import Planck18, z_at_value
from astropy import units as u
import numpy as np
from matplotlib import pyplot as plt
from scipy import interpolate as inter
import astropy.constants as cc# Hubble radius

def Hradius(zed):
    c = cc.c
    return (c/cosmo.H(zed)).to(u.Glyr)

#define cosmology

cosmo = Planck18
# Z and time

redshift = np.arange(-0.9,1089,0.1)

redshift_R = redshift[::-1]

time=cosmo.age(redshift)
# trajectory of a galaxy G1 with z=1.48

TrajectoryG1 = [cosmo.scale_factor(i) * cosmo.comoving_distance(1.48737).to(u.Glyr).value for i in redshift]
### Can't make it work

#lightPathG1 = [cosmo.scale_factor(0) * cosmo.comoving_distance(i).to(u.Glyr).value for i in redshiftG1]
# horizons

hubbleRadius = [Hradius(i).value for i in redshift]

hubbleRadiusMirrored = [-Hradius(i).value for i in redshift]

lightPath = [cosmo.scale_factor(i) * cosmo.comoving_distance(i).to(u.Glyr).value for i in redshift]

lightPathMirrored = [cosmo.scale_factor(i) * -cosmo.comoving_distance(i).to(u.Glyr).value for i in redshift]term1 = [cosmo.scale_factor(i) for i in redshift]

term2 = [cosmo.comoving_distance(i).to(u.Glyr).value for i in redshift_R]

term3 = [-cosmo.comoving_distance(i).to(u.Glyr).value for i in redshift_R]

particleHorizon = [term1*term2 for i in range(len(redshift))]

particleHorizonMirrored = [term1*term3 for i in range(len(redshift))]
#plot

fig, ax1 = plt.subplots()

ax1.plot(time,hubbleRadius, 'g-', label = 'Hubble radius')

ax1.plot(time,hubbleRadiusMirrored, 'g-')

ax1.plot(time,lightPath, 'y-', label = 'Light path')

ax1.plot(time,lightPathMirrored, 'y-')

ax1.plot(time,particleHorizon, 'b-', label = 'Particle horizon')

ax1.plot(time,particleHorizonMirrored, 'b-')
ax1.plot(time,TrajectoryG1, 'k',linestyle='-.', label = 'G1 trajectory')

#ax1.plot(timeG1,lightPathG1, color = 'orange')
ax1.set_ylim(ymin=-50, ymax=50)

ax1.axvline(x=13.7,color='grey',linestyle='--',alpha=0.5)

ax1.axhline(y=0,color='grey',linestyle='--',alpha=0.5)
ax1.fill_between(np.append(time.value, time[::-1].value), np.append(hubbleRadius, hubbleRadiusMirrored[::-1]), color = 'green', alpha = 0.1)
ax1.invert_xaxis()

ax1.legend()

ax1.set_xlabel('Time (Gyr)')

ax1.set_ylabel('Proper distance (Glyr)')

And my plot :

 

[AndyW]

I see that you have successfully plotted the Hubble radius, particle horizon, and galaxy trajectory. To plot the light path, you can use the same method you used for the galaxy trajectory, but with a different redshift value. Since the light from the galaxy is reaching us at t=49 Gyr, you can use the redshift value at that time to calculate the proper distance.

Here is an example code for plotting the light path at t=49 Gyr:

# redshift at t=49 Gyr
redshift_49Gyr = z_at_value(cosmo.age, 49 * u.Gyr)

# calculate proper distance at t=49 Gyr
proper_distance_49Gyr = cosmo.comoving_distance(redshift_49Gyr).to(u.Glyr).value

# plot light path at t=49 Gyr
ax1.plot(49, proper_distance_49Gyr, 'o', color='orange', label='Light path at t=49 Gyr')

# add legend to the plot
ax1.legend()

# show the plot
plt.show()

This should plot the light path at t=49 Gyr as a point on your plot. You can also adjust the code to plot the light path at different time intervals if needed. I hope this helps and good luck with your replication!