Dark Energy WSpline Background Evolution

Author

NumCosmo developers

Published

May 14, 2026

Abstract

This example compares the dark-energy spline model NcHICosmoDEWSpline against the constant-\(w\) model NcHICosmoDEXcdm. It shows how to configure parameters using the dictionary-style API, assign knot values for the spline equation of state, and inspect the impact on background quantities and distances.

Introduction

This example compares two cosmologies with the same background parameters:

  1. NcHICosmoDEXcdm with constant \(w = -1\).
  2. NcHICosmoDEWSpline with a knot-based \(w(z)\) history.

The notebook focuses on:

  • dictionary-style parameter assignment (cosmo["param"] = value),
  • spline-knot configuration using w_i parameters,
  • a side-by-side comparison of \(w(z)\), background functions, and distances.

Import and Initialization

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from numcosmo_py import Nc, Ncm

Ncm.cfg_init()

Build the Cosmological Models

We instantiate a 12-knot WSpline model and a reference XCDM model.

n_knots = 12
z_knots_max = 3.0

cosmo_ws = Nc.HICosmoDEWSpline.new(n_knots, z_knots_max)
cosmo_xcdm = Nc.HICosmoDEXcdm.new()

Set Parameters with the Dictionary API

The scalar cosmological parameters are assigned through dictionary-like access. This is the recommended high-level interface in Python examples.

base_params = {
    "H0": 70.0,
    "Omegac": 0.25,
    "Omegax": 0.70,
    "Omegab": 0.05,
    "Tgamma0": 2.72,
}

for key, value in base_params.items():
    cosmo_ws[key] = value
    cosmo_xcdm[key] = value

# Reference model: constant dark-energy equation of state.
cosmo_xcdm["w"] = -1.0

For the spline model, we assign one value per knot using the same dictionary-style API. The parameter names follow the w_i convention.

alpha_knots = np.array(cosmo_ws.get_alpha().dup_array())
z_knots = np.expm1(alpha_knots)

# Smooth toy profile: close to -1 at low/high z with a moderate excursion near z ~ 1.
w_knots = (
    -1.0
    + 0.18 * np.exp(-((z_knots - 0.9) / 0.55) ** 2)
    - 0.12 * np.exp(-((z_knots - 2.0) / 0.70) ** 2)
)

for i, w_i in enumerate(w_knots):
    cosmo_ws[f"w_{i}"] = float(w_i)

Prepare Background and Distance Calculators

dist = Nc.Distance.new(3.0)
dist.prepare(cosmo_ws)

Sample the Background Functions

z = np.linspace(0.0, 3.0, 240)

def evaluate_background(cosmo: Nc.HICosmo) -> pd.DataFrame:
    dist.prepare(cosmo)
    return pd.DataFrame(
        {
            "z": z,
            "w": [cosmo.w_de(zi) for zi in z],
            "E2": [cosmo.E2(zi) for zi in z],
            "E2Omega_de": [cosmo.E2Omega_de(zi) for zi in z],
            "dE2Omega_de_dz": [cosmo.dE2Omega_de_dz(zi) for zi in z],
            "d2E2Omega_de_dz2": [cosmo.d2E2Omega_de_dz2(zi) for zi in z],
            "Dc": [dist.comoving(cosmo, zi) for zi in z],
        }
    )

ws_df = evaluate_background(cosmo_ws)
xcdm_df = evaluate_background(cosmo_xcdm)

comparison = pd.DataFrame(
    {
        "z": z,
        "w_ws": ws_df["w"],
        "w_xcdm": xcdm_df["w"],
        "E2_rel_diff": ws_df["E2"] / xcdm_df["E2"] - 1.0,
        "E2Omega_de_rel_diff": ws_df["E2Omega_de"] / xcdm_df["E2Omega_de"] - 1.0,
        "Dc_rel_diff": ws_df["Dc"] / xcdm_df["Dc"] - 1.0,
    }
)

comparison.head(8)
z w_ws w_xcdm E2_rel_diff E2Omega_de_rel_diff Dc_rel_diff
0 0.000000 -0.987664 -1.0 0.000000 0.000000 NaN
1 0.012552 -0.986714 -1.0 0.000332 0.000479 -0.000082
2 0.025105 -0.985708 -1.0 0.000677 0.000989 -0.000167
3 0.037657 -0.984643 -1.0 0.001035 0.001531 -0.000253
4 0.050209 -0.983515 -1.0 0.001407 0.002106 -0.000341
5 0.062762 -0.982323 -1.0 0.001793 0.002716 -0.000432
6 0.075314 -0.981063 -1.0 0.002194 0.003363 -0.000524
7 0.087866 -0.979733 -1.0 0.002608 0.004047 -0.000619

Equation of State and Knot Placement

Code 
fig, ax = plt.subplots(figsize=(9, 5))

ax.plot(z, ws_df["w"], lw=2.4, label="WSpline")
ax.plot(z, xcdm_df["w"], lw=1.8, ls="--", label="XCDM ($w=-1$)")
ax.scatter(z_knots, w_knots, s=30, zorder=3, label="WSpline knots")

ax.set_xlabel(r"$z$")
ax.set_ylabel(r"$w(z)$")
ax.set_title("Spline vs. Constant Dark-Energy Equation of State")
ax.grid(alpha=0.25)
ax.legend(loc="best")
plt.show()
Figure 1: Dark-energy equation of state. The markers indicate WSpline knot values.

Background Functions

Code 
fig1, axs1 = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
fig1.subplots_adjust(hspace=0)

axs1[0].plot(z, ws_df["E2"], lw=2.0, label="WSpline")
axs1[0].plot(z, xcdm_df["E2"], lw=1.8, ls="--", label="XCDM")
axs1[0].set_ylabel(r"$E^2(z)$")
axs1[0].grid(alpha=0.25)
axs1[0].legend(loc="best")

axs1[1].plot(z, comparison["E2_rel_diff"], color="tab:red", lw=2.0)
axs1[1].axhline(0.0, color="black", lw=0.9, alpha=0.6)
axs1[1].set_xlabel(r"$z$")
axs1[1].set_ylabel(r"$E^2_{WS}/E^2_{XCDM} - 1$")
axs1[1].grid(alpha=0.25)

fig2, axs2 = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
fig2.subplots_adjust(hspace=0)

axs2[0].plot(z, ws_df["E2Omega_de"], lw=2.0, label="WSpline")
axs2[0].plot(z, xcdm_df["E2Omega_de"], lw=1.8, ls="--", label="XCDM")
axs2[0].set_ylabel(r"$E^2(z)\,\Omega_{de}(z)$")
axs2[0].grid(alpha=0.25)
axs2[0].legend(loc="best")

axs2[1].plot(z, comparison["E2Omega_de_rel_diff"], color="tab:green", lw=2.0)
axs2[1].axhline(0.0, color="black", lw=0.9, alpha=0.6)
axs2[1].set_xlabel(r"$z$")
axs2[1].set_ylabel(r"$(E^2\Omega_{de})_{WS}/(E^2\Omega_{de})_{XCDM} - 1$")
axs2[1].grid(alpha=0.25)

plt.show()
Figure 2: Normalized expansion rate \(E^2(z)\) and relative difference to XCDM
Figure 3: \(E^2(z)\,\Omega_{de}(z)\) and relative difference to XCDM

Dark-Energy Derivatives and Distance Effect

Code 
fig1, axs1 = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
fig1.subplots_adjust(hspace=0)

axs1[0].plot(z, ws_df["dE2Omega_de_dz"], lw=2.0, label="WSpline")
axs1[0].plot(z, xcdm_df["dE2Omega_de_dz"], lw=1.8, ls="--", label="XCDM")
axs1[0].set_ylabel(r"$d(E^2\Omega_{de})/dz$")
axs1[0].grid(alpha=0.25)
axs1[0].legend(loc="best")

axs1[1].plot(z, ws_df["d2E2Omega_de_dz2"], lw=2.0, label="WSpline")
axs1[1].plot(z, xcdm_df["d2E2Omega_de_dz2"], lw=1.8, ls="--", label="XCDM")
axs1[1].set_xlabel(r"$z$")
axs1[1].set_ylabel(r"$d^2(E^2\Omega_{de})/dz^2$")
axs1[1].grid(alpha=0.25)

fig2, ax2 = plt.subplots(figsize=(8, 6))
ax2.plot(z, comparison["Dc_rel_diff"], color="tab:purple", lw=2.0)
ax2.axhline(0.0, color="black", lw=0.9, alpha=0.6)
ax2.set_xlabel(r"$z$")
ax2.set_ylabel(r"$D_{c,WS}/D_{c,XCDM} - 1$")
ax2.set_title("Comoving-Distance Shift Induced by WSpline")
ax2.grid(alpha=0.25)

plt.show()
Figure 4: First and second derivatives of \(E^2\Omega_{de}\)
Figure 5: Relative change in comoving distance

Conclusion

This example highlights WSpline background behavior with an emphasis on clear diagnostics. The parameter assignment uses:

  • scalar parameters are set with dictionary access, and
  • spline knot values are assigned with named entries (w_0, w_1, …, w_{n-1}).

These comparisons provide a practical baseline for testing alternative \(w(z)\) knot configurations and quantifying their impact on background observables.