Observations | Distance Measurements

The Cosmic Distance Ladder

From stellar parallax to Type Ia supernovae: the interlocking chain of methods that maps the scale of the universe and measures $H_0$

1. Introduction: Measuring Distances Across the Universe

The Fundamental Challenge

Measuring distances to astronomical objects is one of the oldest and most consequential problems in all of science. Unlike a laboratory, we cannot stretch a ruler to the nearest star, let alone to a galaxy a billion light-years away. Instead, astronomers have built a cosmic distance ladder — a series of interlocking methods, each calibrated by the rung below it, that extends our reach from the solar neighborhood to the edge of the observable universe.

Every major discovery in observational cosmology — the size of the Milky Way, the existence of external galaxies, the expansion of the universe, and its present acceleration — depended on getting these distances right.

The ladder works because each technique has a limited range. Geometric methods (parallax) are model-independent but restricted to the nearest few kiloparsecs. Stellar standard candles (Cepheids, TRGB) bridge to tens of megaparsecs. Supernovae reach gigaparsec scales. At each overlap, the more distant method is calibrated against the closer one. An error at the base propagates upward — which is precisely why the current $H_0$ tension is so important.

The Distance Modulus

The fundamental relation connecting apparent magnitude $m$, absolute magnitude $M$, and distance $d$ is:

$$\mu = m - M = 5\log_{10}\!\left(\frac{d}{10\;\text{pc}}\right)$$

Equivalently: $d = 10^{(\mu/5)+1}$ pc. The distance modulus $\mu$ encodes how much fainter an object appears relative to its intrinsic luminosity.

In this chapter, we derive the physics behind each major rung, starting with the most fundamental geometric method (parallax) and ascending through Cepheid variables, the tip of the red giant branch (TRGB), and Type Ia supernovae, before confronting the Hubble tension that has dominated cosmology since 2019.

2. Parallax and Geometric Methods

Trigonometric parallax is the only direct, model-independent distance measurement in astronomy. As the Earth orbits the Sun, a nearby star appears to shift its position against the background of distant stars. The half-angle of this apparent shift is the parallax angle $p$.

Derivation: The Parsec from the Arcsecond

Consider a star at distance $d$ from the Sun. The Earth's orbital radius is $a = 1\;\text{AU}$. From simple trigonometry:

$$\tan(p) = \frac{a}{d}$$

For small angles (virtually all astronomical distances), $\tan(p) \approx p$ in radians. Converting to arcseconds ($p'' = p \times 206\,265$):

$$d = \frac{a}{p} = \frac{1\;\text{AU}}{p''/206\,265}$$

We define 1 parsec (pc) as the distance at which a star has a parallax of exactly 1 arcsecond:

$$\boxed{d\;(\text{pc}) = \frac{1}{p''}}$$

Thus 1 pc = 206,265 AU = $3.086 \times 10^{16}$ m = 3.26 light-years. The parsec is defined purely by geometry — no physics assumptions required.

Gaia: Extending the Parallax Frontier

The ESA Gaia mission (launched 2013) has revolutionized astrometry. Gaia DR3 provides parallaxes for ~1.8 billion stars with precisions of:

$\sigma_p \approx 20\;\mu\text{as}$ for $G < 15$ mag (bright stars)
$\sigma_p \approx 500\;\mu\text{as}$ for $G \approx 20$ mag (faint limit)

For a 10% distance accuracy ($\sigma_d/d \approx \sigma_p/p = 0.1$), Gaia reaches $d \approx 1/(10 \times 20 \times 10^{-6}) \approx 5\;\text{kpc}$for bright stars — well into the Galactic disk and halo. With Bayesian inference techniques exploiting prior information on stellar populations, useful distances extend to ~10 kpc.

2.1 The Moving Cluster Method

Star clusters whose members share a common space velocity exhibit a convergent point on the sky (the point toward which proper motions converge). If the cluster's radial velocity $v_r$ is measured spectroscopically and the angular distance$\theta$ to the convergent point is known, then the transverse velocity is:

$$v_t = v_r \tan(\theta)$$

Combined with the proper motion $\mu$ (arcsec/yr), the distance follows from $d = v_t / (4.74\,\mu)$ in parsecs. This method was historically applied to the Hyades cluster ($d \approx 46$ pc), anchoring the lower distance ladder.

2.2 Eclipsing Binary Method

Detached eclipsing binaries provide a nearly geometric distance measurement. From the light curve, one measures the orbital period $P$ and the duration of eclipses. Combined with radial velocity amplitudes $K_1, K_2$, one derives the orbital semi-major axis $a$ via Kepler's third law. The stellar radii$R_1, R_2$ follow from eclipse geometry, giving the surface flux and hence the angular diameter. Comparing angular and physical sizes yields the distance:

$$d = \frac{2R}{\theta_{\text{ang}}}$$

This method has been applied to eclipsing binaries in the LMC ($d \approx 49.6$ kpc) with ~2% precision, providing an independent geometric anchor for the distance scale.

3. Cepheid Period-Luminosity Relation (Leavitt Law)

Classical Cepheids are pulsating yellow supergiants ($F \approx 5000\text{--}7000\;\text{K}$) with periods from ~1 to ~100 days. Their defining property, discovered by Henrietta Swan Leavitt in 1908, is a tight correlation between pulsation period and intrinsic luminosity: brighter Cepheids pulsate more slowly. This makes them superb standard candles.

Derivation: The Pulsation Equation

The pulsation of a Cepheid can be understood as a radial oscillation of the stellar envelope. For a homogeneous sphere of mass $M$ and radius $R$, the fundamental mode period is determined by the dynamical timescale:

$$P \sim \frac{1}{\sqrt{G\bar{\rho}}}$$

More precisely, the pulsation constant $Q$ is defined by:

$$\boxed{P\sqrt{\bar{\rho}} = Q}$$

where $\bar{\rho} = 3M/(4\pi R^3)$ is the mean density and$Q \approx 0.033$ days for Cepheids. Substituting the mean density:

$$P = Q\left(\frac{4\pi R^3}{3M}\right)^{1/2} = Q\left(\frac{4\pi}{3G\bar{\rho}}\right)^{1/2}$$

Since more luminous (larger, less dense) stars have lower $\bar{\rho}$, they have longer periods. From the mass-luminosity relation $L \propto M^\alpha$and the Stefan-Boltzmann law $L = 4\pi R^2 \sigma T_{\text{eff}}^4$, one can show that $P$ scales as a power of $L$, yielding a period-luminosity relation of the form:

$$M_\lambda = a\,\log_{10}(P) + b$$

The physical driver of pulsation is the $\kappa$-mechanism: partial ionization zones of He$^+$ and He$^{++}}$ act as a heat engine, trapping radiation during compression and releasing it during expansion.

Modern Calibration

The modern Leavitt Law in the visual band, calibrated by Gaia parallaxes and HST photometry, takes the form:

$$\boxed{M_V = -2.43\,(\log_{10} P - 1) - 4.05}$$

At a fiducial period of $P = 10$ days ($\log P = 1$), a Cepheid has $M_V = -4.05$. A 50-day Cepheid ($\log P = 1.7$) reaches$M_V \approx -5.75$, making it visible with HST out to ~30 Mpc and with JWST potentially to ~50 Mpc. In the near-infrared (Wesenheit index), the relation tightens further due to reduced sensitivity to reddening:

$$m_H^W = m_H - R(m_V - m_I)$$

The Wesenheit magnitude $m_H^W$ is reddening-free by construction, with$R \approx 0.386$ for $H$-band observations. The SH0ES team uses this formulation to minimize systematic errors from dust.

The Cepheid distance to a galaxy is obtained by measuring apparent magnitudes $m$of Cepheids, determining their periods, computing absolute magnitudes $M$ from the PL relation, and applying the distance modulus:

$$d = 10^{(m - M + 5)/5}\;\text{pc}$$

4. Type Ia Supernovae as Standard Candles

Type Ia supernovae (SNe Ia) are thermonuclear explosions of carbon-oxygen white dwarfs that reach near-Chandrasekhar mass ($\approx 1.4\,M_\odot$) via accretion from a companion. Because the explosion is triggered at a nearly universal mass, the peak luminosity is approximately standardizable. After correction, SNe Ia are the most powerful standard candles available, reaching $z > 1$.

Derivation: The Phillips Relation and Standardization

Mark Phillips (1993) demonstrated that the peak luminosity of SNe Ia correlates with the rate of decline from maximum: brighter supernovae decline more slowly. This is parametrized by $\Delta m_{15}(B)$, the magnitude drop in the $B$-band during the first 15 days after maximum.

The physical origin is the mass of $^{56}\text{Ni}$ synthesized in the explosion. More nickel means:

Higher peak luminosity (more radioactive decay energy: $^{56}\text{Ni} \to {}^{56}\text{Co} \to {}^{56}\text{Fe}$)
Higher ejecta temperature and opacity (Fe-group elements dominate opacity)
Slower photon diffusion time → slower decline rate

After standardization, the absolute magnitude is remarkably uniform:

$$\boxed{M_B \approx -19.3 \pm 0.15\;\text{mag}}$$

This corresponds to a peak luminosity of $\sim 4 \times 10^9\,L_\odot$, briefly rivaling an entire galaxy.

The Tripp Formula

Modern SNe Ia standardization uses the Tripp (1998) formula, which corrects the distance modulus using two empirical parameters derived from the light curve:

$$\boxed{\mu = m_B^* - M_B + \alpha\, x_1 - \beta\, c}$$

where:

$m_B^*$: observed peak apparent magnitude in the $B$-band
$M_B$: fiducial absolute magnitude (the quantity calibrated by Cepheids/TRGB)
$x_1$: stretch parameter — measures light-curve width (broader = brighter, so $\alpha > 0$)
$c$: color parameter — redder SNe are fainter (dust + intrinsic color, so $\beta > 0$)
$\alpha \approx 0.14$, $\beta \approx 3.1$: global nuisance parameters fit jointly with cosmology

4.1 From Distance Modulus to Luminosity Distance

Once the corrected distance modulus $\mu$ is in hand, the luminosity distance follows:

$$d_L = 10^{(\mu - 25)/5}\;\text{Mpc}$$

For a flat $\Lambda$CDM universe, $d_L$ is related to the comoving distance by $d_L = (1+z)\,d_C$, with$d_C = \frac{c}{H_0}\int_0^z \frac{dz'}{E(z')}$ and$E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_\Lambda}$. Fitting $\mu(z)$data to this model yields $H_0$ (given $M_B$) and$\Omega_m$.

5. Tip of the Red Giant Branch (TRGB)

The TRGB method exploits a sharp discontinuity in the luminosity function of old, low-mass red giant stars. As a star ascends the red giant branch (RGB), hydrogen burns in a thin shell around a degenerate helium core. When the core mass reaches$\approx 0.45\,M_\odot$ and the core temperature reaches$\sim 10^8$ K, helium ignites explosively in a helium flash. The star rapidly transitions to the horizontal branch, leaving a sharp cutoff at the tip of the RGB.

Derivation: The Helium Flash Luminosity

The luminosity at the TRGB is set by the core mass at helium ignition. For a degenerate core, the luminosity from the H-burning shell is related to the core mass by the core mass-luminosity relation (Refsdal & Weigert 1970):

$$L_{\text{tip}} \approx 5.9 \times 10^5\,L_\odot\,\left(\frac{M_c}{M_\odot}\right)^6$$

With $M_c \approx 0.45\,M_\odot$ at the helium flash:

$$L_{\text{tip}} \approx 5.9 \times 10^5 \times (0.45)^6 \approx 5.9 \times 10^5 \times 0.00830 \approx 4900\,L_\odot$$

Converting to absolute $I$-band magnitude (where the TRGB is sharpest and least sensitive to metallicity for old, metal-poor populations):

$$\boxed{M_I^{\text{TRGB}} \approx -4.05 \pm 0.10}$$

The key insight: this luminosity is nearly independent of the total stellar mass (for masses $\lesssim 2\,M_\odot$), because the degenerate core reaches the same critical conditions regardless of the envelope mass. Metallicity corrections are small in the $I$-band ($\lesssim 0.1$ mag for$-2.2 < [\text{Fe/H}] < -0.7$).

TRGB as an Independent Standard Candle

The TRGB method offers several key advantages over Cepheids:

Population II indicator: TRGB stars are old ($> 2$ Gyr), metal-poor stars found in galactic halos, far from the dusty, crowded disk regions where Cepheids reside
Reduced crowding systematics: halo fields are far less crowded than disk fields, reducing photometric blending errors
Single-epoch photometry: unlike Cepheids, which require multi-epoch monitoring to determine periods, TRGB requires only a single deep exposure
Independence from Cepheid systematics: provides a fully independent path from geometric anchors to SNe Ia hosts

The Chicago-Carnegie Hubble Program (CCHP, led by Freedman et al.) uses the TRGB to calibrate SNe Ia and finds $H_0 = 69.8 \pm 1.7$ km/s/Mpc — intermediate between the SH0ES Cepheid value and the Planck CMB value.

6. The Hubble Tension

The most pressing crisis in modern cosmology is the persistent discrepancy between “local” measurements of $H_0$ from the distance ladder and “early universe” determinations from the cosmic microwave background. As of 2024, the tension stands at approximately $5\sigma$:

The Two Values of $H_0$

SH0ES (Local / Late Universe)

$$H_0 = 73.0 \pm 1.0\;\frac{\text{km/s}}{\text{Mpc}}$$

Riess et al. (2022). Based on the Cepheid-calibrated distance ladder: geometric anchors (NGC 4258 maser, LMC eclipsing binaries, Milky Way parallaxes) $\to$ Cepheids $\to$ SNe Ia in the Hubble flow.

Planck (Early Universe / CMB)

$$H_0 = 67.4 \pm 0.5\;\frac{\text{km/s}}{\text{Mpc}}$$

Planck Collaboration (2020). Inferred from the angular scale of acoustic peaks in the CMB power spectrum, assuming flat $\Lambda$CDM with 6 parameters. BAO and BBN give consistent values.

Quantifying the Tension

The discrepancy between the two values is:

$$\Delta H_0 = 73.0 - 67.4 = 5.6\;\text{km/s/Mpc}$$

Adding uncertainties in quadrature:

$$\sigma_{\text{comb}} = \sqrt{1.0^2 + 0.5^2} = \sqrt{1.25} \approx 1.12\;\text{km/s/Mpc}$$

The significance is therefore:

$$\boxed{\frac{\Delta H_0}{\sigma_{\text{comb}}} = \frac{5.6}{1.12} \approx 5.0\sigma}$$

A $5\sigma$ discrepancy exceeds the discovery threshold in particle physics. If both measurements are correct and free of unrecognized systematics, new physics beyond $\Lambda$CDM is required.

6.1 Possible Resolutions

New Physics Proposals

Early Dark Energy (EDE)

A scalar field that contributes ~10% of the total energy density near matter-radiation equality ($z \sim 3500$), then decays rapidly. This shrinks the sound horizon $r_s$, so the CMB angular scale $\theta_s = r_s/d_A$is preserved with a larger $H_0$. The required EDE fraction is:

$$f_{\text{EDE}}(z_c) \approx 0.10, \quad z_c \approx 3500$$

Extra Relativistic Species ($\Delta N_{\text{eff}}$)

Additional light particles (sterile neutrinos, dark radiation) increase the radiation density, speeding up the expansion rate before recombination. The required shift is $\Delta N_{\text{eff}} \approx 0.4\text{--}0.5$, but this is in tension with Planck's own constraint of$N_{\text{eff}} = 2.99 \pm 0.17$.

Modified Late-Time Physics

Phantom dark energy ($w < -1$), interacting dark energy-dark matter models, or modified gravity at late times. These generally struggle to resolve the tension without worsening the fit to BAO and SNe Ia data simultaneously.

Systematic Error Hypotheses

Cepheid crowding/blending: In distant SN Ia host galaxies, Cepheids are observed in crowded fields. Unresolved background light could bias Cepheid photometry brighter, making them appear closer, and inflating $H_0$. JWST observations (Riess et al. 2024) have largely ruled this out — JWST's superior resolution confirms the HST Cepheid photometry.

Metallicity dependence: The Cepheid PL relation may have a metallicity dependence that biases distances in metal-rich environments. Current calibrations include metallicity corrections, but the coefficient remains debated.

TRGB calibration: Different groups (Freedman vs. Anand/Tully) obtain different TRGB zero-points depending on the choice of geometric anchor and photometric methodology.

7. Applications: Calibration Chains and Systematic Errors

The distance ladder's power lies in its calibration chain, but this chain is also its vulnerability. Each rung introduces potential systematic errors that propagate upward.

The SH0ES Calibration Chain

Step 1 — Geometric Anchors: Three independent anchors provide absolute distances: (a) NGC 4258 megamaser ($d = 7.576 \pm 0.082$ Mpc from water maser kinematics), (b) LMC eclipsing binaries ($d = 49.59 \pm 0.54$ kpc), (c) Milky Way Cepheid parallaxes (Gaia DR3).

Step 2 — Cepheid PL Calibration: Observe Cepheids in the anchor galaxies/systems. Determine the zero-point of the Leavitt Law. Transfer the PL relation to galaxies hosting both Cepheids and recent SNe Ia (~42 galaxies in SH0ES).

Step 3 — SNe Ia Calibration: Use the Cepheid distances to calibrate$M_B$ for the ~42 SNe Ia in these hosts. This sets the zero-point of the Hubble diagram.

Step 4 — Hubble Flow: Apply the calibrated $M_B$ to ~300 SNe Ia in the Hubble flow ($0.023 < z < 0.15$). Fit the intercept of $\mu(z)$ to obtain $H_0$.

JWST Cepheid Observations

In 2024, the James Webb Space Telescope observed Cepheids in several SH0ES host galaxies at near-infrared wavelengths ($F150W$, $F277W$). JWST's 6.5m aperture and diffraction-limited performance at $\lambda > 2\;\mu\text{m}$dramatically reduce crowding relative to HST. Key findings:

JWST Cepheid photometry agrees with HST photometry to within 0.03 mag
The scatter in the PL relation decreases by ~30% with JWST
The resulting $H_0 = 72.6 \pm 2.0$ km/s/Mpc is fully consistent with the HST-based SH0ES result
Crowding/blending is effectively ruled out as the source of the tension

8. Historical Context

1838

Bessel, Henderson, Struve independently measure the first stellar parallaxes (61 Cygni at ~0.3$''$), confirming the Earth orbits the Sun and providing the first rung.

1908

Henrietta Swan Leavitt publishes her study of 1,777 variable stars in the Magellanic Clouds, noting the period-luminosity correlation for Cepheids: “the brighter variables have the longer periods.”

1912

Leavitt & Pickering publish the definitive PL relation for 25 SMC Cepheids, establishing the Leavitt Law as a distance indicator.

1924

Edwin Hubble identifies Cepheids in the Andromeda Nebula (M31) using the 100-inch Hooker telescope, proving it lies far beyond the Milky Way and settling the “Great Debate.”

1929

Hubble publishes the velocity-distance relation ($v = H_0 d$) using Cepheid and brightest-star distances to 24 galaxies, with an initial (overestimated) value of $H_0 \approx 500$ km/s/Mpc.

1952

Walter Baade discovers the Population I/II distinction, recognizing that Hubble had confused RR Lyrae and classical Cepheids. The revised calibration halved $H_0$ and doubled the size of the universe.

1958–1990

Allan Sandage & collaborators systematically improve the distance ladder, driving $H_0$ from ~180 down to ~50 km/s/Mpc. Meanwhile, de Vaucouleurs consistently obtained ~100, leading to the “factor of two” controversy.

1993

Mark Phillips establishes the luminosity-decline rate relation for SNe Ia, transforming them into precision standard candles.

2001

The HST Key Project (Freedman, Kennicutt, Mould et al.) delivers $H_0 = 72 \pm 8$ km/s/Mpc from Cepheids + secondary indicators, resolving the Sandage-de Vaucouleurs debate.

2011

Nobel Prize in Physics awarded to Perlmutter, Schmidt, and Riess for the discovery of the accelerating expansion via SNe Ia, built upon the Cepheid distance ladder.

2019–

The Hubble tension exceeds $4\sigma$ (now $\sim 5\sigma$), becoming a central problem in cosmology. JWST Cepheid observations (2024) confirm the local measurement.

9. Python Simulation: The Distance Ladder and Hubble Diagram

The simulation below creates two panels. The left panel visualizes the rungs of the cosmic distance ladder, showing the distance range each method covers and how they overlap to form a calibration chain. The right panel plots a mock Hubble diagram with SNe Ia data, comparing the $H_0 = 73.0$ (SH0ES) and $H_0 = 67.4$(Planck) predictions — the highlighted gap between the curves illustrates the Hubble tension.

Python

distance_ladder.py140 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

np.random.seed(42)

fig, axes = plt.subplots(1, 2, figsize=(16, 8))
fig.patch.set_facecolor('#0f0a1a')

# ============================================
# Panel 1: The Distance Ladder Rungs
# ============================================
ax1 = axes[0]
ax1.set_facecolor('#0f0a1a')

# Define the rungs of the distance ladder
rungs = [
    {"name": "Parallax\n(Gaia)", "d_min": 1e-6, "d_max": 1e-2,
     "color": "#38bdf8", "y": 1},
    {"name": "Main Sequence\nFitting", "d_min": 1e-3, "d_max": 5e-2,
     "color": "#818cf8", "y": 2},
    {"name": "Cepheids\n(Leavitt Law)", "d_min": 1e-2, "d_max": 3e1,
     "color": "#a855f7", "y": 3},
    {"name": "TRGB", "d_min": 5e-2, "d_max": 2e1,
     "color": "#c084fc", "y": 4},
    {"name": "Type Ia SNe\n(Phillips)", "d_min": 1e0, "d_max": 3e3,
     "color": "#f472b6", "y": 5},
    {"name": "Tully-Fisher /\nFundamental Plane", "d_min": 5e-1, "d_max": 2e2,
     "color": "#fb923c", "y": 6},
    {"name": "BAO / CMB", "d_min": 1e2, "d_max": 1.4e4,
     "color": "#facc15", "y": 7},
]

for rung in rungs:
    ax1.barh(rung["y"], np.log10(rung["d_max"]) - np.log10(rung["d_min"]),
             left=np.log10(rung["d_min"]), height=0.6,
             color=rung["color"], alpha=0.7, edgecolor='white', linewidth=0.5)
    ax1.text(0.5 * (np.log10(rung["d_min"]) + np.log10(rung["d_max"])),
             rung["y"], rung["name"], ha='center', va='center',
             fontsize=8, fontweight='bold', color='white')

# Draw overlap arrows
for i in range(len(rungs) - 1):
    r1 = rungs[i]
    r2 = rungs[i + 1]
    overlap_min = max(np.log10(r1["d_min"]), np.log10(r2["d_min"]))
    overlap_max = min(np.log10(r1["d_max"]), np.log10(r2["d_max"]))
    if overlap_min < overlap_max:
        mid = 0.5 * (overlap_min + overlap_max)
        ax1.annotate('', xy=(mid, r2["y"] - 0.35), xytext=(mid, r1["y"] + 0.35),
                     arrowprops=dict(arrowstyle='->', color='#4ade80', lw=1.5))

ax1.set_xlabel('Distance (Mpc)', fontsize=13, color='white')
ax1.set_title('The Cosmic Distance Ladder', fontsize=15, color='white', fontweight='bold')
tick_positions = [-6, -4, -2, 0, 2, 4]
tick_labels = [r'$10^{%d}$' % t for t in tick_positions]
ax1.set_xticks(tick_positions)
ax1.set_xticklabels(tick_labels, color='white')
ax1.set_yticks([])
ax1.set_xlim(-7, 5)
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#333')

# ============================================
# Panel 2: Hubble Diagram with H0 tension
# ============================================
ax2 = axes[1]
ax2.set_facecolor('#0f0a1a')

# Generate mock Hubble diagram data
c_km = 3.0e5  # km/s

# Nearby Cepheid-calibrated SNe (SH0ES-like)
z_nearby = np.random.uniform(0.01, 0.15, 25)
z_nearby.sort()
H0_shoes = 73.0
d_nearby = c_km * z_nearby / H0_shoes  # Mpc (Hubble law)
mu_nearby = 5.0 * np.log10(d_nearby) + 25.0
mu_nearby += np.random.normal(0, 0.12, len(z_nearby))

# Distant SNe (Hubble flow)
z_far = np.random.uniform(0.1, 1.2, 50)
z_far.sort()

def luminosity_distance(z, H0, Om):
    """Simple numerical integration for flat LCDM."""
    n = 200
    zz = np.linspace(0, z, n)
    dz = zz[1] - zz[0]
    integrand = 1.0 / np.sqrt(Om * (1 + zz)**3 + (1 - Om))
    integral = np.trapezoid(integrand, zz)
    return (c_km / H0) * (1 + z) * integral

# SH0ES model
mu_shoes = np.array([5.0 * np.log10(luminosity_distance(z, 73.0, 0.31)) + 25.0
                      for z in z_far])
mu_shoes += np.random.normal(0, 0.15, len(z_far))

# Model curves
z_model = np.linspace(0.005, 1.5, 300)
mu_shoes_curve = np.array([5.0 * np.log10(luminosity_distance(z, 73.0, 0.31)) + 25.0
                            for z in z_model])
mu_planck_curve = np.array([5.0 * np.log10(luminosity_distance(z, 67.4, 0.315)) + 25.0
                             for z in z_model])

ax2.plot(z_model, mu_shoes_curve, '-', color='#f472b6', lw=2.5,
         label=r'SH0ES: $H_0 = 73.0$ km/s/Mpc')
ax2.plot(z_model, mu_planck_curve, '--', color='#38bdf8', lw=2.5,
         label=r'Planck: $H_0 = 67.4$ km/s/Mpc')

ax2.errorbar(z_nearby, mu_nearby, yerr=0.12, fmt='o', color='#a855f7',
             markersize=5, elinewidth=0.8, alpha=0.8, label='Cepheid-calibrated SNe')
ax2.errorbar(z_far, mu_shoes, yerr=0.15, fmt='s', color='white',
             markersize=3, elinewidth=0.6, alpha=0.6, label='Hubble-flow SNe Ia')

# Highlight the tension region
ax2.fill_between(z_model, mu_shoes_curve, mu_planck_curve,
                  alpha=0.15, color='#facc15')
ax2.annotate(r'$H_0$ tension', xy=(0.5, 42.5), fontsize=12,
             color='#facc15', fontweight='bold', ha='center')

ax2.set_xlabel('Redshift $z$', fontsize=13, color='white')
ax2.set_ylabel(r'Distance Modulus $\mu$ (mag)', fontsize=13, color='white')
ax2.set_title(r'Hubble Diagram & $H_0$ Tension', fontsize=15,
              color='white', fontweight='bold')
ax2.legend(fontsize=9, loc='lower right', facecolor='#1a1030',
           edgecolor='#444', labelcolor='white')
ax2.set_xlim(0, 1.5)
ax2.set_ylim(32, 45)
ax2.grid(alpha=0.2, color='#555')
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#333')

plt.tight_layout()
plt.savefig('distance_ladder.png', dpi=150, bbox_inches='tight', facecolor='#0f0a1a')
plt.show()
print("Distance ladder plot saved.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Code Description

Left panel: Each horizontal bar represents a distance-measurement technique, spanning its effective range in Mpc on a logarithmic axis. Green arrows indicate overlap regions where calibration transfers from one rung to the next. Parallax anchors the bottom ($\sim \mu\text{pc}$ to $\sim 10$ kpc), Cepheids bridge to $\sim 30$ Mpc, SNe Ia reach to $\sim 3$ Gpc, and BAO/CMB extend to the horizon.

Right panel: A mock Hubble diagram with Cepheid-calibrated (purple) and Hubble-flow (white) SNe Ia. Two model curves are overlaid: the SH0ES prediction ($H_0 = 73.0$, pink) and the Planck prediction ($H_0 = 67.4$, blue). The yellow-shaded region between the curves makes the $\sim 8\%$ offset in distance modulus visually clear — this is the Hubble tension. The computation uses numerical trapezoidal integration of the flat $\Lambda$CDM luminosity distance integral (numpy only, no scipy).

10. Summary: Key Equations

Parallax Distance

$$d\;(\text{pc}) = \frac{1}{p''\;(\text{arcsec})}$$

Pulsation Equation

$$P\sqrt{\bar{\rho}} = Q \approx 0.033\;\text{days}$$

Cepheid Leavitt Law

$$M_V = -2.43\,(\log_{10} P - 1) - 4.05$$

Tripp Formula (SNe Ia)

$$\mu = m_B^* - M_B + \alpha\, x_1 - \beta\, c$$

TRGB Absolute Magnitude

$$M_I^{\text{TRGB}} \approx -4.05 \pm 0.10$$

Distance Modulus

$$\mu = 5\log_{10}\!\left(\frac{d_L}{\text{Mpc}}\right) + 25$$

Luminosity Distance (flat $\Lambda$CDM)

$$d_L(z) = \frac{(1+z)\,c}{H_0}\int_0^z \frac{dz'}{\sqrt{\Omega_m(1+z')^3 + \Omega_\Lambda}}$$

The Hubble Tension

$$H_0^{\text{SH0ES}} = 73.0 \pm 1.0, \quad H_0^{\text{Planck}} = 67.4 \pm 0.5 \quad \Rightarrow \quad \sim 5\sigma\;\text{discrepancy}$$

Bibliography

Textbooks & Monographs

Weinberg, S. (2008). Cosmology. Oxford University Press. — Thorough treatment of distance measures and the Hubble parameter.
Carroll, S.M. (2019). Spacetime and Geometry, 2nd ed. Cambridge University Press. — Clear derivation of cosmological distances in FRW spacetimes.
Ryden, B. (2017). Introduction to Cosmology, 2nd ed. Cambridge University Press. — Excellent pedagogical introduction to the distance ladder.

Key Papers

Leavitt, H.S. & Pickering, E.C. (1912). “Periods of 25 Variable Stars in the Small Magellanic Cloud,” Harvard College Observatory Circular 173, 1–3. — The original period-luminosity relation.
Hubble, E. (1929). “A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae,” PNAS 15, 168–173. — Discovery of the expansion of the universe.
Phillips, M.M. (1993). “The absolute magnitudes of Type IA supernovae,” Astrophysical Journal Letters 413, L105–L108. — The Phillips relation.
Freedman, W.L. et al. (2001). “Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant,” ApJ 553, 47–72. — The HST Key Project final result.
Riess, A.G. et al. (2022). “A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team,” ApJ Letters 934, L7. arXiv:2112.04510. — The definitive SH0ES result.
Planck Collaboration (2020). “Planck 2018 results. VI. Cosmological parameters,” A&A 641, A6. arXiv:1807.06209. — CMB-derived $H_0$.
Freedman, W.L. et al. (2024). “Status Report on the Chicago-Carnegie Hubble Program (CCHP): Three Independent Astrophysical Determinations of the Hubble Constant Using the James Webb Space Telescope,” arXiv:2408.06153. — JWST TRGB and JAGB results.

← Cosmology Home Back to Cosmology →