Observations | Type Ia Supernovae

Type Ia Supernovae & the Discovery of Dark Energy

How exploding white dwarfs became the standard candles that revealed the accelerating expansion of the universe

1. Introduction: Exploding White Dwarfs and the Fate of the Cosmos

The Accelerating Universe

In 1998, two independent teams — the Supernova Cosmology Project (SCP) led by Saul Perlmutter and the High-z Supernova Search Team led by Brian Schmidt and Adam Riess — made one of the most astonishing discoveries in the history of physics: the expansion of the universe is not slowing down under gravity, but accelerating. Their tool was a particular class of stellar explosion known as the Type Ia supernova (SN Ia), which occurs when a carbon-oxygen white dwarf in a binary system reaches a critical mass and detonates in a thermonuclear conflagration visible across billions of light-years.

By comparing the observed brightness of distant SNe Ia with their redshifts, both teams found that supernovae at $z \sim 0.5$ were roughly 25% fainter than expected in a decelerating universe — direct evidence for a mysterious “dark energy” driving cosmic acceleration.

The story of Type Ia supernovae as cosmological probes weaves together stellar astrophysics, nuclear physics, and precision observational cosmology. A white dwarf approaching the Chandrasekhar mass limit undergoes a thermonuclear runaway, synthesizing roughly 0.6 solar masses of radioactive $^{56}\text{Ni}$, whose decay powers the supernova's extraordinary luminosity. Because the explosion physics is remarkably uniform, every SN Ia produces nearly the same peak luminosity — making them “standardizable candles” that can be calibrated to measure distances across the observable universe.

In this chapter, we derive the physics of the explosion from the Chandrasekhar limit through the radioactive decay chain, develop the empirical light-curve standardization technique, construct the Hubble diagram that revealed cosmic acceleration, and show how modern SN Ia surveys constrain the dark energy equation of state.

2. White Dwarf Explosion Physics

2.1 The Chandrasekhar Mass Limit

A white dwarf is supported against gravitational collapse by the quantum-mechanical degeneracy pressure of its electrons. As we add mass (for example, through accretion from a companion star), the electrons are squeezed into ever-smaller volumes and become relativistic. Subrahmanyan Chandrasekhar showed in 1930 that there exists a maximum mass for which electron degeneracy pressure can sustain the star. Above this mass, no stable white dwarf configuration exists.

The Chandrasekhar limit arises from balancing relativistic electron degeneracy pressure against gravity. The relativistic Fermi energy scales as $E_F \sim \hbar c \, n_e^{1/3}$, where $n_e$ is the electron number density. The total kinetic energy of the degenerate electron gas scales as:

$$E_{\text{kin}} \sim N_e \cdot \hbar c \, n_e^{1/3} \sim \frac{\hbar c \, N_e^{4/3}}{R}$$

where $N_e$ is the total number of electrons and $R$ is the stellar radius.

The gravitational potential energy scales as:

$$E_{\text{grav}} \sim -\frac{G M^2}{R}$$

In the relativistic limit, both energies scale as $1/R$. This means there is no equilibrium radius — stability requires that the kinetic energy coefficient exceed the gravitational one. The critical mass at which they become equal defines the Chandrasekhar limit. Writing $M = N_e \mu_e m_p$ (where $\mu_e$is the mean molecular weight per electron), the condition $E_{\text{kin}} + E_{\text{grav}} = 0$ yields:

$$M_{\text{Ch}} = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left(\frac{\hbar c}{G}\right)^{3/2} \frac{1}{(\mu_e m_H)^2}$$

where $\omega_3^0 \approx 2.018$ is the Lane-Emden constant for a polytrope of index $n = 3$. For a carbon-oxygen white dwarf with $\mu_e = 2$:

$$\boxed{M_{\text{Ch}} = \frac{5.83}{\mu_e^2}\,M_\odot \approx 1.44\,M_\odot}$$

The Chandrasekhar mass limit for a carbon-oxygen white dwarf ($\mu_e = 2$).

2.2 Carbon Detonation at the Chandrasekhar Limit

As a white dwarf accretes matter from a companion and approaches $M \to M_{\text{Ch}}$, the central density rises dramatically. At densities $\rho_c \gtrsim 2 \times 10^9$g/cm$^3$ and temperatures $T \gtrsim 7 \times 10^8$ K, carbon fusion ignites. Because the degenerate equation of state decouples pressure from temperature, the star cannot expand to cool itself — a thermal runaway ensues. The carbon burning front propagates through the entire star in approximately 1–2 seconds, consuming carbon and oxygen via:

$$^{12}\text{C} + {^{12}\text{C}} \to {^{24}\text{Mg}} + \gamma \quad (Q = 13.93\;\text{MeV})$$

$$^{12}\text{C} + {^{16}\text{O}} \to {^{28}\text{Si}} + \gamma$$

Subsequent silicon burning produces iron-group elements, predominantly $^{56}\text{Ni}$.

The nuclear energy released ($\sim 1\text{--}2 \times 10^{51}$ erg) is sufficient to completely unbind the white dwarf — there is no remnant. The total nuclear energy can be estimated: burning $1.4\,M_\odot$ of C/O to $^{56}\text{Ni}$releases roughly $q \approx 1\;\text{MeV/nucleon}$, so:

$$E_{\text{nuc}} \sim \frac{M_{\text{Ch}}}{m_p} \times q \approx \frac{1.4 \times 2 \times 10^{33}\;\text{g}}{1.67 \times 10^{-24}\;\text{g}} \times 1.6 \times 10^{-6}\;\text{erg} \approx 2.7 \times 10^{51}\;\text{erg}$$

This exceeds the gravitational binding energy $E_{\text{bind}} \sim GM_{\text{Ch}}^2/R \approx 5 \times 10^{50}$ erg, so the star is entirely disrupted.

2.3 Peak Luminosity from the Nickel Decay Chain

The optical luminosity of a Type Ia supernova is powered not by the explosion itself, but by the radioactive decay chain:

$$^{56}\text{Ni} \xrightarrow{\tau_{1/2} = 6.1\,\text{d}} {^{56}\text{Co}} + e^+ + \nu_e + \gamma \quad (Q_1 = 1.72\;\text{MeV})$$

$$^{56}\text{Co} \xrightarrow{\tau_{1/2} = 77.3\,\text{d}} {^{56}\text{Fe}} + e^+ + \nu_e + \gamma \quad (Q_2 = 3.73\;\text{MeV})$$

The energy deposition rate from $^{56}\text{Ni}$ decay (assuming complete gamma-ray trapping) is:

$$\dot{E}_{\text{Ni}}(t) = \frac{M_{\text{Ni}}}{56\,m_p}\left[\epsilon_{\text{Ni}}\,e^{-t/\tau_{\text{Ni}}} + \epsilon_{\text{Co}}\left(e^{-t/\tau_{\text{Co}}} - e^{-t/\tau_{\text{Ni}}}\right)\right]$$

where $\epsilon_{\text{Ni}} = Q_1/\tau_{\text{Ni}}$ and $\epsilon_{\text{Co}} = Q_2/\tau_{\text{Co}}$ are the specific power outputs, and $\tau = \tau_{1/2}/\ln 2$ is the mean lifetime.

The peak luminosity occurs when the diffusion timescale through the expanding ejecta equals the time since explosion. By Arnett's rule (1982), the luminosity at peak equals the instantaneous rate of radioactive energy deposition:

$$\boxed{L_{\text{peak}} = \dot{E}_{\text{Ni}}(t_{\text{peak}}) \propto M_{\text{Ni}}}$$

Arnett's rule: peak luminosity is directly proportional to the mass of $^{56}\text{Ni}$ synthesized.

For a typical SN Ia with $M_{\text{Ni}} \approx 0.6\,M_\odot$, peak occurs at $t_{\text{peak}} \approx 17\text{--}19$ days after explosion. The energy deposition rate at peak, dominated by $^{56}\text{Ni}$ decay, gives:

$$L_{\text{peak}} \approx \frac{0.6\,M_\odot}{56\,m_p} \times \frac{Q_1}{\tau_{\text{Ni}}} \times e^{-t_{\text{peak}}/\tau_{\text{Ni}}} \approx 1.3 \times 10^{43}\;\text{erg/s}$$

Converting to absolute magnitude using $M_\odot^B = 5.48$:

$$M_B = M_\odot^B - 2.5\log_{10}\!\left(\frac{L_{\text{peak}}}{L_\odot}\right) \approx 5.48 - 2.5\log_{10}\!\left(\frac{1.3 \times 10^{43}}{3.83 \times 10^{33}}\right) \approx -19.3$$

$$\boxed{M_B \approx -19.3 \quad \text{for standard SNe Ia with } M_{\text{Ni}} \approx 0.6\,M_\odot}$$

This corresponds to roughly 4 billion solar luminosities at peak — comparable to the luminosity of an entire galaxy.

3. Light Curve Standardization: The Phillips Relation

3.1 The Width-Luminosity Relation

While all SNe Ia are similar, they are not identical. The key insight of Mark Phillips (1993) was that brighter SNe Ia decline more slowly after maximum light, while fainter ones decline more rapidly. This correlation between peak luminosity and light-curve shape makes SNe Iastandardizable candles.

Phillips introduced the parameter $\Delta m_{15}(B)$, defined as the decline in B-band magnitude during the first 15 days after maximum:

$$\Delta m_{15}(B) \equiv m_B(t_{\max} + 15\,\text{d}) - m_B(t_{\max})$$

The physical origin of this correlation lies in the $^{56}\text{Ni}$ mass: more $^{56}\text{Ni}$ means higher peak luminosity (Arnett's rule) andhigher opacity due to the greater abundance of iron-group elements. Higher opacity increases the diffusion time, causing the light curve to decline more slowly. The empirical relation is approximately:

$$M_B^{\max} \approx -21.73 + 2.70\,\Delta m_{15}(B)$$

Phillips (1993) relation. A SN Ia with $\Delta m_{15} = 1.1$ (typical) gives$M_B \approx -19.3$.

3.2 The SALT2 Stretch-Color Correction

Modern supernova cosmology uses more sophisticated light-curve fitters. The SALT2 (Spectral Adaptive Light-curve Template) framework parameterizes each SN Ia light curve by three quantities: an overall amplitude $m_B^*$ (apparent B-band peak magnitude), a stretch parameter $x_1$ (describing light-curve width), and a color parameter $c$ (measuring reddening). The corrected distance modulus is:

$$\boxed{\mu = m_B^* - M_B + \alpha x_1 - \beta c + \Delta M_{\text{host}}}$$

The Tripp (1998) / SALT2 distance modulus formula. $\alpha$ captures the brighter-slower relation, $\beta$ captures the brighter-bluer relation, and $\Delta M_{\text{host}}$ is a host-galaxy mass step correction.

The nuisance parameters are determined simultaneously with cosmological parameters in a global fit. Typical values from the Pantheon+ analysis are:

$$\alpha \approx 0.15, \qquad \beta \approx 3.1$$

$$M_B \approx -19.25 \pm 0.03\;\text{mag}$$

$$\Delta M_{\text{host}} \approx -0.05\;\text{mag for } M_{\text{host}} > 10^{10}\,M_\odot$$

3.3 Residual Scatter and Systematic Error Budget

The $x_1$ correction accounts for the Phillips relation (brighter-slower), while the $c$ correction accounts for both intrinsic color variation and dust reddening. The host-mass step $\Delta M_{\text{host}}$ captures the empirical observation that SNe Ia in massive galaxies are ~0.05 mag brighter (after standard corrections) than those in low-mass hosts. After applying all corrections, the residual scatter in the Hubble diagram is:

$$\boxed{\sigma_{\text{int}} \approx 0.10\text{--}0.15\;\text{mag}}$$

Intrinsic scatter per supernova after SALT2 standardization, corresponding to a ~5–7% distance uncertainty per object.

This scatter is the floor set by the unknown intrinsic diversity of SN Ia progenitors. With $N$ supernovae, the statistical uncertainty on the mean distance at each redshift bin scales as $\sigma_{\text{int}}/\sqrt{N}$, which is why modern surveys aim for thousands of well-measured SNe Ia.

4. The Hubble Diagram and the Discovery of Acceleration

4.1 The Distance Modulus in FRW Cosmology

The key observable for supernova cosmology is the relationship between the distance modulus $\mu$ and redshift $z$. In a Friedmann-Robertson-Walker universe, the luminosity distance is:

$$d_L(z) = (1+z)\frac{c}{H_0} \int_0^z \frac{dz'}{E(z')}$$

where $E(z) \equiv H(z)/H_0 = \sqrt{\Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}$ for a cosmological constant.

The distance modulus is related to the luminosity distance by:

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

For small redshifts $z \ll 1$, the Hubble law gives $d_L \approx cz/H_0$and all cosmologies converge. The diagnostic power emerges at higher redshifts where the integral over $E(z)$ probes the composition of the universe.

4.2 Expanding the Luminosity Distance

To understand the sensitivity to cosmological parameters, expand $d_L(z)$ as a Taylor series around $z = 0$. Defining the deceleration parameter$q_0 = -\ddot{a}a/\dot{a}^2|_0 = \Omega_m/2 - \Omega_\Lambda$:

$$d_L(z) = \frac{c}{H_0}\left[z + \frac{1}{2}(1 - q_0)z^2 + \mathcal{O}(z^3)\right]$$

A positive $q_0$ (deceleration) makes objects closer (brighter); a negative$q_0$ (acceleration) makes them farther (fainter).

For the concordance cosmology ($\Omega_m = 0.3, \Omega_\Lambda = 0.7$):

$$q_0 = \frac{0.3}{2} - 0.7 = -0.55$$

The universe is accelerating today: $q_0 < 0$.

4.3 The ~0.25 Magnitude Excess

The critical observation was comparing the distance modulus of high-redshift SNe Ia with the prediction from a decelerating (matter-dominated) universe. Consider two benchmark models at $z = 0.5$:

Einstein-de Sitter ($\Omega_m = 1, \Omega_\Lambda = 0$):

$$d_L^{\text{EdS}}(0.5) = \frac{2c}{H_0}\left[(1+z) - \sqrt{1+z}\right] = \frac{2c}{H_0}\left[1.5 - 1.225\right] = \frac{0.550\,c}{H_0}$$

Concordance ($\Omega_m = 0.3, \Omega_\Lambda = 0.7$) — numerical integration:

$$d_L^{\Lambda\text{CDM}}(0.5) \approx \frac{0.650\,c}{H_0}$$

The difference in distance modulus is:

$$\boxed{\Delta\mu = 5\log_{10}\!\left(\frac{0.650}{0.550}\right) \approx 0.36\;\text{mag}}$$

SNe Ia at $z \approx 0.5$ are ~0.25–0.4 mag fainter than expected in a decelerating universe, depending on the reference model. This was the signal detected by both teams in 1998.

4.4 The Constraint Ellipse in the ($\Omega_m, \Omega_\Lambda$) Plane

Each supernova observation constrains a combination of $\Omega_m$ and$\Omega_\Lambda$ through the $\chi^2$ statistic:

$$\chi^2(\Omega_m, \Omega_\Lambda) = \sum_{i=1}^{N_{\text{SN}}} \frac{\left[\mu_{\text{obs},i} - \mu_{\text{th}}(z_i;\,\Omega_m, \Omega_\Lambda)\right]^2}{\sigma_i^2}$$

The resulting confidence contours form elongated ellipses in the ($\Omega_m, \Omega_\Lambda$) plane, oriented roughly along the direction of constant $q_0 = \Omega_m/2 - \Omega_\Lambda$. This is because SNe Ia at moderate redshifts primarily constrain the deceleration parameter. The key result is that the confidence region lies decisively above the $\Omega_\Lambda = 0$line, ruling out a matter-only universe at >$5\sigma$ significance:

$$\Omega_\Lambda > 0 \quad \text{at} \quad >99.7\%\;\text{confidence (3$\sigma$)}$$

Combined with the CMB constraint $\Omega_m + \Omega_\Lambda \approx 1$ (flatness), the intersection gives $\Omega_m \approx 0.3, \Omega_\Lambda \approx 0.7$.

5. Constraining the Dark Energy Equation of State

5.1 The CPL Parameterization

Beyond asking whether dark energy exists, SNe Ia can probe its nature. A cosmological constant has equation of state $w = p/\rho = -1$ exactly. If dark energy is a dynamical field, $w$ could differ from $-1$ and evolve with time. The standard parameterization, due to Chevallier & Polarski (2001) and Linder (2003), is:

$$\boxed{w(z) = w_0 + w_a \frac{z}{1+z}}$$

CPL parameterization. $w_0$ is the present-day value, $w_a$ captures time evolution. For $\Lambda$: $w_0 = -1, w_a = 0$.

5.2 Luminosity Distance with Dynamic Dark Energy

With the CPL parameterization, the dark energy density evolves as:

$$\rho_{\text{DE}}(z) = \rho_{\text{DE},0}\,(1+z)^{3(1+w_0+w_a)}\,\exp\!\left(-\frac{3w_a z}{1+z}\right)$$

The Hubble parameter becomes:

$$E^2(z) = \Omega_m(1+z)^3 + \Omega_{\text{DE}}\,(1+z)^{3(1+w_0+w_a)}\,\exp\!\left(-\frac{3w_a z}{1+z}\right)$$

assuming flatness: $\Omega_{\text{DE}} = 1 - \Omega_m$.

The luminosity distance is then:

$$\boxed{d_L(z;\,w_0, w_a) = \frac{(1+z)\,c}{H_0}\int_0^z \frac{dz'}{E(z';\,w_0, w_a)}}$$

This integral must be evaluated numerically for general $w_0, w_a$.

5.3 Current Constraints on w

The combined analysis of SNe Ia (Pantheon+), baryon acoustic oscillations (BAO), and the cosmic microwave background (CMB) yields tight constraints on the dark energy equation of state. Assuming a constant $w$:

$$\boxed{w = -1.03 \pm 0.03 \quad \text{(SNe + BAO + CMB)}}$$

Exquisitely consistent with $w = -1$ (cosmological constant).

In the $(w_0, w_a)$ plane, the Pantheon+ data alone constrain:

$$w_0 = -0.90 \pm 0.14, \qquad w_a = -0.5 \pm 0.6$$

The DESI BAO 2024 results hint at $w_0 > -1$ and $w_a < 0$, but the evidence is not yet conclusive.

The $\chi^2$ for the equation of state is constructed by comparing the observed distance moduli with the theoretical predictions across redshift bins:

$$\chi^2(w_0, w_a) = \sum_{i=1}^{N_{\text{SN}}} \frac{\left[\mu_{\text{obs},i} - \mu(z_i;\,\Omega_m, w_0, w_a)\right]^2}{\sigma_{\mu,i}^2}$$

The 1$\sigma$ and 2$\sigma$ contours correspond to$\Delta\chi^2 = 2.30$ and $\Delta\chi^2 = 6.17$ for 2 parameters.

6. Modern Applications and Surveys

6.1 The Pantheon+ Catalog

The Pantheon+ analysis (Scolnic et al. 2022, Brout et al. 2022) is the most comprehensive SN Ia cosmological analysis to date, combining 1701 light curves from 1550 spectroscopically confirmed SNe Ia across 18 surveys spanning the redshift range $0.001 < z < 2.26$. Key results include:

Hubble constant: Combined with SH0ES Cepheid calibration: $H_0 = 73.04 \pm 1.04$ km/s/Mpc (5$\sigma$ tension with Planck).
Dark energy: $w = -0.90 \pm 0.14$ (SNe alone); consistent with $w = -1$ when combined with CMB/BAO.
Matter density: $\Omega_m = 0.334 \pm 0.018$ (SNe alone, flat $\Lambda$CDM).
Intrinsic scatter: ~0.10 mag after full SALT2 standardization and bias corrections.

6.2 Dark Energy Survey (DES) Supernovae

The DES 5-year supernova program measured ~1600 photometrically classified SNe Ia in the redshift range $0.1 < z < 1.13$. The DES-SN5YR analysis represents a major advance in photometric classification techniques, using machine learning algorithms trained on spectroscopically confirmed subsamples. Their results give $\Omega_m = 0.352 \pm 0.017$in flat $\Lambda$CDM, consistent with Pantheon+.

6.3 Vera C. Rubin Observatory / LSST Projections

The Legacy Survey of Space and Time (LSST) at the Vera C. Rubin Observatory will transform SN Ia cosmology. Over its 10-year survey, LSST is projected to discover approximately 100,000–500,000 well-measured SNe Ia across $0.1 < z < 1.2$. This enormous statistical sample will reduce the statistical error on $w$ to sub-percent levels, making systematic errors the dominant uncertainty. Key systematic challenges include:

Photometric calibration: Chromatic and achromatic errors in the photometric zeropoints must be controlled to ~1 mmag.
Dust extinction: Disentangling intrinsic SN color from host-galaxy and Milky Way dust reddening remains a major challenge. The color coefficient $\beta$ encodes both effects.
Progenitor evolution: If the SN Ia population changes with redshift (due to evolving star-formation environments, metallicity, or age distributions), the standardized luminosity could shift systematically.
Gravitational lensing: Weak lensing by foreground structures magnifies or demagnifies distant SNe, adding ~0.04$z$ mag of scatter. This is a noise floor that cannot be reduced by better SN physics.

7. Historical Context: The Road to the 2011 Nobel Prize

7.1 The Supernova Cosmology Project (SCP)

Founded in 1988 by Saul Perlmutter at Lawrence Berkeley National Laboratory, the SCP pioneered the “batch” strategy for discovering distant supernovae: by scheduling telescope time for both discovery and follow-up observations in advance, the team could systematically harvest high-redshift SNe Ia. The SCP's landmark paper (Perlmutter et al. 1999) analyzed 42 high-redshift supernovae and concluded that $\Omega_\Lambda > 0$ at the 99% confidence level.

7.2 The High-z Supernova Search Team

Led by Brian Schmidt (Australia) with Adam Riess as lead analyst, this team used a complementary approach. Riess et al. (1998) analyzed 16 high-redshift SNe Ia and applied the multicolor light-curve shape (MLCS) method. Their paper, published slightly before the SCP's, reported the first evidence for $q_0 < 0$ — a decelerating universe was ruled out at the $>3\sigma$ level.

7.3 The 1998 Discovery and 2011 Nobel Prize

The convergence of two independent teams using different analysis methods on largely independent datasets made the result compelling. By early 1998, the evidence was clear: the expansion of the universe is accelerating. The result was named Science magazine's “Breakthrough of the Year” for 1998. In October 2011, Saul Perlmutter, Brian Schmidt, and Adam Riess shared the Nobel Prize in Physics “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae.”

Timeline of Key Events

1930: Chandrasekhar derives the white dwarf mass limit.

1960s: SNe Ia recognized as a roughly homogeneous class of explosions.

1993: Phillips establishes the width-luminosity relation ($\Delta m_{15}$).

1995–1997: Both teams accumulate high-z SN Ia samples.

January 1998: Riess et al. announce evidence for acceleration at the AAS meeting.

1998–1999: Both papers published; result confirmed by independent checks.

2003: WMAP CMB data combined with SNe Ia solidify the concordance model.

2011: Nobel Prize in Physics awarded to Perlmutter, Schmidt, and Riess.

2022: Pantheon+ catalog: 1701 SNe Ia light curves across 18 surveys.

2024: DESI BAO results hint at possible evolving dark energy.

8. Python Simulation: Hubble Diagram and Confidence Contours

The following simulation generates a Pantheon-like Hubble diagram, computes the residual$\mu_{\text{obs}} - \mu(\Omega_m = 1)$ that reveals cosmic acceleration, and plots confidence contours in the ($\Omega_m, \Omega_\Lambda$) plane. Only numpy is used — no scipy.

Python

sne_ia_hubble_diagram.py271 lines

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

# =============================================
# Numerical integration using the trapezoidal rule
# =============================================
def trapz_integrate(y, x):
    """Trapezoidal rule integration (no scipy)."""
    return np.sum(0.5 * (y[:-1] + y[1:]) * np.diff(x))

def luminosity_distance(z, Omega_m, Omega_L):
    """
    Compute d_L in units of c/H0 for a given (Omega_m, Omega_L) cosmology.
    Uses trapezoidal integration over 500 sub-intervals.
    """
    if z <= 0:
        return 0.0
    Omega_k = 1.0 - Omega_m - Omega_L
    z_arr = np.linspace(0, z, 500)
    E_arr = np.sqrt(Omega_m * (1 + z_arr)**3 + Omega_k * (1 + z_arr)**2 + Omega_L)
    E_arr = np.maximum(E_arr, 1e-30)
    integrand = 1.0 / E_arr
    chi = trapz_integrate(integrand, z_arr)  # comoving distance in c/H0

# Handle curvature
    if abs(Omega_k) < 1e-6:
        r = chi
    elif Omega_k > 0:
        sqrtk = np.sqrt(Omega_k)
        r = np.sinh(sqrtk * chi) / sqrtk
    else:
        sqrtk = np.sqrt(-Omega_k)
        r = np.sin(sqrtk * chi) / sqrtk

return (1 + z) * r

def distance_modulus(z, Omega_m, Omega_L, H0=70.0):
    """Compute distance modulus mu = 5*log10(d_L / 10 pc)."""
    dL_cH0 = luminosity_distance(z, Omega_m, Omega_L)
    cH0_Mpc = 3.0e5 / H0
    dL_Mpc = dL_cH0 * cH0_Mpc
    if dL_Mpc <= 0:
        return 0.0
    return 5.0 * np.log10(dL_Mpc) + 25.0

# =============================================
# Generate mock Pantheon-like SN Ia dataset
# =============================================
np.random.seed(2024)

# True cosmology
Om_true = 0.315
OL_true = 0.685

# Low-z sample (z = 0.01 to 0.1): 200 SNe
z_low = np.random.uniform(0.01, 0.10, 200)
# Medium-z sample (z = 0.1 to 0.5): 400 SNe
z_mid = np.random.uniform(0.10, 0.50, 400)
# High-z sample (z = 0.5 to 1.5): 200 SNe
z_high = np.random.uniform(0.50, 1.50, 200)

z_data = np.concatenate([z_low, z_mid, z_high])
z_data.sort()

# Compute true distance moduli and add scatter
mu_true = np.array([distance_modulus(z, Om_true, OL_true) for z in z_data])

# Realistic errors: larger at high-z
sigma_mu = 0.10 + 0.05 * z_data + np.random.uniform(0.0, 0.05, len(z_data))
mu_obs = mu_true + np.random.normal(0, sigma_mu)

# =============================================
# Cosmological models for comparison
# =============================================
models = [
    {"Om": 0.315, "OL": 0.685, "label": "Concordance ($\\Omega_m$=0.31, $\\Omega_\\Lambda$=0.69)",
     "color": "#a855f7", "ls": "-", "lw": 2.5},
    {"Om": 1.0,   "OL": 0.0,   "label": "Einstein-de Sitter ($\\Omega_m$=1, $\\Omega_\\Lambda$=0)",
     "color": "#f472b6", "ls": "--", "lw": 2.0},
    {"Om": 0.315, "OL": 0.0,   "label": "Open CDM ($\\Omega_m$=0.31, $\\Omega_\\Lambda$=0)",
     "color": "#38bdf8", "ls": "-.", "lw": 2.0},
    {"Om": 0.0,   "OL": 0.0,   "label": "Empty (Milne)",
     "color": "#facc15", "ls": ":", "lw": 2.0},
]

z_theory = np.linspace(0.005, 1.8, 400)

# =============================================
# FIGURE 1: Hubble Diagram + Residuals
# =============================================
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(11, 10),
                                gridspec_kw={"height_ratios": [3, 1.2]})
fig.patch.set_facecolor('#0f0a1a')
for ax in [ax1, ax2]:
    ax.set_facecolor('#0f0a1a')
    ax.tick_params(colors='white', labelsize=10)
    for spine in ax.spines.values():
        spine.set_color('#6b21a8')

# Reference model for residuals: Einstein-de Sitter
mu_EdS_data = np.array([distance_modulus(z, 1.0, 0.0) for z in z_data])

# Plot data
ax1.scatter(z_data, mu_obs, s=3, c='white', alpha=0.3, zorder=1, rasterized=True)

# Bin the data for clarity
z_bins = np.linspace(0.01, 1.5, 25)
z_centers = 0.5 * (z_bins[:-1] + z_bins[1:])
mu_binned = np.zeros(len(z_centers))
mu_err_binned = np.zeros(len(z_centers))
for j in range(len(z_centers)):
    mask = (z_data >= z_bins[j]) & (z_data < z_bins[j+1])
    if np.sum(mask) > 0:
        w = 1.0 / sigma_mu[mask]**2
        mu_binned[j] = np.sum(mu_obs[mask] * w) / np.sum(w)
        mu_err_binned[j] = 1.0 / np.sqrt(np.sum(w))

ax1.errorbar(z_centers, mu_binned, yerr=mu_err_binned, fmt='o',
             color='#e9d5ff', markersize=5, elinewidth=1.2, capsize=2,
             zorder=5, label='Binned mock SNe Ia')

# Plot theoretical curves
for model in models:
    mu_theory = np.array([distance_modulus(z, model["Om"], model["OL"]) for z in z_theory])
    ax1.plot(z_theory, mu_theory, label=model["label"],
             color=model["color"], ls=model["ls"], lw=model["lw"])

ax1.set_ylabel('Distance Modulus $\\mu$ (mag)', fontsize=13, color='white')
ax1.set_title('Type Ia Supernova Hubble Diagram', fontsize=16, color='white', fontweight='bold')
ax1.legend(fontsize=8.5, loc='lower right', facecolor='#1a0a2e', edgecolor='#6b21a8',
           labelcolor='white')
ax1.set_xlim(0, 1.8)
ax1.set_ylim(32, 46)
ax1.grid(alpha=0.2, color='#6b21a8')

# Residuals relative to EdS
mu_EdS_theory = np.array([distance_modulus(z, 1.0, 0.0) for z in z_theory])
for model in models:
    mu_theory = np.array([distance_modulus(z, model["Om"], model["OL"]) for z in z_theory])
    ax2.plot(z_theory, mu_theory - mu_EdS_theory, color=model["color"],
             ls=model["ls"], lw=model["lw"])

# Binned residual data
mu_EdS_binned = np.zeros(len(z_centers))
for j in range(len(z_centers)):
    mask = (z_data >= z_bins[j]) & (z_data < z_bins[j+1])
    if np.sum(mask) > 0:
        w = 1.0 / sigma_mu[mask]**2
        mu_EdS_binned[j] = np.sum(mu_EdS_data[mask] * w) / np.sum(w)

ax2.errorbar(z_centers, mu_binned - mu_EdS_binned, yerr=mu_err_binned, fmt='o',
             color='#e9d5ff', markersize=5, elinewidth=1.2, capsize=2)
ax2.axhline(0, color='#f472b6', ls='--', lw=1, alpha=0.5)
ax2.set_xlabel('Redshift $z$', fontsize=13, color='white')
ax2.set_ylabel('$\\Delta\\mu$ (vs EdS)', fontsize=13, color='white')
ax2.set_xlim(0, 1.8)
ax2.set_ylim(-0.8, 1.2)
ax2.grid(alpha=0.2, color='#6b21a8')

# Annotate the acceleration signal
ax2.annotate('SNe fainter than\nexpected: acceleration!',
             xy=(0.6, 0.35), fontsize=10, color='#a855f7',
             arrowprops=dict(arrowstyle='->', color='#a855f7', lw=1.5),
             xytext=(0.2, 0.85))

plt.tight_layout()
plt.savefig('sne_hubble_diagram.png', dpi=150, bbox_inches='tight', facecolor='#0f0a1a')
plt.close()

# =============================================
# FIGURE 2: Confidence contours in (Omega_m, Omega_Lambda) plane
# =============================================
print("Computing confidence contours in (Omega_m, Omega_Lambda) plane...")

# Grid of (Omega_m, Omega_Lambda) values
n_grid = 80
Om_grid = np.linspace(0.0, 1.0, n_grid)
OL_grid = np.linspace(0.0, 1.5, n_grid)
chi2_grid = np.zeros((n_grid, n_grid))

for i in range(n_grid):
    for j in range(n_grid):
        mu_model = np.array([distance_modulus(z, Om_grid[i], OL_grid[j]) for z in z_data])
        # Marginalize over M_B offset (analytical)
        delta = mu_obs - mu_model
        w = 1.0 / sigma_mu**2
        offset = np.sum(delta * w) / np.sum(w)
        chi2_grid[j, i] = np.sum(((delta - offset) / sigma_mu)**2)

# Convert to Delta chi^2
chi2_min = np.min(chi2_grid)
dchi2 = chi2_grid - chi2_min

# Confidence levels for 2 parameters
levels_1sigma = 2.30
levels_2sigma = 6.17
levels_3sigma = 11.83

fig2, ax3 = plt.subplots(1, 1, figsize=(9, 8))
fig2.patch.set_facecolor('#0f0a1a')
ax3.set_facecolor('#0f0a1a')
ax3.tick_params(colors='white', labelsize=11)
for spine in ax3.spines.values():
    spine.set_color('#6b21a8')

# Filled contours
cmap = LinearSegmentedColormap.from_list('purple_fade',
    ['#a855f7', '#6b21a8', '#3b0764', '#0f0a1a'])
ax3.contourf(Om_grid, OL_grid, dchi2,
             levels=[0, levels_1sigma, levels_2sigma, levels_3sigma, 50],
             colors=['#a855f780', '#6b21a850', '#3b076430', '#0f0a1a10'])

# Contour lines
cs = ax3.contour(Om_grid, OL_grid, dchi2,
                 levels=[levels_1sigma, levels_2sigma, levels_3sigma],
                 colors=['#e9d5ff', '#c084fc', '#7c3aed'],
                 linewidths=[2.0, 1.5, 1.0])
ax3.clabel(cs, fmt={levels_1sigma: '$1\\sigma$', levels_2sigma: '$2\\sigma$',
                     levels_3sigma: '$3\\sigma$'}, fontsize=12, colors='white')

# Flat universe line
ax3.plot([0, 1], [1, 0], 'w--', lw=1.5, alpha=0.5, label='Flat: $\\Omega_m + \\Omega_\\Lambda = 1$')

# No acceleration line: q0 = Om/2 - OL = 0
Om_line = np.linspace(0, 1, 100)
ax3.plot(Om_line, Om_line / 2, color='#f472b6', ls='-.', lw=1.5, alpha=0.7,
         label='No accel: $\\Omega_\\Lambda = \\Omega_m/2$')

# Mark concordance model
ax3.plot(0.315, 0.685, '*', color='#facc15', markersize=15, zorder=10,
         label='Concordance (0.31, 0.69)')

# No dark energy line
ax3.axhline(0, color='#38bdf8', ls=':', lw=1.5, alpha=0.5, label='$\\Omega_\\Lambda = 0$')

ax3.set_xlabel('$\\Omega_m$', fontsize=15, color='white')
ax3.set_ylabel('$\\Omega_\\Lambda$', fontsize=15, color='white')
ax3.set_title('SN Ia Confidence Contours', fontsize=16, color='white', fontweight='bold')
ax3.set_xlim(0, 1.0)
ax3.set_ylim(0, 1.5)
ax3.legend(fontsize=10, loc='upper left', facecolor='#1a0a2e', edgecolor='#6b21a8',
           labelcolor='white')
ax3.grid(alpha=0.15, color='#6b21a8')

# Shade the "no dark energy" region
ax3.fill_between(Om_line, 0, Om_line/2, alpha=0.08, color='#f472b6',
                 label='_Decelerating region')
ax3.text(0.7, 0.12, 'Decelerating', fontsize=11, color='#f472b6', alpha=0.7)
ax3.text(0.15, 0.9, 'Accelerating', fontsize=11, color='#a855f7', alpha=0.7)

plt.tight_layout()
plt.savefig('sne_confidence_contours.png', dpi=150, bbox_inches='tight', facecolor='#0f0a1a')
plt.close()

print("Figure 1: sne_hubble_diagram.png")
print("Figure 2: sne_confidence_contours.png")
print()
print("=== Summary Statistics ===")
print(f"Number of mock SNe Ia: {len(z_data)}")
print(f"Redshift range: {z_data.min():.3f} to {z_data.max():.3f}")
print(f"Mean intrinsic scatter: {np.mean(sigma_mu):.3f} mag")

# Find best-fit from grid
idx = np.unravel_index(np.argmin(chi2_grid), chi2_grid.shape)
print(f"Best-fit Omega_m = {Om_grid[idx[1]]:.3f}, Omega_Lambda = {OL_grid[idx[0]]:.3f}")
print(f"Chi^2_min = {chi2_min:.1f} for {len(z_data)} SNe")
print(f"Chi^2/dof = {chi2_min/(len(z_data)-3):.3f}")
print("Done!")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation Description

This simulation generates 800 mock SNe Ia distributed across three redshift bins (low-z, medium-z, and high-z), mimicking the Pantheon+ sample. For each supernova, the true distance modulus is computed by numerically integrating the luminosity distance in the concordance cosmology ($\Omega_m = 0.315, \Omega_\Lambda = 0.685$), then adding realistic Gaussian scatter that increases with redshift.

Figure 1 (upper): The Hubble diagram showing distance modulus vs redshift for the mock data and four theoretical cosmologies. Binned data points with error bars clearly follow the concordance model. (lower): Residuals relative to the Einstein–de Sitter model reveal the ~0.25–0.5 mag excess at$z \sim 0.5\text{--}1$ that constitutes the direct evidence for acceleration.

Figure 2: Confidence contours in the ($\Omega_m, \Omega_\Lambda$) plane, computed via $\chi^2$ on an 80$\times$80 grid with analytical marginalization over $M_B$. The 1$\sigma$, 2$\sigma$, and 3$\sigma$ contours clearly exclude$\Omega_\Lambda = 0$ and cluster around the concordance model (yellow star). The flat-universe line and the no-acceleration boundary are shown for reference.

9. Summary: Key Results

Chandrasekhar Mass

$$M_{\text{Ch}} = \frac{5.83}{\mu_e^2}\,M_\odot \approx 1.44\,M_\odot$$

Arnett's Rule

$$L_{\text{peak}} \propto M_{\text{Ni}}, \qquad M_B \approx -19.3 \;\text{for}\; M_{\text{Ni}} \approx 0.6\,M_\odot$$

SALT2 Standardization

$$\mu = m_B^* - M_B + \alpha x_1 - \beta c + \Delta M_{\text{host}}, \qquad \sigma_{\text{int}} \approx 0.10\text{--}0.15\;\text{mag}$$

Luminosity Distance

$$d_L(z) = (1+z)\frac{c}{H_0}\int_0^z \frac{dz'}{E(z')}$$

CPL Dark Energy

$$w(z) = w_0 + w_a\frac{z}{1+z}, \qquad w = -1.03 \pm 0.03$$

Key Discovery

$$\Delta\mu(z \sim 0.5) \approx 0.25\text{--}0.4\;\text{mag fainter than decelerating} \implies \Omega_\Lambda > 0$$

Bibliography

Key Papers

Riess, A.G. et al. (1998). “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant,” Astronomical Journal 116, 1009–1038. arXiv:astro-ph/9805201.
Perlmutter, S. et al. (1999). “Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae,” Astrophysical Journal 517, 565–586. arXiv:astro-ph/9812133.
Phillips, M.M. (1993). “The absolute magnitudes of Type Ia supernovae,” Astrophysical Journal Letters 413, L105–L108.
Arnett, W.D. (1982). “Type I supernovae. I. Analytic solutions for the early part of the light curve,” Astrophysical Journal 253, 785–797.
Guy, J. et al. (2007). “SALT2: using distant supernovae to improve the use of type Ia supernovae as distance indicators,” Astronomy & Astrophysics 466, 11–21.
Scolnic, D.M. et al. (2022). “The Pantheon+ Analysis: The Full Data Set and Light-curve Release,” Astrophysical Journal 938, 113. arXiv:2112.03863.
Brout, D. et al. (2022). “The Pantheon+ Analysis: Cosmological Constraints,” Astrophysical Journal 938, 110. arXiv:2202.04077.
Chevallier, M. & Polarski, D. (2001). “Accelerating universes with scaling dark matter,” Int. J. Mod. Phys. D 10, 213–224; Linder, E.V. (2003). Phys. Rev. Lett. 90, 091301.
Chandrasekhar, S. (1931). “The Maximum Mass of Ideal White Dwarfs,” Astrophysical Journal 74, 81–82.
DESI Collaboration (2024). “DESI 2024 VI: Cosmological Constraints from BAO,” arXiv:2404.03002.

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