Part VI: The Dark Sector | Chapter 2

Dark Energy & The Accelerating Universe

From the supernova discovery of cosmic acceleration to the cosmological constant problem, quintessence, modified gravity, and the concordance $\Lambda$CDM model

1. Introduction: The Discovery of Cosmic Acceleration

Historical Context

In 1998, two independent teams — the Supernova Cosmology Project (led by Saul Perlmutter) and the High-z Supernova Search Team (led by Brian Schmidt and Adam Riess) — announced one of the most stunning discoveries in the history of science: the expansion of the universe is accelerating. By measuring the apparent brightness of distant Type Ia supernovae and comparing with their redshifts, both teams found that supernovae at $z \sim 0.5$were approximately 25% fainter than expected in a decelerating universe. This implied the existence of a repulsive component — now called dark energy — comprising roughly 68% of the total energy density of the universe.

Perlmutter, Schmidt, and Riess were awarded the 2011 Nobel Prize in Physics “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae.”

The discovery upended the prevailing expectation that the gravitational attraction of matter should be slowing the expansion. In the standard Friedmann framework, the acceleration equation reads:

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$$

Acceleration ($\ddot{a} > 0$) requires either a positive cosmological constant$\Lambda$ or a component with strongly negative pressure: $p < -\rho c^2/3$.

Today, dark energy is characterized by an equation of state parameter$w = p/(\rho c^2)$ with observational constraints centering on$w \approx -1$, consistent with Einstein's cosmological constant. Yet its fundamental nature remains one of the deepest open problems in physics.

2. Type Ia Supernovae as Standard Candles

2.1 The Chandrasekhar Limit and Thermonuclear Explosions

Type Ia supernovae arise from the thermonuclear detonation of a carbon-oxygen white dwarf that approaches the Chandrasekhar mass limit:

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

where $\mu_e = 2$ is the mean molecular weight per electron for C/O composition. At this mass, electron degeneracy pressure can no longer support the star against gravitational collapse.

Because the explosion is triggered at a nearly universal mass, the peak luminosity is approximately standard: $M_B \approx -19.3 \pm 0.3$ mag. This makes SNe Ia powerful cosmological distance indicators visible out to $z \gtrsim 1$.

2.2 The Phillips Relation and Standardization

The raw scatter of $\sim 0.3$ mag in peak brightness is reduced to$\sim 0.15$ mag using the Phillips relation (1993): intrinsically brighter SNe Ia decline more slowly after maximum. The light-curve width parameter$\Delta m_{15}$ (the magnitude decline in the first 15 days past peak) tightly correlates with absolute magnitude, enabling standardization.

Modern standardization uses the SALT2 or SNooPy light-curve fitters, which parametrize the stretch (light-curve width) and colour of each supernova. After correction, the standardized peak magnitude has a dispersion of only $\sigma \approx 0.10$–$0.15$ mag, corresponding to a 5–7% distance precision per supernova.

2.3 Luminosity Distance and the Hubble Diagram

The luminosity distance in a FLRW universe with density parameters $\Omega_m$, $\Omega_\Lambda$, and curvature$\Omega_k$ is:

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

for a spatially flat universe ($\Omega_k = 0$), where the dimensionless Hubble parameter is:

$$E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_\Lambda}$$

The distance modulus relates observable apparent magnitude $m$ and absolute magnitude $M$:

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

The Hubble diagram — a plot of distance modulus$\mu$ versus redshift z — is the primary tool for constraining cosmological parameters with SNe Ia. In 1998, both teams observed that distant supernovae were ~0.25 mag fainter (i.e., further away) than predicted in a matter-only decelerating universe ($\Omega_m = 1$, $\Omega_\Lambda = 0$). The data instead favoured $\Omega_m \approx 0.3$, $\Omega_\Lambda \approx 0.7$.

3. The Cosmological Constant $\Lambda$

3.1 Einstein's Original Motivation

Einstein introduced $\Lambda$ in 1917 to obtain a static cosmological solution from his field equations. He required $\ddot{a} = 0$ and $\dot{a} = 0$, which for a matter-filled universe (p = 0) demands:

$$\Lambda = \frac{4\pi G\rho}{c^2}, \qquad k = \frac{\Lambda a^2}{c^2}$$

Einstein's static universe is unstable: any perturbation causes it to expand or collapse. After Hubble's discovery of expansion, Einstein reportedly called $\Lambda$ his “greatest blunder.”

3.2 $\Lambda$ as Vacuum Energy

In the modern interpretation, $\Lambda$ is equivalent to an energy density of the vacuum that is constant in space and time:

$$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G} \approx 5.96 \times 10^{-27}\;\text{kg/m}^3 \approx 10^{-29}\;\text{g/cm}^3$$

$$p_\Lambda = -\rho_\Lambda c^2$$

The equation of state is $w = p/(\rho c^2) = -1$ exactly. This negative pressure is the source of gravitational repulsion.

3.3 Effect on the Friedmann Equation

Including the cosmological constant, the first Friedmann equation becomes:

$$H^2 = \frac{8\pi G}{3}\left(\rho_m + \rho_r + \rho_\Lambda\right) - \frac{kc^2}{a^2}$$

where $\rho_m \propto a^{-3}$, $\rho_r \propto a^{-4}$, and $\rho_\Lambda = \text{const}$.

3.4 The de Sitter Limit

As the universe expands, matter and radiation dilute while $\rho_\Lambda$ remains constant. In the far future, dark energy completely dominates and the universe approaches de Sitter space:

$$H \to H_\Lambda = c\sqrt{\frac{\Lambda}{3}} = \text{const}$$

$$a(t) \propto e^{H_\Lambda t}$$

Exponential expansion. The universe doubles in size every$t_{\text{double}} = \ln 2 / H_\Lambda \approx 10$ Gyr.

4. The Cosmological Constant Problem

The cosmological constant problem is widely regarded as the most severe fine-tuning problem in all of physics. It arises from the clash between quantum field theory's prediction for the vacuum energy density and the astronomically smaller observed value.

4.1 QFT Prediction: Zero-Point Energy

In quantum field theory, even the vacuum state possesses zero-point energy. For a single scalar field, each mode of wave number $\mathbf{k}$ contributes$\frac{1}{2}\hbar\omega_k$ to the vacuum energy. Summing over all modes up to a UV cutoff $k_{\max}$:

$$\rho_{\text{vac}} = \int_0^{k_{\max}} \frac{\hbar\omega_k}{2} \cdot \frac{4\pi k^2\,dk}{(2\pi)^3} = \int_0^{k_{\max}} \frac{\hbar c\,k}{2} \cdot \frac{4\pi k^2\,dk}{(2\pi)^3}$$

For a relativistic field with $\omega_k = ck$, the integral evaluates to:

$$\rho_{\text{vac}} = \frac{\hbar c}{16\pi^2}\,k_{\max}^4$$

4.2 The Planck Scale Cutoff

If we trust QFT up to the Planck scale, where quantum gravity effects become important, we set $k_{\max} = k_{\text{Pl}} = 1/\ell_{\text{Pl}}$:

$$\ell_{\text{Pl}} = \sqrt{\frac{\hbar G}{c^3}} \approx 1.6 \times 10^{-33}\;\text{cm}$$

$$\rho_{\text{Pl}} \sim \frac{c^7}{\hbar G^2} \approx 5 \times 10^{93}\;\text{g/cm}^3$$

4.3 The 10$^{122}$ Discrepancy

The Worst Prediction in Physics

$$\rho_{\text{QFT}} \sim 10^{93}\;\text{g/cm}^3$$

$$\rho_{\text{obs}} \sim 10^{-29}\;\text{g/cm}^3$$

$$\frac{\rho_{\text{QFT}}}{\rho_{\text{obs}}} \sim 10^{122}$$

This 122-order-of-magnitude discrepancy between the quantum field theory prediction and the observed cosmological constant is the largest known disagreement between theory and experiment in all of science.

4.4 The Radiative Stability Problem

Even if one postulates that the bare cosmological constant cancels the vacuum energy to the required precision (a fine-tuning of 1 part in $10^{122}$), this cancellation is radiatively unstable: quantum corrections from each loop order re-introduce contributions of order $m^4/(16\pi^2)$ for every particle of mass m. Each phase transition in the early universe (electroweak, QCD) shifts the vacuum energy by enormous amounts, requiring re-tuning at every energy scale. No known symmetry principle enforces$\rho_\Lambda = 0$ or the observed small value.

5. The Coincidence Problem

A second, independent puzzle: why are the dark energy and matter densities comparable today?

$$\Omega_m \approx 0.31, \qquad \Omega_\Lambda \approx 0.69$$

These are of the same order of magnitude, yet they evolve very differently with time.

Matter density dilutes as $\rho_m \propto a^{-3}$ while $\rho_\Lambda$remains constant. Their ratio is:

$$\frac{\rho_\Lambda}{\rho_m} = \frac{\Omega_\Lambda}{\Omega_m}\,a^3$$

At early times ($a \ll 1$), matter dominated overwhelmingly; in the far future ($a \gg 1$), dark energy will dominate overwhelmingly. The epoch of$\rho_m \sim \rho_\Lambda$ — which happens to be now — is a fleeting moment in cosmic history.

The transition from deceleration to acceleration occurred at:

$$a_{\text{acc}} = \left(\frac{\Omega_m}{2\Omega_\Lambda}\right)^{1/3} \approx 0.61, \qquad z_{\text{acc}} \approx 0.64$$

Roughly 6 billion years ago, when the universe was about 60% of its current size.

6. Quintessence and Dynamical Dark Energy

The cosmological constant problems motivate exploring models where dark energy is dynamical rather than a fixed vacuum energy. The simplest such model is quintessence: a slowly-rolling scalar field$\phi$ with potential $V(\phi)$.

6.1 Energy Density and Pressure

For a homogeneous scalar field in the FLRW background, the energy density and pressure are:

$$\rho_\phi = \frac{\dot{\phi}^2}{2c^2} + V(\phi)$$

$$p_\phi = \frac{\dot{\phi}^2}{2} - V(\phi) c^2$$

The kinetic term $\dot{\phi}^2/2$ contributes positive pressure, while the potential $V(\phi)$ contributes negative pressure.

6.2 Equation of State

The equation of state parameter for quintessence is:

$$w_\phi = \frac{p_\phi}{\rho_\phi c^2} = \frac{\dot{\phi}^2/2 - V(\phi)}{\dot{\phi}^2/2 + V(\phi)}$$

If the field is slowly rolling ($\dot{\phi}^2 \ll V$), then$w_\phi \to -1$, mimicking a cosmological constant. If kinetic energy dominates, $w_\phi \to +1$ (stiff matter). In general,$-1 \leq w_\phi \leq +1$.

6.3 Klein-Gordon Equation in FLRW

The scalar field obeys the Klein-Gordon equation in the expanding background:

$$\ddot{\phi} + 3H\dot{\phi} + \frac{dV}{d\phi} = 0$$

The $3H\dot{\phi}$ term acts as Hubble friction, damping the field's evolution. For slow roll ($\ddot{\phi} \approx 0$):$\dot{\phi} \approx -V'/(3H)$.

6.4 Tracking Solutions and Common Potentials

A particularly attractive feature of certain quintessence models is the existence of tracking solutions: the field's equation of state follows the background (radiation or matter) for a wide range of initial conditions, then transitions to $w \approx -1$ at late times, partially alleviating the coincidence problem.

Exponential Potential

$$V(\phi) = V_0\,e^{-\lambda\phi/M_{\text{Pl}}}$$

Yields a scaling solution with $w_\phi = w_{\text{bg}}$ (the background equation of state). For $\lambda^2 < 2$, the field can drive acceleration.

Inverse Power-Law Potential

$$V(\phi) = \frac{M^{4+\alpha}}{\phi^\alpha}$$

Proposed by Ratra & Peebles (1988). Exhibits tracking behaviour: the energy density in $\phi$ converges to a common track regardless of initial conditions, eventually dominating and accelerating expansion.

7. Parametrizing the Dark Energy Equation of State

7.1 The CPL Parametrization

Rather than committing to a specific model, observational cosmologists commonly parametrize the dark energy equation of state using the Chevallier-Polarski-Linder (CPL) form:

$$w(a) = w_0 + w_a\,(1 - a) = w_0 + w_a\,\frac{z}{1+z}$$

$w_0$ is the equation of state today, and $w_a$ characterizes its time evolution. For the cosmological constant: $w_0 = -1$, $w_a = 0$.

The dark energy density evolves as:

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

7.2 Current Observational Constraints

From Planck 2018 + BAO + SNe Ia (combined):

$$w_0 = -1.03 \pm 0.08, \qquad w_a = -0.1 \pm 0.4$$

Fully consistent with the cosmological constant ($w_0 = -1$, $w_a = 0$). The Dark Energy Task Force figure of merit is defined as the inverse area of the 95% confidence ellipse in the $(w_0, w_a)$ plane; current experiments achieve FoM $\sim 50$, while next-generation surveys aim for FoM $\gtrsim 500$.

7.3 DESI BAO Results (2024)

The Dark Energy Spectroscopic Instrument (DESI) released its first-year BAO results in 2024, combining 6 million galaxy and quasar redshifts to measure the expansion history. The DESI data alone, and especially when combined with CMB and SNe Ia data, showed intriguing hints of$w_0 > -1$ and $w_a < 0$, suggesting that dark energy may be weakening over time. While the statistical significance remained at the 2–3$\sigma$level (not conclusive), this generated considerable interest as a possible first evidence against a pure cosmological constant.

8. Phantom Dark Energy and the Big Rip

If $w < -1$ (the “phantom” regime), dark energy density increaseswith expansion rather than remaining constant or diluting:

$$\rho_{\text{DE}} \propto a^{-3(1+w)}$$

For $w < -1$, the exponent $-3(1+w) > 0$, so $\rho$grows as the universe expands, leading to a runaway acceleration.

8.1 The Big Rip

Caldwell, Kamionkowski, and Weinberg (2003) showed that phantom energy leads to a future singularity — the Big Rip — at which the scale factor diverges in finite time. All bound structures (galaxies, stars, atoms) are torn apart. The time to the Big Rip is:

$$t_{\text{rip}} - t_0 = \frac{2}{3|1+w|\,H_0\sqrt{\Omega_\Lambda}}$$

For $w = -1.1$: $t_{\text{rip}} - t_0 \approx 100$ Gyr. For$w = -1.5$: $t_{\text{rip}} - t_0 \approx 35$ Gyr.

8.2 Theoretical Problems with Phantom Energy

Phantom dark energy requires a scalar field with negative kinetic energy($\mathcal{L} = -\frac{1}{2}(\partial\phi)^2 - V(\phi)$), which leads to a ghost instability: the vacuum is unstable to spontaneous production of ghost-photon pairs, and the theory violates the null energy condition. These pathologies make most theorists sceptical of fundamental phantom models, though effective phantom behaviour can arise in certain modified gravity theories without ghost degrees of freedom.

9. Modified Gravity as an Alternative to Dark Energy

An alternative explanation for cosmic acceleration is that general relativity itself is modified on cosmological scales, eliminating the need for a dark energy component.

9.1 f(R) Gravity

The simplest modification replaces the Einstein-Hilbert Lagrangian$\mathcal{L} = R$ with a general function of the Ricci scalar:

$$S = \frac{c^4}{16\pi G}\int \sqrt{-g}\,f(R)\,d^4x + S_{\text{matter}}$$

For $f(R) = R - 2\Lambda$, this reduces to GR with a cosmological constant. The Hu-Sawicki model $f(R) = R - \mu R_c\,\frac{(R/R_c)^n}{(R/R_c)^n + 1}$can mimic $\Lambda$CDM expansion while differing in structure growth.

The $f(R)$ field equations are fourth-order in the metric, but can be recast as a scalar-tensor theory (Brans-Dicke with $\omega = 0$) via a conformal transformation, with the scalar degree of freedom $f_R \equiv df/dR$ mediating an additional force.

9.2 DGP Braneworld Model

The Dvali-Gabadadze-Porrati (DGP) model posits that we live on a 4D brane embedded in a 5D Minkowski bulk. Gravity leaks into the extra dimension on scales larger than a crossover scale $r_c = 1/(2H_0)$, weakening gravity at late times and mimicking acceleration. The self-accelerating branch reproduces expansion without a cosmological constant, though it suffers from ghost instabilities.

9.3 Horndeski Gravity

The Horndeski Lagrangian (1974) is the most general 4D scalar-tensor theory with second-order equations of motion (avoiding Ostrogradsky instabilities). It encompasses quintessence, $f(R)$ gravity, galileons, and kinetic gravity braiding as special cases, providing a unified framework for testing deviations from GR.

9.4 Distinguishing Dark Energy from Modified Gravity

The key diagnostic is the growth rate of structure. Both dark energy and modified gravity can produce the same expansion history $H(z)$, but they predict different growth rates for matter perturbations:

$$f(z) \equiv \frac{d\ln D}{d\ln a}$$

For GR with dark energy, the growth rate is well approximated by:

$$f(z) \approx \Omega_m(z)^\gamma$$

where $D$ is the linear growth factor and $\gamma$ is the growth index. For GR: $\gamma \approx 0.55$. For DGP:$\gamma \approx 0.68$. For $f(R)$:$\gamma \approx 0.42$.

Future missions such as Euclid (ESA, launched 2023), the Vera C. Rubin Observatory's LSST, and the Nancy Grace Roman Space Telescope (NASA) will measure both the expansion history and the growth of structure with unprecedented precision, enabling percent-level tests of gravity on cosmological scales.

10. Observational Probes of Dark Energy

No single probe can fully characterize dark energy. The power lies in combining multiple independent probes to break degeneracies between cosmological parameters.

Type Ia Supernovae

Standardizable candles measuring $d_L(z)$. The Pantheon+ sample includes ~1700 SNe out to $z \sim 2.3$. Directly constrains the expansion history.

Baryon Acoustic Oscillations (BAO)

The ~150 Mpc standard ruler imprinted at recombination. BAO measure$d_A(z)$ and $H(z)$ from transverse and line-of-sight clustering. DESI, Euclid, and LSST will map BAO to $z \sim 3$.

Weak Gravitational Lensing

Cosmic shear measures the integrated matter power spectrum weighted by the lensing kernel. Sensitive to both $H(z)$ and the growth of structure. Stage IV surveys (Euclid, LSST, Roman) will measure shapes of billions of galaxies.

Galaxy Cluster Counts

The number density of massive clusters is exponentially sensitive to the growth factor$\sigma_8$. Cluster abundances from X-ray (eROSITA), SZ (SPT, ACT), and optical surveys constrain both $\Omega_m$ and $\sigma_8$.

CMB: Integrated Sachs-Wolfe Effect

In a $\Lambda$-dominated universe, gravitational potentials decay, causing a net blueshift of CMB photons traversing voids and a redshift traversing overdensities. The ISW effect is detected via cross-correlation of the CMB with large-scale structure surveys.

Redshift-Space Distortions (RSD)

Peculiar velocities distort galaxy clustering along the line of sight, allowing measurement of $f\sigma_8(z)$. This directly probes the growth rate, providing the key test to distinguish dark energy from modified gravity.

Multi-Probe Combination and Fisher Forecasting

The Fisher information matrix formalism provides a systematic way to forecast how well future experiments will constrain parameters. For a data vector $\mathbf{d}$ with covariance $\mathbf{C}$, the Fisher matrix is$F_{\alpha\beta} = \frac{\partial \mathbf{d}^T}{\partial p_\alpha} \mathbf{C}^{-1} \frac{\partial \mathbf{d}}{\partial p_\beta}$. Combining probes corresponds to summing their Fisher matrices:$F_{\text{total}} = F_{\text{SNe}} + F_{\text{BAO}} + F_{\text{WL}} + F_{\text{CMB}} + \cdots$

11. The $\Lambda$CDM Concordance Model

The $\Lambda$CDM model is the “Standard Model of Cosmology” — the simplest model consistent with essentially all observations. It describes a spatially flat universe dominated by a cosmological constant and cold dark matter.

11.1 The Six Free Parameters

Parameter	Symbol	Planck 2018 Value	Description
Hubble constant	$H_0$	67.4 km/s/Mpc	Current expansion rate
Baryon density	$\Omega_b h^2$	0.0224	Physical baryon density parameter
CDM density	$\Omega_c h^2$	0.120	Physical cold dark matter density
Spectral index	$n_s$	0.965	Scalar perturbation tilt
Amplitude	$A_s$	$2.1 \times 10^{-9}$	Scalar perturbation amplitude at pivot
Optical depth	$\tau$	0.054	Reionization optical depth

11.2 Key Derived Parameters

$\Omega_\Lambda$

0.685

Dark energy fraction

$t_0$

13.80 Gyr

Age of the universe

$\sigma_8$

0.811

RMS fluctuation at 8 Mpc/h

11.3 Current Tensions

$H_0$ Tension

The CMB-inferred value (Planck): $H_0 = 67.4 \pm 0.5$ km/s/Mpc. The local distance-ladder measurement (SH0ES): $H_0 = 73.0 \pm 1.0$ km/s/Mpc.

The discrepancy exceeds $5\sigma$, suggesting either unresolved systematics or new physics beyond $\Lambda$CDM (e.g., early dark energy, new neutrino interactions, or modified recombination history).

$S_8$ Tension

The parameter $S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$ quantifies the amplitude of matter fluctuations. Planck predicts$S_8 = 0.832 \pm 0.013$, while weak lensing surveys (KiDS, DES, HSC) find $S_8 \approx 0.76 \pm 0.02$.

This 2–3$\sigma$ tension suggests the late-time universe may be less clumpy than predicted, potentially pointing to dark energy dynamics, neutrino masses, or modified gravity.

12. Python Code: Hubble Diagram for Dark Energy Cosmologies

The following Python code computes and plots the Hubble diagram (distance modulus$\mu$ vs. redshift z) for several cosmological models, demonstrating how Type Ia supernova data distinguish between accelerating and decelerating universes.

Python

hubble_diagram_dark_energy.py131 lines

#!/usr/bin/env python3
"""
hubble_diagram_dark_energy.py
Plot the Hubble diagram (distance modulus vs redshift) for different
cosmological models, showing how SNe Ia data distinguish between
an accelerating and a decelerating universe.
"""
import numpy as np
from scipy.integrate import quad
import matplotlib.pyplot as plt

# =============================================
# Define the dimensionless Hubble parameter E(z)
# for different (Omega_m, Omega_Lambda) models
# =============================================
def E_func(z, Omega_m, Omega_L):
    """E(z) = H(z)/H0 for flat + non-flat LCDM."""
    Omega_k = 1.0 - Omega_m - Omega_L
    return np.sqrt(Omega_m * (1 + z)**3
                 + Omega_k * (1 + z)**2
                 + Omega_L)

def luminosity_distance(z, Omega_m, Omega_L):
    """
    Luminosity distance d_L(z) in units of c/H0.
    Handles flat, open, and closed geometries.
    """
    Omega_k = 1.0 - Omega_m - Omega_L
    # Comoving distance integral
    dc, _ = quad(lambda zp: 1.0 / E_func(zp, Omega_m, Omega_L), 0, z)

if abs(Omega_k) < 1e-6:
        # Flat
        chi = dc
    elif Omega_k > 0:
        # Open
        sqrtOk = np.sqrt(Omega_k)
        chi = np.sinh(sqrtOk * dc) / sqrtOk
    else:
        # Closed
        sqrtOk = np.sqrt(-Omega_k)
        chi = np.sin(sqrtOk * dc) / sqrtOk

return (1 + z) * chi

def distance_modulus(z, Omega_m, Omega_L, H0=70.0):
    """
    Distance modulus mu = 5 * log10(d_L / 10 pc).
    d_L is computed in Mpc, so mu = 5 * log10(d_L/Mpc) + 25.
    We use d_L in units of c/H0, then convert.
    """
    dL_cH0 = luminosity_distance(z, Omega_m, Omega_L)
    # Convert c/H0 to Mpc: c/H0 = (3e5 km/s) / (H0 km/s/Mpc) Mpc
    cH0_Mpc = 3.0e5 / H0  # in Mpc
    dL_Mpc = dL_cH0 * cH0_Mpc
    return 5.0 * np.log10(dL_Mpc) + 25.0

# =============================================
# Cosmological models to compare
# =============================================
models = [
    {"Omega_m": 0.31, "Omega_L": 0.69,
     "label": r"$\Omega_m=0.31,\;\Omega_\Lambda=0.69$ (concordance)",
     "color": "#a855f7", "ls": "-"},
    {"Omega_m": 1.0,  "Omega_L": 0.0,
     "label": r"$\Omega_m=1.0,\;\Omega_\Lambda=0$ (EdS, decelerating)",
     "color": "#f472b6", "ls": "--"},
    {"Omega_m": 0.31, "Omega_L": 0.0,
     "label": r"$\Omega_m=0.31,\;\Omega_\Lambda=0$ (open, decelerating)",
     "color": "#38bdf8", "ls": "-."},
    {"Omega_m": 0.0,  "Omega_L": 0.0,
     "label": r"$\Omega_m=0,\;\Omega_\Lambda=0$ (empty / Milne)",
     "color": "#facc15", "ls": ":"},
    {"Omega_m": 0.0,  "Omega_L": 1.0,
     "label": r"$\Omega_m=0,\;\Omega_\Lambda=1$ (de Sitter)",
     "color": "#4ade80", "ls": "--"},
]

z_arr = np.linspace(0.01, 2.0, 300)

# =============================================
# Compute distance modulus for each model
# =============================================
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 10),
                                gridspec_kw={"height_ratios": [3, 1]})

# Reference model for residuals: empty universe (Milne)
mu_milne = np.array([distance_modulus(z, 0.0, 0.0) for z in z_arr])

for model in models:
    mu_arr = np.array([distance_modulus(z, model["Omega_m"],
                       model["Omega_L"]) for z in z_arr])
    ax1.plot(z_arr, mu_arr, label=model["label"],
             color=model["color"], ls=model["ls"], lw=2)
    # Residual relative to empty universe
    ax2.plot(z_arr, mu_arr - mu_milne, color=model["color"],
             ls=model["ls"], lw=2)

# ---- Mock SNe Ia data (Concordance + scatter) ----
np.random.seed(42)
z_data = np.random.uniform(0.01, 1.5, 40)
z_data.sort()
mu_data = np.array([distance_modulus(z, 0.31, 0.69) for z in z_data])
mu_data += np.random.normal(0, 0.15, len(z_data))  # 0.15 mag scatter

ax1.errorbar(z_data, mu_data, yerr=0.15, fmt='o', color='white',
             markersize=4, elinewidth=0.8, alpha=0.7,
             label='Mock SNe Ia data')

mu_data_milne = np.array([distance_modulus(z, 0.0, 0.0) for z in z_data])
ax2.errorbar(z_data, mu_data - mu_data_milne, yerr=0.15, fmt='o',
             color='white', markersize=4, elinewidth=0.8, alpha=0.7)

# ---- Formatting ----
ax1.set_ylabel(r'Distance Modulus $\mu$ (mag)', fontsize=13)
ax1.set_title('Hubble Diagram: Dark Energy Cosmologies', fontsize=15)
ax1.legend(fontsize=9, loc='lower right')
ax1.set_xlim(0, 2.0)
ax1.grid(alpha=0.3)

ax2.set_xlabel('Redshift $z$', fontsize=13)
ax2.set_ylabel(r'$\Delta\mu$ (relative to Milne)', fontsize=13)
ax2.axhline(0, color='#facc15', ls=':', lw=1)
ax2.set_xlim(0, 2.0)
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.savefig('hubble_diagram_dark_energy.png', dpi=150,
            bbox_inches='tight', facecolor='#0f172a')
plt.show()
print("Plot saved to hubble_diagram_dark_energy.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Code Description

The script computes the luminosity distance $d_L(z)$ by numerically integrating$\int_0^z dz'/E(z')$ for five cosmological models: the concordance$\Lambda$CDM ($\Omega_m = 0.31, \Omega_\Lambda = 0.69$), Einstein–de Sitter ($\Omega_m = 1$), open CDM ($\Omega_m = 0.31$, no $\Lambda$), the empty Milne universe, and de Sitter space. The upper panel shows the Hubble diagram with mock SNe Ia data points (0.15 mag scatter). The lower panel shows residuals relative to the empty universe, clearly demonstrating that SNe Ia at$z \sim 0.5$–1 are fainter in the concordance model than in decelerating models, directly revealing cosmic acceleration.

13. Summary: Key Equations and Current Status

Luminosity Distance (flat universe)

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

Distance Modulus

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

Vacuum Energy Density (QFT)

$$\rho_{\text{vac}} \sim \frac{\hbar c\, k_{\max}^4}{16\pi^2}, \qquad \frac{\rho_{\text{QFT}}}{\rho_{\text{obs}}} \sim 10^{122}$$

Quintessence Equation of State

$$w_\phi = \frac{\dot{\phi}^2/2 - V(\phi)}{\dot{\phi}^2/2 + V(\phi)}, \qquad \ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0$$

CPL Parametrization

$$w(a) = w_0 + w_a(1 - a)$$

Big Rip Time (Phantom $w < -1$)

$$t_{\text{rip}} - t_0 = \frac{2}{3|1+w|\,H_0\sqrt{\Omega_\Lambda}}$$

$f(R)$ Gravity Action

$$S = \frac{c^4}{16\pi G}\int \sqrt{-g}\,f(R)\,d^4x$$

Growth Rate Approximation

$$f(z) = \frac{d\ln D}{d\ln a} \approx \Omega_m(z)^\gamma, \qquad \gamma_{\text{GR}} \approx 0.55$$

The State of Dark Energy Research

1. The $\Lambda$CDM model with $w = -1$ remains the simplest and most successful description of all current data, fitting CMB, BAO, SNe Ia, weak lensing, and cluster counts simultaneously.

2. The cosmological constant problem ($10^{122}$ discrepancy) and the coincidence problem remain unsolved, motivating continued theoretical and observational effort.

3. The $H_0$ tension ($5\sigma$) and $S_8$ tension ($2\text{--}3\sigma$) may be pointing toward physics beyond $\Lambda$CDM — or unresolved systematic errors.

4. DESI BAO results (2024) provide tantalizing but inconclusive hints of time-varying dark energy ($w_0 > -1$, $w_a < 0$).

5. Next-generation surveys (Euclid, LSST, Roman, DESI full dataset, CMB-S4) will measure both $H(z)$ and $f\sigma_8(z)$ to percent-level precision, enabling definitive tests of whether dark energy is truly a cosmological constant or something more exotic.

Video Resources

New Developments in Understanding Dark Energy

SVAstronomyLectures

Bibliography

Textbooks & Monographs

Weinberg, S. (2008). Cosmology. Oxford University Press. — Detailed treatment of the cosmological constant, vacuum energy, and dark energy models.
Dodelson, S. & Schmidt, F. (2020). Modern Cosmology, 2nd ed. Academic Press. — Dark energy within the context of precision cosmology, including SNe Ia and BAO constraints.
Carroll, S.M. (2019). Spacetime and Geometry: An Introduction to General Relativity, 2nd ed. Cambridge University Press. — Clear discussion of the cosmological constant and de Sitter spacetime.
Amendola, L. & Tsujikawa, S. (2015). Dark Energy: Theory and Observations, 2nd ed. Cambridge University Press. — Dedicated monograph covering quintessence, modified gravity, and observational probes.
Mukhanov, V. (2005). Physical Foundations of Cosmology. Cambridge University Press. — Treatment of vacuum energy, the cosmological constant problem, and accelerated expansion.

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. — Discovery of cosmic acceleration (Nobel Prize 2011).
Perlmutter, S. et al. (1999). “Measurements of Ω and Λ from 42 High-Redshift Supernovae,” Astrophysical Journal 517, 565–586. arXiv:astro-ph/9812133. — Independent discovery of the accelerating expansion (Nobel Prize 2011).
Einstein, A. (1917). “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, 142–152. — Introduction of the cosmological constant Λ.
Weinberg, S. (1989). “The cosmological constant problem,” Reviews of Modern Physics 61, 1–23. — The seminal review articulating the 120-orders-of-magnitude vacuum energy discrepancy.
Carroll, S.M. (2001). “The Cosmological Constant,” Living Reviews in Relativity 4, 1. arXiv:astro-ph/0004075. — Comprehensive review of the cosmological constant from theory to observation.
Chevallier, M. & Polarski, D. (2001). “Accelerating universes with scaling dark matter,” International Journal of Modern Physics D 10, 213–224; Linder, E.V. (2003). “Exploring the expansion history of the universe,” Physical Review Letters 90, 091301. — The CPL parameterization w(a) = w<sub>0</sub> + w<sub>a</sub>(1-a).
Caldwell, R.R., Dave, R. & Steinhardt, P.J. (1998). “Cosmological Imprint of an Energy Component with General Equation of State,” Physical Review Letters 80, 1582–1585. arXiv:astro-ph/9708069. — Introduction of quintessence as a dynamical dark energy model.
Planck Collaboration (2020). “Planck 2018 results. VI. Cosmological parameters,” Astronomy & Astrophysics 641, A6. arXiv:1807.06209. — Definitive CMB constraints on dark energy parameters.
DESI Collaboration (2024). “DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations,” arXiv:2404.03002. — Latest BAO measurements showing possible hints of evolving dark energy.

← Dark Matter Back to Cosmology →

Parameter	Symbol	Planck 2018 Value	Description
Hubble constant	\(H_0\)	67.4 km/s/Mpc	Current expansion rate
Baryon density	\(\Omega_b h^2\)	0.0224	Physical baryon density parameter
CDM density	\(\Omega_c h^2\)	0.120	Physical cold dark matter density
Spectral index	\(n_s\)	0.965	Scalar perturbation tilt
Amplitude	\(A_s\)	\(2.1 \times 10^{-9}\)	Scalar perturbation amplitude at pivot
Optical depth	\(\tau\)	0.054	Reionization optical depth

Cosmology