Dark Energy

Introduction

Dark energy is a mysterious form of energy that permeates all of space and causes the expansion of the universe to accelerate. Discovered in 1998 through observations of distant Type Ia supernovae, dark energy comprises approximately 68% of the total energy density of the universe.

$$\Omega_\Lambda \approx 0.68, \quad \Omega_m \approx 0.32, \quad \Omega_r \ll 1$$

1. Discovery and Evidence

Supernova Cosmology

Type Ia supernovae serve as standardizable candles. The distance modulus is:

$$\mu = m - M = 5\log_{10}(d_L/10\text{ pc})$$

where $d_L(z)$ is the luminosity distance. For flat ΛCDM:

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

Key Result (1998)

High-redshift SNe Ia appeared ~25% fainter than expected in a matter-dominated universe, indicating accelerated expansion. The deceleration parameter:

$$q_0 = -\frac{\ddot{a}a}{\dot{a}^2}\bigg|_{t=t_0} = \frac{\Omega_m}{2} - \Omega_\Lambda \approx -0.52$$

Negative $q_0$ means acceleration!

Additional Evidence

  • • CMB: Flat universe ($\Omega_{\text{tot}} = 1$) + matter density requires dark energy
  • • BAO: Standard ruler measurements confirm accelerated expansion
  • • Large-scale structure: Growth rate slower than matter-only universe
  • • Age of universe: Old globular clusters require dark energy for consistency

2. The Cosmological Constant

Einstein Field Equations with Λ

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$

The cosmological constant $\Lambda$ has dimensions of (length)⁻². In terms of energy density:

$$\rho_\Lambda = \frac{\Lambda c^2}{8\pi G} \approx 6 \times 10^{-10} \text{ J/m}^3$$

Equation of State

The cosmological constant corresponds to a perfect fluid with equation of state:

$$w = \frac{p}{\rho} = -1$$

This negative pressure drives acceleration. The Friedmann equation becomes:

$$H^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3} - \frac{k}{a^2}$$

Vacuum Energy Interpretation

In quantum field theory, the vacuum has energy density from zero-point fluctuations:

$$\rho_{\text{vac}} = \sum_k \frac{1}{2}\hbar\omega_k = \int_0^{\Lambda_{\text{cutoff}}} \frac{k^3dk}{2\pi^2}\hbar\omega_k$$

With Planck-scale cutoff $\Lambda_{\text{cutoff}} \sim \ell_P^{-1}$:

$$\rho_{\text{vac}} \sim \frac{\hbar c}{\ell_P^4} \sim 10^{113} \text{ J/m}^3$$

This is 122 orders of magnitude larger than the observed dark energy density!

3. The Cosmological Constant Problem

Why is the observed vacuum energy density so incredibly small compared to theoretical predictions? This is considered one of the worst fine-tuning problems in physics.

The Fine-Tuning

$$\frac{\rho_{\Lambda,\text{obs}}}{\rho_{\text{Planck}}} \sim 10^{-123}$$

Every contribution to the vacuum energy (QED, QCD, electroweak, etc.) must cancel to ~120 decimal places.

The Coincidence Problem

Why is $\rho_\Lambda \sim \rho_m$ today? Matter density evolves as $\rho_m \propto a^{-3}$while $\rho_\Lambda = \text{const}$:

$$\frac{\rho_\Lambda}{\rho_m} = \frac{\Omega_\Lambda}{\Omega_m}(1+z)^3 \sim 2(1+z)^3$$

We happen to live at the special epoch when these are comparable. At earlier times, matter dominated; at later times, dark energy will dominate completely.

Proposed Solutions

  • • Anthropic principle: Only universes with small Λ form structures and observers
  • • Supersymmetry: Boson/fermion contributions cancel (but SUSY must be broken)
  • • Multiverse: Landscape of vacua with different Λ values
  • • Modified gravity: Dark energy is geometric, not material

4. Dynamical Dark Energy Models

Equation of State Parameter w(z)

Generalize beyond $w = -1$:

$$w(z) = \frac{p_{DE}(z)}{\rho_{DE}(z)}$$

Common parameterizations:

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

Current constraints (Planck + SNe + BAO): $w_0 = -1.03 \pm 0.03$, $w_a = -0.3^{+0.5}_{-0.7}$

Quintessence

A dynamical scalar field $\phi$ with potential $V(\phi)$. The action is:

$$S = \int d^4x\sqrt{-g}\left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - V(\phi)\right]$$

Energy density and pressure:

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi), \quad p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi)$$

Equation of state:

$$w_\phi = \frac{\frac{1}{2}\dot{\phi}^2 - V(\phi)}{\frac{1}{2}\dot{\phi}^2 + V(\phi)}$$

For $w \approx -1$, need $\dot{\phi}^2 \ll V(\phi)$ (slow-roll).

Example Potentials

$$V(\phi) = M^4\left(\frac{\phi}{\phi_0}\right)^{-\alpha} \quad \text{(Inverse power-law)}$$
$$V(\phi) = M^4 e^{-\lambda\phi/M_P} \quad \text{(Exponential)}$$
$$V(\phi) = M^4\left[1 + \cos\left(\frac{\phi}{f}\right)\right] \quad \text{(Pseudo-Nambu-Goldstone)}$$

K-essence

Non-canonical kinetic term:

$$\mathcal{L} = K(X, \phi), \quad X = -\frac{1}{2}\partial_\mu\phi\partial^\mu\phi$$

Phantom Energy

Models with $w < -1$. Lead to "Big Rip" singularity at finite time:

$$a(t) \to \infty \text{ as } t \to t_{\text{rip}} < \infty$$

All bound structures (galaxies, stars, atoms) eventually torn apart.

5. Modified Gravity Theories

f(R) Gravity

Generalize the Einstein-Hilbert action:

$$S = \frac{1}{16\pi G}\int d^4x\sqrt{-g}\,f(R)$$

Modified field equations:

$$f'(R)R_{\mu\nu} - \frac{1}{2}f(R)g_{\mu\nu} - \nabla_\mu\nabla_\nu f'(R) + g_{\mu\nu}\Box f'(R) = 8\pi GT_{\mu\nu}$$

Example: $f(R) = R + \alpha R^2$ provides accelerated expansion.

Scalar-Tensor Theories (Brans-Dicke)

$$S = \int d^4x\sqrt{-g}\left[\frac{\phi R}{16\pi G} - \frac{\omega_{BD}}{16\pi G\phi}\partial_\mu\phi\partial^\mu\phi - V(\phi)\right]$$

The gravitational "constant" becomes a dynamical field $G_{\text{eff}} = G/\phi$.

DGP Model (Dvali-Gabadadze-Porrati)

Our 4D universe is a brane embedded in 5D bulk spacetime. Gravity leaks into extra dimension at large scales:

$$H^2 = \frac{8\pi G}{3}\rho + \frac{H}{r_c}$$

where $r_c$ is the crossover scale. Self-acceleration without dark energy!

6. Observational Constraints

Distance Measurements

  • • SNe Ia: Probe $d_L(z)$ out to $z \sim 2$
  • • BAO: Standard ruler at sound horizon scale $r_s \approx 150$ Mpc
  • • CMB: Angular diameter distance to last scattering $z \sim 1100$

Growth Rate Measurements

The growth rate of structure:

$$f(z) = \frac{d\ln\delta}{d\ln a} \approx \Omega_m(z)^\gamma$$

where $\gamma \approx 0.55$ for ΛCDM. Modified gravity models predict different $\gamma$.

Current Best Fit (Planck 2018)

$$\Omega_\Lambda = 0.6847 \pm 0.0073$$
$$w = -1.028 \pm 0.031 \quad \text{(constant } w \text{)}$$

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

7. The Future of the Universe

Scenario 1: ΛCDM ($w = -1$)

Exponential expansion continues forever:

$$a(t) \propto e^{H_\Lambda t}, \quad H_\Lambda = \sqrt{\frac{\Lambda}{3}}$$

All distant galaxies eventually recede beyond cosmic horizon. Universe becomes cold, dark, and empty. "Heat death" on timescale ~10¹⁰⁰ years.

Scenario 2: Phantom Energy ($w < -1$)

Big Rip singularity at finite time:

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

For $w = -1.5$, rip occurs in ~20 billion years. Galaxies torn apart, then stars, planets, atoms, nuclei.

Scenario 3: Quintessence ($-1 < w < -1/3$)

Accelerated expansion but slower than ΛCDM. Dark energy density may decrease over time.

Event Horizon

In accelerating universe, there's a cosmological event horizon at comoving distance:

$$r_h = a\int_t^\infty \frac{dt'}{a(t')} = \int_0^{a_{\max}/a_0} \frac{da'}{a'H(a')}$$

For de Sitter space: $r_h = c/H_\Lambda \approx 16$ Gly. We can never observe events beyond this!

8. Open Questions and Future Missions

Fundamental Questions

  • • Is dark energy truly a cosmological constant, or does $w$ evolve?
  • • Why is $\rho_\Lambda$ so small yet non-zero?
  • • Why does dark energy dominate now (coincidence problem)?
  • • Is dark energy related to quantum vacuum energy?
  • • Could dark energy be explained by modified gravity?

Future Experiments

  • • Euclid (ESA): 3D galaxy survey, weak lensing, BAO out to $z \sim 2$
  • • LSST/Vera Rubin: 10-year survey, billions of galaxies, improved SNe statistics
  • • DESI: Spectroscopic survey, precise BAO and growth rate measurements
  • • CMB-S4: Next-generation CMB polarization, improved lensing
  • • Roman Space Telescope: High-precision SNe Ia, weak lensing, BAO

Goal: Measure $w(z)$ to ~1% precision and $w_a$ to ~10%, discriminating between models.