dark energy

1. Phenomenon: The Accelerating Universe

Observations of Type Ia supernovae (SNe Ia) show that the cosmic expansion is accelerating.
In a homogeneous and isotropic FLRW spacetime,

ds^2 = -dt^2 + a^2(t)\left[\frac{dr^2}{1 - k r^2} + r^2 d\Omega^2\right],

the dynamics obey the Friedmann equations:

H^2 \equiv \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\,\rho - \frac{k}{a^2} + \frac{\Lambda}{3},

\frac{\ddot a}{a} = -\frac{4\pi G}{3}\,(\rho + 3p) + \frac{\Lambda}{3}.

Acceleration $\ddot a>0$ requires $\rho + 3p < \Lambda/(4\pi G)$ or, for a fluid with $p = w\rho$ , an effective $w < -1/3$ .

2. Equation of State and Parameterizations

Define the dark-energy equation-of-state (EOS) parameter:

w \equiv \frac{p_{\rm DE}}{\rho_{\rm DE}}.

Cosmological constant: $w=-1$ , $\rho_\Lambda = \text{const}$ .
Quintessence (canonical scalar $\phi$ ): $w \in (-1,1)$ and can vary with time.
Phantom: $w<-1$ (violates NEC).
K-essence / Horndeski / beyond-Horndeski: generalized kinetic terms / modified gravity sectors.

A common phenomenological fit is CPL:

w(a) = w_0 + w_a (1-a) \quad \Longleftrightarrow \quad w(z) = w_0 + \frac{w_a z}{1+z}.

Then the DE density evolves as

\rho_{\rm DE}(z) = \rho_{\rm DE,0} \,(1+z)^{3(1+w_0+w_a)}\,\exp\!\left[-\frac{3w_a z}{1+z}\right].

3. Background Observables

3.1 Hubble Expansion

For a flat universe ( $k=0$ ) with matter, radiation, and DE,

\frac{H^2(z)}{H_0^2} = \Omega_{r0}(1+z)^4 + \Omega_{m0}(1+z)^3 + \Omega_{\rm DE,0}\, \frac{\rho_{\rm DE}(z)}{\rho_{\rm DE,0}} .

3.2 Distances (SNe Ia, BAO, CMB)

The comoving distance:

\chi(z) = \int_0^z \frac{dz'}{H(z')}.

Luminosity distance and distance modulus:

d_L(z) = (1+z)\,\chi(z), \qquad \mu(z) = 5\log_{10}\!\left(\frac{d_L(z)}{10\,{\rm pc}}\right).

BAO provide a standard ruler (sound horizon $r_d$ ) via angles and redshifts; a common isotropic proxy:

D_V(z) \equiv \left[ (1+z)^2 D_A^2(z)\,\frac{cz}{H(z)} \right]^{1/3},

with $D_A(z)=\chi(z)/(1+z)$ .

CMB primarily constrains the distance to last scattering and early-time physics, helping to break degeneracies in late-time parameters.

4. Growth of Structure

In (sub-horizon, linear) Newtonian gauge and GR, matter perturbations $\delta \equiv \delta\rho_m/\rho_m$ follow

\ddot\delta + 2H\dot\delta - 4\pi G \rho_m \delta = 0.

Define the growth rate $f \equiv d\ln D/d\ln a$ where $\delta \propto D(a)$ .
A useful fit in GR+ $\Lambda$ CDM is $f \simeq \Omega_m(a)^\gamma$ with $\gamma \approx 0.55$ .
Redshift-space distortions measure $f\sigma_8(z)$ , testing the consistency of expansion vs growth (and modified gravity).

Modified gravity often appears as scale-/time-dependent effective couplings:

k^2\Psi = -4\pi G_{\rm eff}(k,a)\,a^2 \rho_m \Delta, \qquad \eta \equiv \frac{\Phi}{\Psi} \neq 1,

where $\Phi,\Psi$ are Bardeen potentials. Deviations from $G_{\rm eff}=G$ or $\eta=1$ signal physics beyond GR/ $\Lambda$ .

5. Microscopic Models (Sketch)

5.1 Cosmological Constant $\Lambda$

Vacuum energy with $w=-1$ .
The old CC problem: naive QFT estimates overshoot the observed value by $\sim 10^{60-120}$ .

5.2 Quintessence

Canonical scalar with Lagrangian

\mathcal{L} = -\frac12 (\partial \phi)^2 - V(\phi), \quad \rho_\phi = \frac12 \dot\phi^2 + V, \quad p_\phi = \frac12 \dot\phi^2 - V,

w_\phi = \frac{\frac12 \dot\phi^2 - V}{\frac12 \dot\phi^2 + V}.

Tracker / thawing potentials (e.g., $V \propto \phi^{-\alpha}$ , $V \propto e^{-\lambda \phi/M_{\rm Pl}}$ ) can ease coincidence.

5.3 Modified Gravity (MG)

E.g., $f(R)$ gravity with action $S = \int d^4x \sqrt{-g}\, \frac{M_{\rm Pl}^2}{2} f(R) + \dots$
Equivalent to a scalar-tensor theory (extra scalar d.o.f.); screening mechanisms (chameleon, Vainshtein) recover GR locally.
Horndeski / DHOST give the most general second-order scalar-tensor actions; GW170817 constraints enforce $c_T \simeq c$ today, pruning parameter space.

6. Tensions and Consistency Checks

Hubble tension: early-time (CMB) inference of $H_0$ vs late-time (distance ladder) measurements disagree.
DE dynamics or early dark energy are among proposed explanations, but a coherent fit with all data is nontrivial.
$S_8$ tension: weak-lensing–inferred $S_8 \equiv \sigma_8 \sqrt{\Omega_m/0.3}$ sometimes sits lower than $\Lambda$ CDM+CMB best-fits; could hint at modified growth or systematics.

A robust program is to test background (distances) and perturbations (growth, lensing) jointly.

7. Inference Pipeline (Practical Notes)

Given a parameter vector $\theta = \{\Omega_{m0}, \Omega_{b0}, H_0, w_0, w_a, \sigma_8, n_s, \dots\}$ :

Compute $H(z|\theta)$ and distances $\{\chi, D_A, d_L\}$ .
Integrate linear growth $D(a|\theta)$ or use $f \simeq \Omega_m^\gamma$ .
Build likelihoods for SNe Ia (distance moduli), BAO ( $D_A$ , $H$ , $D_V$ ), CMB (distance to last scattering + priors), RSD ( $f\sigma_8$ ), WL ( $S_8$ ).
Sample posteriors; test internal consistency (e.g., growth vs expansion).

8. Status

Data so far are largely consistent with $\Lambda$ CDM ( $w=-1$ ), but mild deviations remain allowed, and a few tensions persist.
Next-gen surveys (Euclid, Rubin/LSST, Roman, CMB-S4, DESI) will tighten $w_0, w_a$ and stress-test GR via growth and lensing.

Key question: Is cosmic acceleration a new energy component or a sign that gravity itself changes on large scales?