14.3 Cosmic acceleration and dark energy models

Observational Evidence and Dark Energy Concepts

The universe isn't just expanding; it's expanding at an accelerating rate. This discovery, confirmed in the late 1990s through observations of distant supernovae, upended the expectation that gravity should be gradually slowing the expansion down. Dark energy is the name given to whatever is driving this acceleration, and it accounts for roughly 68% of the total energy content of the universe.

Understanding dark energy requires both solid observational evidence and theoretical models that attempt to explain its nature. This section covers the key probes used to measure cosmic acceleration, the physical properties attributed to dark energy, and the leading models proposed to explain it.

Evidence for Universe Expansion

Type Ia Supernovae are the original and most direct evidence for cosmic acceleration. These explosions occur in binary systems when a white dwarf accretes enough mass to reach the Chandrasekhar limit, producing a thermonuclear detonation with a well-characterized peak luminosity. After applying light-curve corrections (the Phillips relation), they function as standardizable candles: you know their intrinsic brightness, so measuring their apparent brightness gives you their luminosity distance. In 1998, two independent teams (the Supernova Cosmology Project and the High-z Supernova Search Team) found that distant Type Ia supernovae at high redshifts appeared dimmer than expected in a matter-dominated, decelerating universe. The conclusion: the expansion rate has been increasing.

Note: Cepheid variables serve a different purpose here. They aren't used to detect acceleration directly but rather to calibrate the local distance ladder that anchors the Type Ia supernova measurements.

Baryon Acoustic Oscillations (BAO) provide an independent geometric probe. In the early universe, pressure waves (sound waves) propagated through the hot plasma of baryons and photons. When the universe cooled enough for atoms to form (recombination, at $z \approx 1100$), these waves froze in place, leaving a characteristic imprint: a slight excess of galaxy pairs separated by about 150 Mpc (comoving) today. This fixed physical scale acts as a standard ruler. By measuring the angular and redshift-space size of this feature in galaxy surveys at different epochs, you can track how the expansion rate has evolved over time. Results consistently support accelerating expansion.

Cosmic Microwave Background (CMB) observations from WMAP and Planck measure the angular size of temperature fluctuations imprinted at recombination. The position of the first acoustic peak in the CMB power spectrum is sensitive to the total energy density and geometry of the universe. CMB data alone constrain the universe to be spatially flat ($\Omega_{\text{total}} \approx 1$). Since matter (both baryonic and dark) accounts for only about 32% of the critical density, the remaining ~68% must come from something else. Combined with supernova and BAO data, the CMB strongly supports a dark-energy-dominated, accelerating universe.

Large-Scale Structure Surveys such as the Sloan Digital Sky Survey (SDSS) map the three-dimensional distribution of galaxies. The clustering patterns, growth rate of structure, and the BAO signal embedded in these surveys all provide additional constraints on the expansion history and the properties of dark energy.

Dark Energy's Role in Expansion

Dark energy is characterized by a few distinctive physical properties:

• It permeates all of space with a roughly uniform energy density. Unlike matter or radiation, it does not clump under gravity or dilute as the universe expands (at least in the simplest models).
• It exerts negative pressure. In general relativity, both energy density and pressure contribute to gravity. Negative pressure produces a repulsive gravitational effect, which is what drives the accelerating expansion.
• It dominates at late times. Radiation density falls as $\rho_r \propto a^{-4}$ and matter density as $\rho_m \propto a^{-3}$, where $a$ is the scale factor. If dark energy density stays roughly constant, it inevitably overtakes matter and radiation as the universe grows. The transition to dark-energy domination occurred at roughly $z \approx 0.4$.

The behavior of dark energy is encoded in its equation of state parameter:

$w = \frac{P}{\rho c^2}$

For the expansion to accelerate, the Friedmann equations require $w < -1/3$. The more negative $w$ is, the stronger the repulsive effect.

Models of Dark Energy

Several theoretical frameworks attempt to explain what dark energy actually is:

Cosmological Constant ($\Lambda$) is the simplest and currently best-fitting model. Einstein originally introduced $\Lambda$ into his field equations, and it corresponds to a constant vacuum energy density with $w = -1$ exactly, at all times. The $\Lambda$CDM model (cosmological constant + cold dark matter) fits essentially all current observational data. The major theoretical problem is the cosmological constant problem: quantum field theory predicts a vacuum energy density roughly $10^{120}$ times larger than what's observed. Why the actual value is so extraordinarily small (but not zero) remains unexplained.

Quintessence models replace the constant $\Lambda$ with a dynamical scalar field that slowly rolls down a potential. In these models, the energy density and equation of state $w$ evolve over cosmic time, with $w$ generally satisfying $-1 < w < -1/3$. One motivation for quintessence is that a time-varying field might naturally reach a small energy density today through "tracker" solutions, potentially easing the fine-tuning problem. However, no compelling particle physics candidate for the quintessence field has been identified.

Modified Gravity Theories take a different approach entirely: rather than adding a new energy component, they modify Einstein's general relativity on cosmological scales. Examples include $f(R)$ gravity (where the Ricci scalar $R$ in the Einstein-Hilbert action is replaced by a function $f(R)$) and brane-world models from extra-dimensional physics. These theories must reproduce general relativity's well-tested predictions on solar system scales while producing acceleration on cosmological scales, which is a significant constraint.

Phantom Energy models allow $w < -1$, meaning the dark energy density actually increases as the universe expands. This leads to increasingly dramatic consequences over time, culminating in a Big Rip: the expansion rate diverges in finite time, eventually tearing apart galaxies, stars, planets, and even atoms. Current data are consistent with $w = -1$ but do not yet rule out phantom behavior at high confidence.

Implications of Dark Energy

The nature of dark energy determines the ultimate fate of the universe:

• Big Freeze ($w = -1$): The universe expands forever at an accelerating rate. Structures become increasingly isolated, stars burn out, and the universe approaches maximum entropy (heat death).
• Big Rip ($w < -1$): Phantom energy causes the expansion rate to diverge. All bound structures are eventually torn apart in finite time.
• Big Crunch: If dark energy were to weaken or change sign in the future, gravitational attraction could eventually halt and reverse the expansion, leading to recollapse. This requires $w$ to evolve significantly from its current value.

Observational challenges in pinning down dark energy are substantial. Its gravitational effects are subtle compared to matter, and different combinations of cosmological parameters can produce similar observational signatures (parameter degeneracies). Breaking these degeneracies requires combining multiple independent probes across a wide range of redshifts.

Next-generation surveys are designed to do exactly this:

• Euclid (ESA) and the Vera C. Rubin Observatory's LSST will map billions of galaxies, measuring weak gravitational lensing, BAO, and galaxy clustering with unprecedented precision.
• These surveys will combine supernova distances, BAO measurements, and weak lensing to constrain $w$ and test whether it varies with time (parameterized as $w(a) = w_0 + w_a(1-a)$).

On the theoretical side, dark energy connects to deep open questions in fundamental physics. The cosmological constant problem sits at the intersection of quantum field theory and general relativity. Some researchers invoke anthropic reasoning within a multiverse framework: if different regions of a multiverse have different values of $\Lambda$, we necessarily observe a value compatible with the formation of structure and observers. Whether this constitutes an explanation or simply restates the problem remains actively debated.