13.4 Implications of dark energy for the universe's fate

Dark Energy and the Universe's Fate

Properties of dark energy

Dark energy is the name given to whatever is causing the universe's expansion to accelerate. It makes up roughly 70% of the total energy content of the universe, yet its fundamental nature remains unknown. What we can do is describe its behavior through two key quantities: its energy density and its equation of state.

• Dark energy density ($\rho_{DE}$)
  • A positive value drives accelerated expansion of the universe
  • If this density stays constant over time, the universe expands exponentially, approaching what's called a de Sitter universe
  • If the density increases over time, expansion accelerates without bound, eventually overpowering gravity at every scale and tearing apart galaxies, stars, and even atoms in a Big Rip scenario
• Equation of state parameter ($w$) relates dark energy's pressure ($P$) to its energy density ($\rho$):

$P = w\rho$

The value of $w$ is what really determines the universe's fate:

• $w = -1$: cosmological constant. The universe accelerates and asymptotically approaches a de Sitter state (exponential expansion).
• $w < -1$: phantom energy. Energy density grows over time, leading to a Big Rip at a finite time in the future.
• $-1 < w < -1/3$: accelerated expansion still occurs, but the expansion rate evolves differently than in the de Sitter case. The acceleration gradually changes rather than locking into a fixed exponential rate.
• $w > -1/3$: expansion decelerates. This would not produce the accelerated expansion we actually observe, so it's ruled out by current data.

Role of the cosmological constant

The cosmological constant ($\Lambda$) is the simplest model of dark energy. It corresponds to a constant energy density filling all of space, with $w = -1$ exactly.

Einstein originally introduced $\Lambda$ into his field equations to produce a static universe, which was the prevailing assumption before Hubble's discovery of cosmic expansion. He later dropped it, but the discovery of accelerating expansion in 1998 brought it back as the leading explanation for dark energy.

A positive cosmological constant drives the following sequence:

1. The universe undergoes continued accelerated expansion.
2. Over time, the expansion approaches a de Sitter state, where the Hubble parameter settles to a constant value:

$H = \sqrt{\frac{\Lambda}{3}}$

1. As expansion accelerates, distant galaxies and galaxy clusters recede faster than light can travel toward us. They cross the cosmic event horizon and become permanently unobservable. Eventually, only gravitationally bound structures (our Local Group) remain accessible.

This is sometimes called the "heat death" trajectory: the observable universe becomes increasingly empty, cold, and dilute.

Observational constraints on dark energy

Three independent lines of evidence constrain dark energy's properties, and the fact that they all converge on the same answer is what makes the case so strong:

• Type Ia supernovae serve as standard candles. Their known intrinsic brightness lets you measure luminosity distances to high-redshift galaxies. This is how accelerating expansion was first discovered in 1998.
• Cosmic microwave background (CMB) anisotropies constrain the total energy density and geometry of the universe. Combined with matter density measurements, they pin down how much dark energy there must be.
• Baryon acoustic oscillations (BAO) are a characteristic scale imprinted in the distribution of galaxies. They act as a standard ruler for measuring the expansion history at different epochs.

Current best-fit values from combined datasets:

• Dark energy density parameter: $\Omega_{\Lambda} \approx 0.7$ (about 70% of the universe's total energy budget)
• Equation of state: $w = -1.03 \pm 0.03$

That value of $w$ is consistent with $-1$, meaning the data so far can't distinguish dark energy from a pure cosmological constant. The implications are:

• Accelerated expansion will continue indefinitely
• The universe will asymptotically approach a de Sitter-like state
• Structures beyond the Local Group (including the Virgo Cluster and everything farther) will eventually redshift beyond detectability and cross our event horizon
• Whether dark energy is truly a cosmological constant or something more dynamic (such as quintessence, a slowly evolving scalar field, or phantom energy with $w < -1$) remains an open question. Next-generation surveys like DESI and the Vera Rubin Observatory are designed to measure $w$ with enough precision to potentially distinguish between these models.