Questions: Cosmological Constant and Dark Energy

Question 1

The cosmological constant Λ can be equivalently described as a contribution to the geometry (left side of Einstein's equations) or as a form of energy (right side). What equation of state does it correspond to when treated as a fluid?

• A. p = 0 (pressureless dust)
• B. p = ρc²/3 (radiation)
• C. p = -ρc² (w = -1)
• D. p = -ρc²/3 (curvature-like)

Answer: (C)

Moving the Λg_μν term to the right side of the Einstein equations, it acts as a stress-energy tensor T^(Λ)_μν = -(Λc⁴/8πG)g_μν, which is that of a perfect fluid with constant energy density ρ_Λ = Λc²/(8πG) and pressure p_Λ = -ρ_Λ c². This gives equation of state parameter w = p/(ρc²) = -1. The negative pressure is the key: in the Friedmann acceleration equation, the combination ρ + 3p/c² = ρ - 3ρ = -2ρ < 0, so the cosmological constant produces gravitational repulsion.

Question 2

The cosmological constant problem arises because the observed value of Λ is much smaller than quantum field theory predicts for the vacuum energy.

• T. True
• F. False

Answer: True

Quantum field theory predicts that the vacuum has a zero-point energy density on the order of the Planck energy density: ρ_QFT ~ M_Pl c²/l_Pl³ ~ 10⁹³ g/cm³. The observed dark energy density is ρ_Λ ~ 10⁻²⁹ g/cm³ — a discrepancy of about 120 orders of magnitude. Even with more conservative cutoffs (the electroweak scale), the discrepancy is about 56 orders of magnitude. Why the vacuum energy is so extraordinarily small compared to any natural quantum field theory scale — but not exactly zero — is the cosmological constant problem, and no satisfactory solution exists.

Question 3

Explain how Type Ia supernovae observations in 1998 provided evidence for accelerating expansion and hence dark energy.

Model answer: Type Ia supernovae are 'standardizable candles' — their peak luminosity can be calibrated from their light-curve shape, allowing their absolute distance to be determined. By comparing this luminosity distance with their redshift z, the expansion history a(t) can be reconstructed. Two independent teams (the Supernova Cosmology Project and the High-z Supernova Search Team) found that distant supernovae (z ~ 0.5-1) were dimmer than expected in a decelerating universe — they were farther away than a matter-dominated model predicts. This implied the expansion has been accelerating for the past ~5 billion years, which requires a component with negative pressure (w < -1/3) dominating the energy budget. The simplest explanation is a cosmological constant with w = -1.

The 1998 discovery earned Perlmutter, Schmidt, and Riess the 2011 Nobel Prize in Physics. The result has been independently confirmed by CMB observations (Planck), baryon acoustic oscillations (SDSS, DESI), and other probes. All evidence is consistent with w = -1, though slight deviations cannot be ruled out.

Question 4

Could dark energy be something other than a cosmological constant? What would distinguish a dynamical dark energy model from Λ observationally?

Model answer: Yes — dark energy could be a dynamical scalar field (quintessence) or some other exotic component with time-varying energy density and equation of state w(z) ≠ -1. A cosmological constant has w = -1 exactly and constant energy density. A dynamical model would have w ≠ -1 or w varying with redshift. Observationally, the distinction requires measuring the expansion history H(z) and the growth rate of cosmic structure with sufficient precision to detect deviations from w = -1. Current data (CMB, supernovae, BAO) are consistent with w = -1 to within about 5%, but next-generation surveys (DESI, Euclid, Rubin/LSST) aim to measure w(z) to percent-level precision. Recent DESI BAO results (2024) hint at possible evolution of w with redshift, though the evidence is not yet conclusive.

The nature of dark energy is one of the most important open questions in physics. If w is exactly -1, dark energy is likely the cosmological constant (vacuum energy), and the cosmological constant problem is the central puzzle. If w ≠ -1 or varies with time, entirely new physics is at work.