Sachs-Wolfe Effect

Sachs-Wolfe effect is a fundamental general-relativistic mechanism generating large-angle temperature anisotropies in the cosmic microwave background (CMB) radiation, resulting from the interaction of CMB photons with spatially varying gravitational potentials Φ(x, t) at the epoch of recombination (z_* ≈ 1090) and along the photons' 13.8-billion-year trajectory to the observer. Derived in 1967 by the American relativists Rainer K. Sachs and Arthur M. Wolfe via a perturbation analysis of Einstein's field equations and the relativistic Boltzmann photon transport equation, the phenomenon is divided into two distinct physical regimes: (1) the non-integrated (ordinary) Sachs–Wolfe effect (SW), which dominates the primary CMB angular power spectrum C_ℓ on super-horizon angular scales (multipoles 2 ≤ ℓ ≲ 30, angular separations θ > 6°) and produces the famous Sachs–Wolfe plateau first detected by the COBE DMR satellite in 1992; and (2) the Integrated Sachs–Wolfe effect (ISW), a secondary anisotropy generated when gravitational potential wells evolve in time (∂Φ / ∂t ≠ 0) during the early radiation-to-matter transition or the late-time dark energy-dominated era (z ≲ 1).

Consider a primordial adiabatic density perturbation on a length scale larger than the causal Hubble horizon at the time of recombination (z_* ≈ 1090), where hydrodynamical baryon acoustic oscillations have not yet had time to operate. Why does a gravitational potential well (where the Newtonian gravitational potential is negative, Φ < 0, and the matter density is higher than average, δρ > 0) appear as a cold spot rather than a hot spot in the CMB sky?

The answer lies in the exact competition between gravitational redshift and gravitational time dilation in an expanding matter-dominated universe:

1. Gravitational Redshift (Climbing Out of the Potential Well: +Φ / c²): A photon emitted from the bottom of an overdense gravitational potential well (Φ < 0) at the surface of last scattering must climb out of that potential well to reach a distant observer, losing energy and experiencing a fractional gravitational temperature shift of (ΔT / T)_grav = +Φ / c² < 0 (a cooling effect).
2. Intrinsic Plasma Overdensity and Gravitational Time Dilation (−2Φ / 3c²): At the same time, because the overdense potential well is compressed, the photon–baryon plasma inside the well is intrinsically hotter than the cosmic mean! Specifically, for adiabatic perturbations in a matter-dominated universe (where scale factor a(t) ∝ t^2/3 and background temperature T ∝ a⁻¹ ∝ t^−2/3), general-relativistic gravitational time dilation slows local proper time inside the potential well by δt / t = +Φ / c². Observing the plasma at a slightly earlier cosmic time δt means its intrinsic temperature at decoupling is hotter by:
   • (ΔT / T)_intrinsic = ¼ (δρ_γ / ρ_γ) = −(δa / a) = −⅔ (δt / t) = −⅔ (Φ / c²) > 0.
3. Net Sachs–Wolfe Cancellation (+1/3 Φ / c²): Summing the gravitational redshift and the intrinsic temperature excess yields the famous Sachs–Wolfe formula:
   • (ΔT / T)_SW = (ΔT / T)_grav + (ΔT / T)_intrinsic = Φ / c² − ⅔ (Φ / c²) = +⅓ (Φ / c²).

Because the +1 gravitational redshift outweighs the −2/3 intrinsic heating, overdense gravitational potential wells (Φ < 0) at the surface of last scattering appear as cold spots (ΔT < 0) on large angular scales, whereas underdense primordial voids (Φ > 0) appear as warm spots! Furthermore, if primordial curvature perturbations drawn from cosmic inflation possess an almost scale-invariant Harrison–Zel'dovich–Peebles power spectrum (P_Φ(k) ∝ k^{n_s − 4} with scalar spectral index n_s ≈ 0.965), the quantity ℓ(ℓ + 1)C_ℓ / (2π) is nearly constant across low multipoles (2 ≤ ℓ ≲ 30)—producing the flat Sachs–Wolfe plateau at ~1,000 μK² measured by COBE, WMAP, and Planck.

As a CMB photon travels from the surface of last scattering (t_*) to Earth (t₀), it continually falls into and climbs out of intervening dark-matter gravitational potential wells Φ(x, t) and supervoids. In a flat, purely matter-dominated Einstein–de Sitter universe (Ω_m = 1, Ω_Λ = 0), linear density perturbations grow at the exact same rate as the cosmic scale factor (δ ∝ a(t)), which causes the Poisson equation ∇²Φ = 4πG a² ρ̄_m δ to keep the gravitational potential strictly constant in time (∂Φ / ∂t = 0). Consequently, the gravitational blueshift a photon gains when falling into a cluster potential well is 100% canceled by the gravitational redshift it loses when climbing back out—producing zero net temperature change!

However, whenever the universe is not purely matter-dominated, gravitational potentials decay or evolve while the photon is inside them, yielding the Integrated Sachs–Wolfe (ISW) effect:

• Line-of-Sight ISW Integral: (ΔT / T)_ISW(n̂) = (1 / c²) ∫_t*^t₀ [ ∂Φ(x, t) / ∂t − ∂Ψ(x, t) / ∂t ] dt = −(2 / c³) ∫₀^χ_* [ ∂Φ(χn̂, η) / ∂η ] dχ (in Newtonian gauge where metric potentials Ψ = −Φ in the absence of anisotropic stress, and η is conformal time).

Cosmologists distinguish four physical regimes of the Sachs–Wolfe and Integrated Sachs–Wolfe effects across cosmic time and spatial scale:

Why is the Late-Time Integrated Sachs–Wolfe effect considered an independent smoking gun for dark energy (or cosmic spatial curvature) alongside Type Ia supernovae and baryon acoustic oscillations?

Because the Late ISW signal (~2–5 μK rms) is buried beneath the much larger primary Sachs–Wolfe fluctuations (~30 μK rms) generated at z = 1090 on the exact same large angular scales (ℓ < 40), it cannot be isolated from the CMB temperature map alone, and its large-angle power is further limited by cosmic variance (ΔC_ℓ / C_ℓ = √[2 / (2ℓ + 1)]). In 1996, Robert G. Crittenden and Neil Turok proposed a decisive solution: cross-correlate the all-sky CMB temperature map T(n̂) with wide-area surveys of low-redshift galaxies, quasars, or AGN (z < 1.5) that trace the late-time gravitational potential wells Φ(n̂, z):

• Zero Cross-Correlation for Primary SW: Because galaxies at z ≈ 0.5 are separated by > 10,000 megaparsecs from the surface of last scattering at z = 1090, their positions are completely uncorrelated with the primary Sachs–Wolfe fluctuations.
• Positive Cross-Correlation for Late ISW (C_ℓ^Tg > 0): In a universe whose expansion is accelerating under a positive cosmological constant Λ, large-scale gravitational potential wells decay as a photon traverses them (|Φ_exit| < |Φ_entry|). Therefore, a photon falling into a z ≈ 0.5 galaxy supercluster gains a gravitational blueshift |Φ_entry| that exceeds the redshift |Φ_exit| it loses upon exit, emerging with a net positive temperature boost (ΔT_ISW > 0). Conversely, a photon traversing a decaying low-redshift supervoid (such as the Eridanus Supervoid or Boötes Void) loses more energy entering the void than it regains exiting, emerging with a net temperature decrement (ΔT_ISW < 0).
• Observational Confirmation: Combined cross-correlations of WMAP and Planck CMB temperature maps with SDSS Luminous Red Galaxies, WISE infrared galaxies, NVSS radio sources, and stacked SDSS superclusters/supervoids have detected this positive C_ℓ^Tg cross-correlation at > 4.0σ to 4.5σ significance, directly demonstrating that cosmic gravitational potentials are decaying at z < 1 due to dark energy.

1. Sachs, R. K., & Wolfe, A. M. (1967). Perturbations of a cosmological model and angular variations of the microwave background. The Astrophysical Journal, 147, 73–90. https://doi.org/10.1086/148982
2. Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., Aymon, J., Boggess, N. W., ... & Wilkinson, D. T. (1992). Structure in the COBE Differential Microwave Radiometer first-year maps. The Astrophysical Journal, 396, L1–L5. https://doi.org/10.1086/186504
3. Crittenden, R. G., & Turok, N. (1996). Looking for Λ with the Rees-Sciama effect. Physical Review Letters, 76(4), 575–578. https://doi.org/10.1103/PhysRevLett.76.575
4. Planck Collaboration, Ade, P. A. R., Aghanim, N., Arnaud, M., Ashdown, M., Aumont, J., ... & Zonca, A. (2016). Planck 2015 results. XXI. The integrated Sachs-Wolfe effect. Astronomy & Astrophysics, 594, A21. https://doi.org/10.1051/0004-6361/201525831