The Integrated Sachs-Wolfe Effect

The intuition: a ball rolling through a sinking valley

Picture a marble rolling down into a valley and back up the far side. If the valley keeps its shape, the marble arrives at the far rim with exactly the speed it had at the near rim — energy in, energy out, no net gain. Now imagine the valley sinks deeper, or shallows out, while the marble is still inside it. The bookkeeping no longer balances: the marble climbs out of a hill that is not the same hill it rolled into, and it pockets (or loses) the difference.

A photon from the cosmic microwave background does exactly this on a cosmic scale. As it streams across the universe for nearly 13.8 billion years it passes through countless large-scale concentrations of matter — superclusters (potential wells) and supervoids (potential hills). Falling into a well blueshifts the photon (it gains energy); climbing out redshifts it back. If the well is static the two cancel perfectly. But if the potential evolves during the crossing — if ∂Φ/∂t ≠ 0 — the cancellation is imperfect and the photon emerges with a tiny net temperature shift. Summed (integrated) along the whole line of sight, that accumulated shift is the integrated Sachs-Wolfe effect.

The deep payoff: in a flat universe made only of matter, the large-scale potentials are frozen — they do not evolve at all — and the ISW effect is exactly zero. So a non-zero late-time ISW is a smoking gun that the universe contains something beyond matter that makes potentials decay. In our universe, that something is dark energy.

The precise mechanism and the governing integral

The full Sachs-Wolfe result (Rainer K. Sachs and Arthur M. Wolfe, 1967) decomposes the temperature anisotropy a CMB photon carries into three pieces: the gravitational redshift at the last-scattering surface, an intrinsic photon-density term, and a line-of-sight integral over the time-evolution of the metric potentials. That last term is the integrated Sachs-Wolfe contribution:

(ΔT/T)_ISW = (2 / c²) ∫ (∂Φ/∂t) dt
           = (2 / c²) ∫ (∂Φ/∂η) dη   (conformal time η)
           = (1 / c²) ∫ (Φ̇ + Ψ̇) dη   (general metric, two potentials)

Here Φ and Ψ are the Newtonian-gauge metric potentials (Φ ≈ Ψ when anisotropic stress is negligible, which holds well after recombination), and the factor of 2 in the single-potential form comes from adding the two equal contributions Φ̇ + Ψ̇ = 2Φ̇. The integral runs along the photon's path from the last-scattering surface to the observer. The key object is the time derivative of the potential: a static potential contributes nothing.

How the potential evolves is fixed by linear perturbation theory. On large scales the relation is

Φ(a) ∝ D(a) / a

where a is the cosmic scale factor and D(a) is the linear growth factor of density perturbations. In a flat matter-only (Einstein-de Sitter) universe, structure grows in lockstep with expansion, D ∝ a, so Φ ∝ a/a = constant and ∂Φ/∂t = 0. Once dark energy begins to dominate, the accelerating expansion suppresses the growth of structure: D grows more slowly than a, so D/a falls and Φ decays. A decaying overdensity potential leaves crossing photons net-blueshifted (hotter); a decaying void potential leaves them net-redshifted (colder).

The key numbers

The ISW lives in a regime of extremely small contrasts, which is exactly why detecting it is hard. The relevant scales:

• CMB temperature: T₀ = 2.7255 K (COBE/FIRAS, 1990, calibrated to ±0.0006 K) — the most perfect blackbody ever measured.
• Primary anisotropies: ΔT/T ~ 10⁻⁵, i.e. fluctuations of order 30 μK rms about the mean.
• Late-time ISW amplitude: ΔT/T ~ 10⁻⁶, a few microkelvin — about ten times smaller than the primary signal and superimposed on it on the largest angular scales.
• Angular scale: the late ISW dominates at low multipoles, ℓ ≲ 30 (angular scales ≳ 6°), because the decaying potentials are the very largest structures — superclusters and supervoids spanning hundreds of millions of light-years.
• Redshift of onset: dark energy (Ω_Λ ≈ 0.69, Ω_m ≈ 0.31 from Planck 2018) overtakes matter in the energy budget at z ≈ 0.3, roughly 3-4 billion years ago, which is when potentials begin to decay and the late ISW switches on.
• Detection significance: cross-correlation analyses cluster around 4σ; cosmic variance caps any single-universe measurement at roughly 7-8σ.
• Supervoid scale: the candidate Eridanus supervoid associated with the CMB "Cold Spot" spans ~1 billion light-years and lies near z ≈ 0.2 (Szapudi et al., 2015), with a central CMB decrement of order ~70 μK.

How it is actually observed

You cannot point a telescope at "the ISW" — a few microkelvin riding on a ~30 μK primordial signal is statistically invisible in any single CMB map. The breakthrough idea, from Robert Crittenden and Neil Turok in 1996, is cross-correlation with large-scale structure. The logic is clean:

• The decaying potentials that source the late ISW are produced by the same low-redshift matter that we can map directly with galaxy and quasar surveys.
• The primary CMB anisotropies were imprinted at z ≈ 1100 and are uncorrelated with structure at z < 2.
• So if you correlate a CMB temperature map against a low-z tracer of mass, the primary CMB averages to zero while the ISW term — which is spatially aligned with that mass — survives as a positive cross-correlation.

The first detections arrived in 2003-2004: Stephen Boughn and Robert Crittenden correlating WMAP against radio (NVSS) and X-ray (HEAO-1) source catalogs, Pablo Fosalba and collaborators using the APM galaxy survey, and Ryan Scranton and the SDSS team using Sloan photometric galaxies. A complementary, model-independent route is stacking: Granett, Neyrinck & Szapudi (2008) stacked the CMB on the locations of 50 superclusters and 50 supervoids identified in SDSS and found the predicted hot/cold pattern. Finally, Planck's 2018 analysis recovered the effect both by external cross-correlation (~4σ) and through its own internal lensing-potential reconstruction (~3σ), tying the measurement entirely to its own data.

ISW in context: who changes the CMB after last scattering

The ISW is one of several "secondary anisotropies" — distortions imprinted on the CMB long after the photons left the last-scattering surface. It helps to see where it sits among its cousins.

Effect	Physical cause	Angular scale	Signature	Probes
Late integrated SW	Decaying large-scale Φ (dark energy)	Large (ℓ ≲ 30)	±few μK, frequency-independent	Dark energy, curvature
Early ISW	Residual radiation makes Φ evolve at z ~ few hundred	ℓ ~ 200 (first peak)	Boosts first acoustic peak	Matter/radiation ratio
Rees-Sciama	Non-linear, collapsing structures evolve Φ	Small (ℓ ≳ few hundred)	Tiny, ~10⁻⁷	Non-linear structure growth
Thermal SZ	Inverse-Compton off hot cluster gas	Arcminute (clusters)	Decrement <217 GHz, excess above	Cluster gas pressure
Kinetic SZ	Doppler from bulk cluster velocity	Arcminute	±, frequency-independent	Peculiar velocities, reionization
CMB lensing	Deflection by intervening Φ	Arcminute-degree	Smears peaks, B-modes	Total matter, Σmν

Two of these — the ISW and CMB lensing — are sourced by the same gravitational potentials, which is why Planck could detect the ISW internally: it reconstructed Φ from the lensing signal and correlated it with the temperature map.

A worked estimate: how much does a supervoid cool a photon?

Let us estimate the temperature decrement a photon picks up crossing a large supervoid in the accelerating universe. We need the depth of the potential and the fractional change in that potential during transit.

The Newtonian potential of a perturbation of size R and density contrast δ is, by Poisson's equation ∇²Φ = 4πGρ̄δ,

Φ ~ (1/2) (H₀ R)² Ω_m δ        (in convenient cosmological units)

Take a supervoid with R ≈ 150 Mpc, density contrast δ ≈ −0.2 (a 20% under-density), Ω_m = 0.31, and H₀ = 70 km/s/Mpc, so H₀R ≈ (70 km/s/Mpc)(150 Mpc) = 1.05 × 10⁴ km/s, giving H₀R/c ≈ 0.035 and (H₀R/c)² ≈ 1.2 × 10⁻³. Then

Φ/c² ~ (1/2)(1.2 × 10⁻³)(0.31)(−0.2) ≈ −3.7 × 10⁻⁵   (a potential hill)

If the photon crosses while the potential decays by a fraction f ~ 0.05-0.1 of its depth (the typical fractional decay over the light-crossing time of the void in a Λ-dominated era), the ISW shift is roughly

(ΔT/T)_ISW ~ 2 × f × (Φ/c²)
           ~ 2 × 0.07 × (−3.7 × 10⁻⁵) ≈ −5 × 10⁻⁶

Converting with T₀ = 2.7255 K:

ΔT ~ (−5 × 10⁻⁶)(2.7255 K) ≈ −1.4 × 10⁻⁵ K ≈ −14 μK

So a single giant supervoid should print a cold spot of order ten microkelvin onto the CMB — the right order of magnitude for the decrement seen toward the Eridanus supervoid. This back-of-the-envelope number is exactly why no individual void is detectable above the ~30 μK primary noise without stacking many of them or cross-correlating against a survey.

Discovery: the people, papers, and missions

The foundational paper is Sachs & Wolfe (1967), "Perturbations of a cosmological model and angular variations of the microwave background," which derived how metric perturbations imprint temperature anisotropies — including the line-of-sight integral that bears their name. Rainer Sachs and Arthur Wolfe wrote it only two years after Penzias and Wilson discovered the CMB, before anyone could measure the anisotropies at all.

The non-linear, small-scale version was identified the following year by Martin Rees and Dennis Sciama (1968) — the Rees-Sciama effect. The crucial reframing of the late-time linear effect as a dark-energy probe, together with the cross-correlation detection strategy, came from Robert Crittenden and Neil Turok (1996).

Then the data arrived. COBE (1992) first measured large-scale anisotropies, which on the largest scales are dominated by the ordinary Sachs-Wolfe plateau. WMAP (launched 2001) produced the first all-sky maps clean enough for cross-correlation, enabling the 2003-2004 detections (Boughn & Crittenden; Fosalba et al.; Scranton et al.; Nolta et al.). Granett, Neyrinck & Szapudi (2008) added the stacking measurement. Planck (launched 2009; legacy release 2018) delivered the definitive ~4σ cross-correlation and the internal lensing-based detection. Looking forward, surveys like the Vera C. Rubin Observatory (LSST), Euclid, and DESI will map low-z structure precisely enough to sharpen the ISW cross-correlation and tighten dark-energy constraints.

Variants and related phenomena

• Early ISW. Just after recombination the universe is not fully matter-dominated; leftover radiation keeps the potentials evolving for a while. This boosts the temperature power spectrum near the first acoustic peak (ℓ ~ 200) and is sensitive to the matter-to-radiation ratio — a small but calculable contribution baked into every CMB likelihood.
• Late-time ISW. The dark-energy probe described here, operating at z ≲ 1 on the largest scales (ℓ ≲ 30).
• Rees-Sciama effect. The non-linear cousin: individual collapsing clusters and superclusters whose potentials evolve non-linearly imprint a tiny (~10⁻⁷) small-scale signal.
• Moving-lens / transverse-velocity effect. A potential moving across the sky also induces a dipole-like temperature shift, a kinematic relative of the ISW probed by future surveys.
• Curvature-driven ISW. Spatial curvature, like dark energy, makes large-scale potentials evolve. An open universe would produce a late ISW even without dark energy; combining the ISW with other probes helps break the geometry-versus-dark-energy degeneracy.

Common misconceptions and subtleties

• "Photons heat up just by falling into wells." No. A static well gives back exactly what it took: blueshift in, redshift out, net zero. The ISW exists only because the potential changes during the crossing. Without time evolution there is no effect, period.
• "The ISW proves dark energy by itself." It is strong corroborating evidence, but spatial curvature can also make potentials evolve. The clean statement is that a detected late-time ISW rules out a flat, matter-only universe; combined with the measured near-flatness from the CMB acoustic peaks, that points to dark energy.
• "It is the same as the ordinary Sachs-Wolfe plateau." The ordinary SW term is the gravitational redshift imprinted at last scattering — a boundary term. The integrated SW is the line-of-sight accumulation along the entire path afterward. They are different pieces of the same 1967 calculation.
• "It is a frequency-dependent distortion like the SZ effect." No. The ISW shifts the photon temperature achromatically — it looks identical at every CMB frequency — which is part of why it cannot be cleaned out by multi-frequency separation the way the thermal SZ decrement can.
• "More data will eventually give a 20σ detection." Cosmic variance forbids it. The late ISW lives on the largest scales, where we have only a few independent samples (one sky, low ℓ). Even a perfect, noiseless experiment tops out near 7-8σ.