Sachs-Wolfe Effect

Three components of the gravitational imprint

The Sachs-Wolfe effect is not one phenomenon but a family of three, each contributing to the temperature pattern we observe on the cosmic microwave background sky:

1. Ordinary (local) Sachs-Wolfe — a single imprint at the surface of last scattering. The photon decouples inside a gravitational potential Φ and must climb out to reach us; what we measure is its redshifted energy plus a correction for the gravitationally dilated emission temperature. In the matter era this gives ΔT/T = (1/3) Φ/c².
2. Late-time integrated Sachs-Wolfe (ISW) — an accumulated, line-of-sight effect. As the photon travels for 13.8 billion years it falls into and climbs out of countless potential wells. For a static potential the two halves cancel exactly. They don't cancel when the potential is evolving, and during the dark-energy era potentials decay. The result is a net blueshift in the direction of overdensities.
3. Early ISW — the same line-of-sight integral, but accumulated shortly after recombination when the universe still contained a non-trivial radiation component. It boosts power around the first acoustic peak (l ≈ 50–200) and contains information about the matter-radiation equality redshift.

The three live at different angular scales, have different signs, and probe different physics. Pulling them apart cleanly is one of the achievements of modern CMB analysis.

The ordinary Sachs-Wolfe formula

Imagine a CMB photon emitted from inside a gravitational potential well of depth Φ at the surface of last scattering, redshift z ≈ 1100. The naive expectation is that climbing out of a well of depth Φ costs an energy Φ per unit mass-equivalent, so the observed temperature shift would be ΔT/T = −Φ/c². The actual answer is one-third of this, with the opposite intuition built in. The careful relativistic accounting goes:

ΔT/T (observed) = (intrinsic δT/T at emission)  +  (gravitational redshift exiting well)
                = +(2/3)(Φ/c²)                  +  (−Φ/c²)
                = −(1/3)(Φ/c²)            (matter era)

The intrinsic-temperature term is the surprise. Clocks at the bottom of a well tick slowly relative to a distant observer, so the locally measured emission temperature corresponds to a slightly different scale factor — and in the matter-dominated era, where δ ∝ a, the density contrast that produced Φ has had longer to grow inside the well. Sachs and Wolfe (1967) worked out the algebra in linear perturbation theory and found a remarkable simplification: in the matter era, the two large terms partially cancel and leave the clean factor of one-third.

The sign is also surprising. A photon climbing out of a well is redshifted (negative ΔT/T proportional to Φ<0, since gravitational potentials are conventionally negative in wells). So overdensities at recombination appear as cold spots on the CMB at large scales, not hot ones. The Sachs-Wolfe plateau in the CMB temperature power spectrum at l < 50 is the imprint of the primordial potential field, with this sign convention baked in.

Why ordinary SW dominates the lowest multipoles

At recombination, the sound horizon — the distance acoustic waves in the baryon-photon plasma can have travelled since the Big Bang — corresponds to about one degree on the sky today, or multipole l ≈ 200. Above that scale, the plasma had time to oscillate; the resulting acoustic peaks dominate the CMB power spectrum from l ≈ 200 upward.

Below the sound horizon scale, no acoustic dynamics had time to act. The only thing the photon-baryon fluid has done at those wavelengths is sit in (or out of) whatever primordial potential the inflation set up. The CMB temperature there is therefore a near-direct map of those primordial potentials, modulated by the (1/3) Φ/c² Sachs-Wolfe factor. The flat plateau visible in any CMB temperature power-spectrum plot at l < 50 is exactly this region. Its amplitude pins down the primordial amplitude As; its near-scale-invariance (slight tilt) pins down ns. Together they are the two most important inflationary observables we have.

The integrated Sachs-Wolfe effect — physics

The line-of-sight integral for the temperature anisotropy in linear theory contains the term

(ΔT/T)_ISW = (2/c²) ∫ Φ̇(x, η) dη                 (along photon path)

where η is conformal time and Φ̇ is the time derivative of the gravitational potential at the photon's location. The key feature is the time derivative. For a static potential, Φ̇ = 0 and the integral is zero — the blueshift the photon picks up falling in is exactly cancelled by the redshift climbing out. This is not an approximation; it is exact in linear theory for a non-evolving Φ.

In a pure matter (Einstein-de Sitter) universe, linear potentials are also exactly constant in time. The growth of density perturbations δ ∝ a is precisely cancelled by the dilution of the metric, leaving Φ time-independent on linear scales. So in EdS, the ISW signal is exactly zero. Detecting ISW is therefore equivalent to detecting departures from a pure matter cosmology — and the cleanest such departure is dark energy.

Why dark energy makes potentials decay

When dark energy starts to dominate, around redshift z ≈ 0.7 in our universe, the expansion rate H(z) grows faster than it would in a matter-only model. Density perturbations, which had been growing as δ ∝ a, slow their growth: in a Λ-dominated universe the growth function D(a) freezes asymptotically to a finite value. The metric, however, keeps diluting because the universe keeps expanding (in fact accelerating). The result is that Φ ~ δ/a starts to decrease — potentials decay.

A photon entering an overdense region (Φ < 0) during this era falls in deeper than it climbs out, because the well shallowed in the time it took the photon to cross. It gains a net blueshift — the line of sight toward a galaxy cluster, say, is slightly hotter than average. The same logic in reverse applies to voids: photons leaving voids exit a less-shallow ridge than they entered, again giving a hot patch in the direction of the void. ISW maps therefore correlate positively with both clusters and voids — the same potential decay, opposite signs of δ.

Detecting ISW via cross-correlation

The ISW contribution to the CMB temperature power spectrum is small — about 5% of the total at the lowest multipoles and effectively invisible in CMB data alone, drowned by cosmic variance and primordial fluctuations. The trick that makes it detectable is cross-correlation with a tracer of the underlying potential field — namely, the galaxy distribution on large scales.

If you weight the CMB temperature by the local galaxy number density on the sky, regions of large-scale overdensity (where potentials are decaying in a Λ-CDM cosmology) should correlate positively with hot CMB patches. The amplitude of that correlation depends on the matter-power normalisation and on Ω_Λ; the very existence of the signal at all argues against Einstein-de Sitter.

Year	Authors	Tracers	Significance
2004	Boughn & Crittenden	WMAP × HEAO-1 X-ray, NVSS radio	~2.5σ (first detection)
2004	Nolta et al.	WMAP × NVSS	~3σ
2004	Fosalba, Gaztañaga & Castander	WMAP × SDSS galaxies	~3σ
2008	Giannantonio et al.	WMAP × 6 tracer catalogues	~4.5σ combined
2014	Planck collaboration	Planck × NVSS, SDSS, WISE	~3σ (CMB lensing route ~4σ)
2016	Stölzner et al.	Planck × multi-tracer Bayesian	~5σ stacked

The community-accepted "ISW detection" headline number is about 4σ when several tracer catalogues are combined — modest by particle-physics standards, but striking given that the signal exists only if Φ̇ ≠ 0 on large scales. Detecting it is detecting a piece of dark energy. Detecting the same signal across multiple independent tracer types (X-ray, radio, optical, infrared) rules out essentially every systematic-error explanation.

Worked example: the amplitude of the SW plateau

The Sachs-Wolfe formula relates large-scale CMB anisotropies to the primordial gravitational potential. In the matter era,

ΔT/T = (1/3) Φ/c²

For typical inflation-generated potentials of amplitude δΦ/c² ~ 3 × 10⁻⁵ on horizon scales (set by the inflationary spectrum normalisation), the Sachs-Wolfe formula predicts

δT/T ~ 1 × 10⁻⁵
δT  ~ (1 × 10⁻⁵) × 2.725 K ≈ 30 μK

This is exactly the amplitude of the temperature fluctuations COBE detected in 1992 and that Planck pinned down to high precision. The "thirty micro-Kelvin plateau" at low multipoles is the Sachs-Wolfe effect, calibrating one number — the primordial power-spectrum amplitude — that propagates into nearly every cosmological constraint we have.

Early ISW and the radiation tail

Strictly, matter-radiation equality occurs at z_eq ≈ 3400, while recombination is at z ≈ 1100. There's a stretch of cosmic history during which photons are streaming freely but the universe still contains a non-negligible radiation component. Radiation makes potentials decay (just like Λ does, for the same reason: it expands the metric without driving structure growth). The result is an early ISW kick imprinted on photons during the first few thousand years after recombination.

The early ISW shows up in the CMB power spectrum as an enhancement of the first acoustic peak and a slight boost around l ≈ 50–200, sitting between the Sachs-Wolfe plateau and the acoustic peaks proper. It is sensitive to ω_r = Ω_r h², the physical radiation density at decoupling, and therefore to N_eff — the effective number of neutrino species — at recombination. The early ISW is one of the routes through which CMB power-spectrum fits constrain neutrino physics.

Variants and closely-related effects

• Rees-Sciutama effect. The non-linear sibling of ISW. When the photon traverses a fully non-linear, collapsing structure (a cluster), the time-derivative of the potential is large and non-Gaussian. The resulting Rees-Sciama signal is much smaller than the linear ISW but is in principle detectable in stacked-cluster analyses.
• Sunyaev-Zel'dovich (tSZ + kSZ). Different physics — inverse Compton scattering off hot cluster electrons rather than gravitational redshift — but produces the other major secondary CMB anisotropy after Sachs-Wolfe. tSZ has a distinctive frequency dependence; ISW is achromatic. The two are cleanly separable.
• CMB lensing. The same gravitational potentials that produce ISW also deflect CMB photons. CMB lensing is sensitive to the integrated Φ along the line of sight (no time derivative), while ISW is sensitive to Φ̇. They are complementary probes of the same field and can be cross-correlated.
• Doppler / dipole effect. The largest-scale anisotropy on the sky, ~3 mK, is a kinematic dipole from our motion through the CMB rest frame. It is much larger than the Sachs-Wolfe plateau and conventionally subtracted before analysis.
• Late-ISW × CMB-lensing reconstruction. A purely CMB-internal route to ISW. Planck used reconstructed CMB lensing maps to bypass the need for external galaxy catalogues and obtained a ~4σ ISW detection from CMB data alone.

Anomalies and tensions

The ISW signal is broadly consistent with ΛCDM, but several individual measurements deviate from prediction by 1.5–2.5σ — most famously a few large supervoids that produce CMB cold spots somewhat colder than the linear ISW prediction. The Eridanus supervoid sits behind the CMB Cold Spot (a ~70 μK cold patch ~5° across in the southern hemisphere). Linear ISW from a ~200 Mpc void of moderate depth predicts a few microkelvin signal; the observed Cold Spot is ten times that. Either there is non-linear ISW we don't fully model, the void is much deeper than catalogued, or the Cold Spot has a different origin (a primordial fluctuation tail, a topological defect, or just an unlucky fluctuation). The question is open.

On the other side, "stacked-void" analyses by Granett, Neyrinck & Szapudi (2008) found ~4σ ISW imprints from the largest supervoids and superclusters — but at amplitudes ~2× higher than ΛCDM predicts. Subsequent re-analyses brought the tension down but did not eliminate it. Whether this is a hint of beyond-ΛCDM physics, residual systematics, or selection effects in void catalogues remains debated.

Common pitfalls

• Mixing up the factor of one-third. The full local SW formula in the matter era is ΔT/T = (1/3) Φ/c². The factor comes from carefully combining the intrinsic-temperature term with the gravitational redshift. Don't write the naive Φ/c² shift; it misses the (2/3) Φ/c² emission-temperature correction.
• Forgetting that ISW = 0 in matter-only cosmologies. The integrated effect requires evolving potentials. A common mistake is to invoke ISW in contexts where the relevant cosmology has Λ = 0 — in which case linear potentials are constant and the ISW signal vanishes identically in linear theory.
• Confusing ISW with the Rees-Sciama effect. Linear ISW comes from potential evolution due to dark energy (or radiation, for early ISW). Rees-Sciama comes from non-linear structure collapse, is much smaller, and is independent of cosmology.
• Treating the CMB Cold Spot as proof of an exotic ISW excess. The Cold Spot might be a deep supervoid producing a stronger-than-linear ISW signal, but it might also be a 1-in-200 primordial fluctuation. Single-feature claims at 2-3σ should not be treated as detections of new physics.
• Cross-correlation contamination by foregrounds. Galaxy catalogues are imperfectly cleaned of stars, dust, and Galactic extinction; CMB maps need careful foreground subtraction. The ISW signal sits at low multipoles where systematics are largest. Robust ISW analyses use multiple tracers and check for spurious correlations with foreground templates.

Where Sachs-Wolfe shows up in practice

• Inflationary normalisation. The Sachs-Wolfe plateau at l < 50 is the cleanest probe of the primordial power-spectrum amplitude As — the single number that fixes the overall amplitude of structure in the universe and that any inflation model must reproduce.
• Dark energy diagnostics. Late-time ISW is one of the very few dynamical tests of dark energy (others: BAO, supernovae, redshift-space distortions, weak lensing growth). Each pins down a different combination of Ω_Λ, w, and Ωm; ISW is most directly sensitive to the deceleration-to-acceleration transition.
• Modified-gravity constraints. ISW is one of the strongest constraints on f(R) and DGP-like modified-gravity alternatives to dark energy — those models generically predict ISW amplitudes that differ from ΛCDM by amounts already excluded by data.
• Neutrino mass. The shape of the SW plateau and the early-ISW contribution at l ~ 50–200 are sensitive to the neutrino mass sum Σm_ν via its effect on potential evolution near matter-radiation equality.
• Texture-defect bounds. Cosmic strings, textures and other defects would imprint their own non-Gaussian Sachs-Wolfe signatures on the CMB; the lack of such features bounds the symmetry-breaking energy scales of any GUT-era defects to roughly 10¹⁶ GeV or lower.