Integrated Sachs-Wolfe Effect

The intuition: a ball rolling through a valley that flattens

Picture a marble rolling across a landscape of valleys and hills. Rolling down into a valley, it speeds up; rolling back up the far side, it slows down by exactly the same amount. If the valley keeps its shape, the marble exits with the same speed it entered. Energy in, energy out, nothing left over.

Now imagine that while the marble is at the bottom of the valley, the valley itself fills in — the far wall sinks lower. The marble had to do less climbing than it expected. It crests the rim with energy to spare. That leftover is the integrated Sachs-Wolfe effect, and the "marble" is a photon of the cosmic microwave background that has been falling and climbing through the universe's gravitational landscape for 13.8 billion years.

The crucial word is integrated. The ordinary Sachs-Wolfe effect is a single transaction stamped on the photon at the moment of last scattering, 380,000 years after the Big Bang. The integrated effect is the running tally of every well the photon crosses on the long way to our telescopes — and it only accumulates a net total where the wells are changing while the photon is inside them. In a universe where structure is just frozen in place, the tally is zero. Dark energy is what makes the wells move.

What it is: the line-of-sight energy integral

A photon's fractional temperature change as it traverses a time-varying gravitational potential Φ is, to linear order in general relativity,

ΔT/T |_ISW = (2/c²) ∫ (∂Φ/∂t) dt        (integrate from last scattering to today)

The factor of 2 is the same factor of 2 that doubles the deflection of starlight by the Sun relative to the naive Newtonian value: both the time-time and space-space parts of the metric perturbation contribute, and for a freely propagating photon they add. Notice what the integrand says: only the partial time derivative of the potential matters. A potential that is deep but constant contributes nothing — the blueshift falling in is undone by the redshift climbing out. Only an evolving potential leaves a residue.

This is the entire physics in one line. To know whether the ISW effect exists, you only need to know one thing about the universe: does ∂Φ/∂t vanish, or not?

Why a matter-only universe gives nothing

Here is the elegant part. In linear perturbation theory, the gravitational potential evolves as

Φ(k, 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-dominated (Einstein-de Sitter) universe, structure grows precisely as fast as the universe expands: D(a) ∝ a. The two effects cancel in the ratio, so Φ is constant in time. Gravity is digging the wells deeper at exactly the rate expansion is stretching them shallower, and the net depth never changes.

That is why ∂Φ/∂t = 0 in an Einstein-de Sitter universe and the integrated Sachs-Wolfe effect is identically zero. The potentials are frozen. A photon pays back on the way out everything it borrowed on the way in.

Break the balance and the effect switches on. Two things break it:

• Radiation early on. Just after recombination, radiation still contributes to the energy budget, so growth is suppressed and potentials decay slightly. This produces the early ISW, which boosts the CMB acoustic peaks around the first peak (l ≈ 200).
• Dark energy late on. Once Λ dominates (z ≲ 0.7), the expansion accelerates, structure growth stalls (D(a) grows slower than a), and the potentials decay. This is the late ISW — the part that fingerprints dark energy.

The math: from growth suppression to a CMB temperature

The late-time decay of the potential follows directly from the growth factor. Defining the normalised growth g(a) = D(a)/a, the ISW source is proportional to dg/da. In matter domination g is flat (dg/da = 0); in the Λ era g declines. A convenient fitting form for the growth rate is

f ≡ d ln D / d ln a ≈ Ω_m(a)^γ,   with γ ≈ 0.55 in GR

so the potential decay rate scales with how far Ω_m has dropped below 1. At z = 0 in the concordance model, Ω_m ≈ 0.31, Ω_Λ ≈ 0.69, and f ≈ 0.53 — meaningfully below the f = 1 of pure matter, which is precisely the deficit that drives the late ISW. The resulting CMB anisotropy in Fourier space is

C_l^ISW ∝ ∫ dk/k  P_Φ(k) [ ∫ dz (dΦ/dz) j_l(k r(z)) ]²

where j_l is a spherical Bessel function and r(z) the comoving distance. The Bessel functions peak at low l for the very large scales where Φ lives, which is why the late ISW is confined to multipoles l ≲ 30 (angular scales bigger than about 6°). Its amplitude is a few percent of the total low-l power — real, but swamped by cosmic variance in the raw spectrum.

How we actually detect it: cross-correlation

You cannot dig the ISW out of the CMB power spectrum alone. At l ≲ 30 there are only of order a few hundred independent modes on the whole sky, so the fractional cosmic-variance error on any single C_l is roughly √(2/(2l+1)) — about 53% at l = 3, and still ~25% out at l = 15. A 5–10% ISW contribution simply disappears into that scatter.

The breakthrough idea, due to Robert Crittenden and Neil Turok in 1996, is that the same large-scale structure that sources the decaying potentials is mappable in the local universe with galaxy surveys. A decaying well around a supercluster produces a hot CMB spot and an overdensity of galaxies at the same place on the sky. So you cross-correlate:

C_l^{Tg} = ⟨ a_lm^T  a_lm^{g*} ⟩      (CMB temperature × galaxy density)

The galaxy term has no cosmic-variance problem the way a single CMB mode does, and the cross-spectrum picks out only the fraction of CMB fluctuation that is physically tied to nearby structure. A positive C_l^{Tg} — hot CMB on galaxy overdensities — is the unambiguous ISW signature, and it is exactly what is observed.

Quantified figures: scales, temperatures, and significance

Quantity	Value	Note
Onset redshift of late ISW	z ≲ 0.7	When Ω_Λ begins to dominate
Dominant angular scale	l ≈ 2–30 (θ ≳ 6°)	Set by horizon-scale potentials
Fraction of low-l power	~5–10 %	Below cosmic variance alone
Typical ISW signal	a few µK	vs. ~70 µK CMB rms anisotropy
Void / cluster stacked imprint	~2–10 µK	Stacking thousands of structures
Cross-correlation significance	~4σ (3–4.5σ across analyses)	Combined galaxy/QSO/radio tracers
Comoving scale of source potentials	~100–300 Mpc	Superclusters and supervoids
γ (growth index, GR)	≈ 0.55	Modified gravity predicts other values

For context, the late ISW is a sub-10-microkelvin perturbation on a 2.725 K background — about one part in a million of one part in ten thousand. That such a whisper is detectable at all, and that it agrees with a parameter-free Λ-CDM prediction, is one of the quiet triumphs of precision cosmology.

Worked example: estimating the void cold spot

Take a supervoid of comoving radius R ≈ 150 Mpc with a fractional underdensity δ ≈ −0.2 at z ≈ 0.5. The associated potential perturbation from the Poisson equation in an expanding universe is roughly

Φ ≈ −(3/2) H₀² Ω_m  δ  R² / a        (order-of-magnitude)

Plugging H₀ ≈ 70 km/s/Mpc = 2.3 × 10⁻¹⁸ s⁻¹, Ω_m ≈ 0.31, δ = −0.2, R ≈ 150 Mpc ≈ 4.6 × 10²⁴ m, the sharp top-hat formula gives |Φ|/c² of order 10⁻⁴; a realistic compensated (under-then-over) density profile, rather than a hard-edged hole, smooths the central potential down to the more representative |Φ|/c² ~ a few × 10⁻⁵. A void is a potential hill, so Φ > 0. If dark energy lowers this hill by a fraction of order f_decay ~ 0.1 during the few-Gyr photon crossing,

ΔT/T ~ 2 × (Φ/c²) × f_decay ~ 2 × (3 × 10⁻⁵) × 0.1 ~ 6 × 10⁻⁶
ΔT   ~ 2.725 K × (−6 × 10⁻⁶) ~ −15 µK   (upper end; cold, because it's a hill that flattened)

So a single large supervoid should imprint a cold spot of order a few to ten microkelvin — far too small to pick out individually against the 70 µK primary anisotropy, which is exactly why analyses stack the CMB on thousands of voids identified in galaxy catalogues to average the signal up out of the noise. (The observed stacked imprints sit at the few-µK level, at or somewhat below this back-of-envelope upper bound.)

Discovery and the major measurements

The theoretical foundation is the 1967 paper of Rainer Sachs and Arthur Wolfe, which contained the full integral; the late-time, dark-energy-driven version was singled out as a probe after the cosmological constant returned to favour. Crittenden and Turok proposed the galaxy cross-correlation method in 1996. The first claimed detections came in 2003–2004, immediately after WMAP delivered a full-sky CMB map:

• Boughn & Crittenden (2003–2004). Cross-correlated WMAP with the NVSS radio-galaxy survey and the HEAO-1 X-ray background, reporting a positive signal — among the first direct evidence of the late ISW.
• Nolta et al. (2004), Fosalba, Gaztañaga & Castander (2003), Scranton et al. (2003). Independent detections cross-correlating WMAP with NVSS and with SDSS galaxies and quasars.
• Giannantonio et al. (2008, 2012). Combined six galaxy/quasar/radio catalogues spanning 0 < z < 2.5 into a joint ISW measurement at roughly 4.5σ — the benchmark multi-tracer result.
• Planck ISW papers (2014, 2016). The ESA Planck mission's own ISW analysis confirmed the cross-correlation at ~3–4σ and produced full-sky ISW reconstruction maps, plus stacking on superstructures.
• Granett, Neyrinck & Szapudi (2008). The much-discussed stacking of 50 superclusters and 50 supervoids from SDSS, which found a hot/cold imprint at a higher amplitude than the simplest Λ-CDM expectation — a result that fed years of debate about whether the largest voids over-imprint.

Why it matters: an independent vote for dark energy

The supernova results of 1998 told us the expansion is accelerating. The CMB acoustic peaks told us the universe is spatially flat. Put those together and you are forced to conclude that roughly 70% of the energy budget is some smooth, negative-pressure component — dark energy. The ISW effect is special because it does not infer dark energy from geometry or from a distance ladder; it watches dark energy acting, in real time, by suppressing the growth of structure and decaying the potentials that CMB photons are passing through right now.

That makes it a complementary and remarkably direct probe. Crucially, it is sensitive to ∂Φ/∂t, which depends on how gravity itself behaves on the largest scales. Modified-gravity alternatives to dark energy generally predict a different growth index γ and therefore a different ISW amplitude and even sign. The ISW is one of the few observables that can, in principle, distinguish "dark energy" from "gravity is wrong on cosmic scales" — which is why upcoming surveys (Euclid, the Vera C. Rubin Observatory's LSST, SKA) treat the ISW as a clean lever on the growth of structure.

Common misconceptions and edge cases

• "The ISW is just gravitational redshift." Pure gravitational redshift through a static well cancels exactly — in, then out. The ISW is the residual from the well changing mid-transit. No change, no effect, no matter how deep the well.
• "Voids make hot spots, clusters make cold spots." It is the opposite. A decaying potential well (cluster) leaves a net blueshift → hot spot; a decaying potential hill (void) leaves a net redshift → cold spot. The mnemonic: structures that are emptying out cool the CMB, structures collapsing heat it — but on linear scales in the Λ era, expansion shallows both, so overdensities go hot and underdensities go cold.
• "It contaminates the primary CMB and biases parameters." The late ISW is a real part of the observed temperature sky, but it sits at l ≲ 30 where it is sub-dominant and is folded into the Λ-CDM prediction; it is a feature, not a contaminant.
• "Early and late ISW are the same thing." They share the ∂Φ/∂t term but have different causes (radiation pressure near recombination vs. dark energy at low z) and live at different scales (early ISW boosts l ≈ 200; late ISW lives at l ≲ 30).
• "It's the same as the Rees-Sciama effect." Rees-Sciama is the nonlinear regime — potentials changing because structures are collapsing/moving, not because the cosmology is accelerating — and it dominates at small scales (l ≳ 500). The linear late ISW is the large-scale, dark-energy piece.
• "A 4σ detection is weak." For a signal that is intrinsically below cosmic variance and only a few microkelvin in amplitude, extracting it at 4σ from cross-correlation is a strong result — and it is corroborated by multiple independent tracer surveys and by void/cluster stacking.