The Rees-Sciama Effect: Nonlinear CMB Temperature Shifts from Evolving Potentials

What the Rees-Sciama effect is

The Rees-Sciama (RS) effect is a secondary anisotropy of the cosmic microwave background — a temperature imprint stamped onto CMB photons long after they left the surface of last scattering at redshift z ≈ 1100. Its cause is gravitational: a photon traversing a region of curved spacetime is blueshifted as it falls into a potential well and redshifted as it climbs out. If the well is perfectly static, these shifts cancel exactly. If the well deepens or shallows while the photon is inside it, the cancellation is incomplete and the photon emerges with a net energy change.

• Linear ISW handles small potential changes driven by the smooth expansion (and, at late times, dark energy).
• Rees-Sciama is the nonlinear piece — the change in potentials caused by structures actively forming: clusters collapsing, voids emptying, halos merging, and masses drifting across the sky.

Rees and Sciama pointed out in 1968 that if large-scale density inhomogeneities existed, their evolution would leave a measurable print on the angular distribution of the microwave background — a remarkably prescient claim made before structure formation theory was mature.

The mechanism and its governing relation

Start from the line-of-sight temperature integral for gravitational potential Φ. In conformal time η, the ISW/RS contribution is

ΔT/T = (2/c²) ∫ (∂Φ/∂η) dη,

integrated from last scattering to today. The factor of 2 is the general-relativistic signature: both the time and space parts of the metric perturbation contribute, doubling the naive Newtonian expectation. The integrand is the time derivative of the potential — a static Φ gives zero.

Linear theory predicts that in a matter-dominated, flat universe Φ is constant, so ∂Φ/∂η = 0 and there is no signal. Two things break this:

• Dark energy makes potentials decay on large, linear scales → the linear ISW.
• Nonlinear structure growth makes potentials evolve on small scales: as a cluster collapses its well deepens; as a void expands its hill flattens. Expanding Φ to second order in the density contrast δ gives terms like Φ ∝ δ² and velocity-squared contributions → the Rees-Sciama effect.

Because these are quadratic in the perturbations, RS is inherently a nonlinear, mode-coupling phenomenon that linear codes cannot capture — it requires N-body simulations or perturbation theory beyond first order.

Characteristic numbers and a worked estimate

The headline number is small: across a wide range of cold-dark-matter models the RS effect gives ΔT/T ~ 10^-7 to 10^-6, i.e. temperature shifts of roughly 0.1 to 1 µK against the 2.725 K CMB. The power spectrum peaks around multipole ℓ ~ 100-300 (angular scales of order a degree), but because the linear ISW falls off steeply, RS dominates the total gravitational signal at small scales, ℓ ≳ 1000.

A back-of-envelope estimate for one structure: the fractional shift scales as ΔT/T ~ (v/c)(Φ/c²) for a mass moving transversely, or ~ (Φ/c²)(t_cross/t_evolve) for a collapsing one. Take a rich supercluster with potential depth Φ/c² ~ 10^-5 and a photon crossing time that is a modest fraction of the collapse time. The product lands near 10^-7 — precisely the observed order.

• Supercluster / void scale imprints: sub-µK, spread over several degrees.
• Individual massive cluster (moving-lens dipole): up to a few µK on arcminute scales.
• Total sky-averaged RS power is ~2-3 orders of magnitude below the primary CMB rms of ~100 µK.

How it is observed and where it appears

The RS signal is far too weak to see photon-by-photon, so it is extracted statistically by cross-correlating the CMB with tracers of the evolving mass distribution:

• Galaxy and cluster catalogs (SDSS luminous red galaxies, DESI) cross-correlated with Planck/WMAP maps — the same technique that detected the linear ISW at the few-sigma level.
• Supervoid and supercluster stacking: averaging CMB patches toward known z ~ 0.5 structures searches for the combined ISW+RS cold/hot signature.
• Moving-lens searches: the transverse-motion (Birkinshaw-Gull) component produces a dipolar hot/cold pattern aligned with a cluster's velocity. The moving-lens effect was first reported with ACT and DESI data in the mid-2020s.

RS also enters as a contaminant and a probe: it biases measured galaxy redshifts slightly, adds a non-Gaussian bispectrum signature to the CMB, and its cross-spectrum with CMB lensing carries information on the total neutrino mass (a characteristic sign-inversion multipole depends nearly linearly on Mν). It is a candidate contributor to the debated CMB Cold Spot via a large foreground supervoid.

How it differs from its close cousins

The RS effect sits inside a family of gravitational CMB effects that are easy to conflate:

• vs. linear ISW: same physics (time-varying Φ) but different regime. ISW is first-order and dark-energy-driven, peaking at the largest scales (ℓ < 30); RS is second-order and structure-driven, extending to small scales. Together they are often written as the single 'ISW-RS' contribution.
• vs. primary Sachs-Wolfe: the ordinary Sachs-Wolfe effect is set by potentials at last scattering; RS is accumulated along the path afterward.
• vs. Birkinshaw-Gull (moving lens): this is the kinetic, transverse-motion slice of RS, giving a clean dipole pattern. Pure RS from collapse/expansion is more monopole/quadrupole-like.
• vs. thermal/kinetic Sunyaev-Zel'dovich: SZ effects are scattering off hot electrons (spectral distortion), not gravitational redshift, and have distinct frequency dependence.

The key discriminant is always: RS is purely gravitational, achromatic (frequency-independent), and vanishes for static potentials.

Significance, open questions, and famous cases

The Rees-Sciama effect matters as both a foreground nuisance and a cosmological probe. Because it is achromatic and correlated with large-scale structure, it must be modeled to avoid biasing precision cosmology from Planck, ACT, SPT, and future CMB-S4 data. Conversely, its cross-correlations offer independent handles on dark energy (via how fast potentials decay), on neutrino mass, and on transverse cluster velocities — a quantity almost impossible to measure by any other means, since spectroscopy only gives line-of-sight motion.

• Cold Spot debate: whether the ~70 µK anomaly in the southern CMB is partly an ISW/RS decrement from the Eridanus supervoid remains contested; the void's estimated imprint may be too small to fully explain it.
• Modeling accuracy: capturing RS demands large N-body simulations (Gpc-scale boxes) because it couples widely separated modes; perturbative treatments still disagree on the small-scale power.
• Clustering dark energy and modified gravity predict altered potential evolution, so the RS spectrum is a testbed for physics beyond ΛCDM.

More than half a century on, Rees and Sciama's 1968 insight has become a precision tool for reading the growth of cosmic structure directly off the microwave sky.

Gravitational (ISW-family) contributions to CMB temperature anisotropy compared by regime and driver

Effect	Regime / driver	Angular scale	Typical ΔT/T
Linear ISW (late-time)	Linear, dark-energy decay of potentials	Large scales, ℓ < 30	~10^-6
Early ISW	Linear, residual radiation near recombination	ℓ ~ 200	~10^-6
Rees-Sciama (intrinsic)	Nonlinear collapse of clusters / evacuation of voids	ℓ ~ 100-300, up to ℓ > 1000	~10^-7 to 10^-6
Birkinshaw-Gull (moving lens)	Transverse bulk motion of a structure	Arcminutes, dipolar pattern	~10^-7 (few µK, massive clusters)
Primary Sachs-Wolfe	Potentials at last scattering (z ≈ 1100)	Large scales, ℓ < 100	~10^-5