Sachs-Wolfe and Integrated Sachs-Wolfe Effects

Every photon in the cosmic microwave background carries the fingerprint of the gravity it was born into. When the universe cooled enough for electrons and protons to combine, the photons trapped in the baryon-photon plasma streamed free — but they did not leave a smooth, featureless glow. Each had to climb out of the gravitational potential well (or ride down the hill) in which its patch of plasma sat, and on the fourteen-billion-year journey to us it crossed one evolving potential after another. Rainer Sachs and Arthur Wolfe worked out exactly what these purely gravitational encounters do to the photon temperature in a single 1967 paper, and their result governs the CMB on the very largest angular scales, where the acoustic ringing of the plasma never had time to develop. The answer splits cleanly into two pieces. One is fixed at last scattering — the ordinary Sachs-Wolfe effect, combining the intrinsic temperature of the plasma with the redshift of climbing out of its well to leave the famous factor $\tfrac13\Psi$ . The other accumulates along the line of sight — the integrated Sachs-Wolfe effect, a tally of how much the potentials changed while each photon was inside them. In a purely matter-dominated universe the potentials are frozen and this integral vanishes exactly; it switches back on only when the expansion is driven by something else — radiation just after recombination (early ISW), and dark energy today (late ISW) — making these gravitational imprints a direct probe of both the primordial spectrum and the accelerating universe.

Key takeaways

The ordinary Sachs-Wolfe effect is the gravitational redshift a CMB photon suffers climbing out of the potential $\Psi$ at last scattering, combined with the plasma's intrinsic temperature; on large scales it nets $\Delta T/T = \tfrac13\Psi$ .
The integrated Sachs-Wolfe effect is the extra shift $\propto\int(\dot\Phi+\dot\Psi)\,\mathrm{d}\eta$ accumulated as the photon crosses potentials that change while it is inside; it vanishes for static potentials.
In linear matter domination the potential is exactly constant, so the ISW switches off — it revives only near radiation-matter equality (early ISW) and under dark energy (late ISW).
On the largest scales ( $\ell\lesssim 30$ ) the ordinary SW effect produces the flat Sachs-Wolfe plateau, whose height fixes the primordial amplitude and tilt $n_s\approx 0.965$ .
The late ISW is a smoking gun for dark energy in a flat universe: absent in a matter-only cosmos, it shows up as a positive cross-correlation between the CMB and large-scale structure.
Key idea: Large-scale CMB anisotropies are gravitational — $\Delta T/T$ is the potential at last scattering (Sachs-Wolfe, $\tfrac13\Psi$ ) plus a line-of-sight integral of the time-varying potential (integrated Sachs-Wolfe), which is nonzero only when the universe is not matter-dominated.

The core insight

Large-scale CMB anisotropies are gravitational: $\Delta T/T$ is set by the potential at last scattering (Sachs-Wolfe, $\tfrac13\Psi$ ) plus a line-of-sight integral of the time-varying potential (integrated Sachs-Wolfe), which is nonzero only when the universe is not matter-dominated.

Where this lives today

The Sachs-Wolfe plateau is an anchor point of precision cosmology. Because the largest scales were never touched by acoustic physics, their amplitude is a clean readout of the primordial spectrum, and the plateau — pinned by COBE, WMAP, and Planck — fixes $A_s$ and, with the acoustic peaks, the tilt $n_s\approx0.965$ . The late ISW is a working tool of its own: buried under cosmic variance in the auto-spectrum, it is extracted by cross-correlating the CMB with wide-area galaxy, quasar, and radio catalogues — from the first WMAP-era detections to the Dark Energy Survey, DESI, and now Euclid. A positive cross-correlation is independent evidence, distinct from supernovae and BAO, that the potentials are decaying, and its shape constrains the dark-energy equation of state and tests modified gravity. The CMB Cold Spot and the low- $\ell$ anomalies are debated partly in ISW terms — is an unusually large supervoid imprinting a cold patch? — keeping this fifty-year-old effect at the center of live controversy.

The physical picture

Picture the last-scattering surface as a rolling landscape of potential wells (overdensities that will grow into galaxies and clusters) and hills (underdensities that will become voids). A photon released from the floor of a well must climb, and climbing costs energy: it emerges redshifted, looking colder. Naively, overdensities should be cold spots at the full depth of their wells. But the plasma at the bottom is also compressed, and compressed plasma is intrinsically hotter. These effects fight, and — for adiabatic fluctuations on super-horizon scales — the fight is not a draw. The intrinsic overheating is $-\tfrac23\Psi$ and the redshift is $+\Psi$ , so two-thirds is cancelled and only $\tfrac13\Psi$ survives. A well ( $\Psi<0$ ) still ends up cold, but only one-third as cold as the bare climb suggests.

Figure

A CMB photon climbs out of a potential well Ψ at last scattering — the compressed plasma is intrinsically hotter by Θ₀ = −⅔Ψ, the climb costs a redshift +Ψ, and the two partly cancel to leave ΔT/T = ⅓Ψ, so a well with Ψ* < 0 becomes a cold spot

Now let the well itself breathe while the photon is inside it. If it grows shallower during the crossing, the photon falls into a deep barrier but climbs out of a shallow one and pockets the difference as a net blueshift; if it deepens, the photon loses energy on balance. This running exchange is the integrated Sachs-Wolfe effect, its size set by how fast the potential changes. The decisive fact is that a linear potential is frozen during matter domination — density grows just fast enough to compensate the dilution of expansion — so entry and exit exactly cancel and the ISW contributes nothing. It wakes up only when something other than matter sets the expansion: leftover radiation just after recombination, and dark energy today. The second figure follows the potential across all three eras.

Figure

The metric potential Φ tracks D(a)/a across cosmic history — it decays while radiation still matters just after recombination (early ISW), holds constant through matter domination so the ISW switches off, then decays again as dark energy takes over, reaching about 0.79 of its matter-era value today (late ISW)

The clean division into a boundary piece (SW) and an integral piece (ISW) is not a modelling choice — it falls straight out of integrating the photon's energy equation by parts.

The core equations

The complete large-scale anisotropy sums terms picked up at last scattering and along the line of sight. In Newtonian gauge it collects an intrinsic temperature piece, a gravitational redshift, a Doppler piece, and the line-of-sight integral.

\frac{\Delta T}{T}(\hat n) = \left[\frac{1}{4}\delta_\gamma + \Psi\right]_{\!*} + \hat n\cdot\mathbf{v}_* + \int_{\eta_*}^{\eta_0}(\dot\Psi+\dot\Phi)\,\mathrm{d}\eta \qquad (1)

where: $\Delta T/T$ — fractional temperature anisotropy in direction $\hat n$ ; $\tfrac14\delta_\gamma$ — intrinsic (monopole) photon temperature perturbation; $\Psi,\Phi$ — Newtonian-gauge metric potentials (time and space curvature); $\mathbf{v}_*$ — peculiar velocity of the fluid at last scattering; $\eta_*,\eta_0$ — conformal time at last scattering and today. The first bracket is the SW term, the integral is the ISW term, and the Doppler term dominates only at intermediate scales. On scales larger than the horizon at recombination the Doppler term is negligible and the bracket collapses to one number. Using the adiabatic super-horizon relation between monopole and potential (derived below), the intrinsic overheating cancels two-thirds of the redshift.

\left.\frac{\Delta T}{T}\right|_{\mathrm{SW}} = \frac{1}{4}\delta_\gamma\Big|_* + \Psi_* = -\frac{2}{3}\Psi_* + \Psi_* = \frac{1}{3}\Psi_* \qquad (2)

where: $\tfrac14\delta_\gamma\big|_* = \Theta_0 = -\tfrac23\Psi_*$ — the intrinsic photon monopole; $+\Psi_*$ — the gravitational redshift climbing out; $\tfrac13\Psi_*$ — the surviving net. Read it as: a well ( $\Psi_*<0$ ) is a cold spot, but at only one-third of the bare gravitational depth. The second contribution integrates over the evolving potentials. Only the explicit (partial) time derivatives appear — the change the photon does not simply outrun by moving through space.

\left.\frac{\Delta T}{T}\right|_{\mathrm{ISW}} = \int_{\eta_*}^{\eta_0}\big(\dot\Psi+\dot\Phi\big)\,\mathrm{d}\eta \qquad (3)

where: $\dot\Psi,\dot\Phi$ — conformal-time derivatives $\partial/\partial\eta$ of the two potentials; $\mathrm{d}\eta$ — element of conformal time, equal to the comoving path length. The integrand vanishes wherever the potentials are static, which is why the ISW is a pure diagnostic of a changing metric. Without anisotropic stress (excellent once neutrinos are subdominant) the potentials are equal and opposite, $\Phi=-\Psi$ , so the integrand is simply $2\dot\Psi$ . Whether the potential evolves is decided by a competition between density growth and the dilution of expansion, packaged in the growth factor.

\Phi(\mathbf{k},a) = \Phi_{\mathrm{prim}}(\mathbf{k})\,T(k)\,\frac{D(a)}{a} \qquad (4)

where: $\Phi_{\mathrm{prim}}$ — primordial potential from inflation; $T(k)$ — transfer function encoding sub-horizon suppression; $D(a)$ — linear growth factor; $D(a)/a$ — the controlling ratio. Read it as: in matter domination $D\propto a$ so $D/a$ is constant and $\Phi$ is frozen; when $D$ grows more slowly than $a$ , $\Phi$ decays and the ISW turns on. Projecting the SW term onto the sky gives the angular power spectrum as a Bessel-weighted integral over the potential spectrum.

C_\ell^{\mathrm{SW}} = \frac{2}{9\pi}\int_0^\infty \frac{\mathrm{d}k}{k}\,P_\Phi(k)\,j_\ell^2\!\big[k(\eta_0-\eta_*)\big] \qquad (5)

where: $C_\ell$ — angular power at multipole $\ell$ ; the prefactor $\tfrac29 = 2\times(\tfrac13)^2$ carries the SW one-third; $P_\Phi(k)$ — dimensionless potential power spectrum; $j_\ell$ — spherical Bessel function projecting $k$ onto scale $\ell$ ; $\eta_0-\eta_*$ — comoving distance to last scattering, about 14000 Mpc. For a scale-invariant input the Bessel integral is analytic and yields the flat plateau, whose height in terms of the curvature power $\Delta^2_{\mathcal R}$ is a single number.

\frac{\ell(\ell+1)}{2\pi}\,C_\ell^{\mathrm{SW}} = \frac{\Delta^2_{\mathcal R}}{25} \approx \text{const} \qquad (n_s=1) \qquad (6)

where: $\Delta^2_{\mathcal R}=A_s(k/k_*)^{n_s-1}$ — dimensionless primordial curvature power; the factor $\tfrac1{25}=(\tfrac13\cdot\tfrac35)^2$ folds the SW one-third with the matter-era relation $\Psi=-\tfrac35\mathcal R$ ; $n_s$ — spectral index. Flat for $n_s=1$ ; the measured tilt $n_s\approx0.965$ gives a gentle downward slope. Finally, the late ISW is too weak to see in the CMB alone, but it correlates with the matter sourcing it, so it is dug out by cross-correlating the temperature map with a galaxy survey.

w_{Tg}(\theta) = \Big\langle \frac{\Delta T}{T}(\hat n_1)\,\delta_g(\hat n_2)\Big\rangle_{\hat n_1\cdot\hat n_2=\cos\theta} \qquad (7)

where: $w_{Tg}$ — angular cross-correlation of CMB temperature and galaxy overdensity; $\delta_g$ — projected galaxy density contrast, a tracer of the low-redshift potential. A positive $w_{Tg}$ at large angles is the fingerprint of decaying potentials, hence of dark energy.

Derivation ADeriving the Sachs-Wolfe factor of one-third

The goal is to show where the celebrated $\tfrac13$ comes from — not from the gravitational redshift alone, which would give the full $\Psi$ , but from a partial cancellation between that redshift and the intrinsic temperature of the adiabatic plasma. We work in Newtonian gauge with no anisotropic stress so $\Phi=-\Psi$ , restrict to a super-horizon adiabatic mode deep in matter domination, combine the photon monopole with the redshift term of equation (1), and check the sign against the expectation that overdensities look cold.

Step 1. Start from the temperature received today along $\hat n$ : $\Delta T/T = \Theta_0(\eta_*) + \Psi(\eta_*) + (\text{Doppler}) + (\text{ISW})$ , with $\Theta_0=\tfrac14\delta_\gamma$ the intrinsic monopole and $\Psi_*$ the redshift climbing out at emission. Why: the observed temperature is the emitted temperature shifted by the potential the photon left behind — the two terms that survive on the largest scales.

Step 2. Impose adiabaticity, tying every species to a common curvature perturbation: $\delta_\gamma=\tfrac43\delta_m$ , so $\Theta_0=\tfrac14\delta_\gamma=\tfrac13\delta_m$ . Why: adiabatic initial conditions from inflation lock the photon temperature to the matter density.

Step 3. Solve the coupled Einstein-fluid equations for a super-horizon mode in matter domination; the energy constraint gives $\Theta_0 = -\tfrac23\Psi_*$ . Why: larger than the horizon, the fluid has not moved, so its density is fixed entirely by the metric — this converts a gravitational quantity into a temperature.

Step 4. Add the redshift term of Step 1: $\Theta_0 + \Psi_* = -\tfrac23\Psi_* + \Psi_* = \tfrac13\Psi_*$ . Why: the intrinsic overheating ( $-\tfrac23\Psi_*$ , positive since $\Psi_*<0$ ) is partly undone by the redshift of the climb, and two-thirds cancels.

Step 5. Check the sign. In a well $\Psi_*<0$ , so $\Delta T/T=\tfrac13\Psi_*<0$ — a cold spot toward the overdensity, matching the observed large-scale CMB-galaxy correlation. Why: the sign is the difference between predicting cold spots on clusters and hot ones, and it confirms the cancellation was assembled correctly.

Go deeper · Derivation B · 1 minSplitting the geodesic into a boundary term and a line-of-sight integral

Derivation BSplitting the geodesic into a boundary term and a line-of-sight integral

The aim is to prove that the total gravitational effect separates exactly into the SW boundary term and the ISW integral, and to show why that integral vanishes for a static potential. The tool is an integration by parts of the photon's energy equation along the null geodesic: we write the rate of change of fractional energy, isolate a total (convective) derivative from the explicit time derivatives, integrate the total derivative to a boundary term, and read off what is left.

Step 1. Along a null geodesic the photon's fractional energy obeys $\dfrac{\mathrm{d}}{\mathrm{d}\eta}\!\left(\dfrac{\Delta T}{T}\right)=\dfrac{\partial\Psi}{\partial\eta}+\dfrac{\partial\Phi}{\partial\eta}+\dfrac{\mathrm{d}\Psi}{\mathrm{d}\eta}-\dfrac{\partial\Psi}{\partial\eta}$ , where $\mathrm{d}\Psi/\mathrm{d}\eta=\partial_\eta\Psi+\hat n\cdot\nabla\Psi$ tracks the potential the moving photon sees. Why: separating the change from moving through space (total derivative) from the change in time (partial derivative) is the whole trick — only the partial pieces are non-boundary.

Step 2. Integrate the total derivative from emission to observation: $\displaystyle\int_{\eta_*}^{\eta_0}\frac{\mathrm{d}\Psi}{\mathrm{d}\eta}\,\mathrm{d}\eta=\Psi(\eta_0)-\Psi(\eta_*)$ . Why: a total derivative always integrates to its endpoints, converting a path integral into a difference of boundary values.

Step 3. Collect the surviving terms. The observer-side value $\Psi(\eta_0)$ is a direction-independent monopole, hence unobservable; the emission-side $-\Psi(\eta_*)$ combines with the intrinsic monopole to give the SW term, and the leftover partial derivatives give $\dfrac{\Delta T}{T}\Big|_{\text{path}}=\Psi_*+\displaystyle\int_{\eta_*}^{\eta_0}(\dot\Phi+\dot\Psi)\,\mathrm{d}\eta$ . Why: this is the split the subject rests on — a boundary term at last scattering (SW) plus an integral over the changing potential (ISW).

Step 4. Take the static limit. If $\dot\Phi=\dot\Psi=0$ the integral is identically zero, so the ISW vanishes. Why: when the well does not change, the energy gained falling in is returned exactly on the way out — no net effect survives a frozen potential.

Step 5. Take the decaying limit. If $\dot\Psi<0$ (a shrinking well) the photon nets a blueshift, giving hot spots correlated with structure. Why: this is the late ISW — decaying potentials under dark energy heat photons crossing superclusters and cool those crossing voids.

Go deeper · Derivation C · 1 minWhy the linear potential is frozen in matter domination

Derivation CWhy the linear potential is frozen in matter domination

The goal is to justify the central claim of the article — that in a purely matter-dominated linear universe the potential is exactly constant, so the ISW switches off — by combining the cosmological Poisson equation with linear growth. We write Poisson in an expanding background, insert the scaling of the mean matter density, replace the density contrast by the growth factor, and read off the potential's time dependence in both the matter and dark-energy eras.

Step 1. The cosmological Poisson equation in Fourier space reads $k^2\Phi=-4\pi G a^2\bar\rho_m\delta_m$ . Since $\bar\rho_m\propto a^{-3}$ , the prefactor $a^2\bar\rho_m\propto a^{-1}$ , giving $\Phi\propto a^{-1}\delta_m$ . Why: the matter dilution in the prefactor would make the potential decay on its own — only growing density can hold it up.

Step 2. Linear theory writes $\delta_m(a)=\delta_{m,0}\,D(a)$ with $D(a)$ the growth factor, so $\Phi\propto D(a)/a$ . Why: all the potential's time dependence is now carried by the ratio $D(a)/a$ — the ISW reduces to a race between growth and expansion.

Step 3. In matter domination $D\propto a$ , so $D/a=\text{const}$ and $\Phi$ is exactly constant, giving $\dot\Phi=0$ and zero ISW. Why: matter grows just fast enough to hold the wells at fixed depth as space stretches — exact only for pressureless matter in a flat universe.

Step 4. When dark energy (or leftover radiation) alters the expansion, $D$ grows more slowly than $a$ , so $D/a$ falls and $\Phi$ decays, driving $\dot\Phi<0$ and reviving the ISW; today $D/a\approx0.79$ for $\Omega_m\approx0.315$ . Why: this is the quantitative origin of the late ISW — a measurable $21\%$ potential decay since matter domination, sourced entirely by dark energy.

Limiting cases

The balance between the SW boundary term and the ISW integral is dictated by which component controls the expansion at a given epoch.

Case	Result	Meaning
Pure matter domination	$\dot\Phi=0\;\Rightarrow\;\text{ISW}=0$	Linear potentials are frozen; only the SW boundary term survives, giving the ideal Sachs-Wolfe plateau.
Radiation influence (early ISW)	$\dot\Phi\neq0$ near $z_{\mathrm{eq}}\approx3400$	Residual radiation makes potentials decay just after recombination, adding power near the first acoustic peak, $\ell\sim200$ .
Dark-energy domination (late ISW)	$\dot\Phi<0$ at $z\lesssim1$	Accelerated expansion decays potentials by $\sim21\%$ , boosting the lowest multipoles $\ell\lesssim10$ .
Large-scale scale-invariant limit	$\ell(\ell+1)C_\ell\approx\text{const}$	For a Harrison-Zel'dovich spectrum $n_s=1$ the SW projection is flat; the plateau height is $\Delta^2_{\mathcal R}/25$ .
Curved or open universe	$\dot\Phi\neq0$ even in matter era	Spatial curvature also breaks the $D\propto a$ balance, so an open matter-only model would still show an ISW.

Common trapThe one-third is a cancellation, not a bare redshift

It is tempting to read $\Delta T/T=\tfrac13\Psi$ as "the gravitational redshift, which happens to be one-third of the potential." It is not. The bare redshift is the full $\Psi$ ; the factor $\tfrac13$ appears only after the intrinsic temperature of the adiabatic plasma, $\Theta_0=-\tfrac23\Psi$ , cancels two-thirds of it. Forget the intrinsic term and you get both the amplitude and, in some sign conventions, the direction wrong. A second trap: the slogan "the ISW is negligible" holds only in exact linear matter domination. It fails precisely where the interesting physics lives — near radiation-matter equality (early ISW) and at low redshift under dark energy (late ISW). The ISW is not a correction to be dropped; it is the switch that turns the CMB into a probe of the expansion history.

Worked examples

1 · The height of the Sachs-Wolfe plateau (fully worked)

Planck measures the primordial curvature amplitude $A_s\approx2.1\times10^{-9}$ . Estimate the rms temperature fluctuation on the Sachs-Wolfe plateau and the plateau height $\mathcal D_\ell=\ell(\ell+1)C_\ell/2\pi$ in $\mu\mathrm{K}^2$ , assuming scale invariance.

Step 1. Relate temperature to curvature. On super-horizon scales $\Delta T/T=\tfrac13\Psi$ , and in matter domination $\Psi=-\tfrac35\mathcal R$ , so $\Delta T/T=-\tfrac15\mathcal R$ — one-fifth of the curvature perturbation.

Step 2. The curvature amplitude is $\Delta_{\mathcal R}=\sqrt{A_s}=\sqrt{2.1\times10^{-9}}=4.58\times10^{-5}$ , so $(\Delta T/T)_{\mathrm{rms}}=\Delta_{\mathcal R}/5=\mathbf{9.2\times10^{-6}}$ .

Step 3. Convert using $T_0=2.725\ \mathrm{K}$ : $\Delta T=(9.2\times10^{-6})(2.725\ \mathrm{K})=\mathbf{25\ \mu K}$ — exactly the scale of the large-angle ripples COBE first mapped.

Step 4. Form the plateau height. For scale invariance $\mathcal D_\ell=\Delta^2_{\mathcal R}/25\times T_0^2=(2.1\times10^{-9}/25)(2.725\times10^6\ \mu\mathrm{K})^2=\mathbf{624\ \mu K^2}$ , whose square root $25\ \mu\mathrm{K}$ agrees with Step 3. The observed low- $\ell$ plateau sits higher, near $1000\ \mu\mathrm{K}^2$ , because the early ISW and the exact projection add power this pure-SW estimate omits. The SW plateau is $\mathcal D_\ell\approx600\ \mu K^2$ , an rms of about 25 μK.

2 · How much have the potentials decayed under dark energy? (fully worked)

In flat $\Lambda$ CDM with $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ , estimate the fractional decay of the linear potential $\Phi\propto D(a)/a$ between matter domination and today, using the Carroll-Press-Turner growth-suppression formula.

Step 1. Write the suppression factor $g\equiv D/a$ , normalized to unity deep in the matter era: $g=\dfrac{5}{2}\Omega_m\big[\Omega_m^{4/7}-\Omega_\Lambda+(1+\tfrac12\Omega_m)(1+\tfrac1{70}\Omega_\Lambda)\big]^{-1}$ .

Step 2. Evaluate: $\Omega_m^{4/7}=0.315^{0.571}=\mathbf{0.517}$ and $(1+\tfrac12\Omega_m)(1+\tfrac1{70}\Omega_\Lambda)=(1.1575)(1.0098)=\mathbf{1.169}$ .

Step 3. Assemble the bracket: $0.517-0.685+1.169=1.001$ , so $g=\tfrac52(0.315)/1.001=\mathbf{0.79}$ .

Step 4. Interpret. Since $\Phi\propto D/a=g$ , the potentials today stand at $0.79$ of their frozen matter-era value — a 21% decay. That decay sources the late ISW: photons crossing the largest superclusters and voids over the last several billion years pick up a net shift from the shrinking wells, imprinting extra power at $\ell\lesssim10$ . The linear potential has decayed ≈21% since matter domination — the direct cause of the late ISW.

Physics of the portrait

The Sachs-Wolfe effect is a rare prediction that waited twenty-five years for its data: written in 1967, when the microwave background was barely two years old, and confirmed when COBE resolved the large-angle sky in 1992.

Origins · 1 minRainer Sachs and how this was discovered

Rainer Sachs

b. 1932 · general relativity and relativistic cosmology

Rainer Kurt Sachs was a mathematical relativist before he was a cosmologist. In the early 1960s he did foundational work on gravitational radiation — the Sachs peeling theorem for how the Weyl tensor falls off along outgoing null rays, and, with Hermann Bondi and others, the Bondi-Metzner-Sachs group of asymptotic symmetries that still anchors modern work on gravitational waves and holography. That fluency with null geodesics is exactly what the CMB problem needed. In 1967, with Arthur M. Wolfe (1939-2014), Sachs traced light rays through the perturbed expanding universe and derived how gravitational potentials imprint temperature fluctuations on the relic radiation, splitting the result into the boundary and line-of-sight integral terms that now bear both their names — decades ahead of the measurements. Wolfe became a leading observer of high-redshift gas through quasar absorption (the damped Lyman-alpha systems are largely his legacy), while Sachs later left physics almost entirely for mathematical biology at Berkeley, modelling how ionizing radiation damages DNA.

History

1965

— Penzias and Wilson detect the cosmic microwave background

1967

— Sachs and Wolfe derive the gravitational imprint on relic photons

1992

— COBE DMR maps the large-angle anisotropy and its Sachs-Wolfe plateau

1996

— Seljak and Zaldarriaga recast the anisotropy as a line-of-sight integral

2008

— CMB-galaxy cross-correlation detects the late integrated Sachs-Wolfe effect

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., Ch. 8-9 (line-of-sight solution and the Sachs-Wolfe terms) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 6 (perturbations and the CMB) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Hu & Sugiyama (1995), ApJ 444, 489 (early ISW and acoustic physics).

Provenance. The SW factor $\tfrac13$ and the relation $\Theta_0=-\tfrac23\Psi$ follow Sachs & Wolfe (1967) and Dodelson Ch. 9; the boundary/integral split is the line-of-sight formalism of Seljak & Zaldarriaga (1996). A verifier should recompute: (1) $\Theta_0=-\tfrac23\Psi$ and hence $\tfrac13\Psi$ ; (2) $\Phi\propto D(a)/a$ from Poisson plus linear growth; (3) the plateau height $\Delta^2_{\mathcal R}/25\approx600\ \mu\mathrm{K}^2$ for $A_s=2.1\times10^{-9}$ ; (4) the growth suppression $D/a\approx0.79$ for $\Omega_m=0.315$ . Standard values ( $z_*\approx1090$ , $z_{eq}\approx3400$ , $n_s\approx0.965$ , $T_0=2.725\ \mathrm{K}$ , $\eta_0-\eta_*\approx14000\ \mathrm{Mpc}$ ) are Planck/textbook.

Prerequisites. Density Perturbations and the Density Contrast · The Primordial Power Spectrum · Acoustic Oscillations in the Photon-Baryon Fluid · Linear Growth of Structure. Leads to. The CMB Anisotropy Power Spectrum · Dark Energy and the Cosmological Constant · The ΛCDM Concordance Model.

Ask the tutor: Ask me to walk through why the potential is frozen in matter domination but decays under dark energy, and we will trace it from the Poisson equation to the late-ISW cross-correlation step by step.

Why is the amplitude ⅓Ψ and not the full gravitational redshift Ψ?

Because two things happen at once inside a well and they partly cancel. The photon climbing out is redshifted by the full potential $+\Psi$ , which alone would make an overdensity look cold at its bare depth. But the plasma at the bottom is adiabatically compressed and therefore intrinsically hotter, contributing a monopole $\Theta_0=-\tfrac23\Psi$ (positive, since $\Psi<0$ in a well). Adding them, $-\tfrac23\Psi+\Psi=\tfrac13\Psi$ : two-thirds of the redshift is undone by the intrinsic heat. The well still appears cold, but at one-third of the naive gravitational depth — a genuine cancellation between gravity and thermodynamics, not a property of the redshift alone.

Why does the ISW vanish during matter domination?

Because the gravitational potential is exactly frozen in that era. The Poisson equation gives $\Phi\propto D(a)/a$ with $D(a)$ the linear growth factor. In a flat, pressureless, matter-dominated universe the growing mode is $D\propto a$ , so $D/a$ is constant and $\dot\Phi=0$ : structure grows at precisely the rate needed to keep each well at fixed depth as the expansion dilutes it. A photon crossing a static well gains energy falling in and loses exactly the same climbing out, so the net shift is zero. The balance is special to matter — add radiation, curvature, or dark energy and $D$ no longer tracks $a$ , the potentials evolve, and the ISW reappears.

What is the difference between the early and late ISW?

Both come from decaying potentials, but at opposite ends of cosmic history and on opposite angular scales. The early ISW happens just after recombination, when leftover radiation still speeds the expansion; potentials on scales entering the horizon then decay a little, and because those scales are small the power lands at intermediate multipoles $\ell\sim200$ , boosting the region around the first acoustic peak. The late ISW happens in the last few billion years, when dark energy decays the potentials on the very largest scales, appearing at $\ell\lesssim10$ . So the early ISW is a modest enhancement near the first peak sensitive to $\Omega_m h^2$ , while the late ISW is a low- $\ell$ diagnostic of dark energy.

How is the late ISW actually detected if it is so weak?

By correlation, not by amplitude. In the temperature power spectrum the late ISW is swamped by cosmic variance at low $\ell$ , so it cannot be isolated from the CMB alone. But it is sourced by the same low-redshift potentials that host today's galaxies, so it is spatially correlated with large-scale structure. Cross-correlating a CMB map with a wide-area galaxy or quasar survey — computing $w_{Tg}=\langle(\Delta T/T)\,\delta_g\rangle$ — yields a signal the primordial SW anisotropy cannot fake. A positive correlation at large angles means photons are slightly hotter toward overdensities, the signature of decaying wells, and this is how WMAP-, DES-, and DESI-era analyses built up several-sigma detections.

Does the ISW ever make overdensities look hot rather than cold?

Yes, and that sign flip is the whole point of the late ISW. The ordinary SW effect makes a large-scale overdensity at last scattering appear cold, because the dominant term is the redshift out of the well. The late ISW works oppositely: as dark energy decays a well while a photon crosses it, the photon emerges with a net blueshift, so late-time superclusters imprint slightly hot spots and voids cold ones. These are different structures at different redshifts — the SW cold spot on the $z\approx1090$ surface, the ISW hot spot on nearby $z\lesssim1$ structure — so they do not contradict each other. The positive CMB-galaxy cross-correlation is exactly this sign.

Why does a slight tilt n_s ≈ 0.965 change the flatness of the plateau?

The plateau is flat only for an exactly scale-invariant spectrum, $n_s=1$ , where $P_\Phi\propto k^{-3}$ and the Bessel projection gives $\ell(\ell+1)C_\ell=\text{const}$ . A spectral index below one puts slightly more power on large scales, so the projected $\ell(\ell+1)C_\ell$ acquires a gentle downward slope across the plateau. The size of that slope measures $n_s$ : Planck's $n_s\approx0.965$ , a few percent below unity, is one of the cleaner confirmations that inflation produced a nearly — but not exactly — scale-invariant spectrum. The plateau is therefore both an amplitude anchor and a tilt gauge.