Rayleigh scattering: blue sky thinking for future CMB observations

Antony Lewis1, ∗

1Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK

Rayleigh scattering from neutral hydrogen during and shortly after recombination causes the CMBanisotropies to be significantly frequency dependent at high frequencies. This may be detectablewith Planck, and would be a strong signal in any future space-based CMB missions. The laterpeak of the Rayleigh visibility compared to Thomson scattering gives an increased large-scale CMBpolarization signal that is a greater than 4% effect for observed frequencies ν & 500GHz. There is asimilar magnitude suppression on small scales from additional damping. Due to strong correlationbetween the Rayleigh and primary signal, measurement of the Rayleigh component is limited bynoise and foregrounds, not cosmic variance of the primary CMB, and should observable over a widerange of angular scales at frequencies 200GHz . ν . 800GHz. I give new numerical calculations ofthe temperature and polarization power spectra, and show that future CMB missions could measurethe temperature Rayleigh cross-spectrum at high precision, detect the polarization from Rayleighscattering, and also accurately determine the cross-spectra between the Rayleigh temperature signaland primary polarization. The Rayleigh scattering signal may provide a powerful consistency checkon recombination physics. In principle it can be used to measure additional horizon-scale primordialperturbation modes at recombination, and distinguish a significant tensor mode B-polarizationsignal from gravitational lensing at the power spectrum level.

I. INTRODUCTION

Neutral hydrogen produced as the universe recombinedat redshift z ∼ 1000 is often modelled as being trans-parent, so that photons only scatter from residual freeelectrons. However neutral hydrogen can also interactwith and scatter radiation. Since recombination onlyhappens once the typical photon energy is well belowthe ionization energy, by the time hydrogen is producedalmost all photons will have wavelengths much largerthan the atomic radius. The classical scattering of long-wavelength photons from the dipole induced in the neu-tral hydrogen is then called Rayleigh scattering, whichhas an asymptotic ν4 scaling with frequency. Higher fre-quencies of the observed CMB anisotropies will thereforebe Rayleigh scattered during and shortly after recombi-nation.

On small, sub-horizon scales Rayleigh scattering leadsto a damping of the anisotropies as photons from hotspots are scattered out of the line of sight, and photonsfrom cold spots are mixed with photons scattering intothe line of sight. Rayleigh scattering therefore gives theobserved small-scale CMB hot spots a red tinge, for thesame reason that sunsets look red. The small-scale polar-ization signal is also reduced by the additional scatteringfor a similar reason. However the large-scale polariza-tion from recombination is due to coherent quadrupolescattering into the line of sight. The additional Rayleighscattering at late times, where the quadrupole is larger,therefore increases the polarization at high frequencies,so the polarized sky is slightly blue on large scales.

The effect on the temperature anisotropies has beenpreviously calculated by Refs. [1, 2] and shown to give

∗URL: http://cosmologist.info

0 200 400 600 800 1000

ν/GHz

100

101

102

103

104

105

∆I/

(10−

26W

m−

2H

z−1sr−

1)

RMS primary anisotropy

RMS Rayleigh anisotropy

FIG. 1: Brightness intensity of the root mean square (RMS)CMB temperature anisotropy at l ≤ 2000 as a function offrequency, for the primary signal (no Rayleigh scattering, solidline) and the Rayleigh scattering contribution (scaling with arelative factor approximately proportional to ν4, dashed line).

several percent effect on the power spectra at 550GHzand above. The Rayleigh scattering effect becomesstronger at higher frequencies, but of course the black-body spectrum is also falling rapidly, so there are notmany observable photons at very high energies. Fig. 1shows the CMB brightness intensity as a function of fre-quency for the primary and Rayleigh anisotropies, whichshows that the Rayleigh signal is most likely to be ob-servable over a range the range of frequencies 200GHz .ν . 800GHz in the absence of foregrounds. This range isspanned by Planck where the signal may be detectable(subject to calibration and foreground issues), and could

arX

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ibili

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Thomson

Rayleigh (normalized)

Rayleigh (857 GHz)

Rayleigh (ν6 normalized)

Rayleigh (ν6 857 GHz)

FIG. 2: Left: Differential optical depth (comoving opacity τ̇ = Γ/(1 + z)) for Rayleigh and Thomson scattering of photonsas a function of conformal time η, with Rayleigh terms scaling ∝ ν4 (solid) and ∝ ν6 (dashed) shown separately for a coupleof observed frequencies. Before the main recombination event the Rayleigh scattering is from neutral helium, which is highlysubdominant to Thomson scattering. The Rayleigh opacity decays rapidly with redshift due to the ν4 ∝ (1 + z)4 redshiftingand the (1 + z)3 dilution of the gas density with the expansion. Right: The corresponding visibility functions. The solid lineshave been normalized; the dashed line shows the relative amplitude of the Rayleigh scattering for 857GHz (which is about theupper limit of observationally relevant frequencies). Here the visibility is defined as τ̇ e−τtot .

also be measured at much higher sensitivity by a next-generation space CMB mission.

Rayleigh scattering is important for a couple of rea-sons: firstly, it is always present, so must be modelledconsistently in any analysis using high-frequency chan-nels for CMB or foreground separation analysis; secondlyit may be able provide new information about the earlyuniverse, potentially tightly constraining the expansionrate and ionization history around recombination, andalso probing additional primordial perturbation modes.In this paper I extend the previous calculation of Ref. [2]to model the Rayleigh temperature signal in more detail,provide a new calculation of the polarization signal, anddiscuss future detectability and measurement.

The outline of this paper is as follows: Sec. II re-views the details of Rayleigh scattering and the scat-tering sources for the CMB; in Secs. III and IV givenumerical results for the temperature and polarizationrespectively, and Sec. V describes the approximate formand contributions to the auto and cross power spectra.Sec. VI discusses the ideal detectability of the signal withboth current data and possible configurations for a futurespace mission, and Sec. VII then studies whether an ac-curate measurement of the Rayleigh signal could be usedto extract more information about the primordial pertur-bations. I assume a linearly-perturbed standard ΛCDMcosmology throughout, and that with suitable sky cutsand observations at many frequencies foregrounds can besubtracted accurately. The intricate work required to as-sess likely realistic levels of foreground residuals and im-plementation of foreground separation technology withnon-blackbody CMB spectra is deferred to the future; in

any case knowledge of the expected foregrounds at therequired scales and level of detail is currently rather lim-ited, so making any clear prediction would be difficult atthis stage.

II. RAYLEIGH SCATTERING

The non-relativistic Rayleigh scattering of photonswith frequency ν from hydrogen in the ground state hascross section given by [3]

σR(ν) =

[(ν

νeff

)4

+638

243

(ν

νeff

)6

+1299667

236196

(ν

νeff

)8

+ . . .

]σT (1)

for ν � νeff , where σT is the Thomson scattering crosssection and νeff ≡

√8/9cRA ≈ 3.1× 106GHz (where RA

is the Rydberg constant, corresponding to the Lymanlimit frequency). The result is derived from the Kramers–Heisenberg formula not including the intrinsic quantummechanical line width of the excited levels, and henceshould not be used for scattering close to or above theLyman-α frequency where resonant scattering becomesrelevant (for z . 1500 this requires observed frequenciesν . 1600GHz). The corrections to the long-wavelengthν4 scaling become non-negligible around recombination,giving a greater than 10% contribution from the ν6 termat observed frequencies ν & 500GHz (see Fig. 2).

Rayleigh scattering is easily included in a line of sightBoltzmann code, with the total scattering rate for pho-

3

tons with frequency ν in the gas rest frame given approx-imately by

Γ(ν) = neσT + σR(ν) [nH +RHenHe] . (2)

Here nH and nHe are the number densities of neutral hy-drogen and helium, and RHe ≈ 0.1 is the relative strengthof Rayleigh scattering on helium compared to hydro-gen [4]. I calculate the ionization history (and hence ne,nH and nHe) using the approximate recombination modelof Ref. [5] calibrated to full multi-level atom codes [6, 7].

The total scattering rate is very insensitive to the he-lium modelling since its contribution to total scattering isvery small before recombination (see Fig. 2), and only apercent level correction once there is a significant amountof neutral hydrogen. Scattering from ionized helium andother constituents can be neglected, and here the relevantfrequencies are well above the 21cm hyperfine transitionenergy (the 21cm signal following recombination is con-sidered in detail in Ref. [8]). Resonant scattering from ex-cited states of hydrogen, and other atoms and moleculesat later times, can also produce interesting frequency-dependent signals [9–13]. The cross sections for reso-nant scattering can be large, but all are suppressed byvery low abundance. The frequency and angular depen-dence of the resonant scattering signal is very differentfrom Rayleigh scattering (which has smooth monotonicfrequency dependence), and should not be a major sourceof confusion in practice. Resonant scattering is not in-cluded in the results of this paper.

I calculate numerical power spectra using a modifiedversion of camb1 [14]. The modifications to include theRayleigh signal are straightforward, but require the evo-lution of a separate Boltzmann hierarchy for each fre-quency of interest, each with different scattering sources,visibility and line of sight integral. The effect of the addi-tional total baryon-photon coupling can be included us-ing a dense sampling of frequencies (or using an effectivefrequency-averaged cross section [15]), but for most pur-poses this effect is small enough to neglect and then onlythe frequencies of interest need to be evolved. Since thesignal is small, for numerical stability Rayleigh-differencehierarchies can be used, giving directly the auto and crosspower spectra of the Rayleigh and primary signals. SinceRayleigh scattering is only important once recombinationstarts, the Rayleigh hierarchies are only evolved once thetight coupling approximation is turned off. Note thatthe frequency dependence of the Rayleigh cross sectiondoes not introduce any additional terms due to boostingfrom the gas rest frame at linear order because the netscattering effect is zero in the background.

Rayleigh scattering is only negligible compared to

1 July 2013 version; modified code on the rayleigh branch of thegit repository (access available on request).

Thomson scattering when

nH

([1 + z]ν

3× 106GHz

)4

� ne. (3)

As recombination happens ne drops rapidly, which in-creases the relative importance of Rayleigh scatteringeven though the frequencies of interest are significantlybelow νeff at the time. For CMB observations Rayleighscattering is potentially important for the higher end ofobservable frequencies, i.e. ν & 200GHz, even though itis only a small fractional change. Detectability is limitedby noise (and foregrounds), not cosmic variance of theprimary anisotropies, since the same perturbation real-ization is being observed at the different frequencies andthe Rayleigh signal is strongly correlated to the primaryCMB [16].

III. RAYLEIGH TEMPERATURE SIGNAL

Neutral hydrogen is only generated once recombina-tion is underway, so the visibility function for Rayleighscattering is peaked at somewhat lower redshift than themain Thomson scattering signal as shown in Fig. 2. How-ever photon frequencies redshift so that ν4 ∝ (1 + z)4,and densities dilute ∝ (1 + z)3 due to expansion, sothe amount of Rayleigh scattering does decay rapidlywith time: the visibility is still well localized aroundthe last scattering surface. In principle it probes slightlydifferent perturbations to the primary signal during re-combination. However the signal is highly correlatedto the primary anisotropies, and since the Rayleigh sig-nal is small the dominant detectable signal is the cor-relation of the Rayleigh contribution with the primaryCMB, though there is also a small uncorrelated compo-nent (see Sec. VII). The Rayleigh scattering contributionoriginates from a somewhat later time that the primaryvisibility peak, so its contribution has acoustic oscilla-tions shifted to slightly lower l. The total power differsfrom the primary signal by both an oscillatory structure,and also a power decrement on small scales since a givenfixed l is damped more.

It is often useful to think of the high frequency obser-vations being the sum of a primary and Rayleigh con-tribution, so that the total power spectrum is a sumof the primary spectrum, twice the Rayleigh-primarycorrelation spectrum, and the Rayleigh-Rayleigh auto-spectrum. The cross-correlation signal can easily be iso-lated in principle by cross-correlating a high frequencyand low-frequency map (with negligible Rayleigh contri-bution). Numerical results for the temperature auto- andcross-spectra are shown in Fig. 3 for various frequencies,compared to an idealized Planck error model. The cross-correlation spectrum dominates the observable signal forcurrent generation observations like Planck.

Rayleigh scattering also increases the total couplingbetween photons and baryons, which affects the pertur-bations at all frequencies, e.g. via the baryon velocity.

4

101 102 103

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10-2

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100

101

102

103

l(l+

1)Cl/

(2πµK

2)

143

217

353

545

857

FIG. 3: Rayleigh contributions to the temperature powerspectra at Planck frequency channel notional central frequen-cies (in GHz, colours). Solid lines are the Rayleigh-primarycross correlation (scaling approximately ∝ ν4), dash-dot linesshow the Rayleigh-Rayleigh power spectra (scaling ∝ ν8).Dotted lines show the naive error per ∆l = l/10 bin in thecross-correlation (no foregrounds). Only the cross-correlationsignal is potentially detectable by Planck.

This effect is very small, ∼ 0.04% (in agreement withRefs. [1, 2, 15]), and can be neglected for current obser-vations (and is anyway not frequency dependent). Theslowing of baryon cooling is also negligible because theenergy transfer in recoil from hydrogen is much lowerthan from a much lighter electron. For further discus-sion and a more detailed semi-analytic discussion of theapproximate form of the Rayleigh scattering temperaturesignal see Ref. [2].

IV. POLARIZATION

Rayleigh scattering is also polarized: in the classicallimit the scattering from the induced dipole has the samedσR ∝ |ε1 · ε2|2 structure as Thomson scattering, whereεi are the polarization vectors. This should be a goodapproximation to energies much larger than those of rel-evance for the CMB since spin-flip scattering events arehighly suppressed even at high energies [17]. Hence theRayleigh polarization can be handled in a Boltzmanncode in exactly the same way as Thomson scattering,e.g. following Refs. [18–20].

Since the Rayleigh visibility peaks at later times, thehorizon size there is larger, and the large-scale polar-ization signal in E-modes from Rayleigh scattering of

101 102 103

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10-2

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l(l+

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EE

Lensing BB

Tensor BB (r=0.02)

FIG. 4: Lensed polarization power spectra at low frequencieswhere Rayleigh scattering is negligible (solid) and 857GHz(dashed). The latter high frequency is chosen to see theRayleigh contribution by eye but may not be observable inpractice. The low-l reionization and lensing signals are hardlychanged, but the 10 . l . 100 polarization power is signif-icantly boosted, along with a significant suppression in thedamping tail at high l.

the quadrupole has more power on large scales, givinga frequency-dependent boost to the power beyond thereionization bump; see Fig. 4. There is also a suppres-sion of power on small scales for the same reason as inthe temperature spectrum. The large-scale bump in thespectrum is due to Thomson scattering at reionizationwhere Rayleigh scattering is negligible, and hence re-mains essentially unchanged. Corresponding fractionaldifferences to the power spectra are shown in Fig. 5. Forthe polarization there are differences at the several per-cent level on both large and small scales.

A quadrupole induced by gravitational waves enteringthe horizon at recombination would also Rayleigh scatter,giving a similar Rayleigh contribution to the BB tensor-mode power spectrum. In contrast the B modes pro-duced by lensing of E modes originate from polarizationat recombination from a wide range of scales, where theRayleigh signal has varying sign. The Rayleigh contribu-tion to the lensed BB spectrum therefore partly averagesout giving a significantly smaller Rayleigh contribution tothe lensing BB power spectrum on large scales as shownin Fig. 4.

V. POWER SPECTRA

The in-principle direct observables are the angularpower spectra between all the fields and frequencies:

CXiY j

l = 〈Xi∗lmY

jlm〉, (4)

where X is T, E, or B, and i labels the frequency. At veryhigh frequencies ν & 800GHz where the Rayleigh scatter-

5

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217x217 217x353 217x545

353x217 353x353 353x545

545x217 545x353 545x545

FIG. 5: Fractional difference between the lensed scalar CMB power spectra CXiYjl for observed frequencies 217, 353 and

545 GHz, compared to the primary (low-frequency) power spectra. Each plot shows the fractional difference ∆Cl/Cl for

temperature (red), E-polarization (blue) and B-polarization (magenta), and ∆CTEl /√CEEl CTTl for the T -E cross-correlation

spectra (green). Each plot is a different pair of frequencies, and the results above and below the diagonal are the same except

for the CTiEjl correlation (green) which is not symmetric. Note that a small fractional difference does not necessarily mean that

the signal is unobservable, since detectability is only limited by noise (and foregrounds); conversely a relatively large fractionaldifference in the polarization is not observable unless the noise is low enough.

ing becomes a substantial effect there are very few CMBphotons and very high foregrounds, and the signal is un-likely to be observable in practice. At lower frequenciesthe Rayleigh scattering is a small effect and can accu-rately be modelled perturbatively as

Xilm ≈ Xlm +

(νiν0

)4

∆X4,lm +

(νiν0

)6

∆X6,lm, (5)

where Xlm is the primary (low frequency) signal, ν0 issome reference frequency, and ∆Xi are the contributionsdue to Rayleigh scattering at frequency ν0 with the cor-responding frequency scaling. This approximation willbreak down when higher terms in the Rayleigh scatter-ing cross section become relevant, and also when theRayleigh optical depth becomes significant, but is a goodapproximation over most of the frequency range of inter-est. With this approximation the power spectra are

CXiY j

l ≈ CXYl +

(1

ν0

)4 [ν4jC

X∆Y j4l + ν4

i C∆Xi4Yl

]+

(1

ν0

)6 [ν6jC

X∆Y j6l + ν6

i C∆Xi6Yl

]+

(νiνjν2

0

)4

C∆Xi4∆Y j4l + . . . (6)

The leading Rayleigh term fits the difference spectrashown in Fig 5 rather well. The last term is not requiredat Planck sensitivity where cross-correlations dominatethe observable signal, but would be detectable with fu-ture missions (see Sec. VII), and is quantitatively moreimportant than the O(ν8) cross-correlation scatteringcontribution. The observed spectra are of course lensed,and the lensed spectra can be calculated for all the crossfrequencies using standard techniques [21].

Since the Rayleigh signal is significant at higher fre-quencies, it is important to model it in cosmological andforeground analyses using those frequencies. The sig-nal is easily simulated exactly from the full set of cross-frequency power spectra, and also for current data ap-proximately using Eq. (5) by writing

xνlm = xlm +( ν

GHz

)4

R4,l xlm +( ν

GHz

)6

R6, xlm + . . .

(7)where xlm = (Tlm, Elm) is a simulation of the blackbodyfields, and Ri is a 2 × 2 matrix that can be computedfrom the (unlensed) cross power spectra such that xνlmhas the required covariance to leading order in the cross-correlation Rayleigh effect. The separate terms can alsobe individually lensed and then combined with the ap-propriate weighting.

6

VI. DETECTABILITY

At high frequencies measurement of the Rayleigh signalis limited by the rapid fall in the blackbody spectrum (sofewer CMB photons and hence larger relative noise), andforegrounds including dust, SZ, molecular lines and thecosmic infrared background (CIB), though the latter isexpected to be only weakly polarized. A detailed treat-ment of likelihood uncertainties from foreground mod-elling is beyond the scope of this paper, but there areseveral reasons why they may not be a major problemfor detecting the Rayleigh signal if many high sensitivitychannels are included at 100GHz . ν . 800GHz:

• The shape of the Rayleigh scattering spectra canbe computed accurately (for fixed cosmological pa-rameters).

• The Rayleigh signal spans the full range of scales,so for example CIB and SZ should only be a smallcontaminant at lower l, and the dust spectrum fallsto higher l.

• The dense frequency coverage and low noise levelsrequired to clean foregrounds for detecting primor-dial B modes should also be adequate to clean thehigher l foregrounds to within small residuals.

• Unlike the foregrounds, the Rayleigh signal isstrongly correlated to the primordial CMB, so across-spectrum between a low frequency map (withlow dust, CIB and noise) and a cleaned high-frequency map (with strong Rayleigh signal) is ex-pected to have small foreground contamination.

• Residual foregrounds should have a power spec-trum shape in l that looks very different from thepredicted Rayleigh contribution and will mainlyserve to increase the effective noise. In particu-lar the Rayleigh signal is oscillatory, with ampli-tude nearly tracking the primary power spectrumon small scales, but with slightly shifted acous-tic scale due to the larger sound horizon size forRayleigh scattering [2].

Conversely it will be important to self-consistently modelthe Rayleigh contribution when doing foreground sepa-ration and data analysis. For a more detailed discus-sion of foreground modelling for future CMB missionssee e.g. Ref. [22–24]. The extent to which the resid-ual foregrounds dominate the noise budget depends onthe density of the frequency sampling and the correla-tion between frequencies of the various components; forthe relatively sparse sampling of Planck, where for ex-ample the CIB has a significant uncorrelated componentbetween channels [25], realistic errors are likely to be sub-stantially larger than those estimated from instrumentalnoise alone. At large scales where the noise levels arevery low (so the difference of maps at a low and high fre-quency would ideally isolate the Rayleigh signal nearly

perfectly), details of non-whiteness of the noise and sys-tematic residuals may also be important to determine theactual level of sensitivity to Rayleigh differences on largescales even if foregrounds can be accurately removed.

Here I simply give ballpark sensitivity numbers assum-ing isotropic noise and negligible foreground residualsfor some simple cases. Since the analysis is only ap-proximate, for simplicity I use the approximate modelof Sec. V with leading ν4 scattering terms, and assumedelta-function frequency bandpasses at band-central fre-quencies. For Gaussian CMB fluctuations the ideal like-lihood function is straightforward. For example considerthe simple case of having one low frequency channel (withnegligible Rayleigh signal) and one high frequency chan-nel, with noise N0

l and Nνl respectively. The covariance

of the measured temperatures T = (T0, Tν) when thenoise dominates the Rayleigh-Rayleigh auto power spec-trum is then approximately⟨

TlmT†lm

⟩≈(Cl +N0

l Cl + CTRνl

Cl + CTRνl Cl + 2CTRνl +Nνl

), (8)

where Cl is the primary power spectrum and CTRνl the(small) Rayleigh-primary cross-correlation. The Fisher

matrix for the fractional measured amplitude of CTRνl ata given multipole assuming it is small is then

σ−2l ≈

(2l + 1)fsky

[Cl(N

νl +N0

l ) +N0l (Nν

l + 2N0l )]

[Cl(Nνl +N0

l ) +N0l N

νl ]

2 .

(9)As expected this blows up (i.e. definite detection) asN → 0 since the primary cosmic variance fluctuationsare the same in both maps. This result can serve as aguide as to whether Rayleigh scattering is important ina given high frequency map, for example see Fig. 3. Thesame form holds for the EE polarization power spec-trum, though for higher noise levels the sensitivity forpolarization is likely to be dominated by the TE correla-tion. More general cases can be considered using the fullFisher matrix

Fij,l ≈(2l + 1)fsky

2Tr

[C−1l

(∂

∂θiCl

)C−1l

(∂

∂θjCl

)](10)

for parameters θi, θj where Cl is the full multi-frequency temperature and polarization covariance ma-trix (a 2Nfreq× 2Nfreq matrix in the case of T and E ob-servations). Since foregrounds are being neglected I shallrestrict to considering the detectability of the primary-Rayleigh cross-spectrum for various high-frequency chan-nels individually rather than super-optimistically consid-ering joint constraints.

I assume fsky = 0.6 and white noise at l ≥ 1000 witha multiplicative (1000/l)0.4 flattening at l < 1000 tocrudely avoid massively overweighting very low l (qual-itatively consistent with the Planck temperature noiseat low l). Fig. 6 shows the corresponding naive sig-nal to noise in the Rayleigh signal for high frequency

7

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PRISM

500 1000 1500 2000

l

0.0

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Fl

EE

FIG. 6: Approximate Fisher signal to noise per l for the Rayleigh-primary cross-spectra in the most sensitive high frequencychannels of Planck, and some individual broad-band high-frequency channels for a couple of configurations for proposed futurespace missions (CORE and PRISM). Noise is assumed to be white on small scales with some flattening at low l, and fsky = 0.6;no foreground residuals are included.

channels in Planck (full mission) and a couple of strawman configurations for future space missions: CORE [26](which has many more detectors at ν < 300GHz thanat high frequencies, and hence is relatively insensitive tothe Rayleigh signal), and the more ambitious PRISM [24](which has many high frequency channels with hundredsof detectors). All are sensitive enough to detect the tem-perature signal at several sigma. PRISM would mea-sure the temperature signal in detail with the powerat each l being measured with small fractional error,and an overall determination of the amplitude of theRayleigh signal to ∼ 0.3%. The Rayleigh-temperaturecross primary-polarization combination dominates thedetectability of polarization cross-spectra, and is easilydetectable in future missions. A CORE-like configura-tion would be marginally able to detect the Rayleigh-primary EE cross-spectrum at a bit under 2σ per chan-nel for 200GHz < ν < 300GHz, and PRISM has thesensitivity to detect it at around 20σ in each of the400GHz < ν < 700GHz channels.

Note that contributions from O(ν6) Rayleigh scatter-ing terms become important to model for high-sensitivityobservations, with PRISM being in principle sensitive tothem at the 10σ–20σ level in the temperature spectra for450GHz . ν . 800GHz. The O(ν8) scattering correc-tions are an additional O(1σ) correction on top of that,which is unlikely to be very important in practice but iseasily included in a full analysis.

VII. ADDITIONAL INFORMATION?

On small scales the Rayleigh scattering functionsmostly as an additional screen just in front of the primarylast-scattering surface, leading to additional scatteringand hence damping of small scale primary anisotropies.The Rayleigh signal is therefore highly (anti-)correlatedto the primary signal, being largely proportional to it,with small additional contributions from anisotropies be-ing sourced, for example by Doppler terms due to pecu-liar motion. The screening effect is much like the op-tical depth suppression from reionization, except thathere the relevant horizon size is that at recombination,leaving a significantly larger range of observable super-horizon modes that are not damped in the same way.The very large-scale E-mode polarization signal is alsohighly correlated to the primary E-mode signal, since itis caused by scattering from nearly the same large-scalequadrupole; see Fig. 8.

On intermediate scales, or with high sensitivity, theRayleigh scattering signal can probe different perturba-tion modes that cannot be isolated from the primaryanisotropies, and hence contains additional informationabout the primordial fluctuations. The Rayleigh-primarycross-correlation signal strongly constrains the cosmolog-ical model around recombination, but does not measurenew fluctuations. The uncorrelated part of the Rayleigh-Rayleigh power spectrum is what contains independentinformation about the perturbations. It is far too smallto be measured by Planck and only marginally by aCORE-like experiment, but more ambitious future ob-

8

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660

800

FIG. 7: The Rayleigh-Rayleigh auto-spectra (solid lines), and the uncorrelated component (dashed lines) for the CMBtemperature (left) and E-polarization (right) at various frequencies (in GHz). Dotted lines show the naive zero-foregrounderror per ∆l = l/10 bin at each frequency for a PRISM-like observation. The temperature spectrum is above the noise levels onintermediate scales, but the polarization signal would require significantly lower noise to be measured well. Thick black linesshow the primary power spectra (and CBBl in the case of the thinner line in the right-hand plot).

servations may be able to measure it if foregrounds canbe modelled to high accuracy.

The Rayleigh anisotropies are never perfectly corre-lated to the primary anisotropies, so in principle eachRayleigh mode from each νn term in the cross sectionhas additional information. A perfect measurement of asmall ν4-scaling effect could double the number of modesthat can be measured (in the same way that polariza-tion doubles the number compared to having just tem-perature). In practice, if the signals are small and alsohighly correlated, extremely low noise levels would be re-quired to achieve this, and in practice it is likely to beimpossible, especially considering the vastly larger fore-grounds that must be distinguished. Nonetheless it isworth briefly considering what extra information mightbe available under marginally more realistic assumptions.

Fig. 7 shows the Rayleigh auto-spectra for the tem-perature and polarization at various frequencies, andalso the uncorrelated component. Even with ambitiousPRISM-like observations the E-polarization auto spec-trum is too small to be useful. However the temperatureauto spectra could be above the instrumental noise, andthe Rayleigh auto-spectrum measured statistically on in-termediate scales. Let’s define the number of modes asbeing

nl ≡ (2l + 1)fskyTr[(

[Cl + Nl]−1Cl

)2], (11)

where Cl is the matrix of theory spectra and Nl is thecorresponding noise contribution. This definition corre-sponds to (two times) the Fisher matrix for an overall

power spectrum amplitude parameter at a given l, andis the sum of the squares of the signal-to-signal-plus-noise eigenmode eigenvalues. With zero noise the pri-mary spectra have nl = (2l + 1) and nl = 2(2l + 1) fortemperature and temperature+polarization respectively.Including a high-frequency spectrum with the Rayleighsignal then adds additional modes if the spectrum is notnoise dominated, and there is a significant uncorrelatedcomponent.

With PRISM sensitivity the low frequency channelshave nl ≈ 2(2l + 1) up to high l since temperature andpolarization would both be measured at high signal tonoise. Fig. 9 shows the number of additional modesas a function of l when adding a high-frequency chan-nel to probe the Rayleigh scattering signal. In totalthe Rayleigh measurement could probe around 10 000new modes, mostly at l . 500. The information is onlarger scales because that is where the noise is lowest,and also because the primary temperature anisotropiesthere come from multiple sources, which the additionalRayleigh measurement can help disentangle. For exam-ple there is no Sachs-Wolfe or Integrated Sachs-Wolfe(ISW) contribution to the Rayleigh signal because thereis no Rayleigh monopole background: it is only generatedby sub-horizon scattering processes, not line-of-sight red-shifting effects.

Some modes contributing to the large-scale CMB alsomeasurable by other means, for example large-scalestructure and CMB lensing can probe the modes con-tributing to the ISW effect independently of the CMB

9

2 40 200 700 1500 2500

l

1.0

0.5

0.0

0.5

1.0r

T

2 40 200 700 1500 2500

l

E

FIG. 8: Correlation coefficient between the Rayleigh andprimary signals (at 545GHz), for temperature (left) and E-polarization (right).

200 400 600 800 1000

l

10

0

10

20

30

40

50

60

Num

ber

of

modes nl

220

265

300

320

395

460

555

660

800

FIG. 9: The number of new perturbation modes per l in prin-ciple measurable by the Rayleigh scattering signal from vari-ous single broad-band high-frequency channels with PRISM-like noise. The information is essentially all in the temper-ature, not the polarization, and nearly the same underlyingmodes are probed by each frequency.

temperature. The Rayleigh signal is however a muchmore direct probe of last scattering than doing model-dependent ISW inference, and most of the additionalmodes in this case are localized at last scattering, notISW. In particular the large-scale Rayleigh temperaturesignal depends directly on the Doppler scattering terms,which are only a subdominant component of the primaryanisotropies. This information is complementary to thelarge-scale primary polarization because the ∼ n̂ · vbterms that source the Rayleigh Doppler signal come frommodes aligned with line of sight n̂, whereas the E polar-ization comes from the quadrupole caused by infall indirections transverse to the line of sight, so the modes

being probed are nearly independent.The number of additional modes is small compared to

the total number available in the primary anisotropies(O(l2max)), even with ambitious observations under sim-plistic assumptions. However they could in principle beuseful since cosmic variance is very limiting on largescales, and there are a number of ‘anomalies’ claimedin the large-scale temperature distribution. If measure-ment of the Rayleigh signal at l . 200 were achiev-able it would be one way to get slightly better statis-tics and help to separate different possible physical mod-els, in addition to the information available in polariza-tion. Orders of magnitude higher sensitivity than PRISMwould be required to get substantial additional informa-tion on smaller scales or from the polarization. How-ever since there is potentially useful information on fairlylarge scales, it may also be possible to exploit degree-resolution observations with a Fourier Transform Spec-trometer (FTS), as proposed by PIXIE [23] and thenPRISM.

VIII. CONCLUSIONS

Rayleigh scattering produces an interesting detectablesignal in the CMB temperature and polarization at fre-quencies & 200GHz. In summary Rayleigh scattering:

• Can easily be modelled accurately using a full set ofcross-frequency power spectra calculated in lineartheory from a Boltzmann code

• Produces a few-percent damping of the tempera-ture power at high frequencies that may be de-tectable with current observations [2].

• Increases the total coupling of baryons andphotons, leading to a very small . 0.04%frequency-independent increase in the small-scaleprimary CMB power spectra and a smaller changein the matter power spectrum [2, 15].

• Boosts the large-scale E-polarization at 10 . l .300 due to the increased horizon size for the laterRayleigh scattering.

• Damps the small-scale polarization in a similar wayto the temperature anisotropies

• Must be modelled for consistent foreground sepa-ration, including significant corrections from the ν6

term in the cross section at higher frequencies.

• Produces temperature and polarization signals thatcould both easily be measured by a future space-based mission; better measurement motivates moresensitivity in the 400GHz . ν . 700GHz frequencyrange.

10

• Enhances the 10 . l . 100 B-mode polarizationsignal from gravity waves at higher frequencies,and has a much smaller effect on the non-linearB-modes from lensing.

• Is strongly correlated to the primary signal, so ameasurement of the cross-correlation may providea robust means of measurement and allow powerfulconstraints on the expansion and ionization historyof the universe around recombination.

• Probes new primordial perturbation modes, thoughdue to the strong correlation of the bulk of theRayleigh signal this requires very high sensitivityand foreground rejection efficiency, and is likely tobe of most use for probing roughly horizon-scalemodes at recombination.

Future work is required to assess likely levels of resid-ual foreground contamination for different possible ob-servation strategies, and hence levels of precision thatmay be achievable in practice. If spectral distortions inthe monopole are studied at high sensitivity over cleandegree-scale patches of sky, the local distortion due toRayleigh scattering may also be non-negligible at the

Jansky level.

IX. ACKNOWLEDGMENTS

I thank Rishi Khatri, Kris Sigurdson, Andrew Jaffe,Anthony Challinor and Duncan Hanson for discussion,Planck colleagues for bringing Rayleigh scattering to myattention, and Guido Pettinari for finding an error in theν8 factor and a helpful comparison of numerical results.I acknowledge support from the Science and TechnologyFacilities Council [grant number ST/I000976/1].

Notes: Since publication in JCAP this versioncorrects the numerical factor in the ν8 term of Eq.(1),and a 0.7% error in the numerical value of νeff used fornumerical calculations, leading to a small quantitativechange in results that does not affect any conclusions.After submission I also became aware of Rayleighscattering calculations by E. Alipour, K. Sigurson, andC. Hirata (in preparation), with results for the CMBpolarization power spectra similar to those reportedhere.

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