Lensing signals from Spin-2 perturbations

# Lensing signals from Spin-2 perturbations

Julian Adamek Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, 24 quai E. Ansermet, CH–1211 Genève 4, Switzerland    Ruth Durrer Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, 24 quai E. Ansermet, CH–1211 Genève 4, Switzerland    Vittorio Tansella Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, 24 quai E. Ansermet, CH–1211 Genève 4, Switzerland
July 27, 2019
###### Abstract

We compute the angular power spectra of the E-type and B-type lensing potentials for gravitational waves from inflation and for tensor perturbations induced by scalar perturbations. We derive the tensor-lensed CMB power spectra for both cases. We also apply our formalism to determine the linear lensing potential for a Bianchi I spacetime with small anisotropy.

## I Introduction

Since many years now, the cosmic microwave background (CMB) is the most precious signal in cosmology. It is being used to determine the parameters which govern the expansion of the Universe, its content and the initial conditions of its fluctuations, see Ade:2015xua () for latest results, Durrer:2015aa () for a historical account and Ruth_book () for a comprehensive monograph on the subject. So far, no gravitational wave background has been detected in the CMB and an upper limit of has been derived for the tensor to scalar ratio Ade:2015tva (). In cosmology, these parameter estimations are of course always model dependent and therefore have to be taken with a grain of salt. The present limit on mainly comes from the contribution of gravitational waves to the temperature anisotropy and from that fact that they induce so called -polarisation, i.e., a rotational component in the polarisation vector field which is absent for purely scalar perturbations. In this paper we study yet another aspect of tensor perturbations. They introduce B-modes (i.e. rotational modes) also in the lensing signal. This effect has already been derived in Dodelson:2003bv (); Li:2006si (); Yamauchi:2012bc () and applied to both, primordial gravitational waves from inflation and topological defects Yamauchi:2012bc (); Yamauchi:2013fra (). In both cases it was found that for realistic parameters the effect is unobservably small. Also the effect of the gravitational wave contributions to the shear in the lensing of large scale structure has been investigated and found to be very small Schmidt:2012ne (); Schmidt:2012nw (), while the effect of the tidal field seems to be more promising Schmidt:2013gwa (). In this work we present an independent, alternative derivation of CMB lensing by tensor modes. As we will see, these modes can describe classical gravitational waves but also other spin-2 perturbations of the metric which do not propagate as waves in the usual sense, e.g. extremely infrared modes which make the Universe locally look like a Bianchi I model. We compare the signal induced by primordial gravitational waves and the one from tensor perturbations which are generated by the formation of large scale structure. This part is new. In a previous paper, the effect from induced tensor perturbations has been considered to second order Saga:2015apa (), while here we incorporate the fully nonperturbative results from relativistic large scale structure simulations Adamek:2014xba (); Adamek:2015eda (). We also discuss the qualitative difference of these tensor modes from ordinary gravitational waves which has not been appreciated in previous literature. Finally, we apply our formalism also to describe lensing in a Bianchi I model, i.e. a homogeneous but anisotropic model of the Universe.

The paper is structured as follows: In Section II we derive the main formulae for CMB lensing by gravitational waves. In Section III we present numerical results for primordial gravitational waves and for gravitational waves induced from large scale structure. In Section IV we study the Bianchi I model and in Section V we discuss our findings and conclude. Some technical aspects are deferred to two appendices.

## Ii CMB lensing by gravitational waves

Notation: We work with a flat Friedmann background and in conformal time, , such that the unperturbed metric is given by

 d¯s2=gμνdxμdxν=a2(t)(−d% t2+γijdxidxj). (1)

Greek indices run from 0 to 3, latin indices run over spacelike coordinates from 1 to 3 while latin indices go from 1 to 2 and denote coordinates on the unit sphere of directions. Throughout this paper we adopt the spherical coordinate system for the metric of the three dimensional slices and we set the spatial curvature to zero,

 γijdxidxj=dχ2+χ2(dθ2+sin2θdϕ2), (2)

where is the comoving distance. The derivative with respect to will be denoted with a dot , so that is the comoving Hubble parameter. is the physical Hubble parameter with present value km/sec/Mpc.

### ii.1 Perturbed photon path and the lensing potentials

We first want to compute the deflection of a light ray in a Friedmann universe with spin-2 perturbations. The perturbation of the line element is given by

 ds2=a2(t)[−dt2+(γij+hij)dxid% xj], (3)

where the tensor modes obey the transverse and traceless conditions. Since we are only interested in deflection of a null-geodesic it is more convenient to consider the conformally related metric

 d~s2=−dt2+(γij+hij)dxidxj. (4)

The observer is placed at and we now consider a photon with unperturbed trajectory given by the null-geodesic where is the photon direction fixed by the angles and is the affine parameter, such that, at the unperturbed level, . The perturbed four-velocity is given by . We define the displacement vector . Solving the spatial parts of the geodesic equations at first order in and in the perturbation, we find

 αa=∫χ∗0dχ[hra(χ,θ0,ϕ0)χ+12χ−χ∗χχ∗∇ahrr(χ,θ0,ϕ0)], (5)

where is the comoving distance at emission, is the gradient on the unit sphere and Born approximation was used.

When considering lensing by density perturbations the first-order deflection angle can be written as the gradient of a single scalar lensing potential Lewis:2006fu (). For tensor perturbations this is no longer true and we have to decompose in its gradient-mode, i.e. E-mode, potential and its curl-mode, i.e. B-mode, potential . We then write

 αa(n)=∇aψ(n)+εba∇bϖ(n), (6)

explicitly

 αθ=∂θψ+1sinθ∂ϕϖ,αϕ=1sinθ∂ϕψ−∂θϖ. (7)

Let us also introduce the spin raising and lowering operators and acting on a helicity tensor field on the sphere as

 ⧸∂s=(scotθ−∂θ−isinθ∂ϕ),⧸∂∗s=(−scotθ−∂θ+isinθ∂ϕ). (8)

With these operators we can write two differential equations for the helicity 0 lensing potentials

 {√2α+=−⧸∂∗(ψ+iϖ)√2α−=−⧸∂(ψ−iϖ), (9)

where we have introduced the helicity basis . Using and the fact that the commutator of the spin raising and lowering operators vanishes on functions, we obtain

 {⧸∂∗α−+⧸∂α+=−√2Δψ⧸∂∗α−−⧸∂α+=−√2Δϖ. (10)

In the next section we will compute in order to solve eq. (10) for the two lensing potentials.

### ii.2 Angular power spectra

It is convenient to work in Fourier space where we define

 hij(x,t)=∫d3k(2π)3[h⊕k(t)e⊕ij(^k)+h⊗k(t)e⊗ij(^k)]e−ik⋅x. (11)

Here we have introduced two time-independent polarization tensors and which can be expressed in terms of vectors of the orthonormal basis as

 e⊕ij(^k)=1√2[e1i(^k)e1j(^k)−e2i(^k)e2j(^k)],e⊗ij(^k)=1√2[e1i(^k)e2j(^k)+e2i(^k)e1j(^k)]. (12)

With these definitions the polarization tensors are normalized as . The polarization tensors depend on the direction since the frame , which reveals the 2 physical d.o.f. of the spin-2 field in eq. (11), depends on . In this frame the spin-2 field has non zero components only in the plane and, for fixed , we can always choose the basis in which

 (e⊕ab)=1√2(100−1),(e⊗ab)=1√2(0110). (13)

We can also write the tensor perturbation in the helicity basis , as

 (14)

where we have defined . We have also set and we shall work with these from now on.111Working with instead of the usual is more convenient and, for parity invariant perturbations, does not make any difference since all the power spectra are equal:
Let us now rewrite the Fourier components of the GWs in the spherical basis of eq. (4). In other words we perform a rotation with Euler angles () to rotate into . After this operation we can write (for more details see Appendix A)

 hrrk=12sin2β(e2iγh+k+e−2iγh−k),hr±k≡1√2(hrθk∓ihrϕk)=sinβ((cosβ∓1)e2iγh+k+(cosβ±1)e−2iγh−k)2√2eiα. (15)

With this and eq. (5) we can now write in Fourier space. We then solve eq. (10) to find expressions for the lensing potentials and in Fourier space in terms of and . We want to compute their angular power spectra in terms of the power spectrum of given by

 (16)

Since we observe the displacement vector on the celestial sphere the natural expansion for the lensing potentials is in terms of spherical harmonics

 ψ(n)=∑ℓmaℓmYℓm(n),ϖ(n)=∑ℓmbℓmYℓm(n). (17)

In order to find the harmonic coefficients, the generalized addition relations for spherical harmonics are required: for a rotation from to with Euler angles , such as the one performed before, we have Ruth_book ()

 √4π2ℓ+1∑m′sYℓm′(θk,ϕk)mY∗ℓm′(θ,ϕ)=sYℓm(β,α)e−isγ. (18)

As it should, this exactly eliminates the Euler angle dependence (coming from the solutions of eq. (10)) in the expressions for and . The dependence on can be integrated out performing the angular integral in k-space and we only have to recast the dependence on the direction of observation into the in order to read out the harmonics expansion coefficients and . For a statistically isotropic field the angular power spectrum is given by where the s are its harmonics coefficients. For the E-mode and B-mode lensing potentials we obtain (see Vittorio-thesis () for more details)

 Cψℓ=2π(ℓ+2)(ℓ−1)ℓ(ℓ+1)∫dkkΔ2h(k)×∣∣ ∣∣∫χ∗0dχTh(k,χ)(ℓ+12(ℓχ−χ∗χ∗+2)jℓ(kχ)k2χ3−jℓ+1(kχ)kχ2)∣∣ ∣∣2 (19)

and

 Cϖℓ=π(ℓ+2)(ℓ−1)ℓ(ℓ+1)∫dkkΔ2h(k)∣∣∣∫χ∗0dχTh(k,χ)jℓ(kχ)kχ2∣∣∣2. (20)

Here we have written the power spectrum in the form

 k32π2Ph(k;χ,χ′)=Δ2h(k)Th(k,χ)Th(k,χ′), (21)

where is the dimensionless primordial power spectrum and is the tensor transfer function. We show numerical results for the E- and B-mode lensing potentials in Section III, Figs. 1 and 2.

The case (quadrupole) of Eq. (19) is peculiar and needs to be discussed carefully. On the one hand, one may note that the -integral is infrared divergent for any initial power spectrum without a blue tilt. On the other hand, even if no infrared divergence was present, the limit generally yields a non-zero result. As will become clear from the discussion in Section IV, the effect of long modes locally looks like an anisotropic Bianchi I model, i.e. a spacetime in which comoving rulers pointing along different axes will expand at different rates. From the point of view of the observer it makes sense to redefine the coordinates such that this effect vanishes locally, i.e. to choose identical rulers in all directions at the time of observation. Due to the evolution of the tensor perturbations the rulers may not remain identical for a long time, but at least observers choose rulers in the different directions such that no lensing occurs in the limit of . To achieve this mathematically, we simply have to subtract the limiting expression for from the full expression given in Eq. (19). Taking the limit inside the -integral also regulates the infrared divergence. We argue that this is the correct prescription for the observational procedure: the observables are only affected by the presence of tensor perturbations which vary over the observed patch of spacetime, while a “constant perturbation” should be absorbed into the choice of coordinates222It is necessary to take care of this subtlety only in tensor perturbations lensing since for lensing from vector perturbations it affects only the dipole and for scalar lensing it affects the monopole. These gauge dependent multipoles are usually not considered.. The regularized expression for the quadrupole is then

 Cψ(reg)2=4π3∫dkkΔ2h(k)∣∣ ∣∣∫χ∗0dχ[3χ∗(j2(kχ)k2χ2Th(k,χ)−Th(k,0)15)−j3(kχ)kχ2Th(k,χ)]∣∣ ∣∣2.

This regularization procedure has not been discussed previously, but when computing the lensing potentials by the total angular momentum method the subtraction which we have introduced here by hand appears as a boundary term, see Yamauchi:2013fra ().

### ii.3 The lensed CMB power spectrum

Having computed the power spectra of the the two lensing potentials from tensor perturbations, we can now determine how gravitational lensing affects the shape of the CMB temperature angular power spectrum , where is the temperature anisotropy field (see Ruth_book ()). In other words we want to compute the lensed CMB power spectrum . The full-sky formalism developed in Hu:2000ee () is preferable to the approach of Lewis:2006fu () since the former is based on a Taylor expansion in the displacement vector which we expect to be very small in the case of lensing by tensor perturbations. We can support this claim by quickly computing the rms deflection angle for lensing by primordial gravitational waves with a tensor/scalar ratio . We find

 (22)

to give which is 20 times smaller than the value for density perturbations Lewis:2006fu (). The harmonic approach developed in Hu:2000ee () for density perturbations is applied to tensor perturbations in Li:2006si () and we follow the same approach. Assuming a small deflection angle we Taylor expand to second order

 ~Θ(n)=Θ(n+α)≃Θ(n)+∇aΘαa+12∇b∇aΘαaαb+... (23)

Recalling the angular decompositions of eq. (17) and defining as the harmonic coefficients of the temperature anisotropies field , eq. (23) yields

 ~θℓm=θℓm+∫dΩ(∇aΘαa+12∇b∇aΘαaαb)Y∗ℓm (24) =θℓm+∑ℓ1m1ℓ2m2θℓ1m1(aℓ2m2I(a)ℓmℓ1m1ℓ2m2+bℓ2m2I(b)ℓmℓ1m1ℓ2m2) +12∑ℓ1ℓ2ℓ3m1m2m3θℓ1m1(aℓ2m2a∗ℓ3m3K(a)ℓmℓ1m1ℓ2m2ℓ3m3+bℓ2m2b∗ℓ3m3K(b)ℓmℓ1m1ℓ2m2ℓ3m3),

where for the last equality we have used and we have defined the following integrals

 I(a)ℓmℓ1m1ℓ2m2=∫dΩ(∇aYℓ1m1)Y∗ℓm∇aYℓ2m2, (25) I(b)ℓmℓ1m1ℓ2m2=∫dΩ(∇aYℓ1m1)Y∗ℓmεba∇bYℓ2m2, K(a)ℓmℓ1m1ℓ2m2ℓ3m3=∫dΩ(∇b∇aYℓ1m1)Y∗ℓm(∇aYℓ2m2)∇bY∗ℓ3m3, K(b)ℓmℓ1m1ℓ2m2ℓ3m3=∫dΩ(∇b∇aYℓ1m1)Y∗ℓmεca(∇cYℓ2m2)εdb∇dY∗ℓ3m3.

Our aim is to compute . For this we substitute eq. (24) into the two-point angular correlation function. After some algebra this leads to

 C~Θℓ=CΘℓ+∑ℓ1ℓ2CΘℓ1(Cψℓ2Π(1a)ℓℓ1ℓ2+Cϖℓ2Π(1b)ℓℓ1ℓ2)+12CΘℓ∑ℓ1(Cψℓ1Π(2a)ℓℓ1+Cϖℓ1Π(2b)ℓℓ1), (26)

where

 Π(1a)ℓℓ1ℓ2=∑m1m2|I(a)|2,Π(1b)ℓℓ1ℓ2=∑m1m2|I(b)|2 (27) Π(2a)ℓℓ1=∑m1(K(a)+K(a)∗),Π(2b)ℓℓ1=∑m1(K(b)+K(b)∗).

Where, for simplicity, we have suppressed the indices in the summands and and only retained the indices on which the result depends on the lhs. To obtain these expressions we made use of the fact that and are uncorrelated Gaussian random variables with zero mean. This implies that the bispectrum vanishes. We have also neglected the trispectrum of these variables since it is of second order and of course any other higher order polyspectra are set to zero. To compute the integrals in eq. (27) we use Gaunt’s formula for the integration of a product of three ’s. We also use and in order to write in terms of and integration by parts to reconstruct . With this we can express the integrals in terms of the Wigner-3j symbols which are defined by

 ∫dΩ(s1Y∗ℓ1m1)(s2Yℓ2m2)(s3Yℓ3m3)= (−1)m1+s1√(2ℓ1+1)(2ℓ2+1)(2ℓ3+1)4π (28) ×(ℓ1ℓ2ℓ3s1−s2−s3)(ℓ1ℓ2ℓ3−m1m2m3).

We also use the identity

 ∑m1m2(ℓ1ℓ2ℓ3m1m2m3)(ℓ1ℓ2ℓ3m1m2m3)=12ℓ3+1.

In the end we find the relatively simple expressions

 Π(1a)ℓℓ1ℓ2=116π(2ℓ+1)(2ℓ1+1)(2ℓ2+1)([ℓ(ℓ+1)−ℓ1(ℓ1+1)−ℓ2(ℓ2+1)](ℓℓ1ℓ2000))2, (29) Π(1b)ℓℓ1ℓ2=116π(2ℓ+1)(2ℓ1+1)(2ℓ2+1)(ℓ1(ℓ1+1)ℓ2(ℓ2+1))([1−(−1)ℓ+ℓ1+ℓ2](ℓℓ1ℓ20−11))2, Π(2a)ℓℓ1=Π(2b)ℓℓ1=−ℓ(ℓ+1)ℓ1(ℓ1+1)2ℓ1+14π.

Finally, combining all the expressions, we arrive at

 C~Θℓ= CΘℓ+∑ℓ1ℓ2CΘℓ12ℓ+1(Cψℓ2Π(1a)ℓℓ1ℓ2+Cϖℓ2Π(1b)ℓℓ1ℓ2) (30) −ℓ(ℓ+1)CΘℓ∑ℓ1ℓ1(ℓ1+1)2ℓ1+18π(Cψℓ1+Cϖℓ1),

which is the first order lensed CMB temperature power spectrum as a function of the unlensed CMB spectrum and the lensing potentials. One can now go on and compute the lensing of CMB polarization. But due to the smallness of the polarization signal this will be even smaller than the temperature signal which we compute here and of which we shall show in the next section that it is smaller than cosmic variance in all circumstances.

## Iii Numerical results

We now evaluate numerically the angular power spectrum of the lensing E-mode potential and the B-mode potential for tensor perturbations that we derived in II.2. There are two types of tensor perturbations relevant for cosmology: primordial gravitational waves from inflation and those induced at second order by scalar perturbations. We treat both of these perturbations in a CDM scenario.

Inflationary cosmology predicts the generation of primordial gravitational waves with a nearly scale invariant spectrum that we can parametrize, following eq. (21), by

 Δ2h(k)=k32π2P(i)h(k)=rAs(kk∗)nt, (31)

where is the amplitude of the scalar perturbation spectrum at the fiducial scale , is the tensor spectral index and is the tensor to scalar ratio. has been measured accurately in the CMB Ade:2015xua (); primordial gravitational waves have not been observed so far, but are limited Ade:2015tva () to . We choose the typical inflationary value and set .333For the value of were assume the second order consistency relation which yields The transfer function in matter or radiation domination is an analytic function of but, if we want to include the effect of domination, we have to solve numerically the evolution equation

 ¨hk+2H˙hk+k2hk=0. (32)

As mentioned before, at second order in perturbation theory the scalar spectrum sources the generation of secondary, or scalar-induced, tensor modes. This, contrary to the primordial gravitational waves, is an unavoidable effect and does not depend on the inflationary model or its tuning. Due to the presence of scalar perturbations, the evolution equation for tensor modes is modified to

 ¨hk+2H˙hk+k2hk=S(k,χ), (33)

where is a source term. At second order is a convolution of two first-order scalar perturbations at different wave numbers Ananda:2006af (); Baumann:2007zm (). The initial value and the time evolution of the first-order scalar spectrum necessary for the computation of the scalar-induced tensor spectrum are obtained from the publicly available Boltzmann code CLASS Lesgourgues:2011re (), while the primordial tensor evolution can be obtained by numerical integration of eq. (32). Furthermore we take advantage of a recent work, by two of us, on relativistic N-body simulations Adamek:2015eda (). This allows us to obtain the fully nonperturbative induced tensor spectrum which, as opposed to , contains the effect of fully nonlinear small scale structure. To simplify the numerical computation we use Limber’s approximation

 ∫k2dkjℓ(kχ)jℓ(kχ′)≃π2χ2δ(χ−χ′), (34)

to reduce the dimensionality of the integrals in eq. (19) and (20). We obtain

 Cψℓ≃4π2(ℓ+2)(ℓ−1)ℓ(ℓ+1)∫χ∗0dχχ{1(ℓ+1)2(2ℓ+3)2[Δ2h(k)T2h(k,χ)]k=(ℓ+1)/χℓ+12ℓ4(2ℓ+1)2(ℓ+12ℓ(ℓχ−χ∗χ∗+2)2−2(ℓχ−χ∗χ∗+2)2)[Δ2h(k)T2h(k,χ)]k=ℓ/χ}, (35)
 Cϖℓ≃π22ℓ5(ℓ+2)(ℓ−1)ℓ(ℓ+1)∫χ∗0dχχ[Δ2h(k)T2h(k,χ)]k=ℓ/χ. (36)

While we have used the Limber approximation (34) for (36) we have also used it for integrals of the type in Eq. (35). Even though it is not derived for this case, we have checked numerically that, for sufficiently large , it is a good approximation also in this case. The Limber approximation is usually reasonable for and it improves as increases. However this is only true if the function integrated with the spherical Bessels is slowly varying. This causes no problem for and , but the transfer function for primordial gravitational waves oscillates very rapidly at small scales and Limber’s approximation gets worse as increases so that we are left with no choice but to compute the double integrals of eq. (19) and (20).

In Fig. 1 we plot the primordial tensor power spectrum and its E-mode and B-mode lensing spectra for a cosmology with , , , and . Primordial tensor modes are constant outside the horizon () and they start to oscillate and decay like once inside the horizon ().

In Fig. 2 the induced tensor power spectrum is shown, together with its lensing spectra. The tilt of the second order spectrum outside the horizon, for , is given by and one can also see the enhancement of power due to non linearities at late time and small scales. The numerical data of Adamek:2015eda () extends to and we present three types of small scale extrapolations to check the robustness of the nonlinear effects in the lensing signals. We find that, for the multipoles shown, the effect is dominated by the scales resolved in the simulations, and therefore its amplitude is nearly independent of the extrapolation to smaller scales. We also show the second order spectrum .

Let us discuss this somewhat more precisely: The Limber’s approximation implies that the main contribution to the lensing potential mode comes from all values of with for . But since the induced tensor power spectrum is decaying with for Mpc, see Fig 2, the dominant contribution for comes from Mpc. Hence for the dominant contribution comes from within the numerical simulation which go up to . This explains the weak dependence of the lensing power spectrum below on the very small scales extrapolations of the induced tensor power spectrum. It is interesting to note that nonlinearities enhance the tensor lensing potential on small scales, by up to several orders of magnitude. We should also point out that we neglect the decaying gravitational wave background which has been induced during the radiation era and which, at second order, becomes relevant at the smallest scales (see Baumann:2007zm () for a discussion of this effect).

In Fig. 3 we plot the time evolution of the primordial and scalar-induced tensor modes. We can see that at small scales and late times non linearities enhance the power of the second order spectrum.

In Fig. 4 we consider the effect of Spin-2 perturbation on the CMB temperature spectrum. We use the result of eq. (30) to evaluate the fractional difference

 Δℓ=C~Θℓ−CΘℓCΘℓ

between the unlensed and the lensed spectrum both for primordial and scalar-induced tensor perturbation.

It is important to note that the scalar-induced tensor power spectrum is not a gravitational wave in the usual sense. Most of its amplitude is frozen in as a scale dependent anisotropy of spacetime and is not oscillating. Such a spectrum of sourced tensor perturbations would actually not show up in interferometric experiments like LIGO ligo:Online () or eLISA elisa:Online () which measure the oscillations of the spacetime geometry as a function of time. However, it can in principle be observed via its lensing effect. Here we have studied the effect of tensor-lensing on the CMB. The final result given in eq. (30) is shown in Fig. 4. We have found that the difference between the tensor-lensed and the unlensed CMB angular power spectrum is smaller than cosmic variance on all scales. The reason for this is twofold. First the gravitational wave deflection angle is about 20 times smaller than the one from scalar perturbations. Secondly, the gravitational wave signal dominates on large scales where lensing deflections of the CMB only have a very small effect. The induced tensor modes might however be measurable with weak lensing or tidal effects on galaxy alignment measurements as proposed e.g. in Schmidt:2012nw (); Schmidt:2013gwa (), or with lensing reconstruction and delensing techniques as proposed in Namikawa:2014lla ().

## Iv Application to anisotropic cosmology

In this section we want to present a different application of our formalism. We consider a homogeneous but anisotropic universe of Bianchi I class, with line element

 ds2=a2(t)[−dt2+e2βi(t)δijdxidxj], (37)

where is chosen such that to ensure that the comoving volume evolves like . The give rise to anisotropic expansion and, evidently, taking restores isotropy and gives the flat Friedmann model. If the are small, i.e. if , we can interpret this metric as a perturbed Friedmann model with . Lensing in this type of geometry has recently been studied in Pitrou:2015iya (). The traceless tensor can be seen as an infrared limit of the spin-2 perturbations we have discussed in the previous sections. However, the transverse condition does not place any constraint on a spatially uniform perturbation, which means that has three additional degrees of freedom. Indeed, we have to pick an element of in order to specify the coordinate system in which is diagonal (three d.o.f.) and then, in this coordinate system, give two of the ’s, e.g.  and , the third being fixed by the traceless condition. As we will see, the three additional degrees of freedom arise in the guise of a prescription of how to take the limit of infinite wavelength. Motivated by this consideration, let us examine the situation where is given by a single mode of finite wavelength. Instead of the stochastic field discussed in the previous sections, we consider a gravitational wave with fixed wave number so that . Inserting this into the expression for the multipole coefficients of the gradient-mode lensing potential gives

 aℓm= il√(ℓ−2)!(ℓ+2)!(ℓ+2)(ℓ−1)(2π)2∫χ∗0dχ(ℓ+12(ℓχ−χ∗χ∗+2)jℓ(k0χ)k20χ3−jℓ+1(k0χ)k0χ2) ×(−2Yℓm(^k0)A+(χ)++2Yℓm(^k0)A−(χ)),

and a corresponding expression for the curl-mode potential. In the limit of infinite wavelength, the curl-mode potential vanishes for all , while the gradient-mode potential is non-zero only for , where we find

 a2m=−110√6π2χ∗∫χ∗0dχ(−2Y2m(^k0)A+(χ)++2Y2m(^k0)A−(χ)). (38)

Note that we take the limit along some fixed direction , and the limiting expression does depend on this choice. This is related to the issue that the polarization tensors in eq. (11) are not well-defined at . We simply define them through a limiting procedure which is, however, not unique. Let us see whether we can obtain by such a limiting procedure. Since the anisotropic expansion is triaxial, we use the sum of two modes whose limits are taken along orthogonal directions. For the first mode, we choose the coordinate frame where . We set the first mode . Next, we rotate our coordinate frame by degrees around the axis such that , and set the second mode . Now, taking the limits , , keeping the directions , fixed, we see from eq. (11) that we recover as desired. Working out the coefficients in the helicity basis we finally arrive at

 a2m=π√685χ∗∫χ∗0dχ[(−2Y2m(^x1)++2Y2m(^x1))β2(χ)+(−2Y2m(^x2)++2Y2m(^x2))β1(χ)],

explicitly

 a20= −√π52χ∗∫χ∗0%dχ(β1(χ)+β2(χ)), (39) a2±1= 0, a2±2= √π302χ∗∫χ∗0dχ(β1(χ