Beyond scalar, vector and tensor harmonics in maximally symmetric three-dimensional spaces

Cyril Pitrou1, ∗ and Thiago S. Pereira2, † 1Institut d’Astrophysique de Paris, CNRS UMR 7095, 98 bis Bd Arago, 75014 Paris, France. 2Departamento de F´ısica, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paran´a,Brazil. (Dated: October 1, 2019) We present a comprehensive construction of scalar, vector and tensor harmonics on max- imally symmetric three-dimensional spaces. Our formalism relies on the introduction of spin-weighted spherical harmonics and a generalized helicity basis which, together, are ideal tools to decompose harmonics into their radial and angular dependencies. We provide a thorough and self-contained set of expressions and relations for these harmonics. Being gen- eral, our formalism also allows to build harmonics of higher tensor type by recursion among radial functions, and we collect the complete set of recursive relations which can be used. While the formalism is readily adapted to computation of CMB transfer functions, we also collect explicit forms of the radial harmonics which are needed for other cosmological ob- servables. Finally, we show that in curved spaces, normal modes cannot be factorized into a local angular dependence and a unit norm function encoding the orbital dependence of the harmonics, contrary to previous statements in the literature.

1. INTRODUCTION ables. Indeed, on cosmological scales, where lin- ear perturbation theory successfully accounts for the formation of structures, perturbation modes, Tensor harmonics are ubiquitous tools in that is the components in an expansion on tensor gravitational theories. Their applicability reach harmonics, evolve independently from one an- a wide spectrum of topics including black-hole other. This fact enormously simplifies the con- physics, gravitational waves, quantum-field the- struction of observables and the assessment of ory in curved spacetimes, and cosmology. In the their statistics. In particular, a decomposition particular context of cosmology, one is usually based on tensor harmonics is essential for the interested in the description of tensor harmonics computation of cosmic microwave background over maximally symmetric manifolds, since these (CMB) fluctuations around a maximally sym- are the spaces in better agreement with observa- metric (but possibly curved) space [5]. The nor- tions. In this work we revisit the construction mal modes which have been introduced in [6,7] of scalar, vector and tensor harmonics in sym- correspond to specific components of those of [5], metric three-dimensional spaces with particular and are consequently an equivalent presentation interest in – but not limited to – cosmological arXiv:1909.13687v1 [gr-qc] 30 Sep 2019 of them. An equivalent covariant formulation of applications. these normal modes is also presented in [8,9]. Scalar harmonics in symmetric spaces are In this article, we review the general con- well known among cosmologists [1], and they are struction of harmonics in maximally symmetric defined as a complete set of eigenfunctions of the three-dimensional spaces, along with the associ- Laplace-Beltrami operator. Vector- and tensor- ated normal modes, and show how they can be valued harmonics can be similarly defined, and systematically built by recursions. In doing so, their explicit forms were gathered in [2–4]. These we collect all explicit expressions of the normal objects are found nearly everywhere in cosmo- modes for scalar, vector and tensor harmonics. logical applications, specially in those related Throughout, we choose to use a modern formu- to large-scale structure and its related observ- lation based on spin-weighted spherical harmon- ics from which even and odd parts and also the ∗[email protected] general structure is more tractable. Hence, this †[email protected] differs from the formulations given in [10, 11]. 2

Section 2.1 is dedicated to definitions and no- K 6= 0 we can further use units for which `c = 1, tation. In particular, we define the harmonics, that is, all lengths are expressed in units of the the helicity basis, the normal modes and the ra- curvature radius and, in the closed case, this im- dial functions of which many properties are col- plies 0 ≤ χ ≤ π. The general case `c 6= 1 can lected in the appendices. Section3 is dedicated be trivially restored from dimensional analysis if to the general construction of these radial func- needed. To emphasize our special choice of units, tions which characterize fully the harmonics and let us introduce a reduced curvature parameter most relations are collected in AppendixD. The 2 reader interested only in the actual expression K ≡ K`c = K/|K| (2.3) of the harmonics can jump directly to Section4 where the explicit expressions of the radial func- which assumes the value +1 (−1) in the closed tions are collected, or to AppendixF if inter- (open) case. ested in the flat case only. The normalization of The Riemann tensor of maximally symmetric harmonics is discussed in Section5, while plane spaces can be written directly in terms of the waves are built in Section6. The formalism is il- space metric and the constant K: lustrated in Section7 for the standard multipole R = K(g g − g g ) . (2.4) expansion of the CMB radiative transfer func- ijkl ik jl il jk tions. Finally the comparison of our results with This greatly simplifies identities involving com- previous references is detailed in AppendixG. mutators of covariant derivatives. One identity The tables of AppendixH gather the most im- that we shall need is portant ancillary notation used throughout.

[∆, ∇k1 · · · ∇kn ]Ti1...ij = 2K [n(n + 1)/2 + nj]

2. DEFINITIONS × ∇k1 · · · ∇kn Ti1...ij (2.5) 2.1. Maximally symmetric spaces where ∇i is the covariant derivative associated We start be recalling some basic properties of with the metric (2.1) (i.e., ∇kgij = 0) and ∆ = j maximally symmetric spaces. A nice and physi- ∇ ∇j is the Laplace-Beltrami operator. In what cist targeted introduction can be found in [12]. follows we shall adopt the unifying notation sin, Maximally symmetric spaces (as opposed to tan and cot for trigonometric functions, defined spacetime) are uniquely fixed by a real parame- as the usual functions when K > 0, and as their ter K, known as constant of curvature. In three hyperbolic counterparts when K < 0. dimensions, and using standard spherical coor- dinates (χ, θ, φ), the metrics of these spaces read 2.2. Helicity basis g dxidxj = dχ2 + r2(χ)[dθ2 + sin2 θdφ2] . (2.1) ij The notion of helicity (or spin) basis is more The radial coordinate χ is implicitly defined by conveniently introduced in terms of an orthonor- the function r(χ), which assumes different values mal triad of basis vectors according to the sign of the parameter K: nχ = ∂χ , (2.6a)  ` sinh(χ/` ) , (K < 0) , n = r−1(χ)∂ , (2.6b)  c c θ θ −1 r(χ) = `c sin(χ/`c) , (K > 0) , (2.2) nφ = r (χ) csc(θ)∂φ , (2.6c) χ , (K = 0) . together with its dual basis Here, ` ≡ 1/p|K| is the curvature radius, c nχ = dχ , (2.7a) which is related to the Ricci scalar by R = 6K. θ Clearly, K distinguishes between open (K < 0), n = r(χ)dθ , (2.7b) closed (K > 0) and flat (K = 0) spaces. When nφ = r(χ) sin(θ)dφ . (2.7c) 3

From this we can form the standard helicity vec- with s = 0. Up to normalisation differences, it tor (spin 1) basis as corresponds to the Legendre tensors introduced in [14] for the cases s = 0, 2. The explicit ex- 1 n± ≡ √ (nθ ∓ inφ) , pressions for j ≤ 3, and a collection of properties 2 (2.8) (which extends those already found in Appendix 1   n± ≡ √ nθ ∓ inφ . A of Ref. [15] for the case s = 0), are given in 2 AppendixC. The set of ˆns with |s| ≤ j form a Ij basis for STF tensors with j free indices at each Given a unit vector n at the origin (χ = 0), the point. Their normalization, used for extraction pair (χ, n) denotes a point reached following a of components along that basis, is given by geodesic of length χ whose tangential direction at the origin is n. It is also obvious from the ±s Ij nˆ nˆ 0 = δss0 djs (2.12) spherical symmetry that the tangential vector of Ij ∓s the geodesic at that point is nχ. Hence it is cus- where tomary to use the symbol n for both nχ and its χ j! 1 dual n . The helicity basis vectors n± also de- djs ≡ 2 , (2j − 1)!! (bjs) pend on the point (χ, n) considered, but they are s (2.13) parallel transported along a radial curve, that is √ |s| (j!)2 b ≡ 2 . js (j + s)!(j − s)! k ± n ∇kni = 0 . (2.9)

Thus, since they depend essentially only on the 2.3. Decomposition of tensor fields direction n, it is customary not to write this dependence explicitly. Any STF tensor field in a maximally sym- We now use the vector basis (2.8) to build metric (three-dimensional) space can be decom- a suitable tensor basis (spin s) for symmetric posed onto the generalized helicity basis using trace-free (STF) tensors. For 0 ≤ |s| ≤ j, we spin-weighted spherical harmonics. This decom- define position can be understood in two steps. At each

±s ± ± point the helicity basis is a basis for STF tensors nˆ ≡ n . . . n ni . . . ni i , (2.10) i1...ij hi1 is s+1 j at that point, hence we can decompose the STF with a similar definition when free indices are tensor field as up. The angle brackets mean that we must form j X s the symmetric trace-free part on the enclosed TIj (χ, n) = sT (χ, n)ˆnIj . (2.14) indices, and this is performed in practice with s=−j (C.1). Analogously to the helicity basis, these The spin functions T (χ, n) are then decom- tensors (which are also parallel transported) de- s posed onto spin-weighted spherical harmonics, pend only on the direction n — a dependence so as to separate its radial and angular depen- which will be omitted from now on. dencies. This leads to In what follows, it will be convenient to in- troduce a multi-index notation j ∞ ` X X X M s TIj (χ, n) = sT`M (χ)sY` (n)ˆnIj . Ij ≡ i1 . . . ij , (2.11) s=−j `≥|s| M=−` (2.15) such that the basis for STF tensors is written Given the rotation property (B.33), this is a de- succinctly asn ˆ±s orn ˆIj . In AppendixB we Ij ±s composition in irreducible components under the summarize how the extended helicity basis (2.10) group of rotations. On the left hand side, we re- is related to spin-weighted spherical harmonics. call that j is the number of free indices given the The generalized helicity basis (2.10) extends multi-index notation (2.11). the multi-index notation reviewed in Ref. [13], The functions sT`M are however constrained which is restricted to using the tensors (2.10) by the fact that the tensor fields must assume a 4 given value at the origin of coordinates (χ = 0). When |m| = 0, 1, 2 these are called respectively Let us consider the tensors Yjm defined at the scalar, vector and tensor harmonics. Ij origin of the system of coordinates, and which We next introduce some derived harmonics are explicitly given in AppendixB, along with which are obtained by STF combinations of their properties. They form a complete basis (for (j − |m|) derivatives of these harmonics. More STF tensors) and we can use them to decompose precisely they are defined as the value of the STF tensor at origin in the form (|m|,m) ∇hi ... ∇i Q (jm) 1 j−|m| ij−|m|+1...ij i j Q ≡ , (2.22) X Ij kj−|m| T (χ, n) = t Yjm . (2.16) Ij χ=0 m Ij m=−j which implies the basic relation for j > |m|

(jm) 1 (j−1,m) Therefore we find that at the origin the coeffi- Q = ∇hi Q . (2.23) Ij k j Ij−1i cient functions sT`M must be It can be checked by using (2.5) that that they

sT`M |χ=0 = δj`tM kjs (2.17) are not divergenceless and do not satisfy (2.19), but they satisfy instead where 2 (jm) [∆+k −K(j−|m|)(j+|m|+1)]QI = 0 (2.24) (2j − 1)!! j k ≡ (∓1)sb j ±s js j! as well as s −1 (jm) (j−1,m) = (∓1) (djsbjs) . (2.18) ∇pQ = −q(jm) Q , (2.25) Ij−1p Ij−1 This can be seen either from property (B.32) (j2 − m2) (ν2 − Kj2) once (2.17) is replaced into (2.15), or from the q(jm) ≡ . (2.26) component extraction by contraction of (2.16) j(2j − 1) k with then ˆs , and using the normalization (2.12) Ij Here we have introduced the notation and the property (B.24). 2 2 In the next section we define the tensor har- ν ≡ k + (1 + |m|)K , (2.27) monics, and in the subsequent one we shall be such that harmonics and derived harmonics can guided by the decomposition (2.15) to define be either characterized by the value of the mode normal modes and radial functions. k or by the related mode ν. If we fix k, then ν is a function of both |m| and k. Conversely if we 2.4. Laplacian and harmonics fix ν, then k is a function of both ν and |m|, and from now on we consider this point of view. The tensor valued eigenfunctions of the Laplacian are defined as 2.5. Comment about notation

2 (∆ + k )Ti1...ij = 0 . (2.19) For simplicity, we often omit to write the de- pendence of the harmonics on k (or ν) to al- We further ask these modes to be STF and leviate the notation. Similarly, wherever not divergence-free tensors, that is needed, the dependence on the position on space, that is on (χ, n), is not written explicitly. Hence, ∇i1 T = 0 . (2.20) i1...ij even though the full expression of an harmonic should be Q(jm)(χ, n; ν), we shall use Q(jm)(ν), Solutions of (2.19) and (2.20) with m free in- Ij Ij dices, and for a given k, are called harmonics of Q(jm)(χ, n) or simply Q(jm), depending on the Ij Ij type m for the mode k, and are denoted as context. Such practice will be used not only for the harmonics, but for any other quantities de- Q(jm) , j = |m| . (2.21) i1...ij pending on χ, n and ν. 5

2.6. Normal modes taken when we present the construction of plane waves in Section6, and the summation on M is In order to find a decomposition of the type needed only when considering general reference (2.15) for the harmonics and their derivations, axis harmonics as detailed in Section 3.5. we follow [6,7] and split the radial and angular Moving forward, let us also define dependence through a new function (jm) (jm) sg˜ ≡ sg djs , (2.34) G(jm)(χ, n; ν) ≡ c α(jm)(χ; ν) Y m(n) , s ` ` s ` s ` such that from (2.12) we get the inverse relation (2.28) with the conventional factor `Q(jm)nˆIj = g˜(jm) G(jm) . (2.35) Ij ∓s ±s ±s ` c ≡ i`p4π(2` + 1) . (2.29) ` From (2.33) and (2.35), we see that the co- (jm) (jm) (jm) efficients sg˜ and sg are used to relate We insist on the fact that sG` depends on ` (jm) (jm) the tensors Q to the functions ±sG — the point considered, that is, on (χ, n), while Ij ` the STF basisn ˆs depends on the choice of n. called normal modes — and vice versa. The nor- Ij (jm) mal modes are the coefficients [with a functional Moreover, the radial functions α` (χ; ν) do not dependence on (χ, n)] of the harmonics in the depend on n, while the spin-weighted spherical generalized helicity basis. Since the coefficients harmonics Y m(n) do 1. The radial functions, to (jm) (jm) s ` sg (and thus sg˜ ) are yet undetermined, be constructed in Section 2.7, are conventionally we can further choose that, for a given (jm) pair normalized when ` = j as [6,7] s (jm) (∓) (jm) (jm) 1 ±sg˜ = 0g˜ , bjs (2.36) sα` = δ`j . (2.30) χ=0 2j + 1 (jm) (j,−m) ±sg˜ = ±sg˜ , Accordingly, it implies that around the origin which, given (2.18), implies that (jm) cj m sG`=j = sY`=j + O(χ) . (2.31) (jm) s (jm) 2j + 1 ±sg = (∓) bjs 0g , (2.37) (j,−m) In general radial functions are non-vanishing = ±sg , only for the conditions and, in particular

j ≥ max(|m|, |s|) , ` ≥ max(|m|, |s|) , (2.32) (jm) s (jm) −sg = (−1) sg , (2.38) and are chosen to be null functions otherwise. (jm) with a similar relation for the sg˜ . The We now search to build a basis for tensor har- (jm) choices (2.37) [that is sg ∝ kjs for the de- monics (and their derivations), with j free STF pendence on s] ensure that indices, in the form `=j (jm) (jm) cj jm Q = 0g˜ Y , (2.39) j Ij Ij (jm) X (jm) χ=0 2j + 1 `Q ≡ g(jm) G nˆs , (2.33) Ij s s ` Ij s=−j for exactly the same reasons detailed after (2.16). (jm) where the sg are numerical coefficients yet to Given the linearity of (2.19), any linear com- be fixed. These harmonics correspond to consid- bination of solutions of the type (2.28) for dif- ering a single (`, M) pair in the otherwise gen- ferent values of ` is also a solution. This is how eral sum of (2.15). The summation on ` will be plane-wave solutions are built, and we discuss this construction in §6. Finally, it is trivial to restore spatial dimensions (` 6= 1) since har- 1 In fact, such separation between radial and angular de- c pendence lies in the heart of the Total Angular Mo- monics, normal modes and radial functions are mentum method – see [6] for more details all dimensionless. 6

2.7. Radial functions We first note that [see e.g. Eq. (A.22) in [9]]

∆[curl (`Q(j,±j))] = curl[∆(`Q(j,±j))] , (3.2) We recall that for simplicity the dependence Ij Ij on χ and ν of the radial functions α(jm) is omit- s ` where the curl is the obvious generalization to ted. We also split them into even and odd parts STF tensors defined by (also called respectively electric and magnetic ra- dial functions) as j p curl TI ≡  ∇ T . (3.3) ` jphi1 I`−1i α(jm) = (jm) ± i β(jm) , (2.40) ±s ` s ` s ` Hence, the curl of an harmonic is also an har- and by construction there is no odd part for s = monic. Furthermore using the divergenceless re- 0, that is lation (2.20) it can be proven that [see, e.g., Eq. (jm) (3.13) of Ref. [9] for the j = 2 case] 0β` = 0 . (2.41) curl curl (`Q(j,±j)) = ν2(`Q(j,±j)) . (3.4) We shall check further that they also satisfy the Ij Ij properties Therefore, we can choose (j,−m) (jm) s = s (2.42) ` (j,±j) ` (j,±j) (j,−m) (jm) curl ( Q ) = ±ν( Q ) . (3.5) sβ = −sβ . Ij Ij In practice this means that we only need to build The choice of sign on the right hand side (which the radial functions for m ≥ 0. could have been ∓ν) amounts to choosing the In most cases ν is real and the electric global normalization of the odd radial function, and magnetic radial functions are real. How- and our choice is made so that we recover the flat ever, when considering super-curvature modes case construction that is recalled in appendixF. on open spaces [16, 17], ν can be complex. Using the property (C.19) of the extended he- In that case one cannot deduce from (2.40) licity basis, and the decomposition (2.33) along that complex conjugation on radial functions with the condition (2.37), we deduce that the amounts to s → −s, and one must rather use divergenceless relation (2.20) leads to the set of (3.35). relations among radial functions for 0 < s < j d α(j,±j) + (j + 1) cotχ α(j,±j) = 3. BUILDING HARMONICS dχ ±s ` ±s ` s s (−λ )(−λ ) ` j α(j,±j) We now proceed to the determination of the 2(j + s)r(χ) ±(s−1) ` radial functions. Indeed, they determine the ( λs)( λs) normal modes from the definition (2.28), and + ` + j (j,±j) + ±(s+1)α` , (3.6) subsequently the harmonics (and derived har- 2(j − s)r(χ) monics) from (2.33). In the next section we first where we defined start by building the radial functions for har- s p monics (j = |m|), and in the subsequent one we ±λ` ≡ (` + 1 ± s)(` ∓ s) (3.7) deduce the radial functions for the derived har- = p`(` + 1) − s(s ± 1) . monics (j > |m|). In all expressions, the value of ` is general. Condition (3.6) is a special case of the diver- gence relation (E.1). In the special case s = 0 it reduces to 3.1. Radial functions of harmonics (j = |m|) d (j,±j) (j,±j) 0 + (j + 1) cotχ 0 We recall that harmonics are divergenceless. dχ ` ` We normalize them with p s `(` + 1) j + 1 (j,±j) (j,±j) = 1` , (3.8) 0g˜ = 1 . (3.1) r(χ) j 7 which is a special case of (E.2). less condition (3.6) at s = 1, leads to Hence, when considering the real and imag- 2 inary parts of radial functions, we see that the d (j,±j) d (j,±j) 2 1β` + 2j cotχ 1β` divergenceless relation brings j relations for the dχ dχ even modes and j−1 relations for the odd modes (j,±j) 2 +1β` cot (χ)[j(j − 1) − 1] (if j ≥ 1). Given (2.41), we conclude that us- [1 − `(` + 1)] + β(j,±j) ing (3.6) we can deduce all radial modes in the 1 ` r2(χ) case j = |m| (that is for all allowed values of s) 2 (j,±j) (j,±j) (j,±j) = −k 1β` . (3.12) once we know 0` and 1β` . These terms are in turn found from the Laplace equation By comparing (3.11) with equation (A.10), we (2.19). Again, using the decomposition (2.33) can now motivate the definition (2.27). More- along with the condition (2.36) and the identi- over, we find that (j,±j) ∝ Φν/rj, where Φν ties (C.22), this leads (when s > 0) to 0 ` ` ` are the hyperspherical Bessel functions – see Ap- pendixA. The normalization which satisfies the d2 d (j,±j) (j,±j) normalization condition (2.30) (this is checked 2 (±sα` ) + 2 cotχ (±sα` ) dχ dχ using (A.5)) and recovers the flat case construc- (j,±j) 2 2 +±sα` cot (χ)(s − j(j + 1)) tion of appendixF is (j,±j) 1 2 s +±sα` 2 (s − `(` + 1)) ν r (χ) (j,±j) (2j − 1)!! (` + j)! ξj Φ` 0` = p j j , (3.13) (j,±j) cotχ (2j)! (` − j)! k r (χ) + α ( λs)( λs) ±(s−1) ` r(χ) − j − ` with the dimensionless constants (j,±j) cotχ s s + α (+λ )(+λ ) ±(s+1) ` r(χ) j ` m Y k 2 (j,±j) ξm ≡ √ . (3.14) = −k ±sα . (3.9) 2 2 ` i=1 ν − Ki

As for the s = 0 case, it is simply We also deduce that the odd radial functions (j,±j) 1−j ν must be such that 1β` ∝ r (χ)Φ` . The d2 d global normalization is deduced from the solu- ( (j,±j)) + 2 cotχ ( (j,±j)) dχ2 0 ` dχ 0 ` tion (3.13) using the curl condition (3.5) con- I`  `(` + 1) tracted withn ˆ which leads to − j(j + 1) cot2(χ) + (j,±j) r2(χ) 0 ` s (j,±j) (j,±j) j cotχ 1β = ∓νr(χ)0 . +2 pj(j + 1)`(` + 1) (j,±j) ` ` (j + 1)(` + 1)` r(χ) 1 ` (3.15) 2 (j,±j) = −k 0` . (3.10) Note that this is a particular case of (E.4) when j = |m|. When combined with the divergenceless condi- The radial functions for larger values of s are tion (3.8), this equation leads to (j,±j) found from (3.8) for 1` , and then from (3.6) for all s > 1, and they satisfy automatically the d2 d Laplace equation (3.9). (j,±j) + 2(j + 1) cotχ (j,±j) dχ2 0 ` dχ 0 `  `(` + 1) (j,±j) 2 2 3.2. Radial functions of derived harmonics +0` k + j(j + 1) cot (χ) − 2 r (χ) (j > |m|) = 0 . (3.11) We now discuss the systematic construction Similarly, the imaginary part of the relation (3.9) of radial functions for the derived harmonics, for s = 1, when combined with the divergence- which must be deduced using the definition 8

(2.22). We start by noticing that the derived When combined with (3.16) and (2.25), we ob- harmonics satisfy the property tain the following relation among derived har- monics ` (jm) mν ` (jm) curl QI = QI (3.16) j j j  (jm)  (j+1,m) ∇ `Q = k `Q p Ij pIj which is inherited from (3.5) and the identity for 2j − 1 (jm) ` (j−1,m) STF tensors [see e.g. Eq. (4.7) of [8]] − q gphi Q 2j + 1 j Ij−1i j mν  (jm)  curl ∇ T = ∇ curl T . (3.17) + r `Q . (3.22) hij+1 Ij i j + 1 hij+1 Ij i j + 1 phij Ij−1ir The derived harmonics are no more divergence- Let us now consider a given m and a given less, as was the case for the j = |m| harmonics. ` ≥ |m|, and use the short notation (j, s) to re- Instead, they satisfy the relation (2.25). As will (jm) fer to the radial function sα` , since we want be shown later, the normalization of the derived to explore the relations between radial functions harmonics which is compatible with (2.30) re- with neighbor values of j and s. Identity (3.22) quires that allows to derive recursive relations among radial

j m functions in the space of the (j, s) parameters, (2|m| − 1)!! Y 0κp g˜(jm) ≡ , the most famous of which connects (j, s) to the 0 (2j − 1)!! k p=|m|+1 (j±1, s) ones. To see how this is possible, we first (3.18) need to contract (3.22) with np and replace the j m (2|m| − 1)!! Y 0κp g(jm) ≡ , harmonics by their expansion (2.33) while using 0 j! k p=|m|+1 the identities derived in Appendix C.3. Then, a relation among radial functions is obtained by where 2 Ij contraction withn ˆ∓s (or equivalently identifica- r ±s 2 2 2 2 tion of then ˆI components), and extraction of m (` − m )(` − s )p 2 2 j sκ ≡ ν − K` . the radial function from the orthogonality rela- ` `2 (3.19) tion (B.13) of spin-weighted spherical harmon- The above normalization and (2.36) also imply ics, along with the relations (3.20). Eventually the useful relations we obtain the central relation [see also Eq. (C5) m in [7]] (jm) (j−1,m) 1 sκj ±sg˜ = ±sg˜ , (2j − 1) k d (jm) iνms (jm) m sα` = − sα` (3.23) (jm) (j−1,m) j sκj dχ j(j + 1) ±sg = ±sg 2 2 , (j − s ) k 1 h m (j−1,m) m (j+1,m)i + −sκ sα + sκ sα , s (3.20) 2j + 1 j ` j+1 ` (j + s) g˜(jm) = ∓ g˜(jm) , ±s ±(s−1) 2(j + 1 − s) which holds for either negative or positive values s of m and s. (jm) (jm) 2(j + 1 − s) Since in the (j, s) plane this links the (j, s) ra- ±sg = ∓±(s−1)g . (j + s) dial functions with the one above [(j − 1, s)] and the one below [(j + 1, s)] for a given ` and m, we From A3 of [15], we see that a general STF tensor hereafter call it the North-South (NS) relation. obeys Other relations can be obtained from (3.22) 2` − 1 by contracting with np and then repeating the ∇ T = ∇ T + g ∇pT ∓ j I` hj I`i 2` + 1 jhi` I`−1ip same procedure. This leads to relations connect- ` p ing the (j, s) radial functions to the (j ∓1, s) and +  curlTI ip . (3.21) ` + 1 jhi` `−1 (j, s + 1) ones. We thus call it the North-South- East (NSE) relation. Similarly, contracting in- p 2 m stead with n and using the same method allows Our definition of sκ` corresponds to the√ one of [7] ± times a factor ν such that νp1 − K`2/ν2 = ν2 − K`2. us to relate the (j, s) radial functions to (j ∓1, s) 9

s and (j, s − 1) radial functions — a relation that 0 1 2 3 4 5 6 7 8 we call North-South-West (NSW). Their exact j expressions are collected in AppendixD. 0 The combination of NS and NSE relations 1 N leads either to a relation between (j, s) radial 0 functions with the (j − 1, s) and (j, s + 1) ones, 2

which we call the North-East (NE) relation, or grad- to a relation between the (j, s) radial functions 3 S N with the (j+1, s) and (j, s+1) ones, which we call 4 ⊕ E the South-East (SE) relation. Similarly combin- 5 grad- S ing the NS and NSW relations leads either to curl s = 0 the North-West (NW) or the South-West (SW) 6 E N relations. All these relations are collected in Ap- pendixD. 7 W

These triangular relations (NW, NE, SW, grad- 8 N S and SE) are the building blocks of all sorts of recursive relations among radial function in the 9 W E (j, s) space. For instance, the NS relation is a 10 div relation combination of the NW and SW. It can also be STF relation curl relation found as a combination of the NE and SE re- 11 W E W E lations. Similarly the NSE relation (resp. the NSW relation) is just the sum of the NE and SE 12 S relations (resp. the NW and SW relations). All 13 N N N the recursive relations are depicted in the (j, s) plane in Fig.1. 14 E W E There is an alternative method to obtain div s = 0 15 triangular relations the triangular relations. Instead of consider- STF s = 0 ing various contractions of the identity (3.22), 16 E W E we can instead extract the radial functions of the divergence relation (2.25), the curl prop- 17 S S S erty (3.16), and the STF construction of derived 18 N modes (2.23). Again, this proceeds by contrac- tions with the generalized helicity basis, extrac- 19 W E tion of the radial functions using (B.13) and re- NW-NE peated use of the properties (3.20). The rela- tions obtained are also gathered in appendixE. FIG. 1: Geographical representation of all recursion relations among radial functions in the (j, s) space Combining the curl relation with the divergence of parameters. Here, ‘STF’, ‘div’ and ‘curl’ denote relation in two different manners leads to the respectively the relations (2.23), (2.25), and (3.16). NW and NE relations. Similarly combining the Moreover, ‘grad-0’ (resp. ‘grad-±’) is obtained by curl relation with the STF relation in two differ- contraction of the gradient identity (3.22) with np p ent manners leads to the SW and SE relations. (resp. n∓). The triangular relations (NW, NE, SW, While this method seems more appealing, it re- and SE) which can be formed from the grad relations quires that we carefully separate the s = 0 cases are collected in appendixD. Shaded squares indicate for which the aforementioned combinations can- radial functions which appear with one derivative in the recursive relation. We depict only the s ≥ 0 not be formed in the same manner. Instead it part in the chart as the negative s are deduced from is found that in the s = 0 case, the curl relation (2.40). Only functions with |s| ≤ j (and |m| ≤ j) are gives the imaginary part of the NE and SE re- non-vanishing. lations. Also the s = 0 case of the divergence relation gives the real part of the NE relation. 10

Finally the s = 0 case of the STF relation gives of these. Given the property (2.42) and the def- the real part of the SE relation. initions (2.40), we only need to build harmonics This indicates that the triangular relations for m ≥ 0 and s ≥ 0, and we now assume these (NW, NE, SW and SE) contain the informa- conditions hold. tion about recursions in the most compact form. Their validity is only restricted by the fact that 1. For a given m, the radial function for s = they should not produce instances with s < 0 0 and j = m is given by (3.13), and it in the (j, s) space, but they can be applied even has no derivative of hyperspherical Bessel if some of the radial functions vanish because of functions. s > j. If we instead use the apparently more di- rect divergence and curl relations, we must treat 2. We then use the NE recursion to obtain the s = 0 case separately. As an illustration, the s = 1 and j = m solution, with un- this is what has been presented in the j = |m| avoidably one derivative of Bessel func- case of Section 3.1. The solution for the s = 0 tion. However, note that this is not pos- case is (3.13). The divergence relation for s = 0, sible in the special case m = 0, and we given by (3.8), gives only the electric function discuss the procedure for this case below. for s = 1. One has then to rely on the curl rela- 3. We can then use the difference of the NE tion for s = 0, which is (3.15), to get the mag- and NW relations to form a North-West- netic function with s = 1. In order to obtain East relation without derivatives whose the s > 1 harmonics, still for j = |m|, one can exact expression is (D.8) and that we note use the divergence relation (3.6), but one could NW-NE hereafter. In the case j = m, the also use more directly the NE relation. Indeed, north component vanishes so it is a rela- given that the north component of the NE rela- tion between (j = m, s−1), (j = m, s) and tions vanishes (since j = |m|), it gives directly (j = m, s + 1). Using it, one can obtain (j = |m|, s + 1) as a function of (j = |m|, s). all radial modes for j = m up to s = j. Finally, we recall that all radial functions are restricted in general to (2.32), and in the closed 4. In order to build the line with j = m + 1, case (K = 1) they are also restricted to the inte- that is the radial functions associated with ger values the first derived harmonics, one needs only to use the NSE relation to deduce ` ≤ ν − 1 . (3.24) (m+1,m) 0α` , and then the NSW relation (m+1,m) to deduce 1α` . This introduces no 3.3. Optimal algorithm derivatives since the NSE and NSW rela- (m+1,m) tion have none. Then all sα` with Given the plethora of recursion relations in 2 ≤ s ≤ j can be found either from the the (j, s) space for radial functions, there are sev- NW-NE relation (D.8), or from the use of eral different ways to deduce the radial modes the NSW relation. This, again, brings no for the derived harmonics for increasing values extra derivatives. of j. However we can judiciously add a condi- tion which selects one method. Since all radial 5. This last method is iterated to obtain all functions are expressed in terms of derivatives of radial functions for increasing values of j. hyperspherical Bessel functions, it is always pos- sible to use equation (A.1) to reduce their form In the special case m = 0, we start from the (00) ν to an expression which involves the hyperspher- known solution 0` = Φ` . Then there is no ical Bessel function and at most its first deriva- need to build the (j = 0, s = 1) solution since tive. However, there are preferred methods for it vanishes, so we must proceed directly by in- the recursive construction of radial function, in creasing the value of j, and build the solution which one never has to rely on (A.1) to reduce for (j = 1, s = 0). In that case, contrary to the order of derivatives. Let us summarize one the procedure mentioned above, we cannot use 11 the NSE relation since the East component van- s ishes, that is, it is outside of its applicability j 0 1 2 3 4 5 6 [see (D.3)]. Instead, we must use the NS rela- (10) 0 tion (3.23) to obtain 0α` , and this brings a derivative of a Bessel function. Finally in or- (10) 1 der to obtain 1α` , we can do as in the general case, and use the NSW relation which involves 2 NE NW-NE no derivative. The rest of the construction to j ≥ 2 then proceeds exactly like in the general 3 NW-NE NW-NE NSE case. NSW In both cases (m = 0 and m > 0), there was 4 NW-NE NW-NE NW-NE NSE only one step of the procedure involving a deriva- NSW tive. Hence, with this method it is possible to 5 NW-NE NW-NE NW-NE NW-NE NSE obtain radial functions up to any desired values NSW of (j, s) for any given m, as illustrated in Fig.2, 6 and with at most one derivative on Bessel func- tions, without ever having to use (A.1) to reduce the order of derivatives. This algorithm has been implemented in a Mathematica notebook avail- able at [18]. Note that the optimal algorithm is FIG. 2: Optimal algorithm: for a given |m|, the first (|m|,|m|) not unique. One could for instance rely on (D.2) step is to start from the solution 0` . Then to relate the s = j to the s + 1 = j + 1 radial all other radial functions are deduced following the functions, thus deducing the radial functions on algorithm described in the Section 3.3. The steps the diagonal of Fig.2. 2, 3, 4, 5 are depicted in resp. red, green, black and blue arrows, and the relation needed to deduce each radial function from the previous ones is written next to the arrow. Here we have illustrated the case |m| = 3.4. Symmetry properties 2, such that we have necessarily j ≥ 2. The index s must also satisfy |s| ≤ j. Following the same algorithm (that is the same set of recursions to travel in the (j, s) space of radial function) it can be checked that the `) that the following property holds properties (jm) (js) sα` (χ; ν) = mα` (χ; ν) . (3.26) (jm) (j,−m) (jm) −sα` (ν) = sα` (ν) = sα` (−ν) (3.25) We have also checked for the first values of m, s, j are always satisfied. It is indeed the case for the and ` that starting radial function (3.13) of the algorithm, (jm) `−j (`m) and it is maintained for all values of (j, s), since sα` = (−1) sαj . (3.27) in all recursions for radial functions the factors m are always multiplied by the sign of s and by Furthermore, it can be checked explicitly on the ν. Hence from the definition (2.40) of the even first values of j, m and s (but for unspecified `) and odd parts we deduce that (2.42) must be that [7] satisfied. d (jm) iνms (jm) In addition to (3.25), there are two other sym- α = − α (3.28) dχ s ` `(` + 1) s ` metry properties. First, we have checked for the 1 h i first values 3 of j, m and s (but for unspecified + κm α(jm) − κm α(jm) . 2` + 1 s ` s `−1 s `+1s `+1

3 In practice we checked it up to j = 4, and for all allowed Combined with (3.23), this is consistent with the values of s and m. j ↔ ` symmetry (3.27). We stress that both the 12 m ↔ s and j ↔ ` symmetries are immediate in (B.33) for spherical harmonics, one finds the flat case, as we demonstrate in AppendixF. ` The m ↔ s symmetry is consistent with the (jm) s (jm) X ` M s R[sG nˆ ] = c` sα D (R)sY nˆ . fact that the relations which relate radial modes ` Ij ` Mm ` Ij M=−` with both the same s and m, that is Eqs. (3.23) (3.30) and (3.28), are obviously invariant under m ↔ s This naturally brings the more general definition m s since sκ` = mκ`. Note that in the m ↔ s sym- for normal modes [11]: metry (3.26), the same ν appears on both sides. Hence, given the relation between ν and k [see ` (jm) X (jm) ` Eq. (2.27)], the symmetry relates radial func- sG` (ν) ≡ sG`M (ν)DMm(Rνˆ), (3.31) tions associated with different k, except in the M=−` flat case. The m ↔ s symmetry can be used as (jm) (jm) M sG`M (ν) ≡ c` sα` (ν)sY` , a shortcut in the algorithm previously described (jm) with the related more general definition for the to, for instance, calculate sα` for |m| > |s| (js) tensor harmonics: from mα` . Hence solving the radial functions in the plane (j, s) for a given m also provides ` (jm) ` (jm) Q (ν) ≡ Rνˆ[ Q (ν)] (3.32) automatically some of the radial functions for Ij Ij ` larger values of m. Conversely, this can be left X (jm) = `M Q (ν) D` (R ) , unused so as to serve as a consistency check. Ij Mm νˆ M=−` j (jm) X (jm) `M Q (ν) ≡ g(jm) G (ν)ˆns . (3.33) Ij s s `M Ij 3.5. General reference axis s=−j We then obtain a relation of the type (2.33) When building the harmonics and the de- j rived harmonics, the central relations were Eqs. (jm) X (jm) `Q (ν) = g(jm) G (ν)ˆns . (3.34) (C.17) and (C.18). They depend on `, s and Ij s s ` Ij j, but not on m. This happens because (2.28) s=−j is implicitly related to a special choice of axis, Evidently, we could redo the general con- which is clearly not the most general construc- struction of radial functions using an arbitrary tion. Indeed, one could perform an active rota- reference axis (instead of our choice for the tion Rνˆ ≡ R(φν , θν , 0) which brings the zenith zenith). Provided we rotate the r.h.s of the con- vector ez into a general direction νˆ with spher- dition Eq. (2.31) [or Eq. (2.39)], it would pro- ical coordinates (θν , φν ), that is Rνˆ[ez] = νˆ. In ceed exactly through the same set of equations order to explore this rotation, let us define the and steps, and one would find exactly the same mode vector radial functions. This is not a surprise, since the latter depend only on ν. ν ≡ ννˆ (3.29) 3.6. Conjugation and parity which contains, at the same time, the informa- tion about the reference axis used to define har- From (2.40) and (3.25) we deduce monics, and the value of the mode ν itself. In (jm) ? (jm) ? Section6 we relate ν to the wave vector of a [sα (ν)] = −sα (ν ) ` ` (3.35) plane wave. (jm) ? = sα` (−ν ) . The harmonics defined with a general direc- tion are related to the ones we have built using From (2.33) and (2.28) with properties (B.11a) the zenith direction. Using the rotation rules and (2.38), one then obtains the conjugation 13 property 4. RADIAL FUNCTIONS FOR SCALARS, h i? VECTORS AND TENSORS `Q(jm)(ν) = `Q(j,−m)(−ν?)(−1)(`+m) . Ij Ij (3.36) We now collect in this section the most com- Furthermore, in the special case of a rota- mon radial functions. We report the results for tion R around the direction ey of angle π [that the even and odd components so we can use is Ry(π) ≡ R(α = 0, β = π, γ = 0)], we find s ≥ 0. Furthermore we assume m ≥ 0 since from (B.35) the negative values are found from (2.42). The ? ` (jm) h` (jm) ? i scalar, vector and tensor cases correspond re- Ry(π)[ Q (ν)] = Q (ν ) , (3.37) Ij Ij spectively to m = 0, 1, 2, with the general re- which we can also relate to (3.36). Rotation strictions (2.32), on which we also add the re- around the y-axis by an angle π, or equivalently striction (3.24) in the closed case. In what fol- a parity inversion of the x and z axis, is equiva- lows, we only report radial functions for j ≤ 2. lent to considering the mode with −m and −ν, For the harmonics (j = m), the radial func- up to a ±1 factor. tions were already derived (even though not for- mulated using spin-weighted spherical harmon- x → −x y → −y z → −z ics) up to m = 2 in Ref. [4]. Derived harmonics, Factor (−1)m yes yes that is with j > m were reported up to j = 2 but Factor (−1)` yes only in the cases s = 0 and s = 2 in [7]. Hence Q(jm) → Q(j,−m) yes yes Ij Ij this section can be used as a complete reference Q(jm)(ν) → Q(jm)(−ν) yes Ij Ij for radial functions. We shall only need two par- ticular cases of the general expression (3.14): TABLE I: Transformation rules for harmonics under the inversion of a single axis. k ξ1 = √ ν2 − K We can also consider a parity transformation (4.1) k2 P, which is defined on tensor fields as ξ2 = . p 2 p 2 j (ν − K) (ν − 4K) P[TIj (χ, n)] ≡ (−1) TIj (χ, −n) . (3.38) Following the same techniques using (B.11b) Since radial functions determine the normal along with modes with (2.28) and then in turn the harmon- s j+s −s ics (and derived harmonics) with (2.33), we also nˆI (−n) = (−1) nˆ (n) , (3.39) (jm) j Ij report the values of the coefficients sg . In one finds case one needs contractions of the type (2.35), (jm) h (j,−m) i? we repeat that these are related to the g˜(jm) us- P[`Q (ν)] = (−1)m `Q (ν?) s Ij Ij ing (2.34), and the first few coefficients needed = (−1)` `Q(jm)(−ν) . (3.40) are Ij It is instructive to combine the previous rotation d00 = 1, with a parity transformation. Indeed, this cor- d = 1, d = 1, (4.2) responds to an inversion of the y-axis only and 10 11 d = 2 , d = 1 , d = 1. we find 20 3 21 2 22 R (π)[P[`Q(jm)(ν)]] = (−1)m `Q(j,−m)(ν) . y Ij Ij Throughout, we abbreviate r(χ) given by (2.2) (3.41) as r. In the expressions reported below, we note The factor (−1)m accounts for a rotation of angle that the radial functions are not invariant under π around the z axis, that is, Rz(π) ≡ R(α = ν → −ν in general, even though it is the case for 0, β = 0, γ = π) which is also an inversion of the hyperspherical Bessel functions of appendix the x and y axis. Hence, we can deduce the A. Indeed there is a prefactor linear in ν in each transformations brought by the inversion of a magnetic radial function, as required by prop- single axis. The results are gathered in TableI. erty (3.25). 14

4.1. Scalar modes (m = 0) The radial functions for the derived harmon- ics are The radial functions of the base scalar har- r  ν  monics are simply the hyperspherical Bessel (21) ξ2 3`(` + 1) d Φ` 0` = (4.8a) functions: k2 2 dχ r  2 (21) ξ2 d d (00) ν , 1` = 2 2 + cot(χ) 0` = Φ` . (4.3) k dχ dχ ν2 1  + − Φν, (4.8b) The radial functions for the derived harmonics 2 r2 ` are given, up to j = 2, by ξ ν d Φν  β(21) = − 2 r ` , (4.8c) 1 ` 2k2 dχ r (10) ξ1 d ν 0 = Φ , (4.4a) p ` k dχ ` ξ (` + 2)(` − 1) d (21) = 2 (rΦν), r 2 ` 2 2 ` ξ `(` + 1) Φν k 2 r dχ (10) = 1 ` , (4.4b) 1 ` k 2 r (4.8d) p ξ  d2  ξ (` + 2)(` − 1) Φν (20) = 2 3 + (ν2 − K) Φν,(4.4c) β(21) = − 2 ν ` . (4.8e) 0 ` 2k2 dχ2 ` 2 ` k2 2 r r  ν  (20) ξ2 3`(` + 1) d Φ` One checks that (4.8a) agrees with (4.4d), in 1 = , (4.4d) ` k2 2 dχ r agreement with (3.26). The constants needed to s ν build the corresponding harmonics are (20) ξ2 3(` + 2)! Φ` 2` = 2 2 . (4.4e) k 8(` − 2)! r (11) (11) 0g = ∓±1g = 1, √ 2 (21) (21) (21) ξ1 (4.9) The constants needed to build the harmonics √ 0g = ∓±1g = 2±2g = . and derived harmonics are 3 ξ2

(00) −1 (10) (10) −1 0g ξ1 = 0g = ∓±1g = ξ1 , (4.5) 4.3. Tensor modes (m = 2) and Finally, we give the radial modes related to √ the base tensor harmonics. They are 3 r3 g(20) = ∓ g20 = g20 = ξ−1. (4.6) 0 2 ±1 2 ±2 2 s ξ 3(` + 2)! Φν (22) = 2 ` , (4.10a) 0 ` k2 8(` − 2)! r2 4.2. Vector modes (m = 1) ξ p(` + 2)(` − 1) d (22) = 2 (rΦν), 1 ` k2 2 r2dχ ` Similarly, for the radial functions built from (4.10b) the vector modes, we find that the base harmon- ξ ν p(` + 2)(` − 1) Φν ics are given by β(22) = − 2 ` , (4.10c) 1 ` k2 2 r  2 r ν (22) ξ2 d d (11) ξ1 `(` + 1) Φ  = + 4 cot(χ) + 2 cot2(χ)  = ` , (4.7a) 2 ` 2 2 0 ` k 2 r 4k dχ dχ 2  ν ξ d(rΦν) +(−K − ν ) Φ , (4.10d) (11) = 1 ` , (4.7b) ` 1 ` 2k rdχ ξ ν d β(22) = − 2 (r2Φν). (4.10e) νξ 2 ` 2k2 r2dχ ` β(11) = − 1 Φν . (4.7c) 1 ` 2k ` Again, one checks that (4.10a) agrees Note that (4.7a) agrees with (4.4b), which cor- with (4.4e), (4.10b) agrees with (4.8d), and roborates (3.26). (4.10c) agrees with (4.8e), in agreement 15 with (3.26). There is an alternative expression 0). This finally proves that (3.18) is the correct (22) for 2 , which is expression needed to enforce (2.30). If we now use these results together with  2  (22) ξ2 d 2 ν 2 ν (B.23), we find that the normalization of har- 2 = (r Φ` ) − (ν − K)Φ` . 4k2 r2dχ2 monics at the origin is

0 The constants needed to build tensor harmonics `=j (jm) `=j (jm )? m QI QI = δmm0 Nj (5.3) are j χ=0 j χ=0 r 2 (22) 1 (22) 2 (22) where 0g = ∓√ ±1g = ±2g = 1. 3 3 3 (2j − 1)!! (4.11) N m ≡ ( g˜(jm))2 j 0 j!

(jm) 2 j! ≡ (0g ) , (5.4) 5. NORMALIZATION (2j − 1)!!

and where the following contraction of indices I In this section we are going to show that ex- j was used on the left hand side of (5.3): pression (3.18) is the correct one to enforce the normalization condition (2.30) in all cases. We (jm) s (jm0) Ij ? (jm) (jm0) g nˆ g nˆs = g g d will also discuss the overall normalization of the s Ij s s s js mm0 tensor harmonics in real space. ≡ Nj , (5.5)

mm m with Nj = Nj . Together with (2.33), the 5.1. Normalization at origin expressions above allow us to write the contrac- tion of harmonics at an arbitrary point. Restor- Following the same algorithm as the one de- ing the dependence with ν of the harmonics and scribed in Section 3.3, one can show that the normal modes, we find radial functions scale like χ|`−j| when χ → 0. 0 Indeed, since Φν ∼ χ` in this limit [see (A.5)], ` (jm) `0 (jm )? 0 ` QI (ν) QI (ν ) (5.6) (j,±j) `−j j j we find from (3.13) that 0 ∼ χ . One j can then check that for the various steps of the mm0 X (jm) (jm0)? 0 = Nj sG` (ν) sG`0 (ν ) . algorithm which increase j and s, this property s=−j is maintained. In practice, to show that such scaling holds one needs to distinguish the cases ` > j, ` = j and ` < j when applying the algo- 5.2. Integral on space rithm. The constant in the scaling can then be determined by ` − j iterations of (3.28) (when We checked by means of integrations by parts keeping only the dominant term as χ → 0), and (on the lowest |m| values) and using (A.1) and using (2.30). For ` ≥ j we find for χ → 0 (A.7) that the harmonics satisfy the normaliza- tion (with again Ij indices contracted) `−j ` (jm) χ (2j − 1)!! Y m Z α ∼ κ . (5.1) (jm) 0 0 (jm0)? s ` (` − j)!(2` + 1)!! s p d3V `M Q (ν) ` M Q (ν0) = (5.7) p=j+1 Ij Ij 0 m 3 (2` + 1) δ(ν − ν ) In the case ` ≤ j, it behaves as δ 0 δ 0 δ 0 N (2π) `` mm MM j 4π (ν2 − Km2)

j−` j (jm) (−χ) (2` − 1)!! Y 3 2 α ∼ κm . (5.2) where d V ≡ r (χ)dχ sin θdθdφ. The case j > s ` (j − `)!(2j + 1)!! s p p=`+1 |m| is deduced from the case j = |m| through the construction (2.23) using repeated applica- This is consistent with ` − j iterations of (3.28) tion of (2.25) when integrating by parts. A sim- (when keeping only the dominant term as χ → ilar method can be used to infer the vanishing 16 of (5.7) when m 6= m0. Note also that in the This is a generalization of the normalization re- closed case, ν takes integer values, hence the lation (A.7) which corresponds only to scalar Dirac delta function must be understood as a modes (see e.g. [16]). Kronecker symbol instead, that is we must read 0 (5.7) with δ(ν − ν ) → δνν0 . From the general definition (3.33) and the identities (B.13) and (5.5), it follows that the normalization (5.7) is equivalent to

j Z X (jm) (jm0)? 0 2 sα` (χ; ν) sα` (χ; ν )r (χ)dχ s=−j Conversely, for a given j, a closure relation 0 π δ(ν − ν ) can be also formulated, showing that we have = δmm0 . (5.8) 2 (ν2 − Km2) built a complete set of basis functions. We find

∞ ` j Z 2 2 X X X 4π (jm) (jm)? (ν − Km )dν `M Q (χ, n; ν)`M Q (χ0, n0; ν) (2` + 1)N m Ij Kj (2π)3 `=0 M=−` m=−j j δ(χ − χ0) = δ2(n − n0)δhk1 . . . δkj i . (5.9) r2(χ) i1 ij

For open cases, the integral runs on ν ≥ 0, 6. PLANE WAVES whereas in the closed case the integrals must be understood as a discrete sum on integer values In flat space, a plane wave is an eigenfunction such that ν ≥ ` + 1. Again, this is equivalent to of the Laplacian which assumes constant values the closure relation for radial functions (which on planes orthogonal to a constant wavevector k. we checked for the lowest values of j) For scalar functions it is simply exp(ik ·x). This idea cannot be generalized to the curved spaces j Z since the notion of a globally constant vector X (jm) (jm)? 0 2 2 sα` (χ; ν)s0 α` (χ ; ν)(ν − Km )dν does not exist. We will nonetheless seek to build m=−j eigenfunctions of the Laplacian in curved spaces π δ(χ − χ0) — which we shall abusively call plane waves — = δ 0 , (5.10) 2 ss r2(χ) which look like flat space plane waves near the origin of coordinate system, i.e., over distances with the same convention that it is a discrete much smaller than the curvature radius. sum on ν ≥ ` + 1 in the closed case. This is a generalization of the closure relation (A.8) which corresponds only to scalar modes [16]. One also verifies immediately that (5.9) is compat- ible with (5.7), since multiplying the former by 6.1. Zenith axis plane waves 0 `0M 0 Q(jm )?(χ, n; ν0) and integrating over space Ij 0 using the latter, yields `0M 0 Q(jm )?(χ0, n0; ν0), ex- Ij Plane waves are defined by summation of har- actly as the r.h.s. of (5.9) indicates. Similarly monics with different values of `. The most gen- (5.10) is obviously compatible with (5.8). eral summation on ` is of the form (restoring the 17 explicit dependence on ν) waves) built in Section 6.1 correspond to a wave vector ν = νez. As detailed in Section 3.5, one ∞,ν−1 m (jm) X ζ (jm) can consider a general direction νˆ with the as- Q (ν) ≡ ` `Q (ν) (6.1) Ij ζm Ij sociated wave vector ν = ννˆ. The associated `≥|m| j plane waves are built in general as and similarly Q(jm)(ν) ≡ R [Q(jm)(ν)] (6.4) ∞,ν−1 Ij νˆ Ij ζm (jm) X ` (jm) ∞,ν−1 ` sG (ν) ≡ sG (ν), (6.2) m m ` X X ζ `M (jm) ` ζj = ` Q (ν)D (R ) . `≥(|m|,|s|) m Ij Mm νˆ ζj ∞ m ` ≥|m| M=−` X ζ (jm) = ` c α (ν) Y (n), ζm ` s ` s `m This form is very similar to the general decom- `≥(|m|,|s|) j position of a STF tensor field (2.15) since it can such that also be written more explicitly as

j m (jm) X (jm) (jm) s (jm) X ζ` (jm) (jm) Q (ν) = sg sG (ν)ˆn . (6.3) Q (ν) ≡ c` sg sα (χ; ν) Ij Ij Ij ζm ` s=−j `Ms j ×D` (R ) Y M (n)ˆns . (6.5) The previous sums on ` run until infinity in the Mm νˆ s ` Ij flat or open case, and are limited by (3.24) in The normal modes associated with these general the closed case. The weights ζm are undeter- ` axis plane waves are mined coefficients. From these definitions we re- cover, near the origin, the same behavior as in ∞,ν−1 ` m (jm) X X ζ` (jm) (2.31). What we hereafter call plane waves cor- sG (ν) = c` sα (χ, ν) ζm ` m m `≥(|m|,|s|) M=−` j responds to the choice ζ` = const. (or ζ` = 1). By contrast, we name pseudo plane waves the ` M ×DMm(Rνˆ)sY` (n) , (6.6) m more general case ζ` 6= const. A pseudo plane is thus specified both by the mode ν and by the and we get a relation of the type (6.3) for the set of ζm: Q(jm)(ν, ζm). mode ν. ` Ij ` m We chose to divide by ζj in the definitions (6.1) and (6.2) so as to maintain the normaliza- 6.3. Extended Rayleigh expansion tion at origin (2.39). All recursion relations that were derived so far for the radial functions in the Eq. (6.5), possibly reshaped using (B.33), is (j, s) space are, in fact, also valid for the G(jm), s ` the generalized Rayleigh expansion for tensor- since the coefficients of all recursions are totally valued plane waves. In the flat case, for standard independent from `. Hence all of these recur- plane waves (i.e., ζm = const.) with j = 0 = m, sive relations are transposed as relations among ` we recover the usual Rayleigh expansion given the summed normal modes G(jm)ζm. This can s j by (F.3). be traced back to the fact that the general re- We can recast Eq. (6.5) in a more covariant lation (3.22), from which all recursive relations form as originate, is satisfied for the tensors ζmQ(jm). j Ij ∞,ν−1 m I (jm) X ζ nˆ j (n)Q (χ, n; ν) = ` (2` + 1) ∓s Ij ζm 6.2. General axis plane waves `≥(|m|,|s|) j g˜(jm) ±s α(jm)(χ; ν)n ˆI` (n)Q(`m)(χ = 0; ν) . In the previous section, we have summed the (`m) ±s ` ∓s I` ±sg˜ harmonics (2.33) on `. They correspond to a (6.7) special choice where the direction used to de- compose the local structure is also the zenith di- This extends Eq. (4.13) of [9] or Eq. (4.1.47) of rection. Hence the plane waves (or pseudo plane [19], which are restricted to the case j = |m| and 18 s = 0. Eq. (6.7) can be understood essentially sume) are orthogonal as we now review. Replac- as a simple Taylor expansion, since derived har- ing (6.4) in (5.7), and using (B.14) and (B.34), monics are precisely made of derivatives of the we find that in the open or flat case, the plane base harmonic. The generalized Rayleigh expan- waves are normalized according to sion is essential for the computation of cosmo- Z logical observables, and we illustrate its use in 3 (jm) (jm0)? 0 3 d VQ (ν) Q (ν ) = δmm0 (2π) Section 7.4 and in Ref. [17]. Ij Ij 2 m ν 3 0 ×Nj 2 2 δ (ν − ν ) . (6.12) 6.4. Parity and conjugation (ν − Km )

The transformation properties of section 3.6 Therefore, we conclude that in the open case the can be extended to pseudo plane waves. For con- plane waves that we defined have orthogonal- jugation, we find ity properties very similar to the flat case plane waves, thus justifying our abusive terminology. ? h (jm) m i However, in the closed case (K = 1), the sum QI (ν, ζ` ) (6.8) j on ` in (6.4) does not extend to infinity, and one = (−1)(j+m) × Q(j,−m)(−ν?, (−1)`ζm?) . Ij ` cannot rely on (B.14). We find instead For π-rotation around axis y, we get Z 3 (jm) (jm0)? 0 2 m d VQ (ν) Q (ν ) = δνν0 δmm0 2π N ? Ij Ij j (jm) m h (jm) ? m? i Ry(π)[QI (ν, ζ` )] = QI (ν , ζ` ) . ν−1 j j X (2` + 1) (6.9) × D` (R−1R ) . (6.13) (ν2 − Km2) mm νˆ0 νˆ Finally for parity transformations, we obtain `=|m| h i? P[Q(jm)(ν, ζm)] = (−1)m Q(j,−m)(ν?, ζm?) Ij ` Ij ` In the case where the modes have the same di- rection (νˆ = νˆ0), and using = (−1)jQ(jm)(−ν, (−1)`ζm) . (6.10) Ij ` ν−1 The combination of parity and rotation (which X (2` + 1) = ν2 − m2 , (6.14) amounts to an inversion of the y-axis) is similar `=|m| to (3.41) and reads

(jm) (j,−m) this reduces to R (π)[P[Q (ν, ζm)]] = (−1)mQ (ν, ζm). y Ij ` Ij `

(6.11) Z 0 d3VQ(jm)(ννˆ) Q(jm )?(ν0νˆ) The rules for an inversion of a single axis are Ij Ij thus exactly the same as in TableI for individual 2 m = 2π δ 0 δ 0 N . (6.15) `Q(jm), except that the factor (−1)` manifests νν mm j Ij m ` m j itself as ζ` → (−1) ζ` along with a global (−1) Eq. (6.13) shows that the plane waves as built factor. in (6.4) are not properly orthogonal in the closed case. In that case one should work directly with `M Q(jm)(ν), which according to (5.7) and (5.9) 6.5. Orthogonality Ij is a proper orthogonal basis.

As for the special case νˆ = ez, the plane In all cases, the closure relation for plane m waves (when ζ` = const., which we now as- waves reads 19

j Z 2 2 2 0 X (jm) (jm)? (ν − Km )dνd νˆ δ(χ − χ ) hk k i Q (χ, n; ν) Q (χ0, n0; ν) = δ2(n − n0)δ 1 . . . δ j , Ij Kj (2π)3N m r2(χ) i1 ij m=−j j (6.16) with the convention that it is a discrete sum on we find for the plane waves harmonics (6.1) that ν ≥ |m| + 1 in the closed case. This relation Z 2 is found from the definition (6.4) with (B.34) d n (jm) (jm0)? QI (χ, n; ν)QI (χ, n; ν) to express the Wigner D-coefficients, and using 4π j j m the orthogonality relation (B.13) to handle the = δmm0 Nj . (6.18) angular integration on νˆ so as to fall back onto If we further integrate (6.18) on the measure (5.9). The closure (6.16) is obviously compatible 4πr2(χ)dχ to complete an integration on the with (6.12) in the open of flat cases, and is also whole volume, we check in the closed case that compatible with (6.13) even though there is no it leads again to (6.15) with ν = ν0, since in the Dirac function on the directions of νˆ and νˆ0 in closed case R d3V = R π 4πr2(χ)dχ = 2π2. its right hand side. This is because the factor 0 Pν−1 ` −1 `=|m|(2`+1)Dmm(Rνˆ0 Rνˆ) effectively plays the role of a Dirac delta when acting on functions 6.7. Discussion on general factorization of with an angular structure limited to ` ≤ ν − 1, normal modes and this is exactly the case for the dependence on the mode direction νˆ of closed space plane It is argued in Appendix C of [7], and this waves as defined by (6.4). point is recalled in Eq. (1.15) of [21] and Eq. (A9.3) of [22], that the normal modes can be factorized in a form which separates clearly the 6.6. Integral on directions intrinsic angular dependence and the orbital one. Restricting the discussion to modes ν = νez for For plane waves, and for the lowest normal simplicity, and omitting the explicit dependence modes [specifically, we checked for j up to 4, and on ν on all functions, this factorization should for all allowed m and s] we have checked that the be of the form following identity holds: c G(jm)(χ, n) =? j Y m(n)F (ν, χ, n) s 2j + 1 s j Z 0 0 (6.19) G(jm)(χ, n; ν) G(j m )?(χ, n; ν) d2n s s with a universal orbital function F such that 4 X (jm) 2 = δ 0 δ 0 |c α (χ; ν)| , mm jj ` s ` |F (ν, χ, n)| = 1 . (6.20) ` 4π Translated to the plane wave harmonics using = δmm0 δjj0 , (6.17) (2j + 1) (2.33), this is c in agreement with equation C8 of [7]. Hence the Q(jm) =? g˜(jm) j Yjm F (ν, χ, n) , (6.21) Ij 0 Ij normalization of plane waves is such that the 2j + 1 dependence on χ, which is there in principle at jm with the Y defined everywhere following the the second line, disappears at the third line. In Ij remark after (B.32). the particular case of j = m = s = 0, and in the Expressions (6.19) and (6.21) are reminiscent flat case, this relation is proven using an addition of what is found in the flat case [Eqs. (F.1) and theorem of spherical Bessel function (e.g. Eq. (A.12) of [20]). Moreover, using properties (5.6) and (6.17), 4 This function is often written as eiδ(~x,~k). 20

(F.2)], where the orbital function is a pure scalar and this differs from the√ correct expression (4.4a) 2 plane-wave exp[i(kez)·(χn)]. If this was the case since obviously x` 6= 1/ ν − K and y` 6= 0. in the curved case, there would be a clear sepa- Another way to show that (6.19) does not ration between orbital and angular momentum. apply in the curved case consists in exhibiting Furthermore, the property (6.17) (which is cor- counterexamples. In the closed case, the sum m rect) would be a trivial consequence of (B.13). on ` in (6.2) (with ζ` = const.) to form plane We argue in this section why this is not possible waves is a finite sum since 0 ≤ ` ≤ ν − 1. in the general curved case, and that the factor- Let us first consider the case j = 0, s = 0, ization (6.19) does not exist. However, we insist m = 0. If ν = 1, then we have only ` = 0 (00) that property (6.17) is still correct since it does and 0α0 (χ; ν = 1) = F (ν = 1, χ, n) = 1 and not imply the factorization property (6.19). there is no issue. However as soon as we consider From the j = m = s = 0 case, one infers ν = 2, we have α(00)(χ; ν = 2) = cos(χ) and 0 0 √ immediately that the universal orbital function (00) 0α (χ; ν = 2) = sin(χ)/ 3, and it is found (00) 1 must be F (ν, χ, n) = 0G . In the flat case, that the orbital function must be F = eik·x, and since by construction we have √ F (ν = 2, χ, n) = cos(χ) + i 3 cos θ sin(χ) . (00) X p ` 0 0G = 4π(2` + 1)i j`(χ)Y` (n) (6.22) (6.25) ` Hence the unit norm condition (6.20) is not we could be tempted to deduce that the radial met. Of course when χ  1, that is for dis- functions can be built exactly like in the flat case, tances much smaller than the curvature scale, i.e. is using (F.5), but with the replacement the norm tends to unity, thus recovering the ν flat case result. Note however that (6.17) still j`(kr) → Φ` (χ). In the flat case, the (usual) (00) 2 2 spherical Bessel functions can be combined by holds since |c0 0α0 (ν = 2)| = 4π cos χ and means of (F.5) and (F.6), which then leads to (00) 2 2 |c1 0α1 (ν = 2)| = 4π sin χ. One could try to the expressions listed in AppendixF. If we in- release this unit norm condition and still look for sist on the idea of using the same combinations a universal orbital function. However, for ν = 2 ν in the curved case, but with Φ` in place of the but for the values (s = 0, j = m = 1), one infers usual j`, we must also use the relations (A.2). But note that these differ from (F.6) by factors F (ν = 2, χ, n) = 1 , (6.26) √ p like ν2 − K`2 and ν2 − K(` + 1)2. Thus, the which is not equal to (6.25). Similarly for (j = results obtained with this method are not the 1, m = s = 0) one infers yet another√ orbital func- radial functions reported in Section4. To be tion, being F (ν = 2) = cos χ + i/ 3 sin χ sec θ. specific, let us attempt to build the radial func- To conclude, not only the orbital function cannot (10) (10) tion 0α` = 0` from the factorization (6.19). be of unit norm, but also it cannot be universal, Starting from that is in practice it cannot depend only on ν 1 and on the position in space (χ, n). At best, (10) =? `Φν − (` + 1)Φν  , (6.23) 0 ` 2` + 1 `−1 `+1 (6.19) can be used for a definition of F for each (jm) set of (s, j, m), that is to define sF orbital and using (A.2), the radial function takes the functions. In the open case, one can also check form numerically (because of the infinite sum in `) d that the orbital function cannot be of unit norm (10) =? x Φν + y cotχΦν , (6.24) 0 ` ` dχ ` ` ` and cannot be universal. We thus conclude that the general factoriza- where the coefficients are tion (6.19) does not exist, and we can rely on the ! 1 ` ` + 1 explicit summation (F.5) only in the flat case. In x` = √ + 2` + 1 ν2 − K`2 pν2 − K(` + 1)2 the curved case, one must determine the radial !modes following the method used in this article `(` + 1) 1 1 (or a related one). The impossibility of the fac- y` = √ − p 2` + 1 ν2 − K`2 ν2 − K(` + 1)2 torizations (6.19) and (6.21) in the curved cases 21 is related to the fact that the norm squared of The helicity bases associated with a direction plane waves is not a constant, and only its aver- and its opposite are related through (3.39). A m age over spheres yields the constant Nj (inde- given point on the manifold is either denoted pendent on χ) as seen on (6.18). This is different by the pair (χ, n) or the pair (χ, −n¯). Plane- from the flat case where it is obvious from (F.2) wave harmonics in the propagation direction are m that the square of the norm of plane waves is Nj linked with those built so far as everywhere. It is important to stress, however, (jm) Q (χ, −n¯) ≡ (−1)j × Q(jm)(χ, n) . (7.2) that while (6.19) does not exist, its use in Ref. [7] Ij Ij was meant only as a heuristic motivation, and all the results are of course correct since they rely In order to be consistent with the construction essentially only on the property (7.29). of derived harmonics we must use, instead of (2.23), the defining property

(jm) 1 (j−1,m) 7. COSMOLOGICAL APPLICATIONS Q = − ∇ Q . (7.3) Ij k hij Ij−1i We are now in position to discuss some phys- The expansion of these new harmonics in terms ical applications of the formalism developed so of the associated normal modes is far. We will focus on the derivation of the Boltzmann hierarchy to describe the evolution j (jm) X (jm) ¯ (jm) ¯ s ¯ of CMB, a key cosmological observable, follow- QIj (χ, −n) ≡ sg sG (χ, n)ˆnIj (n) , ing both the pioneering work of [7] based on nor- s=−j mal modes, and the approach built in [8,9] using (7.4) STF tensors. In the next Section we introduce whereas we recall that the harmonics built with harmonics and normal modes which are adapted observation directions are expanded as (6.3). to the use of the propagating direction (of pho- The normal modes associated with plane waves tons) rather than the observing direction. In are expanded in radial functions in a similar Section 7.2 we summarize the angular decom- fashion to (6.2), as position for CMB temperature and linear polar- (jm) X (jm) ization. We then review the standard derivation sG (χ, n¯) = c¯` sα¯` sY`m(n¯) (7.5) of the Boltzmann hierarchy providing only min- `≥|m| imal ingredients of cosmology in Section 7.3. In where Section 7.4 we address the general method for extracting the multipolar decomposition of all ` `p c¯` ≡ (−1) c` = (−i) 4π(2` + 1) . (7.6) cosmological observables when using plane wave harmonics. When using pseudo plane waves Using (7.5) and (6.2) for plane waves in (7.4) and m (ζ` 6= const.) instead of standard plane waves (6.3), so as to replace in the definition (7.2), we m (ζ` = const.), the results are slightly different, deduce from the properties (3.39) and (B.11b) as detailed in [17]. that the new radial functions are related to the ones built with observed directions by

7.1. Relation to propagation normal modes (jm) (jm) (jm) sα¯` (ν) = −sα` (ν) = sα` (−ν) . (7.7) In the context of CMB, it is often more conve- From the decomposition (2.40) in even and odd nient to rewrite everything in terms of the prop- components, we deduce that the ones built when agating direction of photon, rather than the ob- using propagation directions, are related to those served direction of the incoming photon. Hence, built using observation directions, by let us define the propagation direction as the op- posite of the observed direction: (jm) (jm) s¯` (ν) = s` (ν) (7.8a) ¯(jm) (jm) n¯ ≡ −n . (7.1) sβ` (ν) = −sβ` (ν) . (7.8b) 22

m To summarize, when using propagation direc- mode components Θj (ν, η). From the choice tion, as is common in the context of CMB, one (7.10), with (7.9), (7.11) and (2.35), the temper- ` needs only to add a factor (−1) to c`, factors ature becomes a simple scalar field expanded in of (−1)j in the definition of the harmonics, and normal modes as then the radial functions are exactly the same 3 X Z d ν (jm) up to a global sign for the magnetic (odd) radial Θ = Θm(ν, η) G (ν) , (7.12) (2π)3 j 0 functions. Equivalently, one can use the same jm radial functions but with −ν instead of ν. In fact this is just the parity transformation rule where the propagating direction normal modes (jm) (6.10) since P[Q(jm)(n)] = Q (n¯). are related to the usual ones as specified in § 7.1. Ij Ij The case of linear polarization is analogous, and we decompose the angular dependence of the 7.2. CMB multipole decomposition Stokes parameters Q and U according to (see [19] or Eq. (1.67) in [26]) At each cosmological time, the temperature Q ± iU X I fluctuation field Θ depends both on the position = E ∓ iB  n¯ˆ j . (7.13) 2 Ij Ij ∓2 in space, that is on (χ, n), and on the propa- j gating direction n¯ i [which does not necessarily satisfy (7.1)]. This dependence can be separated The electric and magnetic STF multipoles EIj using a multipolar decomposition as and BIj are decomposed in terms of plane waves as X i1 ij Θ = Θi1...ij n¯ ... n¯ , (7.9) j Z 3 m j 1 X d ν Ej (ν, η) (jm) E = Q (ν) Ij 2 (2π)3 g˜(jm) Ij m=−j 2 where the STF multipoles ΘIj depend only on (χ, n) and on time. However, as argued in [6, j Z 3 m 1 X d ν Bj (ν, η) (jm) B = − Q (ν) . 7], a shortcut consists in fixing the propagating Ij 3 (jm) Ij 2 (2π) 2g˜ direction, m=−j (7.14) n¯i = −ni, (7.10) Exactly like for temperature, we then restrict the when solving for the observed CMB. This is propagating direction according to (7.10). Hence ±2 equivalent to consider, in a given observed di- (Q ± iU)ˆnij is a tensor field on space, which is rection ni, only the propagating directions which tangential to spheres of constant χ. From (2.35) are observed at some time by the observer 5. we find that the expansion in normal modes now

The temperature multipoles ΘIj are ex- reads panded on (general axis) plane wave harmonics X Z d3ν as Q ± iU = (7.15) (2π)3 j jm Z d3ν Θm(ν, η) X j (jm)  m m  (jm) ΘIj = Q (ν) (7.11) × E (ν, η) ± iB (ν, η) G (ν) . (2π)3 g˜(jm) Ij j j ±2 m=−j 0 Finally, the velocity of baryons, V , which we The dynamical evolution for STF multipoles i need to account for the Compton collisions with then translates into evolution equations for each electrons, is decomposed exactly as in (7.11) for i j = 1. This means that the quantity Vin¯ is 5 When considering first order cosmological perturba- decomposed as in (7.12) with only j = 1, which tions, it is enough to consider the background geodesic, in turn defines V m(ν, η). which is a straight line on the maximally symmetric In the closed case, the plane waves as de- spatial background. When considering higher order ef- fects, time-delay and lensing corrections to the trajec- fined by (6.4) are not orthonormal, as shown in tory must also be considered [23–25]. (6.13). Hence we must work directly with the 23

`M Q(jm)(ν), which are orthogonal according to For polarization, the Boltzmann equation is Ij (5.7) and (5.9). The previous expansions on har- even simpler since it is not affected by these grav- monics can be read formally if itational effects, and one has only i 0
ν = (ν, `, M) , (7.16) ∂η +n ¯ ∇i + τ (Q ± iU) = CQ±iU . (7.20) and then one must use the formal replacement [8, We now discuss the individual terms in these 9] equations in the following sections so as to obtain a Boltzmann hierarchy in § 7.3.4. Finally we ∞ ν−1 ` Z d3ν X X X report its formal integral solution in § 7.3.5. → . (7.17) (2π)3 ν=m+1 `=m M=−` To be clear, with the convention (7.16), 7.3.2. Gravitational effects (jm) (jm) (jm) sG (ν) refers to sG (ν) and Q (ν) `M Ij The gravitational term selects only the scalar, refers to `M Q(jm)(ν). Ij vector and tensor modes, that is |m| ≤ 2 and in practice this implies a restriction on the sums on m in (7.12) and (7.15). The effect of the 7.3. Boltzmann equation perturbed metric δgµν on temperature depends on the combinations 7.3.1. General structure i i j δg00 , δg0in¯ , δgijn¯ n¯ , (7.21) When using conformal cosmological time η, the general cosmological metric takes the form since, from the null geodesic equation, one infers

gcosmo = a2(η)(g + δg ) (7.18) 1 1 µν µν µν G = n¯i∇ δg − δg0 n¯in¯j +n ¯in¯j∇ δg . 2 i 00 2 ij j 0i where µ, ν are spacetime indices, a(η) is the scale (7.22) factor, and where the background metric gµν ex- This motivates us to decompose the metric per- tends the purely spatial metric (2.1) with g0i = 0 turbations according to and g = −1. 00 Z 3 Restricting to linear cosmological perturba- d ν (00) δg00 = −2 3 A(ν, η) Q (ν) tions, the general Boltzmann equation dictating (2π) 3 1 the evolution of the distribution function of pho- Z d ν X (1m) δg = B(m)(ν, η) Q (ν) tons reduces to an evolution of the black body 0i (2π)3 i temperature Θ which depends on η, the position m=−1 Z 3 on space, and on the propagating directionn ¯i. d ν (00) δgij = −2 HL(ν, η) Q (ν)gij This equation possesses the general structure (2π)3 Z 3 2 i 0
d ν X (m) (2m) ∂η +n ¯ ∇i + τ Θ = CΘ + G . (7.19) + 2 H (ν, η) Q (ν) . (2π)3 T ij m=−2 Here, C is the collision term accounting for all (7.23) processes with a final photon propagating in the directionn ¯i, whereas the term proportional to It is customary to adopt a gauge in which 0 (0) the Compton interaction rate τ ≡ aneσT (with B (ν) = 0, and we now assume this to be the the background number density of free electrons case. This encompasses both the popular syn- ne and the Thomson cross section σT) accounts chronous and Newtonian gauge. The decompo- for all collisions with an initial photon propagat- sition of (7.22) in terms of normal modes takes ing in directionn ¯i. Furthermore, G accounts for the form the gravitational effects which enter when con- 3 X Z d ν (jm) sidering metric perturbations around a homoge- G = Gm(ν, η) G (ν) , (7.24) (2π)3 j 0 neous and isotropic expanding background. jm 24

(jm) and it follows from (2.35) that factors 0g˜ are found with the literature, in particular with [6,7] m (±1) ±1 brought in the expressions of the Gj . For com- where the contribution of B goes into G1 in- ±1 pleteness we report them here [see App. C of stead of G2 here (see e.g. § 4.3.1 of [27] for a [21] which corrects [6,7]]. The only non vanish- detailed discussion). ing components are (omitting the dependencies 7.3.3. Collisions on ν and η) The collision terms, which account for Comp- 0 0 G0 = HL , ton collisions on electrons, depend only on the 0 G1 = kA , multipoles of temperature and linear polariza- 0 tion, and they are expanded in multipoles with G0 = − g˜(20) H(0) , 2 0 T (7.25) definitions following exactly the decomposition ±1 (21) h (±1) (±1)0 i G2 = − 0g˜ k B − HT , of temperature and linear polarization of the pre- vious section. We find [6,7, 19, 26, 27] ±2 (±2)0 G2 = HT .   ΘCm = τ 0 δjδmΘ0 + δj P (m) + δj V (m) , In each case, the mode k is related to ν through j 0 0 0 2 1 √ (2.27), hence one must distinguish according to E m 0 j (m) Cj = −τ 6δ2 P , the value of m. The relevant factors in these BCm = 0 , expressions are j 1  √  √ √ P (m) ≡ Θm − 6 Em . 2 2 2 2 (20) 2 k − 3K 2 ν − 4K 10 0g˜ = = √ , (7.27) 3k 3 ν2 + K √ (7.26) 2 (21) k − 2K ν 0g˜ = √ = √ √ . 3k 3 ν2 + 2K 7.3.4. Boltzmann hierarchy

In practice, the equations are solved in a specific The only non-trivial part, once the effect of gauge and not all metric perturbations compo- gravitation and collisions are expanded in STF nents are kept [7]. The synchronous gauge cor- multipoles, is the free streaming. It is sufficient (±1) responds to the conditions A = 0 and B = 0, to consider the case of modes aligned with the whereas the Newtonian gauge is found when us- (0) (±1) zenith direction, and we use the expression ing HT = 0 and HT = 0. In fact, the expansion in modes and multi- (jm) d  (jm) n¯i∇ ( G nˆs ) = − G nˆs (7.28) poles of (7.12)(7.15), are exactly like Eq. (55) i s Ij dχ s Ij of [6] and Eq. (23) of [7], with the directional dependence on νˆ explicit, so we can easily com- as well as the recursion relation for the normal pare our results and we find that Eqs. (7.25) dif- modes fer slightly from Eqs. (35-36) of [7]. This is ex- d  (jm) iνms (jm) G = G (7.29) pected since our gravitational effects correspond dχ s j(j + 1) s to Eq. C18 of [21] without the last term. Eqs. 1 h (j−1,m) (j+1,m)i + − κm G + κm G (7.25) also corresponds to what is obtained in 2j + 1 s j s s j+1 s [27], and arise when the observer which defines the temperature anisotropies is chosen to have a which is a consequence of (2.28) and (3.23) velocity proportional to (dη)µ. If a different ob- translated to propagating direction radial func- server is used to define anisotropies, namely, one tions (7.7). Using this property into (7.19) and µ with velocity proportional to (∂η) , then (7.22) (7.20), with (7.12) and (7.15), one finally obtains gets modified as we must consider the entirety the hierarchy (again, omitting the dependence of Eq. C18 in [21]. This explains the variations on ν) 25

m m  0κ 0κ  ∂ Θm = j Θm − j+1 Θm + Gm + ΘCm − τ 0Θm , (7.30) η j 2j − 1 j−1 2j + 3 j+1 j j j m m  2κ 2κ 2mν  ∂ Em = j Em − j+1 Em − Bm + ECm − τ 0Em , η j 2j − 1 j−1 2j + 3 j+1 j(j + 1) j j j m m  2κ 2κ 2mν  ∂ Bm = j Bm − j+1 Bm + Em + BCm − τ 0Bm . η j 2j − 1 j−1 2j + 3 j+1 j(j + 1) j j j

Given the maximal symmetry of the background, that we abbreviate as τ. It is then straightfor- m the evolution of the gravitational sources Gj de- ward to obtain the formal solution to the full pends only on ν. However, their initial condi- hierarchy in the integral form [6,7] tions (set deep in the past) does depend fully on m Z η0 ν. Hence, in practice, the hierarchy (7.30) needs Θj (η0) = dηe−τ × (7.33) to be solved only for various values of ν, and the 2j + 1 0 directional dependence is simply inherited from X Θ m m
(j0m) C 0 + G 0 0¯ (χ) , initial conditions. Let us also comment that we j j j j0 do not necessarily need to use the expansions in normal modes (7.12) and (7.15) with (7.29) to derive the hierarchy. Indeed, this is a shortcut m Z η0 Ej (η0) −τ X E m (j0m) based on using (7.10), and one might prefer us- = dηe Cj0 2¯j (χ) , 2j + 1 0 ing directly the expansions in harmonics (7.11) j0 m Z η0 and (7.14) along with (3.22) to compute the ef- Bj (η0) −τ X E m (j0m) = dηe C 0 β¯ (χ) , fect of free streaming, as in Refs. [8,9, 19, 27]. 2j + 1 j 2 j 0 j0 The hierarchy for multipoles is eventually recov- ered using the orthonormality condition (6.12). where the argument of the radial functions is

7.3.5. Integral solution χ = η0 − η . (7.34)

Finally, and this is crucial, it is customary We can check using (3.28) and (7.8) that to expand the directional dependence of the ob- when the gravitational effects and the collision served temperature (resp. polarization) directly term can be neglected (that is when the evolu- in Y m (resp Y m). Hence, to obtain the corre- tion of multipoles is only due to free streaming), j ±2 j sponding multipoles, one must consider the nor- the functions malization at origin (2.31), and this brings extra (j0m) `p (2j + 1)0¯j (η; ν) , factorsc ¯`/(2` + 1) = (−i) 4π/(2` + 1). Thus, (j0m) for the CMB, we shall use (2j + 1)2¯j (η; ν) , (7.31) ¯(j0m) r (2j + 1)2βj (η; ν) , CMB m m ` 4π Θ` (η0) = Θ` (η0)(−i) , (7.35) are solutions of the hierarchy (7.30) for any j0. 2` + 1 This guides the general resolution of the full hi- with similar relations for the E and B modes. erarchy when collisions and gravitational effects Eventually, one might also prefer to use direc- are taken into account. Let us introduce the op- tions related to observation rather than prop- tical depth from a cosmological time η to today agation for the multipoles observed today, and (η0) this brings extra factors of (−1)` for the temper- Z η0 0 0 0 ature and electric-type polarization multipoles τ(η, η0) ≡ dη τ (η ) , (7.32) `+1 η and (−1) for the magnetic-type ones. 26

7.4. Other cosmological observables The integral solution for the CMB multipoles arises immediately with this method, if one notes Quite generally, all types of cosmological ob- that the Boltzmann equation (7.19) is rewritten servables, such as weak lensing convergence or as shear, lensing field, galaxy number counts, red- d (e−τ Θ) = e−τ [C + G] (7.42) shift drifts, etc., are all of the form of an integral dη Θ on the background past light cone, which can be i written formally as where d/dη ≡ ∂η +n ¯ ∇i, is the derivative along Z η0 the background geodesic. Indeed, the integral X I (jm) O(n) = dη nˆ j S (η, χ, n) (7.36) form of the type (7.36) is s −s Ij 0 jm Z η0 Θ(η ) = e−τ [C + G]dη , (7.43) with (7.34). The sources S(jm) are expanded on 0 Θ Ij 0 plane waves harmonics as and following the aforementioned method, we nˆIj S(jm)(η, χ, n) (7.37) then recover the solution (7.33), up to the dif- −s Ij ference that for CMB we used propagation di- Z d3ν m (jm) rection harmonics and multipoles. Even though = 3 Sj (η; ν)sG (χ, n; ν) . (2π) this derivation appears much faster, one must If we decompose the observable as a sum of the not forget that in the case of CMB the sources effects of each harmonic as depend on the multipoles themselves, and one Z d3ν X must rely on the Boltzmann hierarchy (which O = Om(ν) G(jm)(χ = 0, n; ν) , s (2π)3 s j s can be found by derivation of the integral solu- jm (7.38) tions with respect to η0) to solve for their evolu- then we only need to expand the sources under tion. the integral with the Rayleigh expansion (6.7) The physical interpretation based on this which is equivalent to method is that free streaming builds multipoles with increasing j from the initial multipoles of ∞,ν−1 X ζm sources. The effect of free streaming amounts to G(jm)(χ, n; ν) = (2` + 1) ` (7.39) ±s ζm intersecting plane-wave harmonics with spheres `≥|m| j of increasing radius, and the radial functions pre- (jm) (`m) ×±sα` (χ; ν)±sG (χ = 0, n; ν) , cisely account for the projection effects of the m sources, taking into account the local angular so as to obtain the integral solutions (with ζ` = const. since the decomposition (7.37) is on plane- structure at emission. wave harmonics) m Z η0 0 8. CONCLUSION sOj (ν) X (j m) m = dη sαj (χ; ν)Sj0 (η; ν) . 2j + 1 0 j0≥|m| Thanks to the introduction of the general- (7.40) ized helicity basis, we established in this work Note that in the angular decomposition (7.38) a systematic and comprehensive construction we must use the normalization at χ = 0 given of radial functions, normal modes and ten- by (2.31) but taking into account the rotation sor harmonics in maximally symmetric three- (6.6), that is 6 dimensional spaces. When combined with spin- cj X G(jm)(χ = 0; ν) = Dj (R ) Y M . weighted spherical harmonics, they provide a s 2j + 1 Mm νˆ s j M powerful set of tools adapted to the description (7.41) of symmetric and trace-free tensors, and is suited for separating the radial from the angular depen- 6 In the closed case, and given the formal meaning (7.16), dencies of physical quantities. Furthermore, the (jm) developed framework allows for systematic alge- the normalization at χ = 0 is directly sG`M (χ = j M 0; ν) = δ` cj /(2j + 1)sYj . braic manipulations which greatly benefits from 27 the power of symbolic computational tools. In when studying the properties under par- particular, in this work, we have made intensive ity transformation as in § (3.6). use of xAct [28]. In AppendixG our results are contrasted • Symmetries in the space of (j, m, s, `) val- with earlier literature on vector and tensor ues, namely the m ↔ s and the j ↔ harmonics around maximally symmetric curved ` exchange symmetries [Eqs. (3.26) and spaces. However, our method is not restricted to (3.27)]. vector or tensor harmonics, but can be applied to higher rank harmonics thanks to the full set of • Orthogonality relations (5.8) and (5.10), recursive relations in the (j, s) space. Our results which imply corresponding orthogonality also extend to curved spaces the construction of relations for harmonics. scalar, vector and tensor harmonics presented in [11], and puts on a firmer ground the pioneering Once knowing the radial functions, whose ex- use of normal modes introduced in [7]. However, pressions for j ≤ 2 are gathered in Section4 (or we stress that some of our relations were not fully AppendixF for the flat case), the harmonics are demonstrated but only checked explicitly for all built using (2.28) and (2.33), with the needed (jm) modes up to reasonable values of the eigenvalue coefficients sg given by (2.37) and (3.18), j (typically j ≤ 4 and the associated |m| ≤ j, and the explicit forms of the generalized helic- |s| ≤ j), as was also the case in [7]. Hence, ity bases reported in AppendixC. from a mathematical point of view, our formal- The case of a flat space is very different from ism would benefit from an appropriate general the curved cases. Indeed, we have shown that proof on these relations. Still, for practical phys- the general factorization (6.19), which can be ical applications which depend only on the low- used in the flat case to build systematically all est values of j (but with all allowed values of m radial functions (see AppendixF), does not exist and s), it can be fully trusted since all relations in the curved case, contrary to previous state- were checked with ` being kept general, using the ments in the literature. Our results provide a general properties of hyperspherical Bessel func- systematic algorithm to build recursively the ra- tions. Thus, in a restricted sense, they have been dial functions in curved space by systematic ex- demonstrated. The relations which were checked ploration of the (j, s) space of radial functions. up to j = 4 (and all allowed values of m and s) A Mathematica notebook implementing this al- but with general values of ` are Eqs. (3.28), (5.8), gorithm is available at [18]. (5.10) and (6.17). Relation (3.27), on the other The radial functions are extremely powerful hand, was checked only for j ≤ 4 and ` ≤ 4. for the computation of theoretical expressions The radial functions have very rich proper- for multipoles of observables. Once an observ- ties which fall into four categories. These are able is written as an integral on the background summarized as follows: past light cone, it is sufficient to decompose the angular structure on normal modes, and to use • Recursive relations in the space of (j, s) the Rayleigh expansion in the form (7.39) to ob- values. They can all be deduced from the tain the result. In practical applications, it is triangular relations (NW,NW,SW,SE) de- sometimes preferred to use harmonics which are picted in Fig.1 and whose expressions are decomposed according to a propagation direc- collected in appendixD. They allow us to tion (e.g., photon’s direction in the case of CMB) deduce all radial functions using the al- rather than the observation direction, and the re- gorithm described in § 3.3. Furthermore, lation between both convention is simple, as we Eq. (3.23) is of direct use for the effect of summarized in section 7.1. free streaming on radiation multipoles. Finally, it is worth mentioning that radial functions (and thus harmonics) can also be de- • Sign inversions of either m, s or ν fined for super-curvature modes. They cor- [Eq. (3.25)], which are of direct use respond to values of ν in the complex plane 28

and rely on analytic continuations of the ra- formulas collected in appendixC. TP thanks dial functions built here. In [17] we detail how the Institut d’Astrophysique de Paris for its super-curvature modes can be used to described hospitality during the initial stages of this spatially anisotropic (i.e., Bianchi) space-times project, as well as Brazilian Funding Agencies as super-curvature fluctuations over maximally CNPq (311527/2018-3) and Funda¸c˜aoArauc´aria symmetric space-times. (CP/PBA-2016) for their financial support. Tensorial computations in this work were performed using the tensor algebra package Acknowledgments xAct [28]. CP is indebted to Guillaume Faye for ex- tensive discussions about the compendium of

[1] E. R. Harrison, Rev. Mod. Phys. 39, 862 (1967). Phys. Rept. 465, 61 (2008), 0705.4397. [2] V. D. Sandberg, Journal of Mathematical [20] P. H. F. Reimberg, F. Bernardeau, and Physics 19, 2441 (1978). C. Pitrou, JCAP 1601, 048 (2016), 1506.06596. [3] U. H. Gerlach and U. K. Sengupta, Phys. Rev. [21] T. Tram and J. Lesgourgues, JCAP 1310, 002 D 18, 1773 (1978). (2013), 1305.3261. [4] K. Tomita, Progress of Theoretical Physics 68, [22] R. Durrer, The Cosmic Microwave Back- 310 (1982). ground (Cambridge University Press, Cam- [5] L. F. Abbott and R. K. Schaefer, Astrophys. J. bridge, 2008), ISBN 9780511817205. 308, 546 (1986). [23] W. Hu, Phys. Rev. D 62, 043007 (2000), astro- [6] W. Hu and M. J. White, Phys. Rev. D 56, 596 ph/0001303. (1997), astro-ph/9702170. [24] A. Lewis and A. Challinor, Phys. Rept. 429, 1 [7] W. Hu, U. Seljak, M. J. White, and M. Zal- (2006), astro-ph/0601594. darriaga, Phys. Rev. D 57, 3290 (1998), astro- [25] A. Lewis, A. Hall, and A. Challinor, JCAP ph/9709066. 1708, 023 (2017), 1706.02673. [8] A. Challinor, Phys. Rev. D 62, 043004 (2000), [26] C. Pitrou (2019), 1902.09456. astro-ph/9911481. [27] C. Pitrou, Class. Quant. Grav. 26, 065006 [9] A. Challinor, Class. Quant. Grav. 17, 871 (2009), 0809.3036. (2000), astro-ph/9906474. [28] J. M. Mart´ın-Garc´ıa, xAct, efficient tensor com- [10] L. Lindblom, N. W. Taylor, and F. Zhang, Gen. puter algebra for mathematica (2004), URL Rel. Grav. 49, 139 (2017), 1709.08020. http://www.xact.es. [11] L. Dai, M. Kamionkowski, and D. Jeong, Phys. [29] J. Lesgourgues and T. Tram, JCAP 1409, 032 Rev. D 86, 125013 (2012), 1209.0761. (2014), 1312.2697. [12] S. Weinberg, Gravitation and cosmology: prin- [30] T. Tram, Commun. Comput. Phys. 22, 852 ciples and applications of the general theory of (2017), 1311.0839. relativity (Wiley, 1972). [31] J. N. Goldberg, A. J. MacFarlane, E. T. New- [13] K. S. Thorne, Reviews of Modern Physics 52, man, F. Rohrlich, and E. C. G. Sudarshan, J. 299 (1980). Math. Phys. 8, 2155 (1967). [14] M. Zaldarriaga, U. Seljak, and E. Bertschinger, [32] G. Faye, L. Blanchet, and B. R. Iyer, Class. Astrophys. J. 494, 491 (1998), astro- Quant. Grav. 32, 045016 (2015), 1409.3546. ph/9704265. [33] A. Challinor and A. Lewis, Phys. Rev. D 71, [15] L. Blanchet and T. Damour, Philosophical 103010 (2005), astro-ph/0502425. Transactions of the Royal Society of London Se- [34] C. Pitrou, T. S. Pereira, and J.-P. Uzan, Phys. ries A 320, 379 (1986). Rev. D 92, 023501 (2015), 1503.01125. [16] D. H. Lyth and A. Woszczyna, Phys. Rev. D 52, [35] J. Ben Achour, E. Huguet, J. Queva, and 3338 (1995), astro-ph/9501044. J. Renaud, J. Math. Phys. 57, 023504 (2016), [17] T. S. Pereira and C. Pitrou (2020), submitted. 1505.03426. [18] http://www2.iap.fr/users/pitrou/ RadialFunctions.nb. [19] C. G. Tsagas, A. Challinor, and R. Maartens, 29

ν=1 Appendix A: Hyperspherical Bessel functions Φ0 = 1. The ν = 2 mode allows only for ν=2 global dipolar modulations√ since Φ0 = cos χ ν=2 Hyperspherical Bessel functions are derived and Φ1 = (sin χ)/ 3. in detail in Refs. [1,4,5]. For the convenience of The hyperspherical Bessel functions are nor- the reader, though, we present some key proper- malized similarly to usual spherical Bessel func- ties in this appendix. tions [Eq. (F.8)] ν 0 The Hyperspherical Bessel functions Φ` are Z 0 π δ(ν − ν ) Φν(χ)Φν (χ)r2(χ)dχ = , (A.7) solutions of the following differential equation ` ` 2 ν2 Z π δ(χ − χ0) 1 d 2 d ν ν ν 0 2 r (χ) Φ Φ` (χ)Φ` (χ )ν dν = . (A.8) r2(χ) dχ dχ ` 2 r2(χ)  `(` + 1) In the closed case, the integral on ν must be + ν2 − K − Φν = 0 . (A.1) r2(χ) ` understood as a discrete sum on ν ≥ ` + 1. A class of related functions is given by They satisfy the following recurrence relations ν ν,n Φ` Ψ` ≡ . (A.9) d ` p rn(χ) Φν = ν2 − K`2Φν dχ ` 2` + 1 `−1 One can check that these functions satisfy the (` + 1)p differential equation − ν2 − K(` + 1)2Φν , `+1 2 2` + 1 (A.2) d ν,n d ν,n 1 Ψ + 2(1 + n) cotχ Ψ ν p 2 2 ν dχ2 ` dχ ` cotχ Φ` = ν − K` Φ`−1 2` + 1  `(` + 1) 1 p + ν2 − K(1 + n) − + ν2 − K(` + 1)2Φν , r2(χ) 2` + 1 `+1 2  ν,n +n(n + 1) cot (χ) Ψ` = 0 . (A.10) with In practical numerical computations, hyper- sin νχ Φν ≡ . (A.3) spherical Bessel functions are challenging to 0 ν sinχ compute. The reader interested in fast and ac- curate implementations can check Refs. [29, 30]. A closed expression for a general ` is [4, 16]

` ! 1 Y 1 Appendix B: Spherical Harmonics and Φν = √ (A.4) ` ν2 2 2 helicity basis i=1 ν − Ki  −1 d `+1 × sin`χ cos(νχ) . In this appendix we work in the flat (Eu- sinχ dχ clidean) three-dimensional space, also identified with the tangent space at the origin of the co- Near the origin (χ → 0), they are power-law ordinates (χ = 0) of curved spaces. The unit suppressed (except for ` = 0) as direction vector is n, and we also use the helic- ` √ ity vectors (7.34) along with the general helicity Y ν2 − Ki2 Φν ∼ χ` . (A.5) basis (2.10) and the multi-index notation (2.11). ` (2i + 1) i=1 In the closed case, sin = sin and the variable 1. Spherical harmonics ν must take positive integer values constrained by Spherical harmonics are defined as functions on the unit sphere: 0 ≤ ` < ν . (A.6) s 2` + 1 (` − m)! Y m(θ, ϕ) = eimϕP m(cos θ) , The lowest value ν = 1 corresponds to a con- ` 4π (` + m)! ` stant global perturbation since there is only (B.1) 30 with the associated Legendre polynomials being This allows us to derive a central relation for given by the computation of covariant derivatives on the m m 2 m/2 sphere P` (z) = (−1) (1 − z ) s `+m m ±s (±λ ) m ± ±s d 2 ` D Y nˆ
= ∓ √ ` Y n nˆ × (z − 1) . (B.2) j ±s ` Is ±(s+1) ` j Is dz`+m 2 ( λs) Let us define, for any pair of functions A(n) and ± ∓√ ` Y mn∓nˆ±s , ±(s−1) ` j Is B(n), the Hermitian inner product 2 s Z where the coefficients ±λ where introduced ? 2 ? ` {A|B} = {B|A} ≡ d nA (n)B(n) . (B.3) in (3.7). An explicit form of the spin weighted spheri- The spherical harmonics are orthonormal cal harmonics is n m m0 o s Y |Y 0 = δ``0 δmm0 , (B.4) ` ` m imϕ 2` + 1 (` + m)!(` − m)! sY` = e and complete 4π (` + s)!(` − s)! min(`−s,`+m) ∞ ` X ` − s ` + s  X X m m? 0 2 0 Y` (n)Y` (n ) = δ (n − n ) . (B.5) r r + s − m r=max(0,m−s) `=0 m=−` (cos θ )2r+s−m Given the spherical harmonics, which form ×(−1)`+m−r−s 2 . (B.10) θ 2r+s−m−2` a basis for scalar functions on the sphere, one (sin 2 ) can define spin-weighted spherical harmonics as Useful properties are basis for spin functions on the sphere [31]. These m ? m+s −m are defined as sY` (n) = (−1) −sY` (n),(B.11a) s m ` m sY` (−n) = (−1) −sY` (n), (B.11b) m (` − s)! s sY` = ∂/ Y`m , s ≥ 0 , r (` + s)! m m 2` + 1 −sY (ez) = δms(−1) , (B.11c) s ` 4π (` + s)! −s Y m = (−1)s ∂/ Y , s ≤ 0 , and the s ↔ m interchange property s ` (` − s)! `m (−1)seisφ Y m = (−1)meimφ Y s . (B.12) or by induction as s ` m ` 1 Finally, we also find the orthogonality relation m / m s+1Y` = p ∂sY` , (` − s)(` + s + 1) n m m0 o sY` |sY`0 = δ``0 δmm0 , (B.13) m 1 m s−1Y = − ∂/sY , ` p(` + s)(` − s + 1) ` as well as the closure relation, which generalizes (B.5) where the spin-raising ( ∂/) and spin-lowering ( ∂/) ∞ ` operators are X X m m? 0 2 0 sY` (n)sY` (n ) = δ (n − n ) .  i  ∂/ ≡ −(sin θ)s ∂ + ∂ (sin θ)−s , `=|s| m=−` θ sin θ ϕ (B.14)   s i −s ∂/ ≡ −(sin θ) ∂θ − ∂ϕ (sin θ) . sin θ 2. Relation with helicity basis These operators are related to the covariant derivative D on the sphere. Indeed, it is found In this section, we detail how spherical har- that monics and spin-weighted spherical harmon- √ √ ics are related to the generalized helicity ba- ∂/ = − 2D+ = − 2n− · D , (B.9a) √ √ sis (2.10). This extends the results already col- ∂/ = − 2D− = − 2n+ · D . (B.9b) lected in appendix D of [26]. First, from the 31 general rule for the integration over products of where n [13] 2` + 1 (` − m)!1/2 C ≡ (−1)m , Z d2n `m 4π (` + m)! nI2`+1 = 0 , (B.15) 4π (−1)j(2` − 2j)! a ≡ . Z d2n 1 `mj 2`j!(` − j)!(` − m − 2j)! nI2` = δ(i1i2 . . . δi2`−1i2`) , 4π 2` + 1 Since we used a Cartesian basis in a Euclidean where the parentheses mean full symmetrization space, we also define YI` = Y`m and we have `m I` on enclosed indices, it is possible to show that ?I` m I` the property Y`m = (−1) Y` −m which extends I`  I` hi1 i`i the definition for negative m. The Y satisfy nˆ |nˆJ = ∆`δ . . . δ , (B.16) `m ` j1 j` the orthogonality property 4π`! ∆ ≡ . (B.17) 0 0 ` YI` Y`m ? = ∆−1δm . (B.23) (2` + 1)!! `m I` ` m nˆI` is a special case of (2.10) with multi-index They also allow us to build spin-weighted spher- notation, and is simply the STF product of ` unit ical harmonics, in close analogy to (B.19): direction vectors (see e.g. [13, 32]). Eq. (B.16) m s `m I` ±sY (n) = (∓) b Y nˆ . (B.24) is a particular case of Eq. (C2) in [32]. ` `s I` ∓s If we define This relation is inverted as ` I` −1  I` m (∓)s∆ Y`m ≡ ∆` nˆ |Y` , (B.18) I` ` X m ?I` nˆ∓s = ±sY` (n)Y`m . (B.25) b`s we can relate the spherical harmonics to the gen- m=−` eralized helicity basis (with s = 0) as Using (B.13) and (B.22) we deduce immediately I the useful orthogonality condition for the gener- Y m(n) = ∆−1nˆ nˆI` |Y m =n ˆ Y ` . (B.19) ` ` I` ` I` `m alized helicity basis The inverse relation is n I` ±so ∆` hi1 i`i nˆ |nˆ = δ 0 δ . . . δ (B.26) ` ±s J 0 `` 2 j1 j` ` (b`s) I` X m  I` nˆ = Y (n) Y`m|nˆ , (B.20) ` 4π hi1 i`i = δ``0 d`s δ . . . δ . m=−` (2` + 1) j1 j` ` X m ?I` Furthermore, since the generalized helicity basis = ∆`Y` (n)Y`m . m=−` is a complete basis for STF tensors at each point, we also find the closure relation From the identity ` ` X −1 I` ∓s hi1 i`i (d`s) nˆ nˆ = δ . . . δ . (B.27) X 2` + 1 ±s J` j1 j` Y m(n)Y m ?(n) = , (B.21) ` ` 4π s=−` m=−` In the construction of harmonics of this pa- we get the closure relation per, and more specifically in (2.39), we are not

` working in a Euclidean space. However we can X hi i i YI` Y`m ? = ∆−1δ 1 . . . δ ` . (B.22) still use the object (B.18) if it is understood that `m J` ` j1 j` m=−` it is defined in the tangent space at the origin. I` Finally note that the Y`m are related to the I` Explicitly the Y`m are given by (for m > 0) generalized helicity basis in the zenith direction Since in this special direction (θ = 0) φ is not (`−m) [ 2 ] defined, we choose the convention I` X  i1 i1   im im  Y`m = C`m δ1 + iδ2 ... δ1 + iδ2 1 j=0 nθ = ex, nφ = ey, n± = √ (ex ∓ iey) 2 im+1 i`−2j `−2j+1 `−2j+2 i`−1 i` a`mjδ3 . . . δ3 δ . . . δ , (B.28) 32 which implies that at any point (θ, φ) of the unit consists in actively rotating around the z-axis by sphere, the helicity basis is obtained by a rota- an angle γ, then actively rotating around the y- tion of angle θ around the y axis and a rotation axis by an angle β, and finally rotating around of angle φ around the z axis from the basis at the z-axis by an angle α. The rotated spher- the zenith direction. With this choice we get in ical and spin-weighted spherical harmonics are particular for m ≥ 0 related to the original ones by (see e.g. appendix A of [33] or appendix D.3 of [34]) I` m −1/2 I` Y` ±m = (∓) (∆`d`m) nˆ∓m . (B.29) zen m s X ` m0 s 0 R[sY` nˆIj ] = Dm m(R)sY` nˆIj (B.33) Note that we can recast the value at χ = 0 of m0 harmonics given in (2.39). We find where the Wigner D-coefficients are related to (j ±m) √ (2m − 1)!! spin spherical harmonics through `Q = δj(∓ 2)m ij Ij ` p χ=0 (2m)! r 2` + 1 ξ m m isγ ` m Ij sY` (β, α) = (−1) e D−m s(α, β, γ) . × nˆ∓m . (B.30) 4π ξj zen (B.34) With the formulation (B.29), Eq. (B.24) can In particular, when considering only a rotation be recast as Ry(π) around the y-axis by an angle π (that is r α = 0, β = π, γ = 0) ±m s m 4π ∓s I ` 0 ±sY` = (∓1) (∓1) nˆI nˆ∓m ` `+m 2` + 1 ` zen Dm0m(Ry(π)) = δm −m0 (−1) , (B.35) (B.31) which obviously leads to (B.11a) after complex and conjugation. Using that the generalized helic- R (π)[ Y mnˆs ] = (−1)`−m Y −mnˆs . (B.36) ity basis is a complete basis for STF tensors, we y s ` Ij s ` Ij also obtain by decomposing the generalized he- licity basis in the zenith direction (considered as Appendix C: A compendium of useful a constant tensor) formulae

j X m s jm 1. Explicit expression of the generalized (kjs) sY nˆ = Y . (B.32) j Ij Ij helicity basis s=−j with the factors kjs defined by (2.18). From the general formula to extract the STF In flat case, the definition of the Yjm in the part from a symmetric tensor (see e.g. Eq. (2.2) Ij tangent space at χ = 0 can be trivially ex- of [13]) one infers the general expression for the tended to any point by simple translations. In generalized helicity basis which is the curved case, one uses the relation (B.29) [(`−s)/2] and extend it to any point by parallel transport ±s X ` nˆ = sa g(i i . . . gi i along the radial geodesic reaching this point. I` n i 2 2n−1 2n n=0 Since the generalized helicity basis is also paral- ± ± n . . . n ni . . . n , (C.1) lel transported along radial geodesics, the prop- i2n+1 i2n+s 2n+s+1 i`) erties (B.24) and (B.32) are also valid when Yjm Ij where the parentheses mean full symmetrization andn ˆs are evaluated at a general point with Ij on enclosed indices, and with the coefficients χ 6= 0. (−1)n(2` − 2n − 1)!!(` − s)! a` ≡ . (C.2) s n (2` − 1)!!(2n)!!(` − 2n − s)! 3. Rotations It is instructive to write down explicitly the Let us consider an active rotation of angles first few terms of the generalized helicity ba- R(α, β, γ). With the Euler angle notation, it sis (2.10). For j = 1, we have by convention 33

±1 ± nˆi = ni andn ˆi = ni . For two and three in- Let us now collect relations related to prod- dices, we find respectively ucts of the generalized helicity basis. Applying

1 A3 of [15], we get nˆij = ninj − 3 gij, 1 I hI ji (` − s)(` + s) hI nˆ±1 = (n±n + n n±), (C.3) nˆ ` nj =n ˆ ` + nˆ `−1 gi`ij ij 2 i j i j ±s ±s `(2` + 1) ±s ±2 ± ± nˆ = n n , is I ip ij i j ±  jhi` nˆ `−1 . (C.11) ` + 1 p ±s and For s ≥ 0 we also find 1 nˆ = n n n − (g n + g n + g n ), ijk i j k ij k jk i ki i i(` − s) 5 I` j I`j jhi` I`−1ip nˆ±sn± =n ˆ ∓ p nˆ ±1 1 ± ± ± ±(s+1) ` + 1 ±(s+1) nˆijk = (ni njnk + ninj nk + ninjnk ), 3 (` − s)(` − s − 1) hI − nˆ `−1 gi`ij . 1 ± ± ± `(2` + 1) ±(s+1) − (gijnk + gjkni + gkini ), 15 (C.12) 1 nˆ±2 = (n±n±n + n±n n± + n n±n±), ijk 3 i j k i j k i j k For s > 0 it reads instead nˆ±3 = n±n±n±. (C.4) ijk i j k 1 i(` + s) I ip nˆI` nj = − nˆI`j ∓  jhi` nˆ `−1 ±s ∓ 2 ±(s−1) 2(` + 1) p ±(s−1)

2. Products and contractions (` + s)(` + s − 1) hI`−1 + nˆ gi`ij (C.13). 2`(2` + 1) ±(s−1) In the process of obtaining recursive relations Note that the missing case s = 0 for this relation among radial functions, we encounter a series of is in fact given by (C.12) evaluated in s = 0. products and contractions of generalized helic- Finally, note that the STF part of products ity basis elements which we now collect. The of helicity vectors are the expected relation (for contractions s ≥ 0) 2 2 I p I (` − s ) `−1 `−1 hI ji I j nˆ±s np =n ˆ±s (C.5) ` ` `(2` − 1) nˆ±sn± =n ˆ±(s+1) (C.14) ` − 1 I I`−1p ± `−1 but one should be careful that for s ≥ 1 we also nˆ np = − nˆ± (C.6) 2` − 1 find generalize Eq. (A23) of [15]. For s ≥ 0 we also 1 nˆhI` nji = − nˆI`j . (C.15) find ±s ∓ 2 ±(s−1)

I`−1p ± (` − s)(` − s − 1) I`−1 nˆ±s np = − nˆ . (C.7) `(2` − 1) ±(s+1) 3. Derivatives of helicity basis For s > 0 we obtain a. Simple derivative I`−1p ∓ (` + s)(` + s − 1) I`−1 nˆ±s np = nˆ . (C.8) 2`(2` − 1) ±(s−1) In this section, we collect relations related Repeated application of (C.5)-(C.8) allows to to derivatives of the generalized helicity basis. prove the orthogonality property (2.12). We work on the maximally symmetric curved Defining the Levi-Civita tensor on the space with metric (2.1) whose associated covari- spheres as ant derivative is ∇i. We first find

k ±s ±is φ ±s ij = n kij , (C.9) ∇pnˆ = cot θe nˆ Ij r(χ) p Ij we also have parity inverting relations h ±s ±si + (j − s) cotχ g nˆ − npnˆ phij Ij−1i Ij p ±s is ±s ± ±(s−1)  nˆ = ± nˆI . (C.10) − s cotχ n nˆ . (C.16) hi` I`−1ip ` ` p Ij 34

The first line might seem peculiar at first sight, but it can be absorbed when considering in- stead the derivative of products of spin-weighted j(j + 1) ∇p(Y mnˆ ) = cotχ Y mnˆ spherical harmonics multiplied by the associated ` Ij−1p 2j − 1 ` Ij−1 helicity basis, which is precisely what we always r 1 `(` + 1) (j − 1) m +1 have in all expressions. Indeed, for s > 0 we get + +1Y nˆ r(χ) 2 (2j − 1) ` Ij−1 r m ±s 1 `(` + 1) (j − 1) ∇p(±sY nˆ ) = (C.17) m −1 ` Ij − −1Y nˆ . r(χ) 2 (2j − 1) ` Ij−1 h ±s ±si m +(j − s) cotχ g nˆ − npnˆ ±sY phij Ij−1i Ij ` −s cotχ n±nˆ±(s−1) Y m p Ij ±s ` s c. Curl of helicity basis (+λ`) m ± ±s ∓√ ±(s+1)Y n nˆ 2r(χ) ` p Ij The curl is also deduced by contraction with ( λs) ±√− ` Y mn∓nˆ±s , the Levi-Civita tensor of the expressions in sec- ±(s−1) ` p Ij 2r(χ) tiona. If s > 0 we get and in the particular case s = 0 we find simply  m ±s s m ±s curl ±sY nˆ = ±i cotχ ±sY nˆ ` Ij j ` Ij ∇ (Y mnˆ ) = (C.18) p ` Ij ( λs) (j − s) h i +i + ` Y mnˆ±(s+1) m ±(s+1) ` Ij +j cotχ gphij nˆIj−1i − npnˆIj Y` 2r(χ) j s r (−λ`) (j + s) m ±(s−1) 1 `(` + 1) −i ±(s−1)Y nˆ (C.20) − Y mn+nˆ 2r(χ) 2j ` Ij r(χ) 2 + ` p Ij r 1 `(` + 1) + Y mn−nˆ . and for s = 0 we get simply r(χ) 2 − ` p Ij curl Y mnˆ
= (C.21) From (C.14) and (C.15), it is obvious to consider ` Ij p only the STF part of these relations. `(` + 1)  m +1 m −1 i √ +1Y nˆ + −1Y nˆ . 2r(χ) ` Ij ` Ij

b. Divergence of helicity basis d. Laplacian Furthermore, by contraction with gpij of the expressions in the previous section, we obtain For a generic function f(χ), and using twice relations associated with the divergence of an (C.17) we find for s > 0 helicity basis. For s > 0 it is

2 2 m Ij p m ±s (j − s ) ∆(f ±sY` nˆ±s) = (C.22) ∇ (±sY` nˆI p) = (C.19) j−1 j(2j − 1) + f 00 + 2 cotχ f 0 Y mnˆIj n ±s ` ±s m ±s I × (j + 1) cotχ ±sY` nˆ  2 2  m j Ij−1 + f cot (χ)(s − j(j + 1)) ±sY` nˆ±s ( λs)   + ` (j − s − 1) m ±(s+1) f 2 m Ij ±√ ±(s+1)Y nˆ + (s − `(` + 1)) Y nˆ 2r(χ) (j + s) ` Ij−1 r2(χ) ±s ` ±s s  cotχ √ (−λ`) (j + s − 1) m ±(s−1) s m Ij ±√ ±(s−1)Y nˆ ∓ (j − s) 2(+λ`)±(s+1)Y` nˆ±(s+1) 2r(χ) 2(j − s) ` Ij−1 r(χ)

cotχ (j + s) s m Ij ∓ √ (−λ )±(s−1)Y nˆ . whereas in the s = 0 case it is r(χ) 2 ` ` ±(s−1) 35

In the s = 0 case we find As for the NSE relation, it is (0 < s ≤ j) ∆(fY mnˆIj ) = (C.23) s s ` (−λ`)(−λj) (jm)  00 0 2  m Ij ±sα` = f + 2 cotχ f − f cot (χ)j(j + 1) Y` nˆ (j − s + 1)r(χ) f m I (jm) − `(` + 1)Y nˆ j (j + s) cotχ ±(s−1)α r2(χ) ` ` κm cotχ (s−1) j+1 (j+1,m) p m Ij + ±(s−1)α` − j 2`(` + 1)+1Y nˆ (2j + 1) r(χ) ` +1 m (j + s) (s−1)κj (j−1,m) cotχ p m Ij + j 2`(` + 1) Y nˆ . + ±(s−1)α` r(χ) −1 ` −1 (2j + 1)(j + 1 − s) (j + s)mν ∓i α(jm) . (D.3) j(j + 1) ±(s−1) ` Appendix D: Geographical recursion relations Combining relations NS (3.23) with the NSE or NSW leads to the a set of four triangular re- We collect recursive relations among radial lations (see interpretation of this denomination functions in the (j, s) plane. We illustrate in on Fig.1) Fig.1 the radial functions which are related by these individual relations. • NW relation (for 0 < s ≤ j): The NSW relation, whose derivation is sum- s s (−λ )(−λ ) d marized in section (3.2) is ( for 0 ≤ s ≤ j) ` j (jm) (jm) ±(s−1)α` = ±sα` (D.4) s s (j + s)r(χ) dχ (+λ )(+λ ) ` j α(jm) = (jm) ±s ` +(j + 1 − s) cotχ ±sα` (j + s + 1)r(χ) m (jm) sκj (j−1,m) mν (jm) (j − s) cotχ α + ±sα ± i ±sα . ±(s+1) ` (j + s) ` j ` m (s+1)κ + j+1 α(j+1,m) (2j + 1) ±(s+1) ` • NE relation (for 0 ≤ s < j): m (j − s) (s+1)κj (j−1,m) s s + α (+λ )(+λ ) d (2j + 1)(j + 1 + s) ±(s+1) ` ` j α(jm) = α(jm) (j − s)r(χ) ±(s+1) ` dχ ±s ` (j − s)mν (jm) (jm) ±i ±(s+1)α . (D.1) j(j + 1) ` +(j + 1 + s) cotχ ±sα` (D.5) κm It is understood that when j = s, this is a re- s j (j−1,m) mν (jm) + ±sα ∓ i ±sα . (jm) (j+1,m) (j − s) ` j ` lation linking ±sα` to ±(s+1)α` only. In- deed the coefficients in front of these two terms √ contain j − s, but the coefficients in front of • SW relation (for 0 < s ≤ j + 1) : the other terms contain (j−s) (and also multiply s s (−λ`)(−λj) (jm) d (jm) radial functions that no longer satisfy |s| ≤ j). ±(s−1)α = − ±sα (j + 1 − s)r(χ) ` dχ ` Hence it must be understood that we must di- √ (jm) vide first by j − s before evaluating in j = s +(j + s) cotχ ±sα` (D.6) m and we get sκj+1 (j+1,m) mν (jm) + ±sα ± i ±sα . j (j + 1 − s) ` j + 1 ` (+λ`) (jm) (j+1,m) √ ±jα = ±(j+1)α × (D.2) 2r(χ) ` ` In the case s = j + 1, it reduces to a re- s lation between α(jm) and α(j+1,m). (j + 1)2 − m2 p ±(s−1) ` ±s ` ν2 − K(j + 1)2 . However, both terms contain a factor (j + 1)(2j + 1) √ 1/ j + 1 − s, so it must be understood A recursive application of this relation allows to that the expression is to be multiplied (j,0) (00) ν √ deduce s=jα` from 0` = Φ` . Using (3.26), by j + 1 − s before being evaluated in we then recover (3.13). s = j + 1, and we recover (D.2). 36

• SE relation is (for 0 ≤ s < j): We can check that this latter case corresponds to the real part of the NE relation (D.5). ( λs)( λs) + ` + j (jm) d (jm) The curl relation among radial functions is ±(s+1)α` = − ±sα` (j + 1 + s)r(χ) dχ for 0 < s ≤ j (jm) + (j − s) cotχ ±sα` (D.7) κm   s j+1 (j+1,m) mν (jm) d (jm) (jm) (jm) + ±sα ∓ i ±sα . s α + cotχ α ± imν α (j + 1 + s) ` j + 1 ` dχ ±s ` ±s ` ±s ` ( λs)( λs) Let us also stress that by taking the difference + ` + j (jm) = − ±(s+1)α` of the NW and NE relations, we obtain a useful 2r(χ) s s NW-NE relation which involves no derivative, (−λ`)(−λj) (jm) + α . (E.3) which is (for 0 < s < j) 2r(χ) ±(s−1) ` s s (+λ`)(+λj) (jm) In the s = 0 case it reduces to ±(s+1)α = (D.8) 2r(χ)(j − s) ` p s s (jm) `(` + 1)j(j + 1) (jm) (−λ )(−λ ) mν 1β = −mν0 . (E.4) ` j α(jm) ∓ i α(jm) ` r(χ) ` 2r(χ)(j + s) ±(s−1) ` j ±s ` (jm) s (j−1,m) This latter relation is exactly the imaginary part + s cotχ α + κm α . ±s ` (j2 − s2) s j ±s ` of the NE relation (D.5) or the SE relation (D.7). Note also that combining the curl relation (E.3) Similarly, one could combine the SW and SE re- and the div relation (E.1) allows to remove the lations to get a SW-SE relation without deriva- derivative of the radial function and leads also tives. the NW-NE relation without derivative (D.8). Finally the STF construction of derived har- Appendix E: Divergence, curl and STF monics brings the relation (for 0 < s ≤ j) recursions d (jm) (jm) ±sα` − j cotχ ±sα` (E.5) Following the method of section 3.2, we can dχ obtain recursive relations among radial functions (j + 1) (j+1,m) − κm α in the (j, s) space, from the divergence relation (j + 1)2 − s2 s j+1 ±s ` s s (2.25), the curl property (3.16), and the STF (+λ )(+λ ) = − ` j α(jm) construction of derived modes (2.23). 2r(χ)(j + 1 + s) ±(s+1) ` The divergence relation leads for 0 < s ≤ j s s (−λ )(−λ ) to − ` j α(jm) . 2r(χ)(j + 1 − s) ±(s−1) ` d α(jm) + (j + 1) cotχ α(jm) (E.1) dχ ±s ` ±s ` The case j = s + 1 can also be considered with j method similar to those detailed after (D.6), and + κm α(j−1,m) j2 − s2 s j ±s ` it also reduced to (D.2). s s In the s = 0 case it is (+λ`)(+λj) (jm) = ±(s+1)α` m 2r(χ)(j − s) d (jm) (jm) 0κj+1 (j+1,m) s s 0` − j cotχ 0` − 0` (−λ )(−λ ) dχ j + 1 + ` j α(jm) . 2r(χ)(j + s) ±(s−1) ` s 1 `(` + 1)j (jm) = − 1 . (E.6) In the s = 0 case it reduces to r(χ) (j + 1) ` κm d (jm) (jm) 0 j (j−1,m) This latter relation is nothing but the real part 0` + (j + 1) cotχ 0` + 0` dχ j of the SE relation (D.7). p`(` + 1)(j + 1) We then check that combinations of all the = √ (jm) . (E.2) jr(χ) 1 ` relations of this section can be used to form the 37

s m0m triangular relations (NW, NE, SW and SE re- with the coefficients C`Lj defined in (F.9). lations). Hence this is an alternative derivation The j` are the usual spherical Bessel func- for all recursions among radial functions in the tions satisfying the relations (j, s) space. However, the fact that we need to separate explicitly the real and imaginary parts 0 1 j`(x) = [`j`−1(x) − (` + 1)j`+1(x)] of the triangular relations in the cases s = 0 2` + 1 makes this derivation less direct and we prefer j`(x) 1 = [j`−1(x) + j`+1(x)] (F.6) the method based on the various projections of x 2` + 1 (3.22). with j0(x) ≡ sin(x)/x. They also satisfy the differential equation Appendix F: Radial functions in flat space     1 d 2 d `(` + 1) x j` + 1 − j` = 0 , (F.7) In the flat case, there is a complete separabil- x2 dx dx x2 ity between the angular and the spatial depen- dencies. The spatial dependence is the same for and the normalization condition all modes, and thus the same as for scalar har- Z 2 π δ(a − b) monics, that is, it is a pure Fourier mode. We j`(ax)j`(bx)x dx = . (F.8) 2 a2 choose to align the wavevector k of the Fourier mode with the zenith direction ez. From this The constants in (F.5) are the so-called Gaunt separability, the plane-wave normal modes are coefficients, and are defined as all of the form [6, 11] Z (jm) cj m ike ·r s m1m2m3 2 m1 ? m2 m3 G (r, n) = Y (n)e z , (F.1) C ≡ d Ω(sY )(Y )(sY ) s 2j + 1 s j `1`2`3 `1 `2 `3 r where r = rn, and r is now the radial coordi- (2` + 1)(2` + 1)(2` + 1) = (−1)m1+s 1 2 3 nate, corresponding to plane waves harmonics 4π ! ! (jm) (jm) cj jm ikez·r Q (r, n) = 0g˜ Y e . (F.2) `1 `2 `3 `1 `2 `3 Ij 2j + 1 Ij × , (F.9) s 0 −s −m1 m2 m3 We do not use χ which was in units of curva- ture, since now the curvature length `c is infi- where the 3×2 matrices on the third line are the nite. Note that the Fourier mode magnitude k is well-known Wigner 3-j symbols. From the sym- also no more in units of inverse curvature length. metry properties of the 3-j symbols, we deduce In practice, the trigonometric functions sin(χ), that tan(χ) and cot(χ) need also to be replaced re- s m0m m s0s s m0m s m0m spectively by r, r and 1/r in all expressions. Us- C`Lj = C`Lj , C`Lj = CjL` (F.10) ing the Rayleigh expansion X p which with (F.5) proves rigorously the properties eikez·n = 4π(2` + 1)i`j (kr)Y 0(n) (F.3) ` ` (3.26) and (3.27) in the flat case. ` Let us now collect the explicit forms of the the decomposition of the plane-wave normal radial functions in flat space. We recover results modes is then given by of [6,7] for s = 0 or s = 2 up to the global (jm) X (jm) m sG (r, n) = c` sα` (kr)sY` (n) (F.4) sign inversion for odd modes since we collect re- ` sults when defining harmonics with respect to with the radial functions built as the observed direction (see section 7.1 for prop- (jm) X agation direction harmonics). We also use the α (x) ≡ sCm0mj (x)iL+j−` s ` `Lj L compact notation x ≡ kr and we recall that in L s the flat case ξn = 1 for all n since ν = k, so (4π)(2L + 1) (jm) × , (F.5) the constants sg are directly read from those (2` + 1)(2j + 1) reported in section4. The first radial functions 38 are Appendix G: Comparison with literature

(00) 0` = j`(x), (F.11a) The harmonics built in this paper can be re- (10) lated to the scalar, vector and tensor harmonics  = j0(x), (F.11b) 0 ` ` derived in [4] and [10] in the closed case, and ex- r (10) `(` + 1) j`(x) pressed in the usual orthonormal spherical basis 1 = , (F.11c) ` 2 x (2.6) rather than with the helicity basis. In these (20) 1  00  references, the harmonics and derived harmon- 0 = 3j (x) + j`(x) , (F.11d) ` 2 ` ics are separated into their electric (even parity) r 3`(` + 1) d j (x) and odd parity by considering the contributions (20) = ` ,(F.11e) 1 ` 2 dx x s `Q(j,|m|) ± `Q(j,−|m|) . (G.1) 3(` + 2)! j (x) Ij Ij (20) = ` , (F.11f) 2 ` 8(` − 2)! x2 From the property (2.42), we see that the plus sign selects only the contribution of the electric (even parity) radial modes, whereas the nega- r `(` + 1) j (x) tive sign selects the magnetic (odd parity) radial (11) = ` , (F.12a) 0 ` 2 x modes. To be specific the three vector harmonics defined in Eqs. (12-14) of [10] are proportional to (11) 1 d(xj`(x)) 1 = , (F.12b) respectively the m = 0 harmonics (necessarily of ` 2 xdx 1 even type), the m = 1 magnetic harmonics, and β(11) = − j (x) , (F.12c) 1 ` 2 ` the m = 1 electric harmonics, where the nota- tion used is k ≡ ν −1, such that it takes positive integer values. Similarly the tensor harmonics of r   Eqs. (26-30) are successively proportional to the (21) 3`(` + 1) d j`(x) 0` = , (F.13a) m = 1 magnetic harmonics, the m = 1 electric 2 dx x harmonics, the m = 0 harmonics (necessarily of 0 (21) 00 j`(x) j`(x) j`(x) even type), the m = 2 magnetic harmonics and 1 = j` (x) + − + , ` x x2 2 the m = 2 electric harmonics.   (21) 1 d j`(x) The spectrum of eigenvalues of the Laplacian 1β` = − x , (F.13b) 2 dx x can also be compared for scalar and vector har- p(` + 2)(` − 1) 1 d monics with the exterior calculus approach of (21) = [xj (x)], 2 ` 2 x2 dx ` [35] in the closed case, and we now detail our p(` + 2)(` − 1) j (x) agreement. We still work in units such that β(21) = − ` , (F.13c) 2 ` 2 x `c = 1. The Laplace-de Rahm operator is de- fined as ∆˜ ≡ −(dδ + δd). For scalar functions it matches exactly the Laplace-Beltrami operator (2.19). In the closed case the set of eigenvalues s for scalar harmonics (j = m = 0) is the set of (22) 3(` + 2)! j`(x)  = , (F.14a) 2 2 0 ` 8(` − 2)! x2 k = ν − 1 = L(L + 2) where L ≥ 0 and ν ≥ 1 p are integers. For the derived vector valued har- (22) (` + 2)(` − 1) 1 d monics (j = 1 and m = 0) which correspond 1 = [xj`(x)], ` 2 x2 dx to exact forms, we find that the spectrum is the p (` + 2)(` − 1) j (x) same since β(22) = − ` , (F.14b) 1 ` 2 x 1  j0(x) j (x) ∆˜ ∇ Q(00) = ∇ ∆Q(00) = −k2∇ Q(00) . (G.2) (22) = j00(x) − j (x) + 4 ` + 2 ` , i i i 2 ` 4 ` ` x x2 1 d However for vector harmonics (j = m = 1), β(22) = − [x2j (x)]. (F.14c) 2 ` 2x2 dx ` which correspond to co-exact forms since they 39

are divergenceless, we find Function Definition nˆs (2.10) (11) (11) (11) Ij ˜ 2 (jm) ∆Qi = (∆ − 2K)Qi = −(k + 2K)Qi . sα` (2.28) (jm), β(jm) (2.40) The spectrum of ∆˜ in that case is the set of s ` s ` G(jm) (2.28) k2 + 2 = ν2 with the integer values ν ≥ 2, in s ` G(jm) (6.2) agreement with [35]. s `Q(jm) § 2.4 Ij Q(jm) (6.1) Ij Appendix H: Tables of symbols (jm) (jm) sG`M , sG` (ν) § 3.5 `M Q(jm), `Q(jm)(ν) § 3.5 Ij Ij We gather in the Tables below the most com- (jm) sG (ν), § 6.2 monly used symbols of this work. (jm) Q (ν) § 6.2 Ij YI` (B.18) Variable Definition `m r(χ) (2.2) TABLE III: Main functions and tensors used in build- K (2.3) ing harmonics. The barred version of these functions ν (2.27) are related to the propagating direction, and are de- fined in § 7.1. bjs (2.13) djs (2.13) kjs (2.18) (jm) sg (2.33) (jm) sg˜ (2.34) c`, c¯` (2.29), (7.6) ξn (3.14) q(jm) (2.26) m Nj (5.4) sin, tan, cot § 2.1 m sκ` (3.19) s ±λ` (3.7) m ζ` (6.1) ∆` (B.17)

TABLE II: Main symbols used in the construction of harmonics.