Cosmological constraints on anisotropic Thurston geometries
Ananda Smith Craig J. Copi Glenn D. Starkman

Resolving Twin Jets and Twin Disks with JWST and ALMA:  

The Young WL 20 Multiple System
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§ INTRODUCTION


One of the main assumptions made in the current standard model of cosmology, ΛCDM, is that the Universe is spatially homogeneous and isotropic on large enough scales. 
Known as the Cosmological Principle, this assumption restricts the large-scale geometry of the Universe to one of three types, which can be presented simultaneously in a Friedmann-Lemaître-Robertson-Walker (FLRW) metric. 
Though the Cosmological Principle is well-corroborated by large-scale structure and other cosmological observables, hints of this postulate being violated have emerged in recent decades.

Among the strongest pieces of evidence for the Cosmological Principle has been the high degree of isotropy of the blackbody temperature  of the cosmic microwave background (CMB) radiation, first reported by Penzias and Wilson <cit.>. 
More recently, the isotropy of the statistical properties of the very small amplitude fluctuations in the temperature and polarization of the  CMB has been tested  <cit.>, and several violations of isotropy have been reported.
In particular, a handful of large-angle features in the observed CMB temperature <cit.> were discovered in the first WMAP data release <cit.> and were suggested to indicate significant deviation from statistical isotropy. 
These features have come to be known as the “large-angle anomalies,” and their significance has persisted in  subsequent Planck datasets <cit.>, with recent work arguing these anomalies jointly constitute a > 5σ violation of statistical isotropy <cit.>.

This collection of evidence warrants consideration of cosmological models that deviate slightly from the Cosmological Principle, in particular by violating spatial isotropy. 
In this work, we investigate the consequences of breaking spatial isotropy through geometry – i.e., by equipping the Universe with a background metric that is homogeneous but anisotropic. Relaxing the requirement of isotropy allows the large-scale geometry of the Universe to be of a slightly more general class than the three FLRW geometries. 

Historically, work on the cosmology of anisotropic spaces has centered around the Bianchi models (see <cit.> for review). 
These are 3+1 spacetimes whose homogeneous spatial part corresponds to a 3-dimensional real Lie algebra, falling into one of eleven types within a classification devised by Bianchi in 1898 <cit.>. 
A subset of the Bianchi models have been invoked as potential explanations of CMB anomalies <cit.>, but their deviation from isotropy is strongly constrained. 
This is because many Bianchi spaces require anisotropic expansion – i.e., multiple scale factors – if they are sourced by perfect fluid stress energies. 
This anisotropic expansion leaves potentially detectable imprints on cosmological observables, often characterized by the resulting “shear.”
For instance, in the subset of Bianchi spaces that explicitly contain the three FLRW metrics as special cases, anisotropic expansion is strongly constrained by the anisotropies it induces in the CMB temperature <cit.>. 
Constraints have also been derived from the imprint of shear on nucleosynthesis <cit.>.

This work considers a more recently developed class of homogeneous but not necessarily isotropic geometries known as the Thurston geometries. 
These eight model geometries exhaust the possible local geometries of closed homogeneous 3-manifolds according to Thurston's geometrization theorem <cit.>.
Thurston's classification schema is analogous to Bianchi's, and all but one of the Thurston geometries fall within a Bianchi group <cit.>. 
Out of the eight Thurston geometries, three are the open, closed, and flat FLRW geometries, and the remaining five are anisotropic. 
These latter subset of spaces remain largely unconstrained within a cosmological context (although see <cit.>).

The dynamics and distance measures of spacetimes with Thurston geometries as spatial parts, which we will dub Thurston spacetimes, have recently been investigated <cit.>, albeit when equipped with a single scale factor. 
This requires invoking an anisotropic fluid fine-tuned to prohibit anisotropic expansion. 
The evolution of scale factors under isotropic dust and cosmological constant is provided for each Thurston spacetime in <cit.>, but no accompanying constraints on curvature scales are derived. 

In this paper, we provide strong constraints on the curvature of all five anisotropic Thurston spacetimes for the first time. 
After providing representations of each anisotropic Thurston geometry in Section <ref>, we review the dynamics of the corresponding spacetimes under the presence of perfect fluid dust and cosmological constant in Section <ref>, where we find anisotropic expansion is required. 
Consequently, when coupled with underlying geometric anisotropy, we find that the local flux of CMB photons is distorted, and a present-day observer interprets this flux as that of a blackbody with a non-uniform temperature. 
The amplitudes of the CMB temperature fluctuations induced in these geometries are coupled to the curvature parameter , which is therefore strongly constrained by the isotropy of the CMB. 
We derive this constraint for each of the five anisotropic Thurston spacetimes in Sections <ref>–<ref>, finding that || ≲ 10^-5 in all five geometries, with two even requiring || ≲ 10^-10. 

The GitHub repository associated with this study is publicly available at <https://github.com/cwru-pat/ThurstonGeometry>.
Codes will be deposited there as publicly usable versions become available.



§ THURSTON GEOMETRIES


In 1982 Thurston conjectured <cit.> (and Perelman later proved <cit.>)

    The interior of every compact 3-manifold has a canonical decomposition into pieces which have geometric structures.

In practice, this reduces to a set of eight local three-geometries – three being the well-known and well-studied isotropic FLRW geometries: flat (ℝ^3), spherical (S^3), and hyperbolic (ℍ^3).
The remaining five anisotropic local three-geometries are less well-known and will be discussed below.
Though the conjecture allows for a decomposition of the full space, in this work we restrict to the case where the observable Universe has just one of these eight local geometries. 
Without loss of generality, we take the topology to be the covering space of that geometry.
We refer curious readers to the growing body of work on detecting nontrivial topology in the Universe, e.g., <cit.>.

In the remainder of this section, we present metrics for each of the five anisotropic Thurston geometries. 
We adopt the representations of <cit.>, where the parameter κ found in the spatial part of each metric distinguishes between positive (κ > 0) and negative (κ < 0) spatial curvature. 
All of these local geometries can be made arbitrarily close to flat space in some finite neighborhood by taking κ sufficiently close to zero.



 §.§  and 


The first two anisotropic geometries in Thurston's classification system are  and . 
In the literature, spacetimes equipped with  spatial geometries are often referred to as Kantowski-Sachs spaces <cit.>, and   falls under type III of the Bianchi classification.

 and  are the most straightforward to understand among the five geometries we are considering:  admits positive curvature (κ > 0) along two spatial directions and zero curvature along one, while  is analogous but with its curved directions having negative curvature (κ < 0). 
This description leads naturally to a spatial metric separated into a hyperspherical part and a flat part:

    Σ_3^2 = χ^2 + S_κ (χ)^2 ϕ^2 +  z^2,

where χ∈ [0, ∞), ϕ∈ [0, 2π), z ∈, and

    S_κ(χ) = 
        sin(χ√(κ) )/√(κ),    κ > 0, 
    sinh(χ√(-κ))/√(-κ),    κ < 0.




 §.§ 

The next anisotropic Thurston geometry is , the universal cover of U(ℍ^2), the unit tangent bundle of the hyperbolic plane, and falls within Bianchi types III and VIII. 
We adopt the spatial metric derived in <cit.>,

    Σ_3^2 =  x^2 + cosh(2x√(-κ))  y^2 +  z^2 + 2sinh(x√(-κ)) y z,

where κ < 0 and x, y, z ∈.

This geometry is often referred to as SL(2, ℝ) in the literature since this space is diffeomorphic to .
However, these two spaces are not isomorphic so they are not interchangeable in a physical context.



 §.§ Nil


The next anisotropic Thurston geometry is Nil, the geometry of the Heisenberg group.
It can be thought of as twisted E^2 × and falls within Bianchi type II.
It can be represented with the spatial metric

    Σ_3^2 =  x^2 + (1+κ x^2)  y^2 +  z^2 - 2x√(-κ) y  z,

where x, y, z ∈.



 §.§ Solv


The final anisotropic Thurston geometry is Solv, the geometry of solvable Lie groups.
The Solv geometry falls within Bianchi type VI_0, and can be represented with the spatial metric

    Σ_3^2 = ^2z √(-κ) x^2 + ^-2z√(-κ) y^2 +  z^2,

where x, y, z ∈.



§ EVOLUTION OF ANISOTROPIC THURSTON SPACETIMES


According to general relativity, the evolution of any spacetime is governed by the stress energy content of the universe through Einstein's field equations

    G^μ_ν = 8π G T^μ_ν,

where our choice of c = 1 units persists in all subsequent calculations.
Thus the evolution of spacetimes with anisotropic Thurston spatial geometries is sensitive to the choice of stress energy.
For example, it is possible to equip an anisotropic spacetime with a single scale factor, say a(t), in what are commonly referred to as “shear-free” models in the literature <cit.>.
The stress energy tensor characteristic of the fluid required to achieve this scales like a^-2 and contains off-diagonal elements that are chosen precisely to prevent anisotropic expansion. 
The fine-tuning required for this cancellation to occur makes this approach less compelling from a theoretical perspective and requires the introduction of non-standard sources of stress energy.

Following <cit.>, in this work we take the opposite approach of using known sources of isotropic stress energy and allowing anisotropic expansion.
Here our isotropic stress energy contains perfect fluids in the form of dust (pressureless with energy density ρ) and a cosmological constant (with Λ = 8π Gρ_Λ and p_Λ = -ρ_Λ).
With this content the stress-energy tensor takes the form

    T^μ_ν = ρ u^μ u_ν - Λ/8 π Gδ^μ_ν,

where u^μ is the 4-velocity of the fluid. 
Working in the co-moving frame where u^μ = (1, 0,0,0) yields the diagonal stress energy tensor commonly studied in ΛCDM cosmology

    T^μ_ν = (-ρ - Λ/8 π G, - Λ/8 π G, - Λ/8 π G, - Λ/8 π G) .


In general, the spatial metric of the Thurston geometries has the form

    Σ_3^2 = γ_i j x^i  x^j,

so that the spacetime metric is given by

    s^2 = - t^2 + γ_k ℓα^k_iα^ℓ_j x^i  x^j,

where we allow for (potentially independent) scale factors through the diagonal matrix

    α^k_i = (a_1(t), a_2(t), a_3(t))^k_i.

With anisotropic expansion introduced, the Einstein field equations for the stress energy tensor (<ref>) are quite similar among the five anisotropic Thurston geometries. The diagonal elements of their Einstein field equations all have the form

    
    ^0_0    :     ȧ_1ȧ_2/a_1 a_2       + ȧ_2ȧ_3/a_2 a_3       + ȧ_3ȧ_1/a_3 a_1       + Δ^0_0         =        Λ+ k^(0)κ/a_dom^2 + 8 πG ρ, 

    ^1_1    :     ä_2/a_2       + ä_3/a_3       + ȧ_2ȧ_3/a_2 a_3       + Δ^1_1         =        Λ+ k^(1)κ/a_dom^2, 

    ^2_2    :     ä_3/a_3       + ä_1/a_1       + ȧ_3ȧ_1/a_3 a_1       + Δ^2_2         =        Λ+ k^(2)κ/a_dom^2, 

    ^3_3    :     ä_1/a_1       + ä_̈2̈/a_2       + ȧ_1ȧ_2/a_1 a_2       + Δ^3_3         =        Λ+ k^(3)κ/a_dom^2. 
The dependence of (<ref>) on the choice of geometry lies in the term proportional to κ, where the constants k^(μ) and accompanying scale factor a_dom are listed in Table <ref>, and in extra contributions Δ^μ_ν to the Einstein tensor.
We define Δ^μ_ν to include all of the off-diagonal elements of the Einstein tensor, meaning the accompanying off-diagonal field equations are, given our insistence on a diagonal stress-energy tensor, simply

    Δ^μ_ν = 0.




The evolution of the scale factors are constrained by (<ref>), since the off-diagonal elements of Δ^μ_ν will not necessarily be zero if all of the scale factors evolve independently.
We will show that the solutions to the diagonal field equations (<ref>) equate certain pairs of scale factors in a way that satisfies (<ref>) in the limit that the spatial curvature is small, which is mandated by the Universe being approximately flat on large scales.
To this end, we may define an average scale factor as the geometric mean

    A(t) ≡[ a_1(t) a_2(t) a_3(t) ]^1/3,

the Hubble parameter associated with it

    H(t) ≡Ȧ/A = 1/3( ȧ_1/a_1 + ȧ_2/a_2 + ȧ_3/a_3),

and the dimensionless curvature fraction

    ≡k^(0)κ/3 H_0^2,

where H_0≡ H(t_0) is the current (t=t_0) value of the Hubble expansion parameter.
More precisely, then, the small curvature limit corresponds to ||≪ 1, ensuring the terms proportional to κ on the right-hand side of (<ref>) are subdominant.

With this, taking linear combinations of the spatial equations in <ref>, and following <cit.> we can expand the individual scale factors in powers  to find

    a_i(t) = A(t) [ 1 +  K^(i) F(t) + (^2) ],

where 

    K^(1)   ≡k^(2) + k^(3) - 2 k^(1)/k^(0) , 
    
        K^(2)   ≡k^(3) + k^(1) - 2 k^(2)/k^(0) , 
    
        K^(3)   ≡k^(1) + k^(2) - 2 k^(3)/k^(0)

are given in <ref> and

    F[A(t)] ≡2/5Ω_m∫_A(t_0)^A(t) (1/2, 5/6; 11/6; -a'^3 Ω_Λ/Ω_m)/√(1 + a'^3 Ω_Λ/Ω_m)  a'  .

Here  is the hypergeometric function, the integral is from the time  t_0, and we have made the standard definitions

    Ω_m ≡8π G ρ(t_0)/3 H_0^2 Ω_Λ≡Λ/3 H_0^2.




While these same solutions were obtained in <cit.> and we have followed a similar procedure, there is an important difference.
Namely, in that work, Δ^μ_ν was set to zero by equating scale factors as necessary in each geometry before solving the field equations.  
In U(ℍ^2) and Nil, this meant imposing the restriction a_1 = a_2 = a_3 and a_2 = a_3, respectively.
However, this is inconsistent with (<ref>) given the values of K^(i) in <ref> which indicates that a_1 = a_2 ≠ a_3 to order  in all geometries.
To avoid this contradiction we instead use (<ref>) along with the K^(i) from <ref> to determine which scale factors to equate.
With these solutions, we find that Δ^μ _ν  still vanishes to order  across all geometries. 
In , , and Solv, this is because the only non-trivial entries of Δ^μ _ν exhibit the proportionality

    Δ^0_1 = -a_1^2 Δ^1_0∝(ȧ_̇1̇/a_1 - ȧ_̇2̇/a_2),

which vanishes by the equivalence of a_1 and a_2 to this order.
In the remaining two geometries,  and Nil, the entries of Δ^μ_ν are less obviously zero to working order and take the form

    Δ^μ _ν = √()f(a,b) .

Here f(a,b) contains differences of scale factors a_1 = a_2 ≡ a and a_3 ≡ b.
Thus f must be at least (√()) since it must disappear in the flat limit, i.e., when → 0.
In practice f is found to be () as shown in Appendix <ref>, where more detailed expressions for (<ref>) are listed. 
We proceed knowing that we may neglect Δ^μ _ν to working order in .

Turning to the evolution of the average scale factor A(t), we may square <ref> to find

    H^2
        = 1/9∑_i=1^3 ( ȧ_i/a_i)^2 + 2/9(Λ + k^(0)κ/a_dom^2 + 8π Gρ),

where we have used Δ^μ_ν=0 to the required order and have used the ^0_0 equation from (<ref>) to simplify the second term.
It follows from (<ref>) that

    ∑_i=1^3 ( ȧ_i/a_i)^2 
        = 3 H^2 +  q(t) ∑_i K^(i) + (^2).

for a known function q(t).
The form of q(t) is unimportant since from (<ref>) 

    ∑_i=1^3 K^(i) = 0 ,

independent of geometry.
Combining (<ref>) and (<ref>), we obtain the Friedmann equation

    H^2 = ( Ȧ/A)^2 = H_0^2 ( Ω_m/A^3 + Ω_Λ + /A^2) + (^2).

Expanding in powers of 

    A(t) = A^(0)(t) +  A^(1)(t) + (^2),

and substituting into (<ref>) we find the desired evolution.
As expected, the zeroth order term is the FLRW expansion factor for a matter and cosmological-constant-dominated universe

    A^(0)(t) = ( Ω_m/Ω_Λ)^1/3sinh^2/3(3√(Ω_Λ)/2 H_0 t ).

We can also obtain an explicit form for  A^(1)(t), however, it will not be needed in subsequent calculations.
Finally, combining these results with <ref> we arrive at the fully expanded time evolution for the scale factors

    a_i(t) = A^(0)(t) + [ A^(1)(t) + K^(i) A^(0)(t) F(A^(0)(t)) ] + (^2).


We have thus demonstrated that homogeneous spacetimes with anisotropic Thurston geometries are able to evolve under a perfect fluid stress energy to  𝒪() when multiple scale factors are introduced.
The expansion, combined with geometric anisotropy, induces angular dependence in cosmological observables and distance measures.
In particular, a completely isotropic CMB at last scattering will be observed to be anisotropic by a post-last-scattering observer, such as ourselves.
In the following sections, we show that these fluctuations allow us to put strong bounds on  should our Universe possess these geometries.




§  AND 


We begin by constraining the curvature in spacetimes with  and  spatial geometries, which have the metric

    s^2 = - t^2 + a(t)^2 [χ^2 + S^2_κ(χ) ϕ^2 ]+ b(t)^2  z^2,

where we use the identification a_1(t) = a_2(t) ≡ a(t) and a_3(t) ≡ b(t) in each geometry found in Section <ref>.

Recall for  that

    S_κ(χ) = sinh(√(-κ)χ)/-κ.

Since κ has dimensions of inverse length squared, to expand the metric we require √(-κ)χ≪ 1.
Recalling (<ref>) we have

    √(-κ)χ = √(3 ) H_0 χ≪ 1.

To simplify notation here and throughout we will write all distances in units of H_0^-1.
With this, for  and  we have

    S_κ(χ) = sin(√(-3 )χ)/√(-3),     < 0, 
    sinh(√(3 )χ)/√(3),     > 0.


The photons emitted during recombination travel along null geodesics of the spacetime, which we compute in Section <ref>. 
These trajectories are warped by anisotropic curvature and spatial expansion, meaning that the local photon fluxes at recombination and today computed in Section <ref> are different. 
A present-day observer interprets this photon flux as that of a blackbody with a temperature that is a function of solid angle in the sky, which we confront against the observed CMB angular power spectrum to derive bounds on  in Section <ref>.



 §.§ Null geodesics


We start by computing the trajectories x^μ(λ) = (t(λ), χ(λ), ϕ(λ), z(λ)), parameterized by the affine parameter λ, of photons emitted at recombination that are later detected by an observer at the present time t(λ_0) = t_0 located at the origin of our coordinate system.
Such trajectories obey the geodesic equation

    ^2 x^μ/λ^2 + Γ^μ_αβ x^α/λ x^β/λ = 0.

Much like spherical symmetry allows us to impose that CMB photons follow purely radial geodesics in FLRW spaces, rotational symmetry within the (χ, ϕ)-plane in  and  allows us to set ϕ/λ≡ϕ'(λ) = 0. 
Upon doing this, the geodesic equations for the remaining spatial coordinates read

    χ” + 2a'/aχ' = 1/a^2/λ( a^2 χ' )    = 0, 
    
        z” + 2b'/bz' = 1/b^2/λ( b^2 z' )    = 0,

which of course are solved by

    
    χ'(λ) = c_χ/a(t(λ))^2 

    z'(λ) = c_z/b(t(λ))^2 

where c_χ and c_z are constants.
By demanding the geodesics obey the null condition

    g_αβ x^α/λ x^β/λ = 0,

one recovers the first order equation 

    t'^2 = (a χ')^2 + (b z')^2,

which can in principle be solved for t(λ). 
However, this derivative expression is sufficient for the subsequent analysis based on the 4-velocities of incoming photons. 




 §.§ CMB photon flux



With knowledge of these geodesics, we turn our attention to the distortions induced in the CMB. 
To do this, we first examine the local flux of CMB photons propagating through a point in space at the time of
recombination, i.e., on the last scattering surface,

    Φ(E_r) =  N_r(E_r)/Ω_r  A_r t_r  E_r,

where N_r (E_r) is the number of photons with energies between E_r and E_r +  E_r received within a solid angle Ω_r = (cosθ_r) ϕ_r in an area A_r during a time t_r. 
As these photons propagate from the last scattering surface through an expanding anisotropic geometry, they experience a redshift that will be a function of their direction of propagation, and thus as a function of the direction on the sky from which an observer receives them. 
The flux of CMB photons observed today will differ from that present at recombination due to this effect. 
In particular, an isotropic flux at recombination will be observed today as anisotropic.
This present-day flux can be obtained by performing a change of coordinates from A_r, t_r, and E_r at recombination to A_0, t_0, and E_0, today.
The effect on the flux is captured through the appropriate Jacobian factor, which simplifies to

    Φ(E_0) = Φ(E_r) (Ω_r/Ω_0) ( A_r  t_r/ A_0  t_0) ( E_r/ E_0),

where the factorization of the Jacobian can be trivially confirmed.
To express this flux in a more meaningful way, we must relate the geometric and energy elements at the time observation today with those at recombination.
First, we identify θ as the angle between an incoming photon's trajectory with respect to the z-axis as measured by a local observer.
This is obtained by projecting a photon's 3-velocity v^i(λ) = (χ'(λ), ϕ'(λ), z'(λ)) onto the unit vector pointing in the z-direction,

    cosθ =
        g_iz v^i/√(g_i _j v^i v^j)√(g_zz),

where g_i_j is the spatial part of the metric.
For  and , this reads

    cosθ = b z'/√((a χ')^2 + (b z')^2) = [ 1 + (a χ'/b z')^2]^-1/2.

since ϕ'(λ) = 0.

To evaluate this angle we first use the solutions to the spatial geodesic equations (<ref>) to write

    cosθ = [ 1 + (b/a)^2 (c_χ/c_z)^2 ]^-1/2 .

From the evolution of the scale factors (<ref>) and the fact that (from <ref>) F(λ_0) = F[A^(0)(t_0)]=0, we have that a(t_0) = b(t_0) so that today

    cosθ_0 = [ 1 + (c_χ/c_z)^2 ]^-1/2.

To evaluate cosθ at recombination we, use the K^(i) from <ref> to find that

    (b(t_r)/a(t_r))^2 ≈(A(t_r)[ 1 + 2  F(λ_r)]/A(t_r)[ 1 -  F(λ_r)])^2 ≈ 1 + 6  F(λ_r),

where ≈ denotes equivalence to the displayed order in , which varies between geometries.  For  and , that is to ().
Substituting this into (<ref>) and using (<ref>) to replace the constants we find

    cosθ_r ≈cosθ_0 [ 1 + 3  F(λ_r) sin^2θ_0 ].

Along a radial geodesic, ϕ_r = ϕ_0, so the solid angle Jacobian factor is found to be

    Ω_r/Ω_0 =  (cosθ_r)/ (cosθ_0)≈ 1 - 3  F(λ_r) ( 1 - 3 cos^2θ_0 ) = 1 + 3/2 F(λ_r) ( 1 + 3 cos 2θ_0 ).

The remaining geometric factor arises from the expansion of space and is most easily written in terms of the average scale factor

    A_r  t_r/ A_0  t_0 = [ A(λ_r)/A(λ_0)]^3 .


The energy of a photon measured by a comoving observer with u^μ = (1,0,0,0) is

    E(λ) = |u_μ x^μ/λ| = t'(λ).

From the null condition we know the time derivative (<ref>) so that

    E(λ)^2 = (a χ')^2 + (b z')^2 = c_z^2/b^2[ 1 + (b/a)^2 ( c_χ/c_z)^2 ] = c_z^2/b^2 cos^2θ,

where we have employed (<ref>) in the final equality.
With this, using expressions from above, and simplifying we can show that

    ( E(λ_r)/E(λ_0))^2 ≈[ A(λ_0)/A(λ_r)]^2 [ 1 -  F(λ_r) (1 + 3cos 2θ_0) ]

Finally, this leads to

    E_r/ E_0   ≈A(λ_0)/A(λ_r)[ 1 - 1/2 F(λ_r) (1 + 3cos 2θ_0) ] 
       ≈A(λ_0)/A(λ_r)(  Ω_r/Ω_0)^-1/3,

where we have used (<ref>) to write this in a more suggestive form to simplify subsequent calculations.

The CMB photon flux at the last scattering surface is that of a blackbody with a uniform temperature T_r:

    Φ(E_r) = p E_r^2/exp(E_r/T_r) - 1,

where  p≡1/(2π^2) in natural units (ħ=c=k=1).
To determine the photon flux today we define the temperature today using the average scale factor as

    T_0/T_r = A(λ_r)/A(λ_0) .

Putting all of this together the flux today (<ref>) becomes

    Φ(E_0) = p E_0^2/exp(E_r/T_r ) - 1(Ω_r/Ω_0) ( A_r  t_r/ A_0  t_0) ( E_r^2  E_r/E_0^2  E_0).

Perhaps surprisingly the energy ratio and the Jacobian factors completely cancel, i.e., from (<ref>), (<ref>), and (<ref>) we see that

    (Ω_r/Ω_0) ( A_r  t_r/ A_0  t_0) ( E_r^2  E_r/E_0^2  E_0) ≈ 1.

This means the observed photon flux today is that of a blackbody, albeit with a direction-dependent temperature –
using (<ref>) and (<ref>),

    E_r/T_r = E_0/T_0( E_r/E_0) ( T_0/T_r) ≈E_0/T_0[ 1 - 1/2 F(λ_r) (1 + 3 cos 2θ_0) ] .

Thus the photon flux today in  and  is given by

    Φ(E_0) ≈p E_0^2/exp[E_0/T(Ω_0)] - 1,

where

    T(Ω_0) ≈ T_0 [ 1 + 1/2 F(λ_r) (1 + 3 cos 2θ_0) ] .




 §.§  constraint



The angular dependence of T(Ω_0) means that an isotropic temperature at recombination appears anisotropic to an observer today. 
It is therefore convenient to expand T(Ω_0)  in spherical harmonics.
For (<ref>) this is straightforward since the angular dependence of the perturbation around the mean T_0 is manifestly

    Δ T(Ω_0) ≡ T(Ω_0) - T_0 ≈ a_20 Y_20(Ω_0),

where

    a_20≈
         4 √(π/5) T_0 F(λ_r) .

For Ω_m=0.3, Ω_Λ=0.7, A^(0)(λ_0)=1, A^(0)(λ_r)≡ 1 / (1+z_r), and z_r = 1090 we find F(λ_r) ≈ -1.04 from direct integration of (<ref>).
The fluctuations Δ T(Ω)/T_0 that would be observed today are plotted in <ref> in units of .
The sign of the quadrupole amplitude is opposite  in   (< 0) and  (> 0).  
We also note that taking the Λ→ 0 limit of this result (i.e., in (<ref>)),  changes the form of F(λ_r) and yields the expression for the observed CMB temperature in matter-dominated Bianchi III spacetimes derived in <cit.>.



CMB temperature anisotropies have been measured on large scales (most reliably by the  WMAP <cit.> and Planck <cit.> teams).
To be consistent with these observations the expansion-induced anisotropy must be sufficiently small.
The CMB power spectrum is defined as

    D_ℓ≡ℓ(ℓ+1)/2π C_ℓ         C_ℓ = 1/2ℓ + 1∑_m=-ℓ^ℓ |a_ℓ m|^2.

In this case, only D_2 is non-zero, 

    D_2 ≈48/25 F(λ_r)^2 ^2 T_0^2.

The observed D_2 reported in the most recent Planck data release <cit.> is 

    D_2^obs = 225.9 μ K^2.

A limit is obtained on  by ensuring that the power in the induced quadrupole does not exceed that of the observed one,

    || ≲5/4 √(3) |F(λ_r)|√(D_2^obs)/T_0≈ 3.8 × 10^-6

for  and  spacetimes where we have chosen T_0=2.726K <cit.>.
This is considerably more stringent than the = 0.001 ± 0.002 bound reported by Planck <cit.> for the isotropic FLRW spaces.



§  AND NIL


In the following sections, we will repeat the procedure outlined in Section <ref> for the remaining three anisotropic Thurston spacetimes, starting with the spacetime of . 
This section also contains the constraint on  in Nil since its metric is extremely similar to that of  to the order at which we work.



 §.§ 


As in the previous section, the metric for  can be written in terms of the dimensionless parameter .
Using <ref> we have

    -κ = -3/k^(0) H_0^2  = 12/5 H_0^2 .

Writing all coordinates in units of H_0^-1 we find

    s^2    = - t^2 + a(t)^2  x^2 + 
              + b(t) [ a(t)cosh(4 √(3/5)√() x )  y^2 + b(t)  z^2 + 2 a(t) sinh(2√(3/5)√() x )  y z ].

where we have used the identification a_1(t) = a_2 ≡ a(t) and a_3(t) ≡ b(t).
This identification ensures consistency with an isotropic stress energy to order √().
Expanding the metric to this order and using (<ref>) we see that the scale factors can be replaced with A^(0)(t) leading to

    s^2 ≈ - t^2 + A^(0)(t)^2 [ x^2 +  y^2 +  z^2 + 4 √(3/5)√() x  y z],

where now ≈ denotes equivalence to order √().

The equations for the null geodesics of this spacetime to order √() are

    1/A^(0)(t)^2/λ[A^(0)(t)^2 x'(λ)] - 2√(3/5)√() y'(λ) z'(λ)    ≈ 0, 
    1/A^(0)(t)^2/λ[A^(0)(t)^2 y'(λ)] + 2√(3/5)√() z'(λ) x'(λ)     ≈ 0, 
    1/A^(0)(t)^2/λ[A^(0)(t)^2 z'(λ)] + 2√(3/5)√() x'(λ) y'(λ)    ≈ 0,

accompanied by a null condition (<ref>)

    t'(λ)^2 ≈ A^(0)(t)^2 [x'(λ)^2 + y'(λ)^2 + z'(λ)^2 + 4√(3/5)√() 
     x(λ) y'(λ)z'(λ)] .


The first order solutions in √() to (<ref>–<ref>) can be directly integrated from

    x'(λ)    ≈1/A^(0)(λ)^2[c_x + 2√(3/5) c_y c_z √() S(λ) ], 
    
        y'(λ)    ≈1/A^(0)(λ)^2[c_y - 2√(3/5) c_x c_z √() S(λ) ], 
    
        z'(λ)    ≈1/A^(0)(λ)^2[c_z - 2√(3/5) c_x c_y √() S(λ) ], 
    
        t'(λ)    ≈1/A^(0)(λ),

where
v(λ_0) = 1 requires c_x^2+c_y^2+c_z^2=1, A^(0)(λ) ≡ A^(0)(t(λ)), 

    S(λ)    ≡ H_0 ∫_λ_0^λλ'/A^(0)(λ')^2 = ∫_1^A^(0)(λ) a'/a'^2 √(Ω_m / a'^3 + Ω_Λ)
       = 1/√(Ω_Λ)[  (1/3, 1/2; 4/3; -Ω_m/Ω_Λ) - 1/A^(0)(λ) (1/3, 1/2; 4/3; -Ω_m/A^(0)(λ)^3 Ω_Λ)],

and we have chosen A^(0)(λ_0)=1.

To compute the local photon flux (<ref>) received by a present-day observer at the origin, we need to define local angular coordinates relating to solid angles in the sky. 
For this geometry, we again identify θ as the angle from the z-axis, permitting the use of (<ref>) from the previous section.
Writing the photon 3-velocity as v^i(λ) = (x'(λ), y'(λ), z'(λ)) and expanding we find

    cosθ≈z'(λ) [ x'(λ)^2 + y'(λ)^2 + z'(λ)^2 ] + 2√(3/5)√() x(λ) y'(λ) [ x'(λ)^2 + y'(λ)^2 ]/[ x'(λ)^2 + y'(λ)^2 + z'(λ)^2 ]^3/2.

The velocities are given in (<ref>) and the x equation integrates to

    x(λ) ≈ c_x S(λ).

Plugging all of this in we find the very simple expression

    cosθ≈c_z/√(c_x^2 + c_y^2 + c_z^2),

which is a constant.
In other words, photons propagate along lines of constant θ so that cosθ_0 ≈cosθ_r.

Unlike the previous geometries, however, the polar angle ϕ measured at recombination and today will differ since  lacks rotational symmetry in its xy-plane.
We choose to identify ϕ as the angle from the x-axis within the xy-plane, meaning that

    tanϕ = g_i_yv^i /g_i_xv^i√(g_x_x/g_y_y).

Proceeding as above, expanding, and plugging in the velocities from (<ref>) we find

    tanϕ≈y'(λ)/x'(λ) + 2 √(3/5)√() x(λ) z'(λ)/x'(λ)≈c_y/c_x - 2√(3/5)√()( c_y/c_x)^2 c_z S(λ).

Evaluating (<ref>) and (<ref>) at λ_0 where S(λ_0)=0 we find that these constants take the simple form of spherical coordinates:

    (c_x, c_y, c_z) = (sinθ_0 cosϕ_0, sinθ_0 sinϕ_0, cosθ_0).

Evaluating at recombination gives

    cosθ_r    ≈cosθ_0, 
    tanϕ_r    ≈tanϕ_0 ( 1 - 2 √(3/5)√() S(λ_r) cosθ_0 tanϕ_0 ),

from which we determine the Jacobian factor

    Ω_r/Ω_0≈ 1 - 2 √(3/5)√() S(λ_r) cosθ_0 sin 2ϕ_0.

From (<ref>), we know that the ratio of photon energies measured by observers today and at recombination is trivially

    E(λ_r)/E(λ_0)≈A^(0)(λ_0)/A^(0)(λ_r)

since spatial expansion is isotropic to order √(). 
Similarly,

    A_r  t_r/ A_0  t_0≈[A^(0)(λ_r)/A^(0)(λ_0)]^3.

Thus the photon flux measured by a present-day observer only receives nontrivial modifications from (<ref>) and can be written as

    Φ(E_0) ≈p E_0^2(1 - 2 √(3/5)√()S(λ_r) cosθ_0 sin 2ϕ_0 )/exp(E_0 / T_0) - 1

using (<ref>).

This flux is qualitatively different than the flux (<ref>) for  and . 
Namely, an O(√())  correction appears as a direction-dependent but energy-independent factor, i.e., as a greybody factor to the entire expression, rather than as a  direction-dependent factor on the temperature.
The exponential term in the denominator is unmodified to this order. 
In this case, we cannot simply read off the observed CMB temperature from the exponent, and an effective temperature may instead be obtained from the total observed intensity I integrated over all energies:

    I = ∫_0^∞ E_0  E_0 Φ(E_0).

For a perfect blackbody of temperature T, this evaluates to

    I = pπ^4/15T^4,

which is just the Stefan-Boltzmann law in our choice of units.
With the anisotropic flux (<ref>), the total observed intensity is

    I ≈pπ^4/15T_0^4 (1 - 2 √(3/5)√() S(λ_r) cosθ_0 sin 2ϕ_0 ) .

Comparing (<ref>) and (<ref>), the effective CMB temperature can be identified as

    T(Ω_0) ≈ T_0(1 - 1/2√(3/5)√() S(λ_r) cosθ_0 sin 2ϕ_0 )

to order √().
For Ω_m = 0.3, Ω_Λ = 0.7, A_0(λ_r) = 1 / (1+z_r), and z_r=1090 numerical evaluation of (<ref>) gives S(λ_r) ≈ -3.19.

As in Section <ref>, we can decompose the observed temperature into spherical harmonics to constrain . 
This decomposition for (<ref>) is more involved than in the previous section because the induced fluctuations cannot be represented with finitely many harmonics.
We therefore save the details of this decomposition for <ref>, and quote the results:

    T(Ω) ≈ T_0 { 1 +  √(3π/5)√() S(λ_r) ∑_ℓ = 3
    ℓ odd^∞√(2ℓ+1)√(%s/%s)(ℓ - 2)!(ℓ + 2)![ Y_ℓ  2(Ω) - Y_ℓ  -2(Ω) ] }.

The corresponding normalized temperature fluctuations, Δ T(Ω_0), are shown in Fig. <ref> in units of √().



The resulting octopole has the largest amplitude of the induced modes, and the constraint on  that follows from the induced octopole is

    ≲50/3 S(λ_r)^2D_3^obs/T_0^2≈ 2 × 10^-10

where we have used D_3^obs =936.9μ K^2 again from Planck's best-fit angular power spectrum <cit.>. 
Note that a much stronger constraint is reached here than for  and  because modifications to the photon flux are introduced at 𝒪(√()) instead of 𝒪().

The astute reader will note that the induced temperature fluctuations extend to arbitrarily high ℓ, with D_ℓ∝ℓ^-1, whereas the conventional damping tail of CMB temperature falls as exp(-ℓ^2/ℓ_D^2) (for some cosmological-parameter dependent value of ℓ_D).
This implies that, above some threshold ℓ, the induced D_ℓ will exceed the usual predicted values, and ever-stricter limits on  will result form measurements of the CMB above that threshold. 
For the Plank best-fit FLRW cosmological parameters, that threshold is at ℓ>4000, and thus somewhat above the highest ℓ that have been measured.



 §.§ Nil

The constraint on curvature of the Nil spacetime largely mirrors that of . 
Here

    -κ = -3/k^(0) H_0^2  = 12 H_0^2 ,

a factor of 5 larger than in , so expressing all coordinates in units of H_0^-1 the metric is 

    s^2 = - t^2 + a(t)^2  x^2 + b(t) [a(t) (1 + 12 x^2)  y^2 + b(t) z^2 - 4√(3)a(t) √() x  y z ].

Though this differs from the metric <ref> for , expanding to order √() 

    s^2 ≈ - t^2 + A^(0)(t)^2 [ x^2 +  y^2 +  z^2 - 4√(3)√() x  y z] .

This is the same as the O(√())  metric <ref> for , with the  off-diagonal term flipped in sign and multiplied by √(5).
It is unsurprising, then, that the modes induced in the observed CMB temperature strongly resembles those in ,

    T(θ_0, ϕ_0) ≈ T_0[1 - √(5)√() S(λ_r) ∑_ℓ = 3
    ℓ odd^∞α_ℓ (Y_ℓ  2 - Y_ℓ  -2) ].


The  constraint in the Nil spacetime is

    ≲10/3 S(λ_r)^2D_3^obs/T_0^2≈ 4 × 10^-11,

which is stronger than the  constraint in  by the expected  factor of 5,  the ratio of the k^(0)'s between the two spacetimes.








§ SOLV


The last anisotropic Thurston spacetime is Solv with metric given by 

    s^2 = - t^2 + a(t)^2 [ ^2z√(-κ) x^2 + ^-2z√(-κ) y^2 ] + b(t)^2  z^2,

where we have identified a_1(t) = a_2(t) ≡ a(t) and a_3(t) = b(t).
As in the other geometries, we rewrite κ in terms of  and write the coordinates in units of H_0^-1 to express the metric as

    s^2 = - t^2 + a(t)^2 [ ^2√(3) z √() x^2 + ^-2√(3) z √() y^2 ] + b(t)^2  z^2,



As in  and Nil,  appears with half-integer powers in the metric.
It is natural to guess, then, that modifications to the CMB photon flux will appear at 𝒪(√()), but this is not the case.
To 𝒪(√()), Solv experiences isotropic expansion, and no distortions to the local photon flux are induced. 
Therefore, to see effects in Solv it proves necessary to work to ().
Also in contrast to  and Nil general progress can be made without immediately expanding in powers of √().
The geodesics of (<ref>) are given by

    1/a(t)^2/λ[ a(t)^2 x'(λ)] + 2√(3)x'(λ)z'(λ) = 0, 
       1/a(t)^2/λ[ a(t)^2 y'(λ)] - 2√(3)y'(λ)z'(λ) = 0, 
       1/b(t)^2/λ[ b(t)^2 z'(λ)] - √(3)( a(t)/b(t))^2 [^2√(3) z(λ) √() x'(λ)^2-^-2√(3) z(λ) √()y'(λ)^2]  = 0,

along with a null condition (<ref>) 

    t'(λ)^2 = a(t)^2 [^2√(3) z(λ) √() x'(λ)^2 + ^-2√(3) z(λ) √() y'(λ)^2] + b(t)^2 z'(λ)^2.

The spatial equations are solved exactly to give velocities along null geodesics

    x'(λ)    = c_x ^-2√(3) z(λ) √()/a(λ)^2, 
    
        y'(λ)    = c_y ^2√(3) z(λ) √()/a(λ)^2, 
    
        z'(λ)    = c_z/b(λ)^2 + √(3)c_x x(λ) - c_y y(λ)/b(λ)^2,

which can be substituted into  (<ref>) to find t'(λ).

In deriving the CMB photon flux today, we make the same definitions of angular coordinates (θ,ϕ) as above, permitting the use of (<ref>) and (<ref>).
Using these expression we find that  θ_r and ϕ_r depend on θ_0 and ϕ_0 as

    cosθ_r    ≈cosθ_0 + √(3) S(λ_r) sin^2θ_0 cos 2ϕ_0 - 3 [ 2 F(λ_r) + S(λ_r)^2 ] cosθ_0 sin^2θ_0 , 
    tanϕ_r    ≈tanϕ_0 [ 1 + 2√(3Ω_k) S(λ_r) cosθ_0 + 3 S(λ_r)^2 ( 2 cos^2θ_0 + sin^2θ_0 cos 2ϕ_0 ) ] ,

where ≈ denotes equivalence to (), F(λ) is defined in (<ref>), S(λ) is defined in (<ref>), and we have used the fact that

    H_0 ∫_λ_0^λS(λ') dλ'/A^(0)(λ')^2 = 1/2 S(λ)^2.

The determinant of the Jacobian matrix  gives

    Ω_r/Ω_0≈ 1 + 3 (1 + 3cos 2θ_0) F(λ_r) .

As noted above the leading order correction is ().
The remaining geometric factor is

    A_r  t_r/ A_0  t_0 = a(t_r)^2 b(t_r)/a(t_0)^2 b(t_0)≈(A^(0)(λ_r)/A^(0)(λ_0))^3 [ 1 + 3 A^(1)(λ_r)/A^(0)(λ_r)] .

Similarly the energy factor that appears is

    E_r^2  E_r/E_0^2  E_0≈(A^(0)(λ_0)/A^(0)(λ_r))^3 { 1 - 3 [ (1 + 3 cos 2θ_0) F(λ_r) + A^(1)(λ_r)/A^(0)(λ_r)] }.



Finally, combining the factors (<ref>), (<ref>), and (<ref>), we find they all cancel in the same, perhaps surprising manner as in Section <ref>.
This leads to the flux

    Φ(E_0) ≈p E_0^2/exp(E_r/T_0) - 1,

where

    T(Ω_0) ≈ T_0[1+  (1+3cos 2θ) F(λ_r)].

From this we can write the temperature anisotropy induced in Solv as

    Δ T(Ω_0) ≡ T(Ω_0) - T_0 ≈ T_0  (1 + 3cos2θ)F(λ_r),

twice that found in <ref> for .
Decomposed into spherical harmonics we again see only a Y_20(Ω) contribution

    Δ T(Ω)/ T_0≈ 8  √(π/5)Ω_K F(λ_r)Y_20(Ω_0),

which is plotted in <ref> in units of .
For z_r=1090, Ω_m=0.3, and Ω_Λ=0.7 we have seen that F(λ_r) ≈ -1.04 and S(λ_r) ≈ -3.19.
Finally, the bound on  is a factor of 2 stronger than that found for ,

    ≲5/8√(3) |F(λ_r)|√(D_2^obs)/T_0≈ 1.9× 10^-6.






§ CONCLUSION


In this paper, we have, for the first time, derived powerful constraints on   in spacetimes with homogeneous anisotropic spatial geometries of all five corresponding Thurston types – S^2 ×, ℍ^2 ×, U(ℍ^2), Nil, and Solv – when filled with standard homogeneous perfect-fluid dust and cosmological constant Λ.
We have shown that, in S^2 × and ℍ^2 ×, CMB photons undergo direction-dependent redshift as they propagate from recombination through anisotropically expanding space. 
In U(ℍ^2) and Nil, the underlying anisotropy of the three-geometry distorts the present-day local flux of CMB photons. 
An observer could interpret these effects as a blackbody with an anisotropic temperature, with an amplitude of anisotropy proportional to a power of  √(||). 
We find that, in order for these temperature fluctuations to be no larger than the observed CMB anisotropies, || ≲ 10^-5 in  and  ,
≲ 10^-6 in Solv, and ≲ 10^-10 in  and Nil. 

We have not commented on the impact of primordial perturbations on the observed CMB temperature. 
Though it is theoretically possible for this anisotropy to be exactly canceled out by temperature fluctuations existing at the time of last scattering, it is highly unlikely that we live just at the moment when such cancellation occurs.

We emphasize that these constraints are valid for a Universe filled with perfect fluid dust and cosmological constant, but do not necessarily hold for less conventional stress-energy contents. 
For instance, it has been shown that the CMB temperature maintains its isotropy in certain shear-free models sourced by a finely-tuned anisotropic fluid <cit.>. 
We note however that the angular diameter and luminosity distances are still direction-dependent in the shear-free renditions of anisotropic Thurston spacetimes <cit.>, so it still may be possible to constrain anisotropy less stringently using other cosmological observables (e.g., see <cit.>).

Our powerful constraints come from comparison of observed CMB temperature fluctuations with 
the direction-dependent photon flux that results from the evolution of an isotropic homogeneous blackbody distribution at the time of recombination into an anisotropic photon distribution today.
These constraints are therefore independent of the  physics responsible for  the usual  primordial fluctuations.


We thank Yashar Akrami for early conversations identifying the questions addressed in this paper, and thank Johanna Nagy and John Ruhl for useful conversations about CMB measurements.
A.F.S. was partially supported by NASA ATP Grant RES240737;
G.D.S. by DOE grant DESC0009946. 






§ Δ MU NU IN  AND NIL



Here we list out the entries of Δ^μ _ν in  and Nil.
For μ, ν∈{0,1} 

    Δ^0_0     =  -Δ^1_1     =    -x^23/4k^(0)[ȧ/a - ḃ/b ]^2 + (^2) 

    Δ^0_1     = -a^2Δ^1_0     =     - x 3/2 k^(0)[ȧ/a -ḃ/b ] + (^2),

where we have set a_1 = a_2 = a and a_3 = b. 

The remaining nontrivial elements of Δ^μ _ν are 

    Δ^2_2     =       -x^23 /4k^(0) [(ȧ/a)^2 -(ḃ/b)^2 + 2ä/a - 2b̈/b ]    + (^2) 

    Δ^2_3     =       ζx√(-3/4k^(0))(b/a ) [ - ȧḃ/ab + (ḃ/b)^2 - ä/a + b̈/b  ]    + (^3/2)

    Δ^3_2     =       ζx√(-3/4k^(0))(a/b ) [2(ȧ/a)^2 - 3̇ȧḃ/ab + (ḃ/b)^2 + ä/a - b̈/b  ]    + (^3/2)

    Δ^3_3     =       x^23 /4k^(0) [3(ȧ/a)^2 -4ȧḃ/ab+(ḃ/b)^2 + 2ä/a - 2b̈/b ]    + (^2),

where ζ = +1 in Nil and ζ = -1 in .

From (<ref>), we know a and b differ by an order  correction, meaning (<ref>) and (<ref>) are high enough order in  to be neglected.
We may verify the same is true for the other nontrivial elements  by using (<ref>) to rewrite (<ref>)–(<ref>) as

    Δ^2_2     =      -Δ^3_3     =      -x^23^2/2k^0(K^(1) - K^(3))(3ȦḞ/A + F̈ ) + (^2)

    Δ^2_3     =      -Δ^3_2     =      -ζx√(-3^3/4k^0)(K^(1) - K^(3))(3ȦḞ/A + F̈ ) + (^3/2),

where F = F[A(t)] is given by (<ref>).



§ TEMPERATURE FLUCTUATION AMPLITUDES IN  AND NIL



Here we derive the spherical harmonic representation of the temperature (<ref>) for .
This derivation can easily be extended to that for Nil given their similarities at 𝒪(√()).
We begin with (<ref>) written as

    T(Ω) ≈ T_0 ( 1 - 1/2√(3/5)√() S(λ_r) cosθsin 2ϕ),

where we have dropped subscripts from the angular components for brevity.
From the sin 2ϕ factor we immediately see that only harmonic modes with m=± 2 will contribute, and they will have equal amplitude and opposite sign.
Since T(Ω) is a real function we know that a_ℓ 2 = a_ℓ -2^* and thus a_ℓ 2 is imaginary. 
Thus we only need to compute m=+2.
To begin,

    a_ℓ 2≈ -1/2√(3/5)√() S(λ_r) α_ℓ,

with

    α_ℓ≡∫cosθsin 2ϕ  Y_ℓ 2^*(Ω)  Ω.

Using the standard functional form

    Y_ℓ 2(Ω) = √(2ℓ + 1/4 π)√((ℓ - 2)!/(ℓ + 2)!) P_ℓ^2(cosθ) ^ 2 ϕ,

and recalling that P_ℓ^m(-x)=(-1)^ℓ-mP_ℓ^m(x), we see that α_ℓ=0 for even ℓ.
Meanwhile, we will show below that

    ∫_-1^1 x P_ℓ^2(x)   x = 
            4 ,     ℓ .

The required integral over ϕ is

    ∫_0^2πsin 2ϕ ^-2ϕ ϕ = -π.

Thus

    α_ℓ 
        = 
        -2 √((2ℓ+1)π)√((ℓ-2)!/(ℓ+2)!)   
    
        0


Thus we find

    a_ℓ 2≈√(3 (2ℓ+1) π/5)√(%s/%s)(ℓ - 2)!(ℓ + 2)! S(λ_r) √()   
    
        0

so that

    Δ T(Ω)/√() T_0≈√(3π/5) S(λ_r) ∑_ℓ = 3
    ℓ odd^∞√(2ℓ+1)√(%s/%s)(ℓ - 2)!(ℓ + 2)![ Y_ℓ 2(Ω) - Y_ℓ -2(Ω) ],

as claimed in (<ref>).



The integral in (<ref>) can be evaluated as follows.
From the recursion relation

    x P_ℓ^2(x) = 1/2ℓ+1[ (ℓ+2) P_ℓ-1^2(x) + (ℓ-1) P_ℓ+1^2(x) ]

the required integral is

    ∫_-1^1 x P_ℓ^2(x)   x    = 1/2ℓ+1∫_-1^1[ (ℓ+2) P_ℓ-1^2(x) + (ℓ-1) P_ℓ+1^2(x) ]  x, 
       ≡1/2ℓ+1[ (ℓ+2) J_ℓ-1 + (ℓ-1) J_ℓ+1],

where we have defined for all n≥ 2

    J_n ≡∫_-1^1 P_n^2(x)   x.

This integral can be evaluated by first noting that from the definition of the associated Legendre functions in terms of the Legendre polynomials and Legendre's equation we have

    P_n^2(x) = (1-x^2) ^2 P_n(x)/ x^2 = 2 x  P_n(x)/ x - n(n+1) P_n(x).

Employing the recursion relation

    x  P_n(x)/ x =  P_n-1(x)/ x + n P_n(x)

we thus can write

    P_n^2(x) = 2  P_n-1(x)/ x - n(n-1) P_n(x).

We are left to evaluate

    J_n = ∫_-1^1[ 2  P_n-1(x)/ x - n(n-1) P_n(x) ]  x.

Orthogonality of the Legendre polynomials shows that the second term in the integrand integrates to zero (for n≥ 2).
The first term in the integrand is a total derivative.
Since P_n(-1) = (-1)^n and P_n(1)=1 we arrive at

    J_n = 2 P_n-1(x) |_-1^1 = 2 [ 1 - (-1)^n-1] =  0,     n 
     4,     n

Finally, plugging this into (<ref>) we arrive at the result quoted in (<ref>).



JHEP 


