# Hints of tensions in the cosmic microwave background temperature and
polarization quadrupoles

Jahmour J. Givans  Email<EMAIL_ADDRESS>Center for
Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY
10010, USA  Department of Astrophysical Sciences, Peyton Hall, Princeton
University, Princeton, NJ 08544, USA  Marc Kamionkowski  William H. Miller III
Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles
St., Baltimore, MD 21218, USA

###### Abstract

The large-angular-scale falloff in the autocorrelation function for the cosmic
microwave background (CMB) temperature has long intrigued cosmologists and
fueled speculation about suppressed superhorizon power. Here we highlight an
inconsistency between the temperature quadrupole and the more recently
obtained E-mode polarization quadrupole from Planck PR3. The temperature
quadrupole arises primarily at the CMB surface of last scatter, while the
polarization primarily from the epoch of reionization, but the two still probe
comparable distance scales. Although the temperature quadrupole is
intriguingly low (much greater than a $1\sigma$ fluctuation) compared with
that expected in the standard $\Lambda$CDM cosmological model, the
polarization quadrupole turns out to be somewhat high, at the $1\sigma$ level.
We calculate the joint probability distribution function for both and find a
slight tension: the observed pair of quadrupoles is inconsistent at a
$2.3\sigma$ confidence level. The problem is robust to simple changes to the
cosmological model. If the high polarization quadrupole survives further
scrutiny, then this result disfavors, at comparable significance, new
superhorizon physics. The full-sky coverage and pristine foreground
subtraction of the LiteBIRD satellite will be ideal to help resolve this
question.

††preprint: APS/123-QED

##  I Introduction

The dropoff in the cosmic microwave background (CMB) temperature
autocorrelation function at the largest angular scales has intrigued
cosmologists ever since it was first seen in the Cosmic Background Explorer
(COBE) Differential Microwave Radiometer (DMR) maps [1]. This dropoff is seen
in the correlation function at angular separations $\Delta\theta\gtrsim
60^{\circ}$. In harmonic space, it is manifest as an unusually low temperature
quadrupole ($C_{2}$) [1, 2, 3]; it is about one fifth the value expected from
extrapolation of the power spectrum ($C_{l}$) at higher multipoles $l$.

This fluctuation is nothing to lose sleep over, especially if you take into
account the look-elsewhere effect (of the thousands of multipole moments that
have been measured, you’d expect a few to stray from the curve). But on the
other hand, the quadrupole probes the largest observationally accessible
distance scales in the Universe. Maybe there’s something happening at larger,
superhorizon, distances, and this is just the tip of the iceberg? This
possibility has fueled a number of ideas for new superhorizon physics [4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], and motivates the search for other
ways to access information on these superhorizon scales.

One avenue of investigation [17, 18, 19, 20] has involved study of the
statistical significance of the dearth of large-angle correlations, the
possibility of cross-correlations between different multipole moments that
affect the interpretation. The conclusion, though, is that cosmic variance,
the sample variance that arises from the finite number of $\sim 60^{\circ}$
patches on the sky, prevents the statistical significance to rise much beyond
the $\sim 2\sigma$ level. Another avenue, though, is to seek other observables
that probe the largest distance scales and that may thus provide complementary
information. Ideas along these lines include the additional information
provided by lensing maps [21] and large-scale polarization fluctuations in the
CMB [22, 23]. The latter was considered by the Planck Collaboration in two
different ways: Ref. [20] generalized the analysis of the temperature
autocorrelation function to include temperature-polarization cross-correlation
and polarization autocorrelations—the conclusions of prior work were
essentially unchanged. Ref. [24] took all the polarization and temperature
information to test the possibility that the cosmic curvature power spectrum
might be suppressed at superhorizon scales. Even with polarization, cosmic
variance limits what can be said: the data do not call for a suppression of
power.

Our purpose here is to highlight and discuss the implications of a related
curious feature in the large-angle CMB polarization, as seen in the leftmost
side of Figs. 1 and 2 in Ref. [3], which show, respectively, the CMB
temperature and polarization power spectra (and also the temperature-
polarization cross-correlation). The first Figure shows the longstanding low
CMB temperature quadrupole, the feature that has led over the past thirty
years to the speculation that something interesting may be happening at
superhorizon scales. The second Figure, however, shows something surprising:
The polarization quadrupole is high. The occurrence of a low temperature
quadrupole and a high polarization quadrupole warrants attention. It is
noteworthy because none of the other lowest multipole moments show such a
discrepancy and even moreso if viewed as our best portal to superhorizon
physics.

We quantify the discrepancy by calculating the joint probability distribution
function (PDF) for the two quadrupoles and estimate that they are inconsistent
at a $2.3\sigma$ level, a conclusion robust to changes in the cosmological
model. Given that the temperature quadrupole has now been obtained
independently by three different satellite missions, we surmise that there may
be issues with instrumental systematics and/or foreground removal in Planck.
If not, though, it suggests that new-physics explanations for the low
temperature quadrupole are disfavored. The discrepancy highlights the
importance of LiteBIRD’s improved measurements of these quadrupoles [25].

This paper is organized as follows: in Sections II and III we review the
statistics of CMB multipole coefficients and their power spectra under the
assumptions of $\Lambda$CDM cosmology, or more generally, a cosmological model
with perturbations that are statistically isotropic/homogeneous and
Gaussian.111A recent paper exploring the joint probability of four different
CMB anomalies occurring together has called into question the assumption of
statistical isotropy [26]. Our paper does not address this consideration. In
Section IV we quantify the discrepancy between the temperature and
polarization quadrupoles and their $\Lambda$CDM predictions; we show that the
magnitude of this discrepancy grows when considering their joint probability.
Superhorizon modifications to $\Lambda$CDM which would shift the quadrupole
values are explored in Section V. We summarize our findings with a view toward
LiteBIRD in Section VI.

##  II CMB multipoles and power spectra

The standard $\Lambda$CDM cosmological model tells us that inflation produced
adiabatic, nearly scale-invariant, and Gaussian primordial scalar (i.e.
curvature) perturbations; these perturbations are the seeds that later induce
CMB temperature and E-mode polarization anisotropies. The observed anisotropy
patterns may be expanded in terms of spherical harmonics. For temperature
anisotropies this takes the form,

$T(\mathbf{n})=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}a^{T}_{lm}Y_{lm}(\mathbf{n}),$
(1)

which can be inverted to give harmonic coefficients,

$a^{T}_{lm}=\int d^{2}\mathbf{n}\,T(\mathbf{n})Y_{lm}^{*}(\mathbf{n}).$ (2)

In the case of polarization anisotropies, we first decompose the sky pattern
using Stokes $Q$ and $U$ parameters to get

$(Q\pm iU)(\mathbf{n})=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}a^{(\pm
2)}_{lm}\vphantom{|}{}_{(\pm 2)}Y_{lm}(\mathbf{n}),$ (3)

where $\vphantom{|}{}_{(\pm 2)}Y_{lm}(\mathbf{n})$ are the spin-2 spherical
harmonic functions which are required since polarization is a spin-2 quantity.
The associated harmonic coefficients are

$a^{(\pm 2)}_{lm}=\int d^{2}\mathbf{n}\,(Q\pm
iU)(\mathbf{n})\vphantom{|}_{(\pm 2)}Y^{*}_{lm}(\mathbf{n}),$ (4)

which can easily be converted to E-mode coefficients

$a^{E}_{lm}=-\frac{1}{2}\left(a^{(+2)}_{lm}+a^{(-2)}_{lm}\right).$ (5)

Each $a_{lm}$ is drawn from a Gaussian with zero mean but nonzero variance—or
angular power spectrum—$C_{l}$ given by,

$\left<a_{lm}^{X}a_{l^{\prime}m^{\prime}}^{Y*}\right>=\delta_{ll^{\prime}}\delta_{mm^{\prime}}C_{l}^{XY},$
(6)

where $X,Y\in\\{T,E\\}$.

##  III The origins of the quadrupoles

While both types of CMB anisotropies trace their origins to primordial
curvature perturbations, the timescales over which their respective power
spectra receive their greatest contributions differ. From a distribution of
primordial curvature perturbations $\zeta(\mathbf{k})$ with power spectrum
defined via

$\left<\zeta(\mathbf{k})\zeta^{*}(\mathbf{k^{\prime}})\right>=P_{\zeta}(k)\delta(\mathbf{k}-\mathbf{k^{\prime}}),$
(7)

and dimensionless power spectrum
$\mathcal{P}_{\zeta}(k)=k^{3}P_{\zeta}(k)/2\pi^{2}$, one can calculate the CMB
angular power spectra,

$C_{l}^{XY}=4\pi\int\frac{dk}{k}\,\mathcal{P}_{\zeta}(k)\Delta^{X}_{l}(k,\eta_{0})\Delta^{Y}_{l}(k,\eta_{0}),$
(8)

where $\Delta^{X}_{l}(k,\eta_{0})$ are the photon transfer functions evaluated
today. These can be obtained numerically, e.g., from a Boltzmann code like
CLASS [27] or CAMB [28]. For the quadrupole, the E-mode transfer function can
be approximated by [29]

$\Delta^{E}_{2}(k,\eta_{0})\simeq-\frac{2\sqrt{6}}{3}\int_{0}^{\eta_{0}}d\eta\,g(\eta)\frac{j_{2}[k(\eta_{0}-\eta)]}{[k(\eta_{0}-\eta)]^{2}}j_{2}[k(\eta-\eta_{*})],$
(9)

where $\eta_{0}$ is the conformal time today and $\eta_{*}$ that at
recombination, and $g(\eta)$ is the visibility function for reionization; it
integrates to the reionization optical depth $\tau=\int\,d\eta\,g(\eta)$ and
peaks at the conformal time of reionization. The temperature transfer function
can be approximated by

$\Delta^{T}_{2}(k,\eta_{0})\simeq-\frac{2}{9}j_{2}[k(\eta_{0}-\eta_{*})].$
(10)

It is determined primarily by the quadrupole at the surface of last scatter
and has an $O(\tau)$ correction (not shown) from reionization and also a small
contribution from the integrated Sachs-Wolfe effect. Fig. 1 shows
contributions to the CMB TT (top) and EE (bottom) power spectra as functions
of $\eta$ and $\eta^{\prime}$, using exact results from CLASS [27].

Even though the peaks of recombination and reionization transfer functions
occur at substantially different redshifts ($z_{*}\sim 1100$ and $z_{re}\sim
6$, respectively), the comoving distances to their surfaces are comparable.
Consequently, the two probe a comparable range of physical separations, as
shown in Fig. 2.

Figure 1: Logarithm of the conformal-time integrands of the temperature (top)
and E-mode polarization (bottom) quadrupoles. Figure 2: The CMB quadrupole
transfer functions for the temperature (top), temperature-polarization
(middle), and polarization (bottom) power spectra.

##  IV Discrepancy between quadrupoles and $\Lambda$CDM

###  IV.1 Noiseless, full-sky observations

We can estimate the level of discrepancy of the measured moments with the
$\Lambda$CDM expectation analytically. Our aim here is to assess how much more
unusual the low temperature quadrupole is, in light of the high polarization
quadrupole. To do so, we make the simplifying assumption of noiseless
measurements taken over a full-sky map. The complications induced by noise and
Planck’s $86\%$ ($50\%$) sky coverage in temperature (polarization) [30] will
reduce the statistical significances but leave the basic conclusions
unaltered. We will assess more precisely the impact of imperfect measurements
and $f_{\rm{sky}}<1$ in the next subsection.

If primordial perturbations are Gaussian, then each spherical-harmonic
coefficient $a_{lm}$ derived from a full-sky map is a Gaussian random variable
with zero mean and variance $C_{l}$. The estimators $\widehat{C_{l}^{TT}}$ and
$\widehat{C_{l}^{EE}}$ for the temperature/polarization moments derived from a
noiseless full-map are thus $\chi^{2}$-distributed variables with $2l+1$
degrees of freedom. Given the cross-correlation $C_{l}^{TE}$ between the
temperature and polarization spherical-harmonic coefficients, though, the
estimators $\widehat{C_{l}^{TT}}$ and $\widehat{C_{l}^{EE}}$ are correlated
for any given $l$. The joint PDF for $\chi^{2}$ variables correlated in this
way can be related to the Wishart distribution [31, 32] and is derived in
Appendix A. In our case, the variables can be regarded as $X=5C_{2}^{TT,{\rm
observed}}/C_{2}^{TT,{\rm expected}}$ and $Y=5C_{2}^{EE,{\rm
observed}}/C_{2}^{EE,{\rm expected}}$, with $n=5$. The cross-correlation
coefficient is $\rho=C_{2}^{TE}/\left[C_{2}^{TT}C_{2}^{EE}\right]^{1/2}$. From
Planck Legacy Archive data222http://pla.esac.esa.int/pla/#home used by Figs. 1
and 2 in Ref. [3], we estimate this to be $\rho\simeq
3/\sqrt{(1000)\times(0.055)}\simeq 0.5$. Again, from these data, we infer a
measured value of $Y\simeq 18$ (the polarization quadrupole is roughly three
times its expected value, $2l+1=5$) and $X\simeq 1.1$, 0.22 times the expected
value

First, let’s re-visit the temperature quadrupole. The cumulative distribution
for a $\chi^{2}$-distributed variable $X$ with 5 degrees of freedom below one
fifth the mean value is about $4.5\%$, implying that a departure from the mean
of this magnitude will occur (taking into account the possibility of an
unusually high fluctuation) in roughly 1 out of every 11 trials, about a
$1.7\sigma$ fluctuation. The fluctuation in the polarization quadrupole is
more significant—roughly a $2.7\sigma$ departure, a 1-in-150 occurrence.

If, however, we take into account the joint PDF for the two quadrupoles, the
observed values of the two quadrupoles together occur in roughly one in 25,000
realizations, a $4.1\sigma$ fluctuation.

###  IV.2 Imperfect measurements on a cut sky

What we obtain from Planck is not the idealized picture of the microwave sky
heretofore discussed. The measured power spectra contain contributions from
residual foregrounds and instrumental systematics, and they are made on a
masked sky map. Any statements regarding the statistical significance of CMB
anomalies must account for these limitations. Here we introduce a simple model
to approximate the effects of partial-sky coverage and instrumental noise.

To begin, we recall that for a full-sky map and no noise, the PDF for the
observed temperature/polarization quadrupoles are given by $P(X,Y)$ with
$X=5\,C_{2}^{TT,{\rm obs}}/C_{2}^{TT,{\rm exp}}$ and $Y=5\,C_{2}^{EE,{\rm
obs}}/C_{2}^{EE,{\rm exp}}$, with $n=5$ and
$\rho=C_{2}^{TE}/\left[C_{2}^{TT}C_{2}^{EE}\right]^{1/2}$.

Instrumental noise provides additional contributions $C_{l}^{TT,n}$ and
$C_{l}^{EE,n}$ to the temperature and polarization power spectra,
respectively, but none to the cross-correlation. In practice, the temperature
noise in Planck is negligible for the quadrupole and so we neglect it. With
these assumptions (still assuming full-sky coverage), the $a_{2m}^{T}$ are
still distributed with variance $C_{l}^{TT}$. The measured $a_{2m}^{E}$
include both signal and noise and are distributed with variance
$C_{2}^{EE}+C_{2}^{EE,n}$. The joint PDF for the observed quadrupoles is then
$P(X,Y)$ where now $Y=5(C_{2}^{EE,{\rm
obs}}+C_{2}^{EE,n})/(C_{2}^{EE}+C_{2}^{EE,n})$. Note that here,
$C_{2}^{EE,{\rm obs}}=C_{2}^{EE,{\rm meas}}-C_{2}^{EE,n}$; i.e., it is the
true quadrupole we infer from the measured quadrupole after subtracting the
expected noise contribution. The cross-correlation coefficient is now reduced
to $\rho=C_{2}^{TE}/\left[C_{2}^{TT}(C_{2}^{EE}+C_{2}^{EE,n})\right]^{1/2}$.

Now consider the effects of fractional sky coverage. This would be easy if
both the temperature and polarization maps had the same $f_{\rm sky}$. In this
case, we would replace $n=5\to 5\,f_{\rm sky}$ and also replace $5\to
5\,f_{\rm sky}$ in the definitions of $X$ and $Y$. However, given that $f_{\rm
sky}$ is different for polarization and temperature, a bit more thought is
required. To treat this, we first consider the temperature quadrupole. The PDF
for this can simply be taken to be a $\chi^{2}_{f_{\rm sky}^{T}(2l+1)}$
distribution. We can then infer from Eq. (17) the conditional PDF, for our
given $C_{2}^{TT}$, for $C_{2}^{EE}$ with $n$ scaled by $f_{\rm sky}^{E}$.

To be more precise, we first note that

$P_{n}(X)\equiv\int_{0}^{\infty}\,dY\,P_{n}(X,Y;\rho),$ (11)

is a $\chi_{n}^{2}$ distribution for $X$. We then note that the conditional
probability for $Y$, given some value $X$, is

$P_{n}(Y;X,\rho)=\frac{P_{n}(X,Y;\rho)}{P_{n}(X)}.$ (12)

The unnormalized joint PDF for the temperature quadrupole $C_{2}^{TT,{\rm
obs}}$ observed over $f_{\rm sky}^{T}$ of the sky and polarization quadrupole
$C_{2}^{EE,{\rm obs}}$ observed over $f_{\rm sky}^{E}$ of the sky shown in
Fig. 3 is

$\frac{P_{5\,f_{\rm sky}^{T}}\left(5\,f_{\rm sky}^{T}C_{2}^{TT,{\rm
obs}}/C_{2}^{TT,{\rm exp}}\right)}{P_{5\,f_{\rm sky}^{E}}\left(5\,f_{\rm
sky}^{E}C_{2}^{TT,{\rm obs}}/C_{2}^{TT,{\rm exp}}\right)}P_{5\,f_{\rm
sky}^{E}}\left(5\,f_{\rm sky}^{E}\frac{C_{2}^{TT,\rm obs}}{C_{2}^{TT,\rm
exp}},5\,f_{\rm sky}^{E}\frac{C_{2}^{EE,\rm obs}+C_{2}^{EE,n}}{C_{2}^{EE,\rm
exp}+C_{2}^{EE,n}};\rho\right).$ (13)

Figure 3: The unnormalized PDF of Eq. (13), ranging from lower values in red
to higher values in purple. The black point corresponds to the observed
temperature and polarization quadrupole values.

The two-tailed probability of getting a temperature quadrupole at least as
extreme as we observe, given Planck’s limited sky coverage, is 12.8% or a
$1.5\sigma$ fluctuation. Table 3 of Ref. [30] gives the probability of
obtaining a polarization quadrupole as anomalous as measured as 29.6%, or
$1\sigma$, based on FFP10 simulations [33]. From this result, we can solve for
$Y$ using the equation

$\int_{Y}^{\infty}\chi^{2}(5f_{\rm sky}^{E},Y)\,dY=0.296/2$ (14)

which gives $Y=4.6$. The amount of noise is thus $C_{2}^{EE,n}=4.1\times
10^{-15}$. We now have all the ingredients required to calculate the joint
probability. Integrating Eq. (13) and dividing by its normalization factor
gives a two-tailed probability of 2.1% or $2.3\sigma$.

##  V A large-scale modification to $\Lambda$CDM

Taken at face value, the results of Section IV are in minor conflict with the
standard-cosmological-model expectations. We thus consider a modification to
$\Lambda$CDM in which we modify the matter power spectrum $P_{\mathrm{m}}(k)$
at $k\lesssim 10^{-3}$ Mpc-1. This appears in the low-$l$ CMB power spectra
but not in large-scale structure surveys which probe higher $k$. For
simplicity, we do not consider any effects that modifying the power spectrum
will have on the inferred cosmological parameters.

The full CMB quadrupole transfer functions are shown in Fig. 2. From here we
can see that even though temperature and E-mode polarization probe different
epochs, the two extend over comparable distance scales. This provides us a $k$
range over which to invoke new physics and subsequently alter the matter power
spectrum.

We first consider decreasing $P_{\mathrm{m}}(k)$ by $P_{\mathrm{m}}\rightarrow
P_{\mathrm{m}}/17$ for $k<3\times 10^{-4}$ Mpc-1. This is one way ensure that
our modified $\Lambda$CDM model correctly predicts $C_{2}^{TT,\rm obs}$. The
complete set of model predictions are $C_{2}^{TT}=3.18\times 10^{-11}$,
$C_{2}^{TE}=5.52\times 10^{-14}$, and $C_{2}^{EE}=3.17\times 10^{-16}$; this
implies that the probability of our model generating an E-mode quadrupole at
least as discrepant as $C_{2}^{EE,\rm obs}$ is 4.1%. Put another way,
observations and theory would be in tension at $2\sigma$. If instead we
appropriately boost the large-scale power of our model to agree with
$C_{2}^{EE,\rm obs}$, the probability of having an anomalous temperature
quadruople comparable to or greater than what we see is 2.6% or a $2.2\sigma$
discrepancy. We conclude that the tension between the two quadrupoles is not
easily resolved by an increase/decrease of large-scale power.

Figure 4: Same as Fig 2 but for the octupole moment.

##  VI Conclusions

The low CMB temperature quadrupole has been a curiosity for three decades [1].
Although the inconsistency with its expected value is not that statistically
significant, the fact that the power on the largest observable scale is low
has been interpreted as a possible hint of new superhorizon physics. The
independent confirmation of a low quadrupole by WMAP [2] and then Planck [3]
suggests that this low quadrupole is robust.

More recently, however, Planck has found the CMB polarization quadrupole to be
somewhat high. Here we have calculated the joint PDF for the two quadrupoles
and found that they are inconsistent with the $\Lambda$CDM expectation at the
$2.3\sigma$ level. We also argued that the inconsistency is robust to simple
changes to the cosmological model.

What to make of this result? It is obvious to wonder whether the Planck
polarization result may be skewed high by some instrumental systematic and/or
unusual foreground contamination. The former was mitigated a few years ago
with the development of the SRoll2 algorithm [34], while the latter was
considered more recently with the updated SRoll3 neural network [35]. Results
using the SRoll2 algorithm included a lower measured value for $C_{2}^{EE}$
with a PTE of 38%, while the PTE for $C_{2}^{TE}$ decreased to 58.1% [36]. The
temperature-polarization joint probability was not calculated, but we expect
it to be similar to the PR3 results calculated in this work.

Fortunately, though, COBE, WMAP, and Planck will not be our only sources of
quadrupole-scale CMB measurements for long. The forthcoming LiteBIRD
experiment will target precise polarization measurements on the largest
angular scales, with unprecedented frequency coverage to enable foreground
removal over the entire sky. The anomaly should also motivate the search for
some type of calibration of the polarization quadrupole, analogous to that
suggested for the temperature quadrupole in Ref. [37] (the peculiar-velocity-
induced temperature quadrupole discussed in there does not produce a
polarization quadrupole). If the marginally high polarization quadrupole
persists, it will then disfavor explanations for the low quadrupole that
involved suppressed large-scale power. But will it then simply be chalked up
to a statistical fluctuation? Or perhaps some other new physics?

## Acknowledgements

We thank J. Dunkley for useful discussions. This work was supported at Johns
Hopkins by NSF Grant No. 2112699 and the Simons Foundation. JG acknowledges
support from Princeton’s Presidential Postdoctoral Research Fellowship.

##  Appendix A Derivation of the joint $\chi^{2}$ probability distribution
function

Let $x$ and $y$ be Gaussian random variables with unit variance, zero mean,
and covariance ${\left\langle xy\right\rangle}=\rho$. If we have $n$
realizations $x_{i}$ and $y_{i}$ (for $i=1,\ldots,n$) of these pairs, then the
joint probability distribution function (PDF) for these $2n$ random variables
is

$p(\vec{x},\vec{y})=\frac{1}{(2\pi)^{n}(1-\rho^{2})^{n/2}}\exp\left[-\frac{\vec{x}^{2}+\vec{y}^{2}-2\rho\vec{x}\cdot\vec{y}}{2(1-\rho^{2})}\right],$
(15)

where $\vec{x}$ and $\vec{y}$ are $n$-dimensional vectors with components
$x_{i}$ and $y_{i}$, respectively.

We now define two variables $X\equiv\vec{x}^{2}$ and $Y\equiv\vec{y}^{2}$
which are correlated $\chi^{2}$ variables with joint PDF,

$P_{n}(X,Y;\rho)=\int d^{n}x\int
d^{n}y\,p(\vec{x},\vec{y})\delta_{D}(X-\vec{x}^{2})\delta_{D}(Y-\vec{y}^{2}),$
(16)

where $\delta_{D}(x)$ is the Dirac delta function. The probability
distribution is a function of the vector norms $\vec{x}^{2}$ and
$\vec{y}^{2}$, and also the dot product $\vec{x}\cdot\vec{y}$, and we are
integrating over all $\vec{x}$ and all $\vec{y}$. The integration over
$\vec{x}$ can be written as $S_{n-1}\int x^{n-1}\,dx$, where
$S_{n-1}=2\pi^{n/2}/\Gamma(n/2)$ is the volume of the unit $n$-sphere. The
integration over $\vec{y}$ can then be written as $S_{n-2}\int\,dy_{1}\int
y_{\perp}^{n-2}\,dy_{\perp}$, where $y_{1}$ is the component of $\vec{y}$
along the direction of $\vec{x}$ and $\vec{y}_{\perp}=(y_{2},\ldots,y_{n})$
the components perpendicular to $\vec{x}$. We then write
$x^{n-1}\,dx=(x^{2})^{(n/2)-1}\,d(x^{2})/2$,
$y_{\perp}^{n-2}\,dy_{\perp}=(y_{\perp}^{2})^{(n/2)-(3/2)}\,d(y_{\perp}^{2})/2$,
and the argument of the second Dirac delta function in Eq. (16) as
$(Y-y_{1}^{2})-y_{\perp}^{2}$. We then find

$\displaystyle P_{n}(X,Y;\rho)$
$\displaystyle=\frac{(XY)^{\frac{n}{2}-1}}{2^{n}(1-\rho^{2})^{n/2}\left[\Gamma(n/2)\right]^{2}}\exp\left[-\frac{X+Y}{2(1-\rho^{2})}\right]$
(17) $\displaystyle\times G_{n}\left(\frac{\rho\sqrt{XY}}{1-\rho^{2}}\right),$

where

$G_{n}(\alpha)\equiv\frac{1}{\sqrt{\pi}}\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}\int_{-1}^{1}\,d\mu\,(1-\mu^{2})^{\frac{n}{2}-\frac{3}{2}}e^{\alpha\mu}.$
(18)

Mathematica can write the integral in terms of a hypergeometric function,
which provides little insight but may facilitate numerical evaluation.

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