# On the prior dependence of constraints on the tensor-to-scalar ratio

Marina Cortês  Andrew R. Liddle  and David Parkinson

###### Abstract

We investigate the prior dependence of constraints on cosmic tensor
perturbations. Commonly imposed is the strong prior of the single-field
inflationary consistency equation, relating the tensor spectral index $n_{\rm
T}$ to the tensor-to-scalar ratio $r$. Dropping it leads to significantly
different constraints on $n_{\rm T}$, with both positive and negative values
allowed with comparable likelihood, and substantially increases the upper
limit on $r$ on scales $k=0.01{\rm\,Mpc}^{-1}$ to $0.05{\rm\,Mpc}^{-1}$, by a
factor of ten or more. Even if the consistency equation is adopted, a uniform
prior on $r$ on one scale does not correspond to a uniform one on another;
constraints therefore depend on the pivot scale chosen. We assess the size of
this effect and determine the optimal scale for constraining the tensor
amplitude, both with and without the consistency relation.

##  1 Introduction

The inflationary proposal [1] is a compelling scenario for the origin of
primordial density fluctuations. Its prescription for the spectrum of density
perturbations has passed the scrutiny of high precision measurements of the
Cosmic Microwave Background (CMB), being in agreement with small deviations
from a purely scale-invariant spectrum [2]. In all models of inflation, a
small but potentially non-negligible stochastic background of primordial
gravitational waves is also generated alongside the density fluctuations. This
prediction remains to be confirmed, with only an upper bound on the density of
gravitational waves so far.

Although there are currently no useful theoretical lower bounds on the tensor
amplitude from inflation models (since we have classes of models that predict
undetectable levels of tensors [3]), determining the amplitude of
gravitational waves or, failing that, imposing a stringent upper bound is
still formally a requirement for completion of the underlying physical picture
behind the inflation mechanism. The amplitude of gravitational waves
determines the energy scale at which inflation took place, shedding light on
the particle physics sector in those regimes, and reveals the total excursion
of the scalar field during inflation. This provides information on symmetries
that may have been broken at the time, thereby probing energy scales many
orders of magnitude larger than those at the Large Hadron Collider.

One of the predictions of the simplest models of inflation is that the
spectrum of tensor perturbations depends on that of scalars. The spectra have
a common origin in the same function, the potential of the single scalar
field, and are therefore related. This prediction characterizes the simplest
models of inflation (single field and slowly rolling), and yields a set of
consistency relations all valid to a given order in the slow-roll regime [4].
These models give a good description of current data [2].

Nevertheless, other assumptions on the nature of tensor perturbations are
possible. More general classes of inflation models include those with multiple
scalar fields [5], where the consistency equation becomes an inequality [6],
and those with deviations from the slow-roll mechanism. In addition there are
models in which density fluctuations are seeded through a physical mechanism
making use of the known duality between expanding and contracting cosmologies.
In collapse-type models the rapidly expanding horizon in an inflationary era
is associated to contraction in a matter-dominated era [7]. In ekpyrotic
models the slow contraction takes place during domination by a stiff fluid
with equation of state $w>1$ [8]. Alternatively, tensor perturbations might be
seeded by cosmic defects [9], or one might simply ask what can be learnt from
observations without imposing specific theoretical preconceptions. There are
therefore a variety of possible assumptions as to how the tensor perturbations
might behave, and some examples have been investigated in the literature.

Finelli, Rianna, and Mandolesi [10] tested the validity of the consistency
equation by sampling freely from the plane $16\,\epsilon-r$. They probed
tensors at a scale close to the scalar pivot, $k=0.01{\rm\,Mpc}^{-1}$, and not
at $k=0.002{\rm\,Mpc}^{-1}$. Camerini et al [11] dropped the assumption of
single field inflation and searched at $k=0.002{\rm\,Mpc}^{-1}$ for
compatibility with blue tilted tensors in current data. The applied prior on
$n_{\rm T}$ was $-1<n_{\rm T}<20$. This apparently strongly conservative prior
still misses a large fraction of allowed red tilted models at $n_{\rm T}<-1$.
Also they probed solely at $k=0.002{\rm\,Mpc}^{-1}$ and thus didn’t detect the
weakening of constraints as we move towards smaller scales. Valkenburg,
Krauss, and Hamann [12] considered a prior density on $r$ which differs from
the common choice of a flat prior. They select for a theoretically motivated
uniform prior on the energy scale of inflation. They also probe at a single
scale, $k=0.002{\rm\,Mpc}^{-1}$. Even assuming the consistency equation they
detect that constraints on $r$ reflect this change on the tensor prior. Zhao,
Baskaran and Zang, [13] searched for the best multipole to probe tensors at.
Here they dropped the consistency equation assumption and probed tensors at
more than one scale. They then draw forecasts under this setup, for a variety
of ground- and space-based upcoming B-mode experiments. The chosen fiducial
model had small tensor amplitude and as result they don’t detect the large
variation allowed at the short wavelengths. In Ref. [14] Gjerløw and Elgarøy
relax the consistency equation relation but retain the assumption of a nearly
scale invariant tensor spectrum $n_{\rm T}\sim n_{\rm S}-1$. Finally Powell
[15] investigates the sensitivity of upcoming B-mode experiments for detecting
the tensor tilt, both for blue and red $n_{\rm T}$.

In our work we allow for full variability in these sets of assumptions, and
make a systematic exploration of how constraints on tensor fluctuations
respond to changes of prior. We relax the imposition of the simplest inflation
models, vary the cosmological scale probed, and study different prior
densities on both $r$ and $n_{\rm T}$. This permits us to unveil the
underlying variation of constraints on tensor parameters, that result from
shifts in one’s set of priors, and we observe a larger range of behaviour of
the tensor spectra than any of the studies in the literature so far.

##  2 Methodology

We assume throughout a flat $\Lambda$CDM model and parameterize our set of
primordial spectra as

$\displaystyle A_{\rm S}^{2}(k)$ $\displaystyle=$ $\displaystyle A_{\rm
S}^{2}(k_{0})k^{n_{\rm S}(k_{0})-1+\alpha(k_{0})\ln k/k_{0}}\,,$ (2.1)
$\displaystyle A_{\rm T}^{2}(k)$ $\displaystyle=$ $\displaystyle A_{\rm
T}^{2}(k_{0})k^{n_{\rm T}}\,,$ (2.2)

where $\alpha\equiv dn_{\rm S}/d\ln k$, and $k_{0}$ is the pivot scale where
observables are specified. We define the ratio of tensor to scalar amplitude
of perturbations as

$r(k)\equiv 16\frac{A_{\rm T}^{2}(k)}{A_{\rm S}^{2}(k)}\,.$ (2.3)

The usual assumption when fitting primordial fluctuations for parameter
estimation is the imposition of the first consistency relation between the
scalar and tensor power spectra,

$r=-8n_{\rm T}\,.$ (2.4)

For inflationary models the first and second derivatives of the scalar field
potential are usually expressed in terms of the slow-roll parameters [16],

$\epsilon=\frac{m_{{\rm
Pl}}^{2}}{16\pi}\left(\frac{V^{\prime}}{V}\right)^{2}\quad;\quad\eta=\frac{m_{{\rm
Pl}}^{2}}{8\pi}\,\frac{V^{\prime\prime}}{V}\,,$ (2.5)

where prime indicates $d/d\phi$. In the context of the slow-roll expansion and
a single field, we can write the observables in terms of the derivatives of
the potential [16]

$\displaystyle n_{\rm S}-1$ $\displaystyle=$
$\displaystyle-6\epsilon+2\eta\,;$ (2.6) $\displaystyle n_{\rm T}$
$\displaystyle=$ $\displaystyle-2\epsilon\,;$ (2.7) $\displaystyle r$
$\displaystyle=$ $\displaystyle 16\epsilon\,,$ (2.8)

from which the first consistency relation immediately follows. A consequence
of the consistency equation Eq. (2.4) is to enforce the amplitude of tensors
to be a decreasing function of wavenumber, since $r$ is by construction always
positive.

In multi-field models, tensor perturbations are still given by the usual
formula but extra scalar perturbations can be generated by conversion of
isocurvature perturbations in the additional degrees of freedom. The
consistency equation then weakens to an inequality, $r<-8n_{\rm T}$ with
$n_{\rm T}$ still constrained to be negative [6]. Other models may give yet
further variation, e.g. collapsing Universe inflation models such as the pre-
big-bang models [7] give a positive tensor spectral index.

In this paper we carry out an extensive exploration of the prior dependence of
constraints on the tensor-to-scalar ratio. The prior dependence takes two
forms: (a) restrictions on the parts of the $r$–$n_{\rm T}$ plane that are
considered, e.g. by enforcing the above consistency equation or inequality,
and (b) the prior probability distribution adopted on the permitted parameter
region. For the latter, commonly a uniform prior is chosen at whichever
‘pivot’ scale has been selected to specify the parameters, but this picks out
a special scale for which this is being assumed true. A related question is to
ask on what scale the tensor-to-scalar ratio is actually best constrained by a
given dataset, given prior assumptions.

In order to explore these dependencies, we will consider three different
cases:

  1. 1.

No consistency relation, i.e. $r$ and $n_{\rm T}$ able to vary independently
including positive $n_{\rm T}$.

  2. 2.

Single-field consistency equation imposed, i.e $r=-8n_{\rm T}$.

  3. 3.

Multi-field consistency inequality imposed, i.e. $r\leq-8n_{\rm T}$.

In each case we need to consider the effect of imposing priors at different
scales; we choose $0.002{\rm\,Mpc}^{-1}$ as used in WMAP papers [17, 2] and
$k=0.02{\rm\,Mpc}^{-1}$ which is about the scale at which scalar perturbations
are optimally constrained [18].

##  3 Constraints on the tensor amplitude

[ caption = Uniform priors assumed in this article., label = prior, ]llcc
[a]at scale $k=0.002{\rm\,Mpc}^{-1}$ [b]at scale $k=0.02{\rm\,Mpc}^{-1}$ &
lower upper
Physical baryon density $\Omega_{\rm b}h^{2}$ 0.005 0.1
Physical dark matter density $\Omega_{\rm cdm}h^{2}$ 0.01 0.99
Sound horizon $\theta$ 0.5 10
Optical depth $\tau$ 0.01 0.8
Scalar amplitude $\log(10^{10}A_{\rm S}^{2})$ 2.7 4
Scalar index $n_{\rm S}$ 0.5 1.5
Scalar running $dn_{\rm S}/d\ln k$ $-0.2$ 0.2
SZ-amplitude $A_{\rm SZ}$ 0 2
Tensor-to-scalar ratio $r$ 0 $1\tmark[a]$
0 $16\tmark[b]$
Tensor spectral index $n_{\rm T}$ $-4$ $4\tmark[a]$
$-4$ $8\tmark[b]$

To probe parameter space we use the Markov Chain Monte Carlo method as
implemented in the CosmoMC package [19], and assume our cosmology can be
described by the set of parameters in Table LABEL:prior where we specify the
uniform priors imposed. All our MCMC runs use the combination of datasets from
WMAP-7 year [2], matter power spectrum measurements from SDSS-DR7 [20], and
$H_{0}$ from the Hubble Science Telescope (HST) [21]. Though the non-CMB
experiments (SDSS and HST) cannot detect the tensor perturbations directly,
these extra data are necessary to break some of the parameter degeneracies
between $r$ and other cosmological parameters.

$\begin{array}[]{cc}\includegraphics[width=220.0pt]{noce_lpnt44_r01_mar4_0002_infl_line_2D.ps}&\includegraphics[width=220.0pt]{noce_lpnt48_r16_mar4_002_2D.ps}\\\
\end{array}$

Figure 1: The $n_{\rm T}$–$r$ two-dimensional distributions obtained at
different scales, without imposition of the consistency equation, and uniform
priors over the regions plotted. Left panel: constraints obtained at
$k=0.002{\rm\,Mpc}^{-1}$; right panel: constraints at scale
$k=0.02{\rm\,Mpc}^{-1}$. The colour scale represents the mean likelihood of
samples, going to higher likelihood towards redder regions, and the contours
denote the marginalized distributions at 68% and 95% confidence limits. The
straight line shows the consistency equation.

###  3.1 No consistency relation

We begin by discussing the most general case where $r$ and $n_{\rm T}$ vary
freely, limited only by priors that we have chosen to be so wide that the
posterior distributions are limited by the data, weak though they are in
constraining the tensors, particularly their scale dependence. As well as
displaying these ‘model independent’ constraints, we use this case to explore
the effect of pivot scale transformations.

Figure 1 (left panel) shows combined constraints on $n_{\rm T}$ and $r$, with
uniform priors imposed at $k=0.002{\rm\,Mpc}^{-1}$; similar results were
obtained by Camerini et al [11]. The consistency equation would have
restricted models to the line, and obviously dropping this assumption greatly
increases the parameter space that is available. As a byproduct, this figure
shows how far we are from a meaningful test of the consistency relation, which
would require observational constraints tightly associated to a point on the
consistency equation line or excluding it entirely.

We find that the data do constrain $n_{\rm T}$, albeit very weakly, both from
above and from below. The constraints weaken as $r$ becomes small, because
$A_{\rm T}^{2}$ is very small there and $n_{\rm T}$ is irrelevant if the
amplitude is too small to detect on all scales probed. Large very positive
$n_{\rm T}$ would eventually contribute to the temperature modes at large
$\ell$ and conversely large negative $n_{\rm T}$ would disrupt the Sachs–Wolfe
effect beyond cosmic variance.

Note that the data allow the spectral tilt of tensor perturbations to be
positive, contrary to standard inflationary predictions. Indeed the one-
dimensional marginalized confidence limits on $n_{\rm T}$ are approximately
symmetric about zero:

$n_{\rm T}(k=0.002{\rm\,Mpc}^{-1})=0.0^{+1.1\,+2.0}_{-0.9\,-2.0}\,,$ (3.1)

where uncertainties quoted are 68% and 95%.

We now wish to investigate the constraints if the tensors are specified at a
shorter scale, $k=0.02{\rm\,Mpc}^{-1}$. The right panel of Figure 1 shows the
$n_{\rm T}$–$r$ plane when we change the pivot scale to
$k=0.02{\rm\,Mpc}^{-1}$ and otherwise maintain the same setup, i.e. the same
datasets and priors on remaining cosmology. The principal difference is that
the priors on parameters are now assumed uniform at this scale; we also
widened the ranges of $r$ and $n_{\rm T}$ in order to be able to reach regions
constrained by the data. Motivated by the fact that $\epsilon=1$ signals the
end of slow-roll inflation, and that $r=16\epsilon$ from the consistency
equation, we allow for $0<r<16$ and choose the prior on $n_{\rm T}$ such that
95% limits are well within the prior range $-4<n_{\rm T}<8$.

One might have expected that at least the constraints on $n_{\rm T}$ would
have been unchanged, since it is unaffected by a transformation of pivot
scale. However we see that this is not the case; the distribution is shifted
significantly to the right. The reason is that a uniform prior on one scale
does not transform to a uniform prior on another, and because the tensors are
so weakly constrained (at least in absence of consistency relations) the
effect of this is quite dramatic. Figure 2 (left panel) shows the prior
induced on parameters at $0.02{\rm\,Mpc}^{-1}$ by a uniform prior at
$k=0.002{\rm\,Mpc}^{-1}$. We stress there is no data at all in this figure; we
just draw uniform points on one scale and analytically transform $r$ to the
second scale ($n_{\rm T}$ does not change). As we see, the resulting prior is
extremely non-uniform. Note that it is the region where the prior is
compressed that leads to a high prior (since the fraction of the prior in each
interval of $n_{\rm T}$ is constant).

$\begin{array}[]{cc}\includegraphics[width=220.0pt]{nt_r_prior_chain_2D.ps}&\includegraphics[width=220.0pt]{ntr_ovrl.ps}\\\
\end{array}$

Figure 2: Left panel: The induced prior at $k=0.02{\rm\,Mpc}^{-1}$ from a
uniform prior at $k=0.002{\rm\,Mpc}^{-1}$. Right Panel: superposition of two-
dimensional distributions of $n_{\rm T}$–$r$ at $k=0.02{\rm\,Mpc}^{-1}$. The
dashed lines represent the marginalized distributions of $n_{\rm T}$–$r$
obtained by running a chain with uniform prior $k=0.02{\rm\,Mpc}^{-1}$. The
solid lines and colour code represent the distributions of $n_{\rm T}$–$r$
obtained in a chain run with uniform prior at $k=0.002{\rm\,Mpc}^{-1}$ and
shifted to $k=0.02{\rm\,Mpc}^{-1}$.

The consequence is shown in the right panel of Figure 2, which overlays the
chain run at $k=0.02{\rm\,Mpc}^{-1}$ with the chain run at the pivot scale
$k=0.002{\rm\,Mpc}^{-1}$ and transposed to the smaller scale
$k=0.02{\rm\,Mpc}^{-1}$. The dashed lines are the posterior assuming a uniform
prior, matching Fig. 1 (right panel), and hence also indicate the likelihood.
The posterior of the transformed chain is obtained by multiplying this
likelihood with the transformed prior of Fig. 2 (left panel), which shifts the
preferred region downwards and leftwards yielding the solid lines. The choice
of scale to impose the uniform prior clearly substantially modifies the
constraints on each parameter.

Note that on this shorter scale the constraint on $r$ is considerably weakened
with the $2\sigma$ constraints increased by a factor of more than ten (see
below in Eq. (3.1)). The primordial amplitude of tensor fluctuations is
allowed to be as large as six times that of scalar fluctuations, and steeply-
rising tensor spectra of tilt $n_{\rm T}=5$ are allowed at the $95\%$ level.
We show the TT and BB CMB spectra for such a model in Fig. 3. Even at $n_{{\rm
T}}=0$, corresponding to scale-invariant tensors, there is a significant
change in the upper limit on $r$ due to this change in prior, investigated in
more detail later in this section.

Figure 3: Angular power spectra for a model selected from the 2-dimensional
$n_{\rm T}-r$ distribution of Fig. 1 (right panel), allowed at 95% confidence
level, and comparison with the single-field inflationary prediction. Parameter
values are $n_{\rm T}=5$, $n_{\rm S}=0.956$, $r=0.05$ and $dn_{\rm S}/d\ln
k=-0.009$, all specified at scale $k=0.02{\rm\,Mpc}^{-1}$. Solid blue and pink
lines show TT and BB total power modes. Yellow long-dashed shows the scalar
contribution to TT power, and green short-dashed the tensor contribution to
TT. Blue dotted line shows the corresponding BB total power for the
corresponding single-field model that would have been obtained by imposition
of the consistency equation.

The 1D (95%) marginalized limits on $n_{\rm T}$ and $r$ at the two scales are:

$\displaystyle-2.0<$ $\displaystyle n_{\rm T}$ $\displaystyle<2.0\,,\quad
r<0.35\quad{\rm at}\quad k=0.002{\rm\,Mpc}^{-1}\,,$ $\displaystyle 0.76<$
$\displaystyle n_{\rm T}$ $\displaystyle<3.6\,,\quad r<6.6\quad{\rm at}\quad
k=0.02{\rm\,Mpc}^{-1}$ (3.2)

which reveals a significant shift towards more positive values in the
preferred $n_{\rm T}$ at smaller scales, with the best fit changing by about
$1\sigma$. At this scale the interesting region for inflation, $-1\lesssim
n_{\rm T}<0$, lies around the $1\sigma$ lower limit. Naive examination of
these contours would lead to the conclusion that data are favouring a larger
value of $n_{\rm T}$ at the smaller scale, and give evidence of non-zero
running of the tensor spectral index, given the stark difference in confidence
regions. However, none of the models fitting the data in the left panel of
Fig. 1 fail the prior imposed in the right panel, i.e. no model with $r$,
$n_{\rm T}$ at $k=0.002{\rm\,Mpc}^{-1}$ (left panel) is outside the confidence
limits at $k=0.02{\rm\,Mpc}^{-1}$ (right panel in the same figure). This means
that the same models are represented in the two distributions. It is rather
the effect of change in prior, i.e. the change in sampling at the new scale,
which is causing the change in the posterior. This is true for any parameter
that transforms under cosmological scale, but is more relevant if in addition
the parameter is poorly constrained as in the case of tensor spectra.

###  3.2 With consistency relations

We now turn to examining constraints on $r$ under different assumptions for
the inflation model. We assess the effect of imposing the single-field
equality, $r=-8n_{\rm T}$, and that of restricting to the multi-field region,
$r\lesssim-8n_{\rm T}$, obtained by clipping out from the full case the models
that don’t obey the inequality. We then obtain constraints for these models at
different scales, and their response to variations in the prior imposed at
each scale. Figure 4 shows the one-dimensional marginalized constraints on $r$
grouped in two different ways.

$\begin{array}[]{ccc}\includegraphics[width=132.00134pt]{r_1D_ce_multi_noce_0002.ps}&\includegraphics[width=132.00134pt]{r_1D_ce_multi_noce_002.ps}&\includegraphics[width=132.00134pt]{r_1D_ce_multi_noce_transported_002.ps}\\\
\end{array}$
$\begin{array}[]{ccc}\includegraphics[width=132.00134pt]{r_1D_ce_0002_002_transported.ps}&\includegraphics[width=132.00134pt]{r_1D_multi_0002_002_transported.ps}&\includegraphics[width=132.00134pt]{r_1D_noce_0002_002_transported.ps}\\\
\end{array}$

Figure 4: One-dimensional constraints on $r$. In the upper panels they are
grouped by scale and in the lower by class of models.

In the upper row of Fig. 4 the grouping is by scale: the first two panels show
constraints obtained by sampling from the likelihood under a uniform prior at
$k=0.002{\rm\,Mpc}^{-1}$ and $k=0.02{\rm\,Mpc}^{-1}$, and the last column
shows the chain transported from $k=0.002{\rm\,Mpc}^{-1}$ to
$k=0.02{\rm\,Mpc}^{-1}$, which results in a non-uniform prior at the new
scale. We see that at the scale $k=0.002{\rm\,Mpc}^{-1}$ used by WMAP, the
constraint is only modestly dependent on the prior. By constrast, at
$k=0.02{\rm\,Mpc}^{-1}$ the constraint on $r$ changes greatly under the
different prior assumptions, the centre panel showing the effect of the choice
of model type, and the right panel then showing further modification when the
alternative prior, induced from a uniform prior at $k=0.002{\rm\,Mpc}^{-1}$,
is used instead.

The lower row shows the same results grouped by model type. In the single-
field case (left panel) the constraint on $r$ does not much change when we
alter the pivot scale or vary our prior. This is a consequence of $n_{\rm T}$
being less than zero by assumption, preventing the tensors from growing
towards shorter scales, in combination with the observed near scale-invariance
of the scalar spectrum across the scales we are considering. As we have
already seen, the data alone do not significantly constrain the tensor
amplitude at $0.02{\rm\,Mpc}^{-1}$, without this additional model assumption.
Interestingly, sampling under the consistency equation assumption at
$k=0.002{\rm\,Mpc}^{-1}$ actually gives weaker bounds on $r$ than the case of
freely-varying $r$ and $n_{\rm T}$, as it happens that the models not
satisfying the consistency equation typically cannot fit the data with $r$
values as high as in the single-field case.

The multi-field constraints on $r$ are given by the middle panel in the bottom
row and are tighter than in single-field. That is because here we are sampling
from the region below the consistency equation line, which corresponds to
smaller $r$, as in Figs. 1. Also, we see that the region passing the multi-
field inequality at the tensor pivot, Fig. 1 (left panel), is much larger than
the region at the smaller scale $k=0.02{\rm\,Mpc}^{-1}$ (right panel). At
$k=0.002{\rm\,Mpc}^{-1}$ about 50% of the models are multi-field, against a
scarce 2% at $k=0.02{\rm\,Mpc}^{-1}$. This also results in lower statistics at
$k=0.02{\rm\,Mpc}^{-1}$ in our procedure for selecting multi-field models from
the full case, and accounts for the fluctuations seen in the middle bottom
panel of Fig. 4 (red line). The same panel shows that constraints for multi-
field models are, as in single-field, rather robust to changes in our prior.
The variation here is more evident however: $n_{\rm T}$ is more negative than
in the single-field case, so going to smaller scales leads to smaller $r$ and
tightens the contours as compared to the case at the tensor pivot (green, red
and blue lines in bottom middle panel).

Overall, Fig. 4 illustrates how constraints on tensor quantities with present-
day data are rather sensitive to one’s choice of prior and cosmological scale
probed. This means that at current sensitivities, in addition to uncertainty
in the data, one must admit an uncertainty arising from our prior assumptions,
for which the variation we see in Fig. 4 is an indication. This is not the end
of story, as priors leading to more extreme variations could be envisaged too,
and we have investigated only the transformation of a uniform prior.

##  4 Transformation of observables with cosmological scale

We now return to the issue of choice of scale to impose constraints on the
tensors. This issue in fact splits into two separate ones:

  1. 1.

On what scale is the prior probability distribution imposed (for example the
choice of a uniform prior on some specific scale)?

  2. 2.

For a given choice of prior, on what scale are the tensors optimally
constrained?

###  4.1 Scale transformations and the choice of priors

Our discussion above focussed on the first of these. Only when parameters
transform linearly with scale will the prior density be preserved when
transforming between cosmological scales $k$, because linear transformations
then amount to a rotation in the 2D plane of the parameters considered. This
is the case for $\ln A_{\rm T}^{2}$ and $n_{\rm S}$, for example, which
transform linearly with scale,

$\displaystyle\ln A_{\rm T}^{2}(k)$ $\displaystyle=$ $\displaystyle\ln A_{\rm
T}^{2}(k_{0})+n_{\rm T}(k_{0})\ln\frac{k}{k_{0}}\,,$ (4.1) $\displaystyle
n_{\rm S}(k)$ $\displaystyle=$ $\displaystyle n_{\rm S}(k_{0})+\frac{dn_{\rm
S}}{d\ln k}(k_{0})\ln\frac{k}{k_{0}}\,.$ (4.2)

For parameters that do not transform linearly this is not usually true. In
this case the density of models in parameter space is modified as parameters
do not keep their relative proportions when we shift with $k$. This is what
happens with $r$, where the transformation is exponential, depending on
$n_{\rm S}$, $n_{\rm T}$ and scalar running according to

$r(k)=r(k_{0})\left(\frac{k}{k_{0}}\right)^{n_{\rm T}(k_{0})-\left[n_{\rm
S}(k_{0})-1\right]-\alpha(k_{0})\ln{k/k_{0}}}.$ (4.3)

As we saw in Fig. 2 (left panel), this can substantially modify the prior
relative to assuming a uniform prior at the new scale. The net effect on the
prior in $r$, when going towards smaller scales, is a compression in $r$
regions with negative $n_{\rm T}$, raising its prior density, and expansion of
regions that correspond to positive $n_{\rm T}$ lowering their prior density.
If the parameters were well constrained data could overcome this change in
prior, but unfortunately they are not.

For our choice of smaller scale, $k=0.02{\rm\,Mpc}^{-1}$, this differentiated
sampling amounts to the posterior preferring a best-fit $n_{\rm T}$ different
by one-sigma from the one at $k=0.002{\rm\,Mpc}^{-1}$, as shown in Fig. 1. We
would expect an even larger variation in the posterior if we were to sample
$r$ at smaller scales. This change is a consequence of the variation in prior
alone, since we are not including any extra information when we sample at
smaller scales.

From a purely theoretical point of view a prior imposed at one scale is as
plausible as that imposed at the next. Unless the theoretical framework
selects for a particular scale as preferred to specify parameters at, our
results for the significantly different constraints on $r$ and $n_{\rm T}$ at
each scale are equally reasonable: a large positive tilt of the tensor power
spectrum is just as plausible as a nearly scale invariant spectrum.

One might wonder whether sampling from a log prior on $r$, or $A_{\rm T}^{2}$,
would help.111One might even suspect that $A_{\rm T}^{2}$ is already
effectively being sampled from a log prior by inheritting the prior on
$\log(A_{\rm S}^{2})$ in combination with the uniform prior on $r$. But since
$A_{\rm S}^{2}$ is tightly constrained by data its prior is fairly flat in the
region where the likelihood peaks, so the fact that $r$ has a uniform prior
means that $A_{\rm T}^{2}$ is effectively being sampled from a uniform prior
as well. However with a log prior any well-motivated lower cut-off is likely
to be at an extremely low value of $r$, and already puts most of the prior
parameter range well below future observational sensitivity. While this
doesn’t prevent a detection, in absence of a detection any observational
limits are going to be dominated by the prior rather than the data.

###  4.2 Variation of constraints with pivot — implications for inflationary
models

Figure 5: Variation of the 95% confidence limits on $r$ with scale. We compare
the variation of chains which have the consistency equation imposed (blue),
with those that have not (red). The prior on $r$ is uniform at
$0.002{\rm\,Mpc}^{-1}$ in both cases.

We now turn to the question of the scale on which constraints should be
imposed once the prior is fixed. Figure 5 shows the change in 95% confidence
limits on the tensor-to-scalar ratio when we probe at different $k$ scales. We
compare this effect on chains which have the consistency equation imposed with
those that haven’t.

The non-inflationary runs have a quite well-defined pivot point, outside of
which constraints rapidly deteriorate. The strongest constraints are obtained
at $k=0.002{\rm\,Mpc}^{-1}$, with the WMAP constraints on tensors coming
mainly from the amplitude of TT modes at low multipoles [17, 2]. Due to cosmic
variance the constraints are dominated by statistical uncertainty up to
multipoles $\ell\sim 10$ corresponding to $k\sim 0.002{\rm\,Mpc}^{-1}$. The
upper limits quickly go up as we move away from this pivot, towards smaller
scales, and allow for $r$ to become as large as 6 at $k\simeq
0.02{\rm\,Mpc}^{-1}$.

However, for the consistency equation runs, the situation is very different.
Constraints on $r$ appear robust to variation with scale in this case. This is
because enforcing the consistency relation imposes a strong correlation on
$n_{\rm S}$ and $r$, since it makes the assumption these can be described
jointly in terms of the slow-roll parameters, given in Eq. (2.5) via Eq.
(2.6). This imposition mimics apparent robust constraints on tensor quantities
at all angular scales, when in fact it is the prior rather than the data that
is ruling out sizable tensor mode contributions at all but the very largest
scales.

Figure 6: An overlay of two $n_{\rm S}$–$r$ distributions shown at
$k=0.02{\rm\,Mpc}^{-1}$, the scalar pivot scale. Green contours show
constraints when the consistency equation is enforced and red when it is not.
Relaxing the consistency equation prior decorrelates $n_{\rm S}$ and $r$ and
as a result, in the absence of the inflationary prior, $n_{\rm S}$ actually
becomes better determined.

Enforcing the consistency equation on small scales, around
$k=0.02{\rm\,Mpc}^{-1}$, excludes significant parameter space that would
otherwise be available in the $n_{\rm S}$–$r$ plane, as shown in Fig. 6. It
may be that some of this space is available in models with non-inflationary
sources for primordial gravitational waves, or in the more general set of
inflationary proposals (multiple-field, higher-order corrections) that deviate
from single-field slow roll. These are expected to have their own internal
consistency relations that do not necessarily comply with the single-field
one.

##  5 Conclusions

The simplest models of single-field slow-roll inflation predict a simple
connection between the scalar and tensor contributions to the spectra of
perturbations, given by the consistency equation. Aside from those there is a
wide variety of models that predict deviations from the simplest models, as
well as non-inflationary proposals for the origin of perturbations.

The tensor-to-scalar ratio, and therefore connections between the scalar and
tensor perturbations, are a powerful discriminator between models. We have
shown that, with the current state of knowledge, extraction of constraints on
$r$ depends significantly on our prior assumptions for the form of scalar and
tensor perturbations. We show that enforcing the consistency relation leads to
a reduction of available parameter space by a factor of 10 or larger when
quoting constraints at the usual scales around $k=0.01{\rm\,Mpc}^{-1}$ to
$0.05{\rm\,Mpc}^{-1}$ for the $n_{\rm S}$–$r$ plane. As a result, constraints
obtained under the assumption of this relation should be used for studying
models of single-field slow roll alone, for which it is valid. For other
inflation proposals an analysis based on the imposition of the consistency
equation can lead to artificial inferences.

In particular, the allowed values for the tensor-to-scalar ratio can be
significantly different from those one would expect from traditional fits.
Even when combining multiple datasets, data still allows for tensor mode
amplitudes which are several times larger than the amplitude of scalar modes
at scales around $k=0.02{\rm\,Mpc}^{-1}$. Furthermore, the ekyprotic- and
collapse-type models prediction for positive spectral index of tensor
perturbations, $n_{\rm T}>0$, is as valid as the inflationary equivalent which
predicts $n_{\rm T}<0$.

We conclude that constraints on tensors presently have significant prior
dependence, and must be interpreted with care in light of the particular
models to be studied. Even the apparently innocuous assumption of placing
uniform priors at one scale rather than another can significantly modify the
constraints obtained, whether or not the consistency equation or inequality is
imposed.

## Acknowledgments

We thank Eiichiro Komatsu and Antony Lewis for helpful discussions. M.C.
thanks the Astronomy Centre at the University of Sussex for hospitality during
this work. M.C. was supported by the Director, Office of Science, Office of
High Energy Physics, of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. A.R.L. was supported by the Science and Technology Facilities
Council [grant numbers ST/F002858/1 and ST/I000976/1]. D.P. was supported by
the Australian Research Council through a Discovery Project grant.

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##  Appendix A Wave equations in a general spacetime

This Appendix takes a deeper dive into the derivation of wave equations in a
general spacetime. Subsections A.1–A.5 give the equations of motion that lead
to wave equations for spins $s=0$, $\tfrac{1}{2}$, 1, $\tfrac{3}{2}$, and 2.
Subsection A.6 gives a general expression for derivatives of shear needed in
the wave equations for gravitational waves.

Massless waves are described by independently evolving chiral fields with all-
right-handed or all-left-handed bivector and spinor indices. For example, spin
$\tfrac{3}{2}$ waves have a bivector index and a spinor index, and those
indices are either both right-handed or both left-handed, equation (158).
Similarly, spin 2 waves have two bivector indices, and those indices are
either both right-handed or both left-handed, equation (196). Equations of
motion yield linear differential equations that relate components of adjacent
boost (and spin) weight to each other. The right- and left-handed differential
equations involve purely right- or purely left-handed Lorentz connections.
Right- and left-handed Lorentz connections are defined by (the tilde on
${\tilde{\Gamma}}$ signifies a right- or left-handed chiral component)

${\tilde{\Gamma}}_{klp}\equiv\tfrac{1}{2}\left(\Gamma_{klp}\mp
i\,\varepsilon_{kl}{}^{mn}\Gamma_{mnp}\right)\ .$ (234)

In a completely general spacetime (not necessarily a conformally separable
spacetime), the most general right- or left-handed linear differential
operators with the properties that (1) they involve purely right- or left-
handed Lorentz connections, and (2) they raise and lower the boost/spin weight
by one, are the radial ${}_{s\sigma}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}$ and angular ${}_{s\varsigma}{\cal D}_{\pm}$ operators
defined by

$\displaystyle{}_{s\sigma}{\cal D}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}$ $\displaystyle\equiv\partial_{\overset{\scriptstyle v}{\scriptstyle
u}}\pm\sigma\,\Gamma_{\overset{{\scriptstyle uvv}}{{\scriptstyle
vuu}}}+\left\\{\begin{array}[]{ll}\mp\sigma\,\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle-}{\scriptscriptstyle+}v}}{{\scriptstyle{\scriptscriptstyle+}{\scriptscriptstyle-}u}}}+(s\pm\sigma+1)\,\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}v{\scriptscriptstyle-}}}{{\scriptstyle{\scriptscriptstyle-}u{\scriptscriptstyle+}}}}&\mbox{right}\\\
\mp\sigma\,\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}{\scriptscriptstyle-}v}}{{\scriptstyle{\scriptscriptstyle-}{\scriptscriptstyle+}u}}}+(s\pm\sigma+1)\,\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle-}v{\scriptscriptstyle+}}}{{\scriptstyle{\scriptscriptstyle+}u{\scriptscriptstyle-}}}}&\mbox{left}\end{array}\right.\
,$ (235c) $\displaystyle{}_{s\varsigma}{\cal D}_{\pm}$
$\displaystyle\equiv\partial_{\pm}\mp\varsigma\,\Gamma_{\overset{{\scriptscriptstyle-}{\scriptscriptstyle+}{\scriptscriptstyle+}}{{\scriptscriptstyle+}{\scriptscriptstyle-}{\scriptscriptstyle-}}}+\left\\{\begin{array}[]{ll}\pm\varsigma\,\Gamma_{\overset{{\scriptstyle
uv{\scriptscriptstyle+}}}{{\scriptstyle
vu{\scriptscriptstyle-}}}}+(s\pm\varsigma+1)\,\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}vu}}{{\scriptstyle{\scriptscriptstyle-}uv}}}&\mbox{right}\\\
\pm\varsigma\,\Gamma_{\overset{{\scriptstyle
uv{\scriptscriptstyle+}}}{{\scriptstyle
vu{\scriptscriptstyle-}}}}+(s\pm\varsigma+1)\,\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}uv}}{{\scriptstyle{\scriptscriptstyle-}vu}}}&\mbox{left}\end{array}\right.\
.$ (235f)

Note that the boost and spin weights of the various terms of each operator
agree, as they must, per equations (32).

Explicit calculation shows that the index $s$ of the operators (235) denotes
the spin of the field, with $s=0,\tfrac{1}{2},1,\tfrac{3}{2},2$ for
respectively scalar, spinor, electromagnetic, gravitino, and gravitational
fields. The index $\sigma$ denotes the boost weight of the field, which ranges
over the $2s+1$ components $-s,-s{+}1,...,s$, while $\varsigma=\pm\sigma$
denotes the spin weight of the field, with $+$ right-handed, $-$ left-handed.
For the most part, the spin $s$ and boost/spin weight $\sigma$ and $\varsigma$
indices can be suppressed, because they equal the spin and boost/spin weight
of the field they are acting on. Equations for the various spins are (243),
(244) & (245), (246) & (247), (249) & (250), and (251) & (252).

The operators ${\cal D}$ defined by equations (235) do not commute with each
other, but (primed) operators defined by

$\displaystyle{}_{s\sigma}{\cal D}^{\prime}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}$ $\displaystyle\equiv{}_{s\sigma}{\cal
D}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}+\left\\{\begin{array}[]{ll}-\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle-}{\scriptscriptstyle+}v}}{{\scriptscriptstyle+}{\scriptscriptstyle-}{\scriptstyle
u}}}+\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle-}v{\scriptscriptstyle+}}}{{\scriptstyle{\scriptscriptstyle+}u{\scriptscriptstyle-}}}}&\mbox{right}\\\
-\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}{\scriptscriptstyle-}v}}{{\scriptscriptstyle-}{\scriptscriptstyle+}{\scriptstyle
u}}}+\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}v{\scriptscriptstyle-}}}{{\scriptstyle{\scriptscriptstyle-}u{\scriptscriptstyle+}}}}&\mbox{left}\end{array}\right.\
,$ (236c) $\displaystyle{}_{s\varsigma}{\cal D}^{\prime}_{\pm}$
$\displaystyle\equiv{}_{s\varsigma}{\cal
D}_{\pm}+\left\\{\begin{array}[]{ll}\Gamma_{\overset{{\scriptstyle
uv{\scriptscriptstyle+}}}{{\scriptstyle
vu{\scriptscriptstyle-}}}}-\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}uv}}{{\scriptstyle{\scriptscriptstyle-}vu}}}&\mbox{right}\\\
\Gamma_{\overset{{\scriptstyle vu{\scriptscriptstyle+}}}{{\scriptstyle
uv{\scriptscriptstyle-}}}}-\Gamma_{\overset{{\scriptstyle{\scriptscriptstyle+}vu}}{{\scriptstyle{\scriptscriptstyle-}uv}}}&\mbox{left}\end{array}\right.\
,$ (236f)

have the property that, in the conformally separable spacetimes, the radial
and angular ${\cal D}$ and ${\cal D}^{\prime}$ operators do commute,

${}_{s\varsigma}{\cal D}^{\prime}_{\pm}\,{}_{s\sigma}{\cal
D}_{{\overset{\scriptstyle v}{\scriptstyle u}}}-{}_{s\sigma}{\cal
D}^{\prime}_{{\overset{\scriptstyle v}{\scriptstyle u}}}\,{}_{s\varsigma}{\cal
D}_{\pm}=0\ ,$ (237)

in both right- and left-handed versions. The existence of radial/angular
operators ${\cal D}^{\prime}$ that commute with angular/radial operators
${\cal D}$ is crucial to forming wave equations for individual components of
particular boost and spin weight. In the conformally separable spacetimes, the
${\cal D}$ and ${\cal D}^{\prime}$ operators are related by

${}_{s\sigma}{\cal D}^{\prime}_{k}=\rho^{-1}{}_{s\sigma}{\cal D}_{k}\,\rho$
(238)

in both right- and left-handed versions, with $\rho$ the conformal factor,
equation (6).

In conformally separable spacetimes, the raising and lowering operators
${}_{s\sigma}\eth_{k}$ defined by equations (49) are related to the
${}_{s\sigma}{\cal D}_{k}$ operators by, in the right-handed case,

$\sqrt{2}\rho\,{}_{s\sigma}{\cal D}_{k}=(\pm)f_{s}\,\bar{\rho}^{\mp
2\sigma-1}\,{}_{\sigma}\eth_{k}\,\bar{\rho}^{\pm 2\sigma+1}\,f_{s}^{-1}\ ,$
(239)

where $f_{s}$ is given by equation (56), and the initial $(\pm)$ sign is: $+$
for $k=v$ or $+$, or $k=u$ outside the horizon; $-$ for $k=-$, or $k=u$ inside
the horizon; while the $\pm$ sign multiplying $2\sigma$ is $+$ for indices
$k=v$ or $+$, and $-$ for $k=u$ or $-$. The left-handed relation is the
complex conjugate of the right-handed relation (239), obtained by replacing
$f_{s}\rightarrow f_{s}^{\ast}$ and $\bar{\rho}\rightarrow\bar{\rho}^{\ast}$.

Wave equations are obtained by taking the differential equations linear in the
operators ${\cal D}$, equations (235), that follow from the equations of
motion, applying operators ${\cal D}^{\prime}$, equations (236), and
differencing the resulting second order differential equations in such a way
that the commutation (237) leads to cancellation of terms. The resulting
equations can be expressed in terms of the difference of the radial and
angular wave operators $\square$ defined by equations (59). For arbitrary
spins $s$ and boost weights $\sigma$, the difference
$\square_{\overset{\scriptstyle vu}{\scriptstyle
uv}}-\square_{\overset{\raisebox{0.0pt}[1.80835pt][0.0pt]{$\scriptstyle+-$}}{\raisebox{0.0pt}[1.80835pt][0.0pt]{$\scriptstyle-+$}}}$
of radial and angular wave operators, expressed in terms of the ${\cal D}$ and
${\cal D}^{\prime}$ operators, and then in terms of the $\eth$ raising and
lower operators (49) are, for right-handed ($\varsigma=+\sigma$) modes,

$\displaystyle\left({}_{\sigma}\square_{\overset{\scriptstyle vu}{\scriptstyle
uv}}-{}_{\sigma}\square_{\overset{\raisebox{0.0pt}[1.80835pt][0.0pt]{$\scriptstyle+-$}}{\raisebox{0.0pt}[1.80835pt][0.0pt]{$\scriptstyle-+$}}}\right){\hat{\psi}}_{\sigma}$
$\displaystyle=2\rho^{2}f_{s}^{-1}\left[\left(-\,{}_{s,\sigma\mp 1}{\cal
D}^{\prime}_{{\overset{\scriptstyle v}{\scriptstyle u}}}\,{}_{s,\sigma}{\cal
D}_{{\overset{\scriptstyle u}{\scriptstyle v}}}+{}_{s,\sigma\mp 1}{\cal
D}^{\prime}_{\pm}\,{}_{s,\sigma}{\cal
D}_{\mp}\right)+2(\sigma\mp\tfrac{1}{2})(\sigma\mp
1){\tilde{C}}_{0}\right]f_{s}{\hat{\psi}}_{\sigma}$
$\displaystyle=\Bigl{[}\bar{\rho}^{-(\pm
2\sigma-1)}\left(-\operatorname{\textrm{sgn}}(\Delta_{x})\,{}_{\sigma\mp
1}\eth_{{\overset{\scriptstyle v}{\scriptstyle u}}}\,\bar{\rho}^{2(\pm
2\sigma-1)}\,{}_{\sigma}\eth_{{\overset{\scriptstyle u}{\scriptstyle
v}}}-{}_{\sigma\mp 1}\eth_{\pm}\,\bar{\rho}^{2(\pm
2\sigma-1)}\,{}_{\sigma}\eth_{\mp}\right)\bar{\rho}^{-(\pm 2\sigma-1)}$
$\displaystyle\quad\ \ +4\rho^{2}(\sigma\mp\tfrac{1}{2})(\sigma\mp
1){\tilde{C}}_{0}\Bigr{]}{\hat{\psi}}_{\sigma}\ .$ (240)

The equivalent result for left-handed modes ($\varsigma=-\sigma$) is obtained
by flipping angular indices $+\leftrightarrow-$, and complex-conjugating
$f_{s}\rightarrow f_{s}^{\ast}$ and $\bar{\rho}\rightarrow\bar{\rho}^{\ast}$,

$\displaystyle\left({}_{\sigma}\square_{\overset{\scriptstyle vu}{\scriptstyle
uv}}-{}_{\sigma}\square_{\overset{\raisebox{0.0pt}[1.80835pt][0.0pt]{$\scriptstyle-+$}}{\raisebox{0.0pt}[1.80835pt][0.0pt]{$\scriptstyle+-$}}}\right){\hat{\psi}}_{\sigma}$
$\displaystyle=2\rho^{2}(f_{s}^{\ast})^{-1}\left[\left(-\,{}_{s,\sigma\mp
1}{\cal D}^{\prime}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}\,{}_{s,\sigma}{\cal D}_{{\overset{\scriptstyle u}{\scriptstyle
v}}}+{}_{s,\sigma\pm 1}{\cal D}^{\prime}_{\mp}\,{}_{s,\sigma}{\cal
D}_{\pm}\right)+2(\sigma\mp\tfrac{1}{2})(\sigma\mp
1){\tilde{C}}_{0}\right]f_{s}^{\ast}{\hat{\psi}}_{\sigma}$
$\displaystyle=\Bigl{[}(\bar{\rho}^{\ast})^{-(\pm
2\sigma-1)}\left(-\operatorname{\textrm{sgn}}(\Delta_{x})\,{}_{\sigma\mp
1}\eth_{{\overset{\scriptstyle v}{\scriptstyle
u}}}\,(\bar{\rho}^{\ast})^{2(\pm
2\sigma-1)}\,{}_{\sigma}\eth_{{\overset{\scriptstyle u}{\scriptstyle
v}}}-{}_{\sigma\pm 1}\eth_{\mp}\,(\bar{\rho}^{\ast})^{2(\pm
2\sigma-1)}\,{}_{\sigma}\eth_{\pm}\right)(\bar{\rho}^{\ast})^{-(\pm
2\sigma-1)}$ $\displaystyle\quad\ \
+4\rho^{2}(\sigma\mp\tfrac{1}{2})(\sigma\mp
1){\tilde{C}}_{0}\Bigr{]}{\hat{\psi}}_{\sigma}\ .$ (241)

###  A.1 Spin 0 waves

In a general spacetime, the d’Alembertian operator that goes in the scalar
wave equation (93) is

$D^{k}D_{k}\varphi=\left(-\,{}_{1,-1}{\cal
D}^{\prime}_{v}\,\partial_{u}-{}_{1,+1}{\cal
D}^{\prime}_{u}\,\partial_{v}+{}_{1,-1}{\cal
D}^{\prime}_{+}\,\partial_{-}+{}_{1,+1}{\cal
D}^{\prime}_{-}\,\partial_{+}\right)\varphi\ ,$ (242)

which holds true for both right- and left-handed versions of the ${\cal
D}^{\prime}$ operators (236). In conformally separable spacetimes, equation
(242) can be recast as

$D^{k}D_{k}\varphi=2\left(-\ {}_{0,\mp 1}{\cal
D}^{\prime}_{{\overset{\scriptstyle v}{\scriptstyle u}}}\ {}_{0,0}{\cal
D}_{{\overset{\scriptstyle u}{\scriptstyle v}}}\,+\,{}_{0,\mp 1}{\cal
D}^{\prime}_{\pm}\ {}_{0,0}{\cal
D}_{\mp}\,+\,{\tilde{C}}_{0}\,+\,\tfrac{1}{12}R\right)\varphi\ ,$ (243)

which again holds true for both right- and left-handed versions of the ${\cal
D}$ and ${\cal D}^{\prime}$ operators. Note that whereas the index $s$ on
${}_{s\sigma}{\cal D}^{\prime}_{k}$ is $s=1$ in the general equation (242),
the index is $s=0$ on ${}_{s\sigma}{\cal D}_{k}$ and ${}_{s\sigma}{\cal
D}^{\prime}_{k}$ in the conformally separable version (243).

###  A.2 Spin $\tfrac{1}{2}$ waves

In a general spacetime, the Dirac equation (95) for the components
$\psi_{\sigma}$ of the spinor field, equation (120), expressed with respect to
a Newman-Penrose tetrad in terms of the differential operators defined by
equations (235), are, for the right-handed components,

$\displaystyle{1\over\sqrt{2}}(\bm{D}\psi)_{\overset{\scriptscriptstyle{\Uparrow}{\downarrow}}{\scriptscriptstyle{\Downarrow}{\uparrow}}}$
$\displaystyle={}_{\mp 1/2}{\cal D}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}\,\psi_{\overset{\scriptscriptstyle{\Downarrow}{\downarrow}}{\scriptscriptstyle{\Uparrow}{\uparrow}}}-{}_{\pm
1/2}{\cal
D}_{\mp}\,\psi_{\overset{\scriptscriptstyle{\Uparrow}{\uparrow}}{\scriptscriptstyle{\Downarrow}{\downarrow}}}$
$\displaystyle=0\ ,$ (244a)

and for the left-handed components,

$\displaystyle{1\over\sqrt{2}}(\bm{D}\psi)_{\overset{\scriptscriptstyle{\Uparrow}{\uparrow}}{\scriptscriptstyle{\Downarrow}{\downarrow}}}$
$\displaystyle=-\,{}_{\mp 1/2}{\cal D}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}\,\psi_{\overset{\scriptscriptstyle{\Downarrow}{\uparrow}}{\scriptscriptstyle{\Uparrow}{\downarrow}}}-{}_{\mp
1/2}{\cal
D}_{\pm}\,\psi_{\overset{\scriptscriptstyle{\Uparrow}{\downarrow}}{\scriptscriptstyle{\Downarrow}{\uparrow}}}$
$\displaystyle=0\ .$ (245a)

###  A.3 Spin 1 waves

In a general spacetime, Maxwell’s equations (116) for the components
${\tilde{F}}_{\sigma}$ of the electromagnetic field, equation (120), expressed
with respect to a Newman-Penrose tetrad in terms of the differential operators
defined by equations (235), are, for the right-handed components
($\varsigma=+\sigma$),

$\displaystyle D^{k}{\tilde{F}}_{\overset{{\scriptstyle k}{\scriptstyle
v}}{k{\scriptstyle u}}}$
$\displaystyle=\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,{\tilde{F}}_{\mp 1}\,\pm\,{}_{0}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,{\tilde{F}}_{0}-{}_{\pm 1}{\cal
D}_{\mp}\,{\tilde{F}}_{\pm 1}$
$\displaystyle={\tilde{j}}_{\overset{\scriptstyle v}{\scriptstyle u}}\ ,$
(246a) $\displaystyle D^{k}{\tilde{F}}_{\overset{{\scriptstyle
k}{\scriptscriptstyle+}}{k{\scriptscriptstyle-}}}$
$\displaystyle=\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,{\tilde{F}}_{\mp 1}\,\pm\,{}_{0}{\cal
D}_{\pm}\,{\tilde{F}}_{0}-{}_{\pm 1}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,{\tilde{F}}_{\pm 1}$ $\displaystyle={\tilde{j}}_{\pm}\
,$ (246b)

and for the left-handed components ($\varsigma=-\sigma$),

$\displaystyle D^{k}{\tilde{F}}_{\overset{{\scriptstyle k}{\scriptstyle
v}}{k{\scriptstyle u}}}$
$\displaystyle=\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,{\tilde{F}}_{\mp 1}\,\pm\,{}_{0}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,{\tilde{F}}_{0}-{}_{\mp 1}{\cal
D}_{\pm}\,{\tilde{F}}_{\pm 1}$
$\displaystyle={\tilde{j}}_{\overset{\scriptstyle v}{\scriptstyle u}}\ ,$
(247a) $\displaystyle D^{k}{\tilde{F}}_{\overset{{\scriptstyle
k}{\scriptscriptstyle-}}{k{\scriptscriptstyle+}}}$
$\displaystyle=\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,{\tilde{F}}_{\mp 1}\,\pm\,{}_{0}{\cal
D}_{\mp}\,{\tilde{F}}_{0}-{}_{\pm 1}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,{\tilde{F}}_{\pm 1}$ $\displaystyle={\tilde{j}}_{\pm}\
.$ (247b)

The units of the currents $j_{k}$ are Heaviside; in Gaussian units the
currents would be multiplied by $4\pi$. Conservation of electric current is
expressed by

$D^{k}j_{k}=-\,{}_{+1}{\cal D}^{\prime}_{u}\,j_{v}-{}_{-1}{\cal
D}^{\prime}_{v}\,j_{u}+{}_{-1}{\cal
D}^{\prime}_{+}\,j_{\scriptscriptstyle-}+{}_{+1}{\cal
D}^{\prime}_{-}\,j_{\scriptscriptstyle+}=0\ .$ (248)

###  A.4 Spin $\tfrac{3}{2}$ waves

Wave equations for the spin $\tfrac{3}{2}$ gravitino field $\bm{\Psi}$ follow
from the Jacobi identity (169). In a general spacetime, the left hand sides of
the Jacobi identity (169) in terms of the differential operators defined by
equations (235), are, for the right-handed components,

$\displaystyle\frac{1}{\sqrt{2}}\bigl{(}\bm{D}\tilde{\Psi}_{\overset{{\scriptstyle
v}{\scriptstyle u}}{{\scriptstyle u}{\scriptstyle
v}}}\bigr{)}_{\overset{\scriptscriptstyle{\Uparrow}{\downarrow}}{\scriptscriptstyle{\Downarrow}{\uparrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptstyle v}\scriptscriptstyle{\Downarrow}{\downarrow}}{{\scriptstyle
k}{\scriptstyle
u}\scriptscriptstyle{\Uparrow}{\uparrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptscriptstyle-}\scriptscriptstyle{\Downarrow}{\downarrow}}{{\scriptstyle
k}{\scriptscriptstyle+}\scriptscriptstyle{\Uparrow}{\uparrow}}}$
$\displaystyle=\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,\tilde{\Psi}_{\mp 3/2}\pm{}_{\mp 1/2}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,\tilde{\Psi}_{\mp 1/2}\mp{}_{\pm 1/2}{\cal
D}_{\mp}\,\tilde{\Psi}_{\pm
1/2}+\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}}\,\tilde{\Psi}_{\pm 3/2}\ ,$ (249a)
$\displaystyle\frac{1}{\sqrt{2}}\bigl{(}\bm{D}\tilde{\Psi}_{\overset{{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle
u}{\scriptscriptstyle-}}}\bigr{)}_{\overset{\scriptscriptstyle{\Uparrow}{\downarrow}}{\scriptscriptstyle{\Downarrow}{\uparrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptstyle v}\scriptscriptstyle{\Uparrow}{\uparrow}}{{\scriptstyle
k}{\scriptstyle u}\scriptscriptstyle{\Downarrow}{\downarrow}}}$
$\displaystyle=\mp\,2\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,\tilde{\Psi}_{\mp 1/2}\pm{}_{\pm 1/2}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,\tilde{\Psi}_{\pm 1/2}-{}_{\pm 3/2}{\cal
D}_{\mp}\,\tilde{\Psi}_{\pm 3/2}\ ,$ (249b)
$\displaystyle-\frac{1}{\sqrt{2}}\bigl{(}\bm{D}\tilde{\Psi}_{\overset{{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle
u}{\scriptscriptstyle-}}}\bigr{)}_{\overset{\scriptscriptstyle{\Downarrow}{\uparrow}}{\scriptscriptstyle{\Uparrow}{\downarrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptscriptstyle+}\scriptscriptstyle{\Uparrow}{\uparrow}}{{\scriptstyle
k}{\scriptscriptstyle-}\scriptscriptstyle{\Downarrow}{\downarrow}}}$
$\displaystyle=\mp\,2\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,\tilde{\Psi}_{\mp 1/2}\pm{}_{\pm 1/2}{\cal
D}_{\pm}\,\tilde{\Psi}_{\pm 1/2}-{}_{\pm 3/2}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,\tilde{\Psi}_{\pm 3/2}\ ,$ (249c)

and for the left-handed components,

$\displaystyle-\frac{1}{\sqrt{2}}\bigl{(}\bm{D}\tilde{\Psi}_{\overset{{\scriptstyle
v}{\scriptstyle u}}{{\scriptstyle u}{\scriptstyle
v}}}\bigr{)}_{\overset{\scriptscriptstyle{\Uparrow}{\uparrow}}{\scriptscriptstyle{\Downarrow}{\downarrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptstyle v}\scriptscriptstyle{\Downarrow}{\uparrow}}{{\scriptstyle
k}{\scriptstyle u}\scriptscriptstyle{\Uparrow}{\downarrow}}}=\pm
D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptscriptstyle+}\scriptscriptstyle{\Uparrow}{\downarrow}}{{\scriptstyle
k}{\scriptscriptstyle-}\scriptscriptstyle{\Downarrow}{\uparrow}}}$
$\displaystyle=\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,\tilde{\Psi}_{\mp 3/2}\pm{}_{\mp 1/2}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,\tilde{\Psi}_{\mp 1/2}\pm{}_{\mp 1/2}{\cal
D}_{\pm}\,\tilde{\Psi}_{\pm
1/2}-\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}}\,\tilde{\Psi}_{\pm 3/2}\ ,$ (250a)
$\displaystyle\frac{1}{\sqrt{2}}\bigl{(}\bm{D}\tilde{\Psi}_{\overset{{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptstyle
u}{\scriptscriptstyle+}}}\bigr{)}_{\overset{\scriptscriptstyle{\Uparrow}{\uparrow}}{\scriptscriptstyle{\Downarrow}{\downarrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptstyle v}\scriptscriptstyle{\Uparrow}{\downarrow}}{{\scriptstyle
k}{\scriptstyle u}\scriptscriptstyle{\Downarrow}{\uparrow}}}$
$\displaystyle=\pm\,2\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,\tilde{\Psi}_{\mp 1/2}\pm{}_{\pm 1/2}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,\tilde{\Psi}_{\pm 1/2}-{}_{\mp 3/2}{\cal
D}_{\pm}\,\tilde{\Psi}_{\pm 3/2}\ ,$ (250b)
$\displaystyle\frac{1}{\sqrt{2}}\bigl{(}\bm{D}\tilde{\Psi}_{\overset{{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptstyle
u}{\scriptscriptstyle+}}}\bigr{)}_{\overset{\scriptscriptstyle{\Downarrow}{\downarrow}}{\scriptscriptstyle{\Uparrow}{\uparrow}}}=D^{k}\tilde{\Psi}_{\overset{{\scriptstyle
k}{\scriptscriptstyle-}\scriptscriptstyle{\Uparrow}{\downarrow}}{{\scriptstyle
k}{\scriptscriptstyle+}\scriptscriptstyle{\Downarrow}{\uparrow}}}$
$\displaystyle=\pm\,2\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,\tilde{\Psi}_{\mp 1/2}\pm{}_{\mp 1/2}{\cal
D}_{\mp}\,\tilde{\Psi}_{\pm 1/2}-{}_{\pm 3/2}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,\tilde{\Psi}_{\pm 3/2}\ .$ (250c)

###  A.5 Spin 2 waves

In a general spacetime, the Weyl evolution equations (197) in terms of the
differential operators defined by equations (235), are, for right-handed
components,

$\displaystyle D^{k}{\tilde{C}}_{\overset{{\scriptstyle
k}{\scriptscriptstyle-}{\scriptstyle v}{\scriptscriptstyle+}}{{\scriptstyle
k}{\scriptscriptstyle+}{\scriptstyle u}{\scriptscriptstyle-}}}$
$\displaystyle=-\,2\,\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,{\tilde{C}}_{\mp 1}+{}_{0}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,{\tilde{C}}_{0}-{}_{\pm 1}{\cal
D}_{\mp}\,{\tilde{C}}_{\pm
1}+\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle-}}}\ ,$ (251a) $\displaystyle
D^{k}{\tilde{C}}_{\overset{{\scriptstyle k}{\scriptstyle u}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle k}{\scriptstyle v}{\scriptstyle
u}{\scriptscriptstyle-}}}$
$\displaystyle=-\,2\,\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,{\tilde{C}}_{\mp 1}+{}_{0}{\cal
D}_{\pm}\,{\tilde{C}}_{0}-{}_{\pm 1}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,{\tilde{C}}_{\pm
1}+\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
uu}}{{\scriptscriptstyle+}{\scriptstyle vv}}}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptstyle u}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle v}{\scriptstyle
u}{\scriptscriptstyle-}}}\ ,$ (251b) $\displaystyle
D^{k}{\tilde{C}}_{\overset{{\scriptstyle k}{\scriptstyle v}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle k}{\scriptstyle u}{\scriptstyle
u}{\scriptscriptstyle-}}}$
$\displaystyle=-\,3\,\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,{\tilde{C}}_{0}+{}_{\pm 1}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,{\tilde{C}}_{\pm 1}-{}_{\pm 2}{\cal
D}_{\mp}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptstyle v}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle u}{\scriptstyle
u}{\scriptscriptstyle-}}}\ ,$ (251c) $\displaystyle
D^{k}{\tilde{C}}_{\overset{{\scriptstyle k}{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle k}{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}$
$\displaystyle=-\,3\,\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,{\tilde{C}}_{0}+{}_{\pm 1}{\cal
D}_{\pm}\,{\tilde{C}}_{\pm 1}-{}_{\pm 2}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\ ,$ (251d)

and for left-handed components,

$\displaystyle D^{k}{\tilde{C}}_{\overset{{\scriptstyle
k}{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle-}}{k{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle+}}}$
$\displaystyle=2\,\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,{\tilde{C}}_{\mp 1}+{}_{0}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,{\tilde{C}}_{0}+{}_{\mp 1}{\cal
D}_{\pm}\,{\tilde{C}}_{\pm
1}+\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle+}}}\ ,$ (252a) $\displaystyle
D^{k}{\tilde{C}}_{\overset{{\scriptstyle k}{\scriptstyle u}{\scriptstyle
v}{\scriptscriptstyle-}}{k{\scriptstyle v}{\scriptstyle
u}{\scriptscriptstyle+}}}$
$\displaystyle=2\,\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,{\tilde{C}}_{\mp 1}+{}_{0}{\cal
D}_{\mp}\,{\tilde{C}}_{0}+{}_{\pm 1}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,{\tilde{C}}_{\pm
1}+\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
uu}}{{\scriptscriptstyle-}{\scriptstyle vv}}}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptstyle u}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptstyle v}{\scriptstyle
u}{\scriptscriptstyle+}}}\ ,$ (252b) $\displaystyle
D^{k}{\tilde{C}}_{\overset{{\scriptstyle k}{\scriptstyle v}{\scriptstyle
v}{\scriptscriptstyle-}}{k{\scriptstyle u}{\scriptstyle
u}{\scriptscriptstyle+}}}$
$\displaystyle=-\,3\,\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,{\tilde{C}}_{0}-{}_{\pm 1}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,{\tilde{C}}_{\pm 1}-{}_{\mp 2}{\cal
D}_{\pm}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptstyle v}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptstyle u}{\scriptstyle
u}{\scriptscriptstyle+}}}\ ,$ (252c) $\displaystyle
D^{k}{\tilde{C}}_{\overset{{\scriptstyle k}{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{k{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}$
$\displaystyle=-\,3\,\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,{\tilde{C}}_{0}-{}_{\mp 1}{\cal
D}_{\mp}\,{\tilde{C}}_{\pm 1}-{}_{\pm 2}{\cal D}_{{\overset{\scriptstyle
u}{\scriptstyle v}}}\,{\tilde{C}}_{\pm 2}$
$\displaystyle={\tilde{J}}_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\ .$ (252d)

###  A.6 Derivatives of shear

The Newman-Penrose formalism makes a 2+2 split of the tangent space of
spacetime into a radial subspace (null indices $v$ and $u$) and an angular
subspace (angular indices $+$ and $-$). Conformally separable black-hole
spacetimes are shear-free, meaning that the eight Lorentz connections of the
form $\Gamma_{aza}$, where $a$ and $z$ are from opposite spaces, are all zero.
Although the shears all vanish in the unperturbed background, their
derivatives yield spin $\pm 2$ components of the Riemann tensor (compare [46]
p. 431, last of eqs. (3) and (4)):

$\displaystyle{}_{s,\varsigma\pm 1}{\cal
D}^{\prime}_{\pm}\left(\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,\psi_{s\sigma\varsigma}\right)-{}_{s,\sigma\pm 1}{\cal
D}^{\prime}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}\left(\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,\psi_{s\sigma\varsigma}\right)-\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle-}{\scriptstyle u}{\scriptstyle
u}}}\,{}_{s\varsigma}{\cal
D}_{\pm}\,\psi_{s\sigma\varsigma}+\Gamma_{\overset{{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,{}_{s\sigma}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,\psi_{s\sigma\varsigma}$
$\displaystyle=R_{\overset{{\scriptstyle v}{\scriptscriptstyle+}{\scriptstyle
v}{\scriptscriptstyle+}}{{\scriptstyle u}{\scriptscriptstyle-}{\scriptstyle
u}{\scriptscriptstyle-}}}\,\psi_{s\sigma\varsigma}$
$\displaystyle\quad\mbox{right}\ ,$ (253a) $\displaystyle{}_{s,\varsigma\mp
1}{\cal
D}^{\prime}_{\mp}\left(\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,\psi_{s\sigma\varsigma}\right)-{}_{s,\sigma\pm 1}{\cal
D}^{\prime}_{{\overset{\scriptstyle v}{\scriptstyle
u}}}\left(\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,\psi_{s\sigma\varsigma}\right)-\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptstyle v}}{{\scriptscriptstyle+}{\scriptstyle u}{\scriptstyle
u}}}\,{}_{s\varsigma}{\cal
D}_{\mp}\,\psi_{s\sigma\varsigma}+\Gamma_{\overset{{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,{}_{s\sigma}{\cal D}_{{\overset{\scriptstyle
v}{\scriptstyle u}}}\,\psi_{s\sigma\varsigma}$
$\displaystyle=R_{\overset{{\scriptstyle v}{\scriptscriptstyle-}{\scriptstyle
v}{\scriptscriptstyle-}}{{\scriptstyle u}{\scriptscriptstyle+}{\scriptstyle
u}{\scriptscriptstyle+}}}\,\psi_{s\sigma\varsigma}$
$\displaystyle\quad\mbox{left}\ .$ (253b)

Equations (253) are valid in an arbitrary spacetime for arbitrary spin $s$ and
boost/spin weight $\sigma$ and $\varsigma$. The case relevant to gravitational
waves, equation (227), has $\psi_{s\sigma\varsigma}={\tilde{C}}_{0}$, for
which $s=\sigma=\varsigma=0$.

##  Appendix B A relation among differential operators

In the conformally separable spacetimes considered in this paper, the
differential operators $\eth_{k}$ defined by equations (49), acting on any
arbitrary (not necessarily separable) function of the coordinates
$\\{x,t,y,\phi\\}$, satisfy the following cubic relations:

$\displaystyle\bar{\rho}\eth_{l}\bar{\rho}^{-1}\eth_{k}^{2}$
$\displaystyle=\bar{\rho}^{-1}\eth_{k}\bar{\rho}\left[\eth_{k}\eth_{l}-\frac{1}{\bar{\rho}}\left(\pm_{k}\operatorname{\textrm{sgn}}(\Delta_{x})R^{2}\sqrt{|\Delta_{x}|}\eth_{l}\pm_{l}ia\sqrt{\Delta_{y}}\eth_{k}\right)\right]\
,$ (254a) $\displaystyle\bar{\rho}\eth_{k}\bar{\rho}^{-1}\eth_{l}^{2}$
$\displaystyle=\bar{\rho}^{-1}\eth_{l}\bar{\rho}\left[\eth_{k}\eth_{l}-\frac{1}{\bar{\rho}}\left(\pm_{k}\operatorname{\textrm{sgn}}(\Delta_{x})R^{2}\sqrt{|\Delta_{x}|}\eth_{l}\pm_{l}ia\sqrt{\Delta_{y}}\eth_{k}\right)\right]\
,$ (254b)

where the index $k$ is either of $v$ or $u$, and the index $l$ is either of
$+$ or $-$. The $\pm_{k}$ sign is $+$ or $-$ as $k=v$ or $u$, while the
$\pm_{l}$ sign is $+$ or $-$ as $l=+$ or $-$. The relations (254) lead to the
expressions (146) for the boost weight 0 component ${\tilde{F}}_{0}$ of the
electromagnetic field.

##  Appendix C Comparison to Chandrasekhar [46] notation

Chandrasekhar’s null directions $\bm{l}$, $\bm{n}$, $\bm{m}$, $\bar{\bm{m}}$,
p. 41 eq. (283), correspond to

$\bm{l}=\bm{\gamma}_{v}\ ,\quad\bm{n}=\bm{\gamma}_{u}\
,\quad\bm{m}=\bm{\gamma}_{+}\ ,\quad\bar{\bm{m}}=\bm{\gamma}_{-}\ .$ (255)

Chandrasekhar’s null indices 1,2,3,4 are

$1=v\ ,\quad 2=u\ ,\quad 3=+\ ,\quad 4=-\ .$ (256)

Chandrasekhar’s spin coefficients $\gamma_{klm}$, p. 37 eq. (253), coincide
with the Lorentz connections $\Gamma_{klm}$,

$\gamma_{klm}=\Gamma_{klm}\ .$ (257)

Chandrasekhar’s electromagnetic fields $\phi_{i}$ are, p. 51 eq. (324),

$\displaystyle\phi_{0}$ $\displaystyle=F_{+1}$
$\displaystyle\equiv{\tilde{F}}_{v+}\ ,$ (258a) $\displaystyle\phi_{1}$
$\displaystyle={\tilde{F}}_{0}$
$\displaystyle\equiv\tfrac{1}{2}(F_{vu}-F_{+-})\ ,$ (258b)
$\displaystyle\phi_{2}$ $\displaystyle={\tilde{F}}_{-1}$ $\displaystyle\equiv
F_{u-}\ ,$ (258c)

and his gravitational fields $\Psi_{i}$ are, p. 43 eq (294),

$\displaystyle-\Psi_{0}$ $\displaystyle={\tilde{C}}_{+2}$ $\displaystyle\equiv
C_{v+v+}\ ,$ (259a) $\displaystyle-\Psi_{1}$ $\displaystyle={\tilde{C}}_{+1}$
$\displaystyle\equiv C_{vuv+}=C_{-+v+}\ ,$ (259b) $\displaystyle-\Psi_{2}$
$\displaystyle={\tilde{C}}_{0}$ $\displaystyle\equiv
C_{v+-u}=\tfrac{1}{2}(C_{vuvu}-C_{vu+-})=\tfrac{1}{2}(C_{+-+-}-C_{vu+-})\ ,$
(259c) $\displaystyle-\Psi_{3}$ $\displaystyle={\tilde{C}}_{-1}$
$\displaystyle\equiv C_{uvu-}=C_{+-u-}\ ,$ (259d) $\displaystyle-\Psi_{4}$
$\displaystyle={\tilde{C}}_{-2}$ $\displaystyle\equiv C_{u-u-}\ .$ (259e)

Chandrasekhar’s operators $\mathscr{D}_{n}$, $\mathscr{D}^{\prime}_{n}$,
$\mathscr{L}_{n}$, $\mathscr{L}^{\prime}_{n}$, p. 383 eq. (3), are related to
the raising and lowering operators (49) of the present paper by

$\mathscr{D}_{n}={1\over R^{2}\sqrt{\Delta_{x}}}\,{}_{-2n}\eth_{v}\
,\quad\mathscr{D}^{\prime}_{n}=-{1\over
R^{2}\sqrt{\Delta_{x}}}\,{}_{2n}\eth_{u}\
,\quad\mathscr{L}_{n}=-\,{}_{n}\eth_{-}\
,\quad\mathscr{L}^{\prime}_{n}={}_{-n}\eth_{+}\ .$ (260)