\section{Introduction}
\label{sec:intro}

The recent detection of gravitational waves (GWs) has consecrated gravitational wave interferometers (GWIs) as a vital experimental tool for astronomy, cosmology, and fundamental physics~\cite{LIGOScientific:2018mvr,Abbott:2020niy}.  The GWs detected thus far are a product of cataclysmic transient events, such as binary black hole mergers.  These signals are strong, with gravitational strain of the order of \(h\sim10^{-21}\), but very short, from a fraction of a second to several seconds.  Much weaker signals can be detected if they are coherent over a longer time, such as the continuous GWs (CWs) emitted by rapidly spinning neutron stars~\cite{Riles:2017evm} or ultra-compact Galactic binaries~\cite{Nelemans:2001hp}.  In the former case, recent searches for this type of signal have been performed in~\cite{Pisarski:2019vxw,Dergachev:2020fli,Steltner:2020hfd}.  Having detected no CWs, this set the upper limit \(h\sim10^{-25}\) on the maximum strain for this type of signal at frequencies of about \(f\sim10^2\)~Hz.

Another important source of CWs is due to the scattering of ultra-light bosons off black holes via a mechanism known as superradiance~\cite{Brito_2020}.  Bosons with masses \(m\ll1\)~eV, are predicted in theories beyond the Standard Model of particle physics, and are an excellent candidate for the cosmological dark matter dubbed ultra-light dark matter (ULDM)~\cite{Preskill:1982cy,Abbott:1982af,Dine:1982ah,Turner:1983he,Nelson:2011sf,Ferreira:2020fam}.  In particular, spin-2 ULDM is especially interesting because it arises as a modification of gravity itself, even though it is in the guise of an additional particle, the dark matter~\cite{Marzola:2017,Aoki:2017cnz}.  Searches for CWs produced by superradiance have been carried out only for spin-0 bosons~\cite{Palomba:2019vxe,Ng:2020ruv}, whereas no limits on the properties of spin-2 ultra-light bosons with GWI data exist yet.

In this work we show that, if ULDM has spin two, it interacts with GWIs in a way that, owing to its quasi-monochromaticity and persistence, closely resembles CWs.  The spin-2 ULDM-CW signal can be detected by existing Earth-based facilities such as advanced LIGO~\cite{TheLIGOScientific:2014jea} / advanced Virgo~\cite{TheVirgo:2014hva} (HLV) in their entire accessible frequency range, approximately corresponding to masses \(4\times10^{-14}~\mathrm{eV}\lesssim m \lesssim4\times10^{-11}~\mathrm{eV}\).  Furthermore, planned facilities such as LISA~\cite{Baker:2019nia}, DECIGO~\cite{Seto:2001qf}, and the BBO~\cite{Harry:2006fi} will extend this range down to \(m\sim\mathrm{few}\times10^{-19}~\mathrm{eV}\).  The spin-2 ULDM-CW signal is produced by the coherent oscillations of the ULDM field which is universally coupled to Standard Model fields and is unrelated to superradiance; this is similar to dark photon dark matter, where the ULDM carries additional interactions~\cite{Pierce:2018xmy,Miller:2020vsl}\footnote{Other types of direct interactions between ULDM and matter have also been considered, see, e.g., \cite{Arvanitaki:2014faa,Morisaki:2018htj,Grote:2019uvn,Michimura:2020vxn}} (notice however that in the spin-2 case the interaction can not be tuned away).  Moreover, regardless the spin, if the ULDM field only interacts gravitationally, the signal is undetectable by GWIs~\cite{Aoki:2016kwl}.  Our findings demonstrate that, in case of a null result, GWIs can place some of the most stringent bounds on the spin-2 Yukawa fifth force strength \(\al\) in the frequency ranges accessible by GWIs.

This paper is structured as follows.  In Section~\ref{sec:maths} we compute the strength and shape of the expected signal from spin-2 ULDM for the frequency ranges of interest for GWIs.  In Section~\ref{sec:res} we present our results, and in Section~\ref{sec:out} we put them into context and give an outlook for future work.  We work with units in which \(c=k_B=\hslash=1\), and use latin indices \((i,j,\ldots)\in[1,3]\) for spatial tensor components.

\section{The shape and strength of the signal}
\label{sec:maths}

The behaviour of the spin-2 ULDM in sufficiently small regions inside the local dark matter halo is described by the oscillating tensor field~\cite{Marzola:2017}
\begin{align}
    \Mij(t) &= \frac{\sqrt{2\rhoDM}}{m}\cos{(mt+\Upsilon)}\epij(\vr) \,, \label{eq:Mij}
\end{align}
where \(\rhoDM\) is the observed local dark matter energy density, for which we assume   \(\rhoDM=0.3\)~GeV/cm\(^3\)~\cite{Piffl:2014mfa,Evans:2018bqy,2015ApJ...814...13M}, and \(\Upsilon\) is a random phase.  The five polarisations of the spin-2 field are encoded in the \(\epij(\vr)\) tensor, which has unit norm and zero trace, is symmetric and is direction-dependent via the unit vector \(\vr\)~\cite{Maggiore:1900zz}.  The solution \Eq{eq:Mij} assumes a single frequency \(2\pi f=m\) and a coherent polarisation structure.  The latter is justified for scales shorter than the characteristic scale of the inhomogeneities of the ULDM field, which is given by the de~Broglie wavelength \(\ldB \deq 2\pi/mv = 1/fv\) where \(v\sim 10^{-3}\) is the effective velocity of the ULDM.  Thus, owing to the fact that \(\ldB\) is much larger than the physical size of the GWIs and the distance between the HLV sites, we can safely neglect gradients (see~\cite{Armaleo:2020yml} for further discussion).  The coherence of the oscillation frequency is instead guaranteed up to a coherence time that is given by\footnote{Notice the definition of the coherence time differs from the one that is commonly used in the ULDM literature by a factor~4.  We adopt the definition used in the GW literature here.} \(\tcoh \deq 4\pi/mv^2 = 2/fv^2\).  Given that a typical GWI observation run will last for much longer than \(\tcoh\), a more precise description of the ULDM field would be a superposition of plane waves, see~\cite{Pierce:2018xmy,Miller:2020vsl}; we neglect this for our order of magnitude estimates\footnote{This solution is also valid provided that the energy, or frequency, scales of the system is well below the ultra-violet cutoff of the effective field theory, that is \(f \sim m/2\pi \ll (M_\text{P}m^2)^{1/3}\)~\cite{Akrami:2015qga}; this is easily verified for all the values of the spin-2 mass \(m\) we consider in this work.}.

In the ULDM reference frame \((\vp,\vq,\vr)\) the polarisations of the spin-2 field can be described as \(\epij(\vr) \deq \sum_\kappa \vep_\kappa {\cal Y}^\kappa_{ij}(\vr)\)~\cite{Maggiore:1900zz,Armaleo:2019gil}, where the summation runs over the five amplitudes \(\left\{ \vepCross, \vepPlus, \vepL,\vepR, \vepS \right\}\) that obey \(\sum_\kappa \vep_\kappa^2=1\)---the overall amplitude is fixed by the requirement that \(\Mij\) makes up all of the dark matter.  The five polarisation matrices are given by
\begin{align}
    {\cal Y}^\times_{ij} &\deq \frac{1}{\sqrt2} \left(p_i q_j + q_i p_j\right) \,, & {\cal Y}^+_{ij} &\deq \frac{1}{\sqrt2} \left(p_i p_j - q_i q_j\right) \,, & \nn\\
    {\cal Y}^L_{ij} &\deq \frac{1}{\sqrt2} \left(q_i r_j + r_i q_j\right) \,, & {\cal Y}^R_{ij} &\deq \frac{1}{\sqrt2} \left(p_i r_j + r_i p_j\right) \,, & \nn\\
    {\cal Y}^S_{ij} &\deq \frac{1}{\sqrt6} \left(3 r_i r_j - \delta_{ij}\right) \,. &&& \nn
\end{align}
Notice that, unlike for CWs, there is no propagation along the \(\vr\) direction, which in our case serves merely as reference for the decomposition in tensor, vector, and scalar helicities according to their behaviour under a rotation about \(\vr\) (see also Appendix~\ref{app:not}).

Spin-2 ULDM couples to Standard Model fields \(\Psi\) as~\cite{Marzola:2017}
\begin{align}\label{eq:int}
    S_\text{int}[g,\Mij,\Psi] & \deq -\frac{\al}{2\mpl} \int\!\dd^4x\,\sqrt{-g} \Mij T_\Psi^{ij} \,,
\end{align}
where \(T_\Psi^{ij}\) is the stress tensor of the fields \(\Psi\) and \(\mpl\) is the reduced Planck mass.  At leading (linear) order in \(\alpha\) the interaction \Eq{eq:int} can be absorbed into a redefinition of the metric \(g_{ij}\to g_{ij} + \al  M_{ij}/\mpl\)~\cite{Armaleo:2020yml}.  Therefore, the effect of spin-2 ULDM on the detector can be equivalently described by the gravitational effect of an oscillating metric perturbation \(\hij\) given by
\begin{align}
    \hij(t) &= \frac{\al}{\mpl}\Mij(t) = \frac{\al\sqrt{2\rhoDM}}{m\mpl}\cos{(mt+\Upsilon)}\epij(\vx) \,.
\end{align}
The parameter \(\al\) is idiosyncratic for spin-2 ULDM because it is required by the self-consistency of the model, such as in bigravity~\cite{Babichev:2016bxi}.  This parameter defines the inverse ULDM self-interaction strength: there is no ULDM at all with \(\al\rar0\) because the ULDM field becomes infinitely strongly coupled in this limit.  Furthermore, spin-2 ULDM is ineluctably coupled universally to standard matter fields, so that ULDM will appear as a Yukawa-like fifth force modification of the gravitational potential \(\Phi\) in the weak field regime, for which \(\al\) quantifies the strength: \(\Phi\rar\Phi\left[1+\al^2\exp(-mr)\right]\).  The strength of this fifth force for different values of the mass \(m\) (or, equivalently, frequency) is constrained by several experiments and tests of gravity~\cite{Murata:2014nra,Sereno:2006mw}: we call this maximal coupling \(\al=\al_Y\).

In the reference frame of the detector, \((\vx,\vy,\vz)\), the response function \(D^{ij}\) is given by the differential change in the length of the detector arms directed along the unit vectors \(\vn\) and \(\vm\) as \(D^{ij} = (n^i n^j - m^i m^j)/2\)~\cite{Maggiore:1900zz}.  The signal is the combination of the variation of the metric perturbation and the response function:
\begin{align}
    h(t) &\deq D^{ij} h_{ij}(t) = \frac{\al\sqrt{\rhoDM}}{\sqrt2 m\mpl}\cos{(mt+\Upsilon)}\Delta\vep \deq  h_s\sin{(mt)} + h_c\cos{(mt)}\,, \label{eq:signal}
\end{align}
where we defined \(\Delta\vep \deq \,\epij (n^i n^j - m^i m^j)\), and introduced the sine \(h_{s}\) and cosine \(h_{c}\)     amplitudes.  This is the central equation of the paper.

The theoretical spin-2 ULDM-CW signal \Eq{eq:signal} presents two key features.  First, the signal is inversely proportional to the spin-2 boson mass \(m\).  This inverse linear scaling is also found in dark photon dark matter, where the spin-1 ULDM field carries a additional charges such as baryon number \(B\) or baryon minus lepton number \(B-L\), through which the ULDM directly interacts with the mirrors of the detector~\cite{Pierce:2018xmy,Miller:2020vsl}.  The inverse linear dependence should be compared with the generic inverse \emph{quadratic} dependence obtained by pure gravitational interaction~\cite{Aoki:2016kwl}.  In other words, in absence of non-gravitational interactions, the signal strength decays much more rapidly with increasing mass (or frequency).  This makes it practically impossible to detect such a signal with future GWIs, let alone existing ones.  Second, the spin-2 ULDM-CW signal has a unique geometric structure that sets it apart from other CWs.  Explicitly we have
\begin{align}
    \Delta\vep  &= \sqrt{2} \vepCross\left[\left(\vp\cdot\vn\right) \left(\vq\cdot\vn\right) - \left(\vp\cdot\vm\right) \left(\vq\cdot\vm\right)\right] + \frac{\vepPlus}{\sqrt{2}}\left[\left(\vp\cdot\vn\right)^2 - \left(\vq\cdot\vn\right)^2 -\left(\vp\cdot\vm\right)^2 + \left(\vq\cdot\vm\right)^2\right] \nn\\
                &~~+ \sqrt{2} \vepL\left[\left(\vq\cdot\vn\right) \left(\vr\cdot\vn\right) - \left(\vq\cdot\vm\right) \left(\vr\cdot\vm\right)\right] + \sqrt{2}\vepR\left[\left(\vp\cdot\vn\right) \left(\vr\cdot\vn\right) - \left(\vp\cdot\vm\right) \left(\vr\cdot\vm\right)\right] \nn\\
                &~~+ \sqrt{\frac{3}{2}}\,\vepS\left[\left(\vr\cdot\vn\right)^2 - \left(\vr\cdot\vm\right)^2\right] \label{eq:signal_rpq} \\
                &=\frac{\cos2\phi}{\sqrt{2}}  \left[\vepPlus\left(\cos^2\theta+1\right) + \vepR\,\sin2\theta + \sqrt3\,\vepS\,\sin^2\theta\right] -\sqrt{2}\sin2\phi\left(\vepCross\,\cos\theta + \vepL\,\sin\theta\right) \,, \label{eq:signal_xy}
\end{align}
where in obtaining the last expression we have set \(\vn=\vx\) and \(\vm=\vy\), which we can always do for a single L-shaped detector, and we have defined the ULDM reference frame in terms of the detector's frame as \(\vr = (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\), \(\vp = (\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)\), \(\vq = (-\sin\phi,\cos\phi,0)\); the origins of the two frames are connected by the vector \(r\vr\).

Before moving on to our results, a comment is in order here.  The detector is moving with respect to the ULDM, this motion being the result of three contributions: (1) the Earth is rotating about its axis with equatorial velocity of approximately \(v\sim10^{-6}\) (this only applies to Earth-bound detectors); (2) the Earth is moving along its orbit around the Sun with speed \(v\sim10^{-4}\); (3) the Solar System is moving through the dark matter halo at a speed of \(v\sim10^{-3}\) causing what is known as the dark matter wind.  Therefore, in principle we should Lorentz-boost the ULDM frame to the reference frame of the detector.  However, owing to the smallness of the velocities in question, the effect of the boost on \(r\vr\) amounts to less than a percent correction to the theoretical signal and can be safely neglected.  The relative acceleration of the two frames also induces a Doppler frequency shift \(\Delta f_\text{Doppler}\) that affects the spin-2 ULDM-CW signal, and that needs to be accounted for when designing a data analysis pipeline~\cite{Miller:2020vsl,Frasca:2005ey,DAntonio:2018sff}.

All-sky searches for CWs with Earth-bound GWIs resort to semi-coherent methods because it is not computationally feasible to analyse the data from the entire observation campaign in a fully coherent way\footnote{In the case of space-based detectors such as the upcoming LISA interferometer, owing to the sparse sampling frequency of around 1~Hz, compared to the HLV sampling of about \(10^4\)~Hz, this is not an issue.}~\cite{Brady:1998nj,Krishnan:2004sv,Antonucci:2008jp,PhysRevD.90.042002}.  In semi-coherent methods the whole data set is broken into shorter time chunks of length \(\tchunk\), each of which is then analysed coherently but separately.  One of the advantages of this approach is that, by choosing \(\tchunk<\tdop\deq1/\Delta f_\text{Doppler}\) the Doppler frequency shift can be neglected\footnote{To be more precise, within each chunk the instantaneous Doppler shift that would contribute to \(\dot{f}\) can be neglected, i.e., the frequency is held constant.  Nevertheless, in CW searches, in order to identify viable source candidates for the follow-up steps in the hierarchical semi-coherent analysis, the predicted Doppler shift for each chunk and each location in the sky needs to be corrected for.  Being there no ``sky location'' for ULDM searches, this is not a concern.}.  Moreover, one should ensure that \(\tchunk<\tcoh\) in order to have a stable ULDM configuration within a given chunk.  The sensitivity for a coherent analysis over the whole observation campaign time \(\tobs\) scales as \(\tobs^{-1/2}\).  In semi-coherent methods, assuming that all \(N\) chunks last the same time \(\tchunk\) and all together they cover the whole observation run such that \(\tobs=N\tchunk\), the sensitivity scales instead as \(N^{-1/4}\tchunk^{-1/2} = \tobs^{-1/4}\tchunk^{-1/4}\).  Thanks to the coherence of the signal, even within the limitation of the semi-coherent methods, the actual sensitivity attained by the HLV collaboration for CW searches is more than a factor \(10^{-3}\) smaller than the design sensitivity \(h_0\) for transient events~\cite{Pisarski:2019vxw,Dergachev:2020fli,Steltner:2020hfd}.

The semi-coherent techniques have been adapted and optimised, taking into account the coherence time and the geometry of the signal, for dark photon dark matter searches~\cite{Miller:2020vsl}.  They can therefore be tailored for spin-2 ULDM-CW searches by replacing the average over the different polarisations of ULDM waves (which for the spin 1 dark photon case amounts to a factor \(\sqrt{2}/3\)~\cite{Pierce:2018xmy,Miller:2020vsl}) with \(\sqrt{\langle\Delta\varepsilon^2\rangle} = \sqrt{2/5}\).  We define the effective theoretical strain amplitude \(h\) for the spin-2 ULDM-CW signal as the root mean square average, taken over all the polarisation angles and the random phase \(\Upsilon\), of the sine and cosine amplitudes of \Eq{eq:signal}:
\begin{align}\label{eq:signalaver}
    h &\deq \langle h_s^2+h_c^2\rangle^{1/2} = \frac{\al\sqrt{\rhoDM}}{\sqrt{5}m\mpl} \,.
\end{align}

\section{Results}
\label{sec:res}

In order to estimate the values of \(\al\) accessible with GWIs, we compare the expected theoretical signal \(h\) of \Eq{eq:signalaver} with the design sensitivities of a number of current and planned GWIs (Fig.~\ref{fig:signal}).  We find that the HLV detectors can nominally detect spin-2 ULDM for \(\al\gtrsim10^{-4}\) depending on the frequency (Fig.~\ref{fig:signal}).  We expect that a dedicated semi-coherent search for the spin-2 ULDM-CW signal will improve the range of detectable \(\al\) by a few orders of magnitude, potentially down to \(\al\sim10^{-7}\) or less for frequencies of tens of Hz, corresponding to masses around the \(10^{-13}\)~eV mark; this is shown in Fig.~\ref{fig:signal} as the dotted line ``HLV opt''---the details on how we obtained this curve can be found in Appendix~\ref{app:opt}.  In this frequency range, from \(f\sim10\)~Hz (\(m\sim4\times10^{-14}\)~eV) to \(f\sim10^3\)~Hz (\(m\sim4\times10^{-12}\)~eV) and beyond the planned experiments Einstein Telescope (ET)~\cite{Hild:2010id} and Cosmic Explorer (CE)~\cite{Evans:2016mbw} should reach sensitivities of order \(h_0\sim10^{-22}\text{---}10^{-23}\), further improving the chances to detect spin-2 ULDM.

\begin{figure}[htbp]
\centering
	\includegraphics[width=1.0\textwidth]{sensitivitiesVSalpha}
	\caption{Design sensitivity \(h=h_0\) for several current and planned GWIs, as a function of frequency (solid lines).  The dotted line ``HLV opt'' is the optimised sensitivity obtained with a semi-coherent method tailored for spin-2 ULDM-CW searches (Appendix \ref{app:opt}).  Overlaid as dashed lines are the signal strains \(h\) of \Eq{eq:signalaver} for different values of the parameter \(10^{-4}\leq\al\leq10^{-10}\).  The dot-dashed black line is the spin-2 ULDM-CW strain corresponding to the maximal values of \(\al\) allowed by fifth force constraints, \(h=h(\al_Y)\) with \(\al_Y\) obtained from~\cite{Murata:2014nra,Sereno:2006mw}; the region above this line is excluded.}\label{fig:signal}
\end{figure}

Future facilities will also be able to probe much lower values of the ULDM mass.  In the intermediate frequency range \(0.1~\mathrm{Hz}\lesssim f \lesssim1\)~Hz, corresponding to \(4\times10^{-16}~\mathrm{eV}\lesssim m \lesssim4\times10^{-15}\)~eV, the BBO and DECIGO detectors are expected to attain sensitivities of order of \(h_0\sim10^{-23}\text{---}10^{-24}\)~\cite{Seto:2001qf,Harry:2006fi}.  This means these GWIs could detect a spin-2 ULDM-CW signal for \(\al\lesssim10^{-8}\) at those frequencies.  In the low frequency range the planned space-based interferometer LISA will reach a sensitivity of \(h_0\sim10^{-21}\) for \(f\sim10^{-2}\)~Hz (\(m\sim4\times10^{-17}\)~eV), which means that it could detect spin-2 ULDM with \(\al\sim10^{-7}\) and below.  These limits would be much improved with a dedicated pipeline for these interferometers, as is the case for HLV.  We collect all the sensitivities as compiled in~\cite{Schmitz:2020syl} and compare them to the theoretical signal in Fig.~\ref{fig:signal}---notice that strictly speaking these sensitivities are valid only for the standard tensor modes of GWs, namely the \(\vepCross\) and \(\vepPlus\) in our notation, but the differences are small and not relevant for our order of magnitude estimates~\cite{Zhang:2019oet}.

\section{Conclusion and outlook}
\label{sec:out}

GWIs are a unique tool to understand the nature of gravity.  In this work we have shown that GWIs have the potential to test the properties of gravity \emph{and} dark matter by detecting or constraining spin-2 ULDM.  In particular, we expect that the existing HLV facilities could detect spin-2 ULDM for values of the coupling parameter \(\al\) as little as \(\al\sim10^{-7}\) for frequencies of \(f\sim\mathrm{few}~\times100\)~Hz (that is, a Yukawa range \(\lambda\deq1/2\pi f\sim10^4\)~m).  A null result would place the most stringent limits the strength of the Yukawa-like fifth force modification of the inverse-square law of gravitational interaction, quantified by \(\al\), provided that the fifth force is carried by the dark matter.  Looking forward, future GWIs in the same frequency range can push this limit even further by up to two orders of magnitude, whereas planned facilities such as DECIGO and the BBO (\(f\sim0.1\)~Hz), and the milli-Hertz space-based LISA interferometer are expected to attain \(\al\lesssim10^{-7}\text{---}10^{-8}\) in their respective frequency ranges; limits which can be significantly improved with a dedicated pipeline for the spin-2 ULDM-CW signal.

Our results complement our previous studies on the bounds on the spin-2 ULDM coupling \(\al\) coming from PTAs~\cite{Armaleo:2020yml} and individual pulsar timing data~\cite{Armaleo:2019gil}, which cover the frequency range \(10^{-9}~\mathrm{Hz}\lesssim f \lesssim10^{-3}\)~Hz, and for which comparable limits on \(\al\) were obtained.

Our findings should be compared with existing limits on spin-2 ULDM coming from superradiance.  By measuring the spin and mass of known black holes and other astrophysical objects, the mass ranges  \(6.4\times10^{-22}~\mathrm{eV}\lesssim m \lesssim7.7\times10^{-21}~\mathrm{eV}\), \(1.8\times10^{-20}~\mathrm{eV}\lesssim m \lesssim1.8\times10^{-16}~\mathrm{eV}\) and \(2.2\times10^{-14}~\mathrm{eV}\lesssim m \lesssim2.8\times10^{-11}~\mathrm{eV}\)  are excluded, or else these black holes and celestial objects would not be there~\cite{Stott:2020gjj}.  These bounds are valid provided that any additional interactions that the bosons might possess are small enough not to interfere with the onset and development of superradiance.  In particular, these limits are valid if \(10^{-30}\,\mathrm{eV}/m\ll\al\ll1\)~\cite{Brito:2020lup}, which is verified for most of the parameter space we are considering.  Therefore, the limits we can obtain from GWIs will at the same time independently exclude some of the parameter space that is probed by superradiance, as well as test regions not accessible by it.  Spin-2 ULDM can also be detected thanks to the CW signal that superradiance produces, which is physically unrelated to and distinct from the signal we have described in this work~\cite{Brito:2020lup}; no such searches have been carried out yet for spin-2 ULDM.

In order to fully take advantage of GWI data to test spin-2 ULDM, a dedicated pipeline should be developed.  As we have shown in Sec.~\ref{sec:maths} the signal \Eq{eq:signal} has a peculiar geometric structure that is explicitly given in \Eq{eq:signal_rpq}.  Moreover, the ULDM signal is expected to be coherent for a time \(\tcoh = 2/fv^2\).  An optimised analysis molded onto the shape of this signal can not only improve the sensitivity of GWIs to spin-2 ULDM-CW, but also discriminate between ULDM and other sources of CWs at different frequencies, such as fast-spinning Galactic neutron stars (at high frequencies) or ultra-compact Galactic binaries (in the milli-Hertz band), CW coming from superradiance, and other variants of ULDM, furthering our grasp of dark matter and gravity.

\acknowledgments

The Authors would like to thank C.~Palomba for valuable discussion on the semi-coherent methods used in GW full-sky searches, R.~Brito and V.~Cardoso for an update on the status of spin-2 superradiance, and N.~Tamanini, A. Klein and R. Sturani for useful correspondence on CW searches with LISA.  FU is supported by the European Regional Development Fund (ESIF/ERDF) and the Czech Ministry of Education, Youth and Sports (MEYS) through Project CoGraDS - \verb|CZ.02.1.01/0.0/0.0/15_003/0000437|.  The work of DLN and JMA has been supported by CONICET, ANPCyT and UBA.


\section{Introduction}

Semantic change -- that is, change in the meaning of individual words \cite{campbell_1998} -- is a continuous, inevitable process stemming from numerous reasons and influenced by various factors. Words are continuously changing, with new senses emerging all the time. \cite{campbell_1998} presents no less than 11 types of semantic change, that are generally classified in two wide categories: narrowing and widening.
Most linguists found structural and psychological factors to be the main cause of semantic change, but the evolution of technology and cultural and social changes are not to be omitted.

Measuring semantic divergence across languages can be useful in theoretical and historical linguistics -- being central to models of language and cultural evolution -- but also in downstream applications relying on cognates, such as machine translation.

\textbf{Cognates} are words in sister languages (languages descending from a common anscestor) with a common proto-word. For example, the Romanian word \emph{victorie} and the Italian word \emph{vittoria} are cognates, as they both descend from the Latin word \emph{victoria} (meaning \emph{victory}) -- see Figure \ref{fig:cognates}. In most cases, cognates have preserved similar meanings across languages, but there are also exceptions. These are called deceptive cognates or, more commonly, false friends. Here we use the definition of cognates that refers to words with similar appearance and some common etymology, and use "true cognates" to refer to cognates which also have a common meaning, and "deceptive cognates" or "false friends" to refer to cognate pairs which do not have the same meaning (anymore). The most common way cognates have diverged is by changing their meaning. For many cognate pairs, however, the changes can be more subtle, relating to the feeling attached to a word, or its conotations. This can make false friends even more delicate to distinguish from true cognates.

\begin{figure}[ht]
\center
\includegraphics[width=200pt]{resources/example_paper_new.pdf}
\caption{\label{fig:cognates}Example of cognates and their common ancestor}
\end{figure}

Cognate word pairs can help students when learning a second language and contributes to the
expansion of their vocabularies. False friends, however, from the more obvious differences in meaning to the more subtle, have the opposite effect, and can be confusing for language learners and make the correct use of language more difficult.
Cognate sets have also been used in a number applications in natural language processing, including for example machine translation \cite{kondrak2003cognates}. These applications rely on properly distinguishing between true cognates and false friends.

\subsection{Related work}

Cross-lingual semantic word similarity consists in identifying words that refer to similar semantic concepts and convey similar meanings across languages \cite{vulic_and_moens_2}. Some of the most popular approaches rely on probabilistic models \cite{vulic_and_moens} and cross-lingual word embeddings \cite{soegard_et_al}.

A comprehensive list of cognates and false friends for every language pair is difficult to find or manually build - this is why applications have to rely on automatically identifying them. There have been a number of previous studies attempting to automatically extract pairs of true cognates and false friends from corpora or from dictionaries. Most methods are based either on ortographic and phonetic similarity, or require large parallel corpora or dictionaries \cite{inkpen2005automatic,st2017identifying,nakov2009unsupervised,chen2016false}. We propose a corpus-based approach that is capable of covering the vast majority of the vocabulary for a large number of languages, while at the same time requiring minimal human effort in terms of manually evaluating word pairs similarity or building lexicons, requiring only large monolingual corpora.

In this paper, we make use of cross-lingual word embeddings in order to distinguish between true cognates and false friends. There have been few previous studies using word embeddings for the detection of false friends or cognate words, usually using simple methods on only one or two pairs of languages \cite{castro2018high,torres2011using}.

\begin{figure*}[!ht]
    \centering
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/fr_es_scores3.png}
        \caption{Es-Fr}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/es_it_scores3.png}
        \caption{Es-It}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/ro_es_scores3.png}
        \caption{Es-Ro}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/es_pt_scores3.png}
        \caption{Es-Pt}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/fr_it_scores3.png}
        \caption{Fr-It}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/ro_fr_scores3.png}
        \caption{Fr-Ro}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/fr_pt_scores3.png}
        \caption{Fr-Pt}
    \end{subfigure}
    \begin{subfigure}{0.23\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/ro_it_scores3.png}
        \caption{It-Ro}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/it_pt_scores3.png}
        \caption{It-Pt}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/ro_pt_scores3.png}
        \caption{Ro-Pt}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/fr_en_scores3.png}
        \caption{En-Fr}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/it_en_scores3.png}
        \caption{En-It}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/es_en_scores3.png}
        \caption{En-Es}
    \end{subfigure}
    \begin{subfigure}{0.23\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/ro_en_scores3.png}
        \caption{En-Ro}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/pt_en_scores3.png}
        \caption{En-Pt}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/en_la_scores3.png}
        \caption{En-La}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/fr_la_scores3.png}
        \caption{Fr-La}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/es_la_scores3.png}
        \caption{Es-La}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/it_la_scores3.png}
        \caption{It-La}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/pt_la_scores3.png}
        \caption{Pt-La}
    \end{subfigure}
    \begin{subfigure}{0.20\textwidth}
        \includegraphics[width=\linewidth]{resources/cognates/histograms3/ro_la_scores3.png}
        \caption{Ro-La}
    \end{subfigure}
    \caption{\label{fig:exp1}Distributions of cross-language similarity scores between cognates.}
\end{figure*}

\subsection{Contributions}
The contributions of our paper are twofold: firstly, we propose a method for quantifying the semantic divergence of languages; secondly, we provide a framework for detecting and correcting false friends, based on the observation that these are usually deceptive cognate pairs: pairs of words that once had a common meaning, but whose meaning has since diverged.

We propose a method for measuring the semantic divergence of sister languages based on cross-lingual word embeddings. We report empirical results on five Romance languages: Romanian, French, Italian, Spanish and Portuguese.
For a deeper insight into the matter, we also compute and investigate the semantic similarity betweeen modern Romance languages and Latin. We finally introduce English into the mix, to analyze the behavior of a more remote language, where words deriving from Latin are mostly borrowings.

Further, we make use of cross-lingual word embeddings in order to distinguish between true cognates and false friends. There have been few previous studies using word embeddings for the detection of false friends or cognate words, usually using simple methods on only one or two pairs of languages \cite{castro2018high,torres2011using}.

Our chosen method of leveraging word embeddings extends naturally to another application related to this task which, to our knowledge, has not been explored so far in research: false friend correction. We propose a straightforward method for solving this task of automatically suggesting a replacement when a false friend is incorrectly used in a translation. Especially for language learners, solving this problem could result in a very useful tool to help them use language correctly.

\section{The Method}

\subsection{Cross-lingual Word Embeddings}

Word embeddings are vectorial representations of words in a continuous space, built by training a model to predict the occurence of a given word in a text corpus given its context. Based on the distributional hypothesis stating that similar words occur in similar contexts, these vectorial representations can be seen as semantic representations of words and can be used to compute semantic similarity between word pairs (representations of words with similar meanings are expected to be close together in the embeddings space).

To compute the semantic divergence of cognates across sister languages, as well as identify pairs of false cognates (pairs of cognates with high semantic distance), which by definition are pairs of words in two different languages, we need to obtain a multilingual semantic space, which is shared between the cognates. Having the representations of both cognates in the same semantic space, we can then compute the semantic distance between them using their vectorial representations in this space.

We use word embeddings computed using the FastText algorithm, pre-trained on Wikipedia for the six languages in question. The vectors have dimension 300, and were obtained using the skip-gram model described in \cite{bojanowski_et_al} with default parameters.

The algorithm for measuring the semantic distance between cognates in a pair of languages $ (lang1, lang2) $ consists of the following steps:

\begin{enumerate}
\item Obtain word embeddings for each of the two languages.
\item Obtain a shared embedding space, common to the two languages. This is accomplished using an alignment algorithm, which consists of finding a linear transformation between the two spaces, that on average optimally transforms each vector in one embedding space into a vector in the second embedding space, minimizing the distance between a few seed word pairs (for which it is known that they have the same meaning), based on a small bilingual dictionary. For our purposes, we use the publicly available multilingual alignment matrices that were published in \cite{smith2017offline}.
\item Compute semantic distances for each pair of cognates words in the two languages, using a vectorial distance (we chose cosine distance) on their corresponding vectors in the shared embedding space.
\end{enumerate}

\subsection{Cross-language Semantic Divergence}

We propose a definition of semantic divergence between two languages based on the semantic distances of their cognate word pairs in these embedding spaces. The semantic distance between two languages can then be computed as the average the semantic divergence of each pair of cognates in that language pair.

We use the list of cognates sets in Romance languages proposed by \cite{ciobanu_and_dinu_lrec}. It contains 3,218 complete cognate sets in Romanian, French, Italian, Spanish and Portuguese, along with their Latin common ancestors. The cognate sets are obtained from electronic dictionaries which provide information about the etymology of the words. Two words are considered cognates if they have the same etymon (i.e., if they descend from the same word).

The algorithm described above for computing semantic distance for cognate pairs stands on the assumption that the (shared) embedding spaces are comparable, so that the averaged cosine similarities, as well as the overall distributions of scores that we obtain for each pair of languages can be compared in a meaningful way. For this to be true, at least two conditions need to hold:

\begin{enumerate}
\item The embeddings spaces for each language need to be similarly representative of language, or trained on similar texts - this assumption holds sufficiently in our case, since all embeddings (for all languages) are trained on Wikipedia, which at least contains a similar selection of texts for each language, and at most can be considered comparable corpora.

\item The similarity scores in a certain (shared) embeddings space need to be sampled from a similar distribution. To confirm this assumption, we did a brief experiment looking at the distributions of a random sample of similarity scores across all embeddings spaces, and did find that the distributions for each language pair are similar (in mean and variance). This result was not obvious but also not surprising, since:
\begin{itemize}
    \item The way we create shared embedding spaces is by aligning the embedding space of any language to the English embedding space (which is a common reference to all shared embedding spaces).
    \item The nature of the alignment operation (consisting only of rotations and reflections) guarantees monolingual invariance, as described in these papers: \cite{artetxe2016learning,smith2017offline}.
\end{itemize}
\end{enumerate}

\subsubsection{The Romance Languages}

We compute the cosine similarity between cognates for each pair of modern languages, and between modern languages and Latin as well. We compute an overall score of similarity for a pair of languages as the average similarity for the entire dataset of cognates. The results are reported in Table \ref{table:exp1}.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{l | l l l l l}
\hline
& Fr & It & Pt & Ro & La\\
\hline

Es & 0.67 & 0.69 & 0.70 & 0.58 & 0.41\\
Fr & & 0.66 & 0.64 & 0.56 & 0.40 \\
It & & & 0.66 & 0.57 & 0.41 \\
Pt & & & & 0.57 & 0.41\\
Ro & & & & & 0.40 \\
\hline

\end{tabular}
\end{center}
\caption{\label{table:exp1}Average cross-language similarity between cognates (Romance languages).}
\end{table}

We observe that the highest similarity is obtained between Spanish and Portuguese (0.70), while the lowest are obtained for Latin. From the modern languages, Romanian has, overall, the lowest degrees of similarity to the other Romance languages. A possible explanation for this result is the fact that Romanian developed far from the Romance kernel, being surrounded by Slavic languages.
In Table \ref{table:exp3} we report, for each pair of languages, the most similar (above the main diagonal) and the most dissimilar (below the main diagonal) cognate pair for Romance languages.

\begin{table*}[!ht]
\begin{small}
\begin{center}
\begin{tabular}{l | l l l l l}
\hline
& Es & Fr & It & Ro & Pt\\
\hline

Es & -- & ocho/huit(0.89) & diez/dieci(0.86) & ocho/opt(0.82) & ocho/oito(0.89) \\
Fr & caisse/casar(0.05) & -- & dix/dieci(0.86) & décembre/decembrie(0.83) & huit/oito(0.88) \\
It & prezzo/prez(0.06) & punto/ponte(0.09) & convincere/convinge(0.75) & convincere/convencer(0.88) \\
Ro & miere/mel(0.09) & face/facteur(0.10) & as/asso(0.11) & -- & opt/oito(0.83) \\
Pt & prez/preço(0.05) & pena/paner(0.09) & preda/prea(0.08) & linho/in(0.05) --\\
\hline

\end{tabular}
\end{center}
\caption{\label{table:exp3}Most similar and most dissimilar cognates}
\end{small}
\end{table*}

The problem that we address in this experiment involves a certain \textit{vagueness of reported values} (also noted by \cite{eger_et_al} in the problem of semantic language classification), as there isn't a gold standard that we can compare our results to. To overcome this drawback, we use the degrees of similarity that we obtained to produce a language clustering (using the UPGMA hierarchical clusering algorithm), and observe that it is similar with the generally accepted tree of languages, and with the clustering tree built on intelligibility degrees by \cite{ciobanu_and_dinu_intelligibility}. The obtained dendrogram is rendered in figure \ref{fig:dendrogram}.

\begin{figure}[ht]
\center
\includegraphics[width=0.8\linewidth]{resources/comp/dendrogram.png}
\caption{\label{fig:dendrogram}Dendrogram of the language clusters}
\end{figure}

\subsubsection{The Romance Languages vs English}

Further, we introduce English into the mix as well. We run this experiment on a subset of the used dataset, comprising the words that have a cognate in English as well\footnote{Here we \textit{stretch} the definition of \textit{cognates}, as they are generally referring to sister languages. In this case English is not a sister of the Romance languages, and the words with Latin ancestors that entered English are mostly borrowings.}. The subset has 305 complete cognate sets.

The results are reported in Table \ref{table:exp2}, and the distribution of similarity scores for each pair of languages is rendered in figure \ref{fig:exp1}. We notice that English has 0.40 similarity with Latin, the lowest value (along with French and Romanian), but close to the other languages. Out of the modern Romance languages, Romanian is the most distant from English, with 0.53 similarity.

Another interesting observation relates to the distributions of scores for each language pair, shown in the histograms in \ref{fig:exp1}. While similarity scores between cognates among romance languages usually follow a normal distribution (or another unimodal, more skewed distribution), the distributions of scores for romance languages with English seem to follow a bimodal distribution, pointing to a different semantic evolution for words in English that share a common etymology with a word in a romance language. One possible explanation is that the set of cognates between English and romance languages (which are pairs of languages that are more distantly related) consist of two distinct groups: for example one group of words that were borrowed directly from the romance language to English (which should have more meaning in common), and words that had a more complicated etymological trail between languages (and for which meaning might have diverged more, leading to lower similarity scores).

\begin{table}[!ht]
\begin{center}
\begin{tabular}{l | l l l l l l}
\hline
& Fr & It & Pt & Ro & En & La\\
\hline

Es & 0.64 & 0.67 & 0.68 & 0.57 & 0.61 & 0.42 \\
Fr & & 0.64 & 0.61 & 0.55 & 0.60 & 0.40 \\
It & & & 0.65 & 0.57 & 0.60 & 0.41 \\
Pt & & & & 0.56 & 0.59 & 0.42 \\
Ro & & & & & 0.53 & 0.40 \\
En & & & & & & 0.40 \\
\hline

\end{tabular}
\end{center}
\caption{\label{table:exp2}Average cross-language similarity between cognates}
\end{table}

\subsection{Detection and Correction of False Friends}

In a second series of experiments, we propose a method for identifying and correcting false friends.
Using the same principles as in the previous experiment, we can use embedding spaces and semantic distances between cognates in order to detect pairs of false friends, which are simply defined as pairs of cognates which do not share the same meaning, or which are not semantically similar \textit{enough}.

This definition is of course ambiguous: there are different degrees of similarity, and as a consequence different potential degrees of \textit{falseness} in a false friend. Based on this observation, we define the notions of \textit{hard false friend} and \textit{soft false friend}.

A \textit{hard false friend} is a pair of cognates for which the meanings of the two words have diverged enough such that they don't have the same meaning anymore, and should not be used interchangibly (as translations of one another). In this category fall most known examples of false friends, such as the French-English cognate pair \textit{attendre} / \textit{attend}: in French, \textit{attendre} has a completely different meaning, which is \textit{to wait}. A different and more subtle type of false friends can result from more minor semantic shifts between the cognates. In such pairs, the meaning of the cognate words may remain roughly the same, but with a difference in nuance or connotation. Such an example is the Romanian-Italian cognate pair \textit{amic} / \textit{amico}. Here, both cognates mean \textit{friend}, but in Italian the conotation is that of a closer friend, whereas the Romanian \textit{amic} denotes a more distant friend, or even aquaintance. A more suitable Romanian translation for \textit{amico} would be \textit{prieten}, while a better translation in Italian for \textit{amic} could be \textit{conoscente}.

Though their meaning is roughly the same, translating one word for the other would be an inaccurate use of the language. These cases are especially difficult to handle by beginner language learners (especially since the cognate pair may appear as valid a translation in multilingual dictionaries) and using them in the wrong contexts is an easy trap to fall into.

Given these considerations, an automatic method for finding the appropriate term to translate a cognate instead of using the false friend would be a useful tool to aid in translation or in language learning.

As a potential solution to this problem, we propose a method that can be used to identify pairs of false friends, to distinguish between the two categories of false friends defined above (\textit{hard false friends} and \textit{soft false friends}), and to provide suggestions for correcting the erroneous usage of a false friend in translation.

\textbf{False friends} can be identified as pairs of cognates with high semantic distance. More specifically, we consider a pair of cognates to be a false friend pair if in the shared semantic space, there exists a word in the second language which is semantically closer to the original word than its cognate in that language (in other words, the cognate is not the optimal translation). The arithmetic difference between the semantic distance between these words and the semantic distance between the cognates will be used as a measure of the \textit{falseness} of the false friend. The word that is found to be closest to the first cognate will be the suggested "correction".

The algorithm can be described as follows:
\begin{algorithm}
\caption{Detection and correction of false friends}
\label{Detection and correction of false friends}
\begin{algorithmic}[1]
    \State Given the cognates pair $(c_1, c_2)$ where $c_1$ is a word in $lang_1$ and $c_2$ is a word in $lang_2$:
    \State Find the word $w_2$ in $lang_2$ such that for any $w_i$ in $lang_2$, $distance(c_2, w_2) < distance(c_2, w_i)$
    \If{$w_2 \neq c_2$}
    \State $(c_1, c_2)$ is a pair of false friends
    \State Degree of falseness $= distance(c_1, w_2) - distance(c_1, c_2)$ \\
    \Return $w_2$ as potential correction
    \EndIf
\end{algorithmic}
\end{algorithm}

We select a few results of the algorithm to show in Table \ref{table:exp3}, containing examples of extracted false friends for the language pair French-Spanish, along with the suggested correction and the computed degree of falseness.

Depending on the application, the measure of \textit{falseness} could be used by choosing a threshold to single out pairs of false friends that are \textit{harder} or \textit{softer}, with a customizable degree of sensitivity to the difference in meaning.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{l l l l}
\hline
FR cognate & ES cognate & Correction & Falseness \\
\hline
prix & prez & premio & 0.67 \\
long & luengo & largo & 0.57 \\
face & faz & cara & 0.41 \\
change & caer & cambia & 0.41 \\
concevoir & concebir & dise{\~n}ar & 0.18 \\
majeur & mayor & importante & 0.14 \\
\hline

\end{tabular}
\end{center}
\caption{\label{table:exp3}Extracted false friends for French-Spanish}
\end{table}

\subsubsection{Evaluation}

In this section we describe our overall results on identifying false friends for every language pair between English and five Romance languages: French, Italian, Spanish, Portuguese and Romanian.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{l | l l l}
\hline
& Accuracy & Precision & Recall\\
\hline

Our method & 81.12 & 86.68 & 75.59 \\
(Castro et al) & 77.28 &  &  \\
WN Baseline & 69.57 & 85.82 & 54.50 \\
\hline

\end{tabular}
\end{center}
\caption{\label{table:exp1}Performance for Spanish-Portuguese using curated false friends test set}
\end{table}

We evaluate our method in two separate stages. First, we measure accuracy of false friend detection on a manually curated list of false friends and true cognates in Spanish and Portuguese, used in a previous study \cite{castro2018high}, and introduced in \cite{torres2011using}. This resource is composed by 710 Spanish-Portuguese word pairs: 338 true cognates and 372 false friends. We also compare our results to the ones reported in this study, which uses a method similar to ours (using a simple classifier that takes embedding similarities as features to identify false friends) and shows improvements over results in previous research. The results are show in Table \ref{table:exp1}.

For the second part of the experiment, we use the list of cognates sets in English and Romance languages proposed by \cite{ciobanu_and_dinu_lrec} (the same that we used in our semantic divergence experiments), and try to automatically decide which of these are false friends.
Since manually built false friends lists are not available for every language pair that we experiment on, for the language pairs in this second experiment we build our gold standard by using a multilingual dictionary (WordNet) in order to infer false friends and true cognate relationships. We assume two cognates in different languages are true cognates if they occur together in any WordNet synset, and false friends otherwise.

\begin{table}[!ht]
\begin{center}
\small
\begin{tabular}{l | l l l}
\hline
& Accuracy & Precision & Recall\\
\hline

EN-ES & 76.58 & 63.88 & 88.46 \\
ES-IT & 75.80 & 41.66 & 54.05 \\
ES-PT & 82.10 & 40.0 & 42.85 \\
EN-FR & 77.09 & 57.89 & 94.28 \\
FR-IT & 74.16 & 32.81 & 65.62 \\
FR-ES & 73.03 & 33.89 & 69.96 \\
EN-IT & 73.07 & 33.76 & 83.87 \\
IT-PT & 76.14 & 29.16 & 43.75 \\
EN-PT & 77.25 & 59.81 & 86.48 \\
\hline

\end{tabular}
\end{center}
\caption{\label{table:exp4}Performance for all language pairs using WordNet as gold standard.}
\end{table}

We measure accuracy, precision, and recall, where:
\begin{itemize}
\item a \textit{true positive} is a cognate pair that are not synonyms in WordNet and are identified as false friends by the algorithm,
\item a \textit{true negative} is a pair which is identified as true cognates and is found in the same WordNet synset,
\item a \textit{false positive} is a word pair which is identified as a false friends pair by the algorithm but also appears as a synonym pair in WordNet,
\item and a \textit{false negative} is a pair of cognate words that are not synonyms in WordNet, but are also not identified as false friends by the algorithm.
\end{itemize}

We should also note that in the WordNet based method we can only evaluate results for only slightly over half of cognate pairs, since not all of them are found in WordNet. This also makes our corpus-based method more useful than a dictionary-based method, since it is able to cover most of the vocabulary of a language (given a large monolingual corpus to train embeddings on).

To be able to compare results to the ones evaluated on the manually built test set, we use the WordNet-based method as a baseline in the first experiment. Results for the second evaluation experiments are reported in Table \ref{table:exp4}. In this evaluation experiment we were able to measure performance for language pairs among all languages in our cognates set except for Romanian (which is not available in WordNet).

\section{Conclusions}

In this paper we proposed a method for computing the semantic divergence of cognates across languages. We relied on word embeddings and extended the pairwise metric to compute the semantic divergence across languages. Our results showed that Spanish and Portuguese are the closest languages, while Romanian is most dissimilar from Latin, possibly because it developed far from the Romance kernel. Furthermore, clustering the Romance languages based on the introduced semantic divergence measure results in a hierarchy that is consistent with the generally accepted tree of languages. When further including English in our experiments, we noticed that, even though most Latin words that entered English are probably borrowings (as opposed to inherited words), its similarity to Latin is close to that of the modern Romance languages. Our results shed some light on a new aspect of language similarity, from the point of view of cross-lingual semantic change.

We also proposed a method for detecting and possibly correcting false friends, and introduced a measure for quantifying the \textit{falseness} of a false friend, distinguishing between two categories: hard false friends and soft false friends. These analyses and algorithms for dealing with false friends can possibly provide useful tools for language learning or for (human or machine) translation.

In this paper we provided a simple method for detecting and suggesting corrections for false friends independently of context. There are, however, false friends pairs that are context-dependent - the cognates can be used interchangibly in some contexts, but not in others. In the future, the method using word embeddings could be extended to provide false friend correction suggestions in a certain context (possibly by using the word embedding model to predict the appropriate word in a given context).

\section*{Acknowledgements}
Research supported by BRD --- Groupe Societe Generale Data Science Research Fellowships.


\section{Introduction}
\label{sec:intro}

The \emph{average sensitivity} $\mathbb{AS}(f)$ of a Boolean function $f:\mathbb{Z}_2^n \to \{+1,-1\}$ is $n$ times the probability that $f(x) \ne f(y)$, where $x$ is chosen uniformly at random and $y$ is chosen uniformly from $x$'s neighbors at Hamming distance $1$.  Similarly, the \emph{noise sensitivity} $\mathbb{NS}_\varepsilon(f)$ is the probability that $f(x) \ne f(y)$ where $x$ is uniformly random and $y$ is formed by flipping each bit of $x$ independently with probability $\varepsilon$.  Sensitvity is a basic structural characteristic of Boolean functions, with applications to computational complexity, pseudorandomness, machine learning, and the theory of social choice~\cite{LinialMN:1993,KhotKMO:2007,ODonnell:2004,HarshaKM:2014,DiakonikolasSTW:2014,Kalai:2010,odonnell-book}.

Many cases of interest focus on threshold functions $f(x)$ defined from a smooth function $p: \mathbb{R}^n \rightarrow \mathbb{R}$ and a threshold value $\theta \in \mathbb{R}$,
\[
f(x) = \begin{cases}
+1 & \text{if $p(x) \geq \theta$} \, , \\
-1 & \text{if $p(x) < \theta$} \, .
\end{cases}
\]
where we define a Boolean function by restricting $p$, and $f$, to the hypercube $\{\pm 1\}^n \subset \mathbb{R}$. In particular, polynomial threshold functions $f(x) = \mathop{\mathrm{sgn}}(p(x))$, where $p(x)$ is a polynomial of degree $d$, have played a dominant role in this setting.
Gotsman and Linial~\cite{gotsman-linial} conjectured that the sensitivity of such functions is maximized when $p$ is a symmetric polynomial whose roots slice the hypercube at $d$ Hamming weights near $n/2$,
\[
p(x_1,\ldots,x_n) = q\!\left( \sum_i x_i \right)
\quad \text{where} \quad
q(x) = \prod_{h = \lfloor (n-d+1)/2 \rfloor}^{\lfloor (n+d-1)/2 \rfloor} \left( x - h - \frac{1}{2} \right) \, .
\]
This implies that
\[
\mathbb{AS}(f) = O(d \sqrt{n})
\quad \text{and} \quad
\mathbb{NS}_\varepsilon(f) = O(d \sqrt{\varepsilon}) \, ,
\]
where the constant in the $O$ depends neither on $d$ nor $n$.  This is known in the case $d=1$, i.e., where $f(x)$ is a halfspace~\cite{peres}.  However, for $d > 1$ it has remained open for some time.

The first nontrivial bounds for threshold functions of degree $d > 1$ were obtained quite recently~\cite{DiakonikolasSTW:2014,HarshaKM:2014}, showing
\[
\mathbb{AS}(f) = 2^{O(d)} n^{1-\alpha} \log n
\quad \text{and} \quad
\mathbb{NS}_\varepsilon(f) = 2^{O(d)} \varepsilon^\alpha \log(1/\varepsilon) \, ,
\]
where $\alpha = O(1/d)$.  These bounds work by dividing polynomials into two classes: ``juntas'' where a few variables are highly influential, and ``regular'' polynomials where no variable has large influence.  The regular case is handled using anticoncentration bounds and the invariance principle of~\cite{MOO05}, showing that the distribution of values of $p(x)$ is close to what it would be if $x$ were drawn from the Gaussian distribution as opposed to the uniform distribution on the hypercube.

Using different reasoning~\cite{kane-correct-exponent}, it was recently shown that
\[
\mathbb{AS}(f) = \sqrt{n} \,(\log n)^{O(d \log d)} \,2^{O(d^2 \log d)}
\quad \text{and} \quad
\mathbb{NS}_\varepsilon(f) = \sqrt{\varepsilon} \,(\log (1/\varepsilon))^{O(d \log d)} \,2^{O(d^2 \log d)} \, .
\]
While this dependence on $d$ is somewhat regrettable, these results show that the Gotsman-Linial conjecture holds, up to polylogarithmic factors, for each fixed $d$.

Even when our ultimate questions pertain to sensitivity on the hypercube, working with functions defined on $\mathbb{R}^n$ can permit techniques from analysis to be brought to bear on the problem. This has motivated interest in continuous notions of noise sensitivity, most notably \emph{Gaussian sensitivity},
which is obtained by placing a Gaussian measure on $\mathbb{R}^n$ and applying Gaussian noise.  In this setting, a simple and elegant argument~\cite{kane-surface-area} shows that the Gaussian analog of $\mathbb{NS}_\varepsilon(f)$ is indeed $O(d \sqrt{\varepsilon})$.

In this article, we introduce a notion of \emph{spherical sensitivity} for functions defined on the unit $n$-sphere.  By analyzing how spherical harmonics are carried to Boolean harmonics, i.e., Fourier basis functions over the hypercube $\mathbb{Z}_2^n$, we give a transfer theorem bounding the Boolean sensitivity in terms of the spherical sensitivity.  Our results hold in expectation, when the function, or equivalently the hypercube,
is randomly rotated in $\mathbb{R}^n$.  In essence, we show that the distribution of angles induced by Boolean noise on the hypercube can be modeled by the effect of Brownian motion on the sphere, or equivalently diffusion driven by the spherical heat equation.

As an application, by bounding the spherical sensitivity of polynomial threshold functions, we establish the Gotsman-Linial conjecture on average, in the following sense: for any polynomial $p$ of degree $d$, if we apply a random rotation $R$ and then restrict to the hypercube, the expected average sensitivity and noise sensitivity of the resulting Boolean threshold function $f(x) = \mathop{\mathrm{sgn}}(Rp(x))$ are $\Exp_R \mathbb{AS}(f) = O(d \sqrt{n})$ and $\Exp_R \mathbb{NS}_\varepsilon = O(\varepsilon \sqrt{n})$ respectively.

\section{Spherical harmonics and the heat equation}

Here and in later sections we repeat information from classic texts~\cite{szego,vilenkin} and two excellent reviews~\cite{frye-efthimiou,gallier}.  Recall that the Laplace operator on $\mathbb{R}^n$ is defined as
\[
\Delta_{\mathbb{R}^n}
= \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2} \, .
\]
The Laplace-Beltrami operator on $S_{n-1}$ consists of the contribution to $\Delta_{\mathbb{R}^n}$ arising from the dependence of a function on angular variables rather than the distance from the origin.  It can be defined by writing $\Delta_{\mathbb{R}^n}$ in polar coordinates,
\[
\Delta_{\mathbb{R}^n} = \frac{\partial}{\partial r^2} + \frac{n-1}{r} \frac{\partial}{\partial r} + \frac{1}{r^2} \Delta_{S_{n-1}} \, .
\]
A function $h:\mathbb{R}^n \to \mathbb{R}$ is \emph{harmonic} if $\Delta_{\mathbb{R}^n} h = 0$.  In particular, for each $\ell \ge 0$, there is a linear subspace of harmonic homogeneous polynomials of degree $\ell$ of dimension (where we assume $n \ge 3$)
\begin{equation}
\label{eq:dim}
d_\ell
= {n + \ell - 1 \choose \ell} - {n + \ell - 3 \choose \ell - 2}
= \frac{n+2\ell-2}{n-2} {n + \ell -3 \choose \ell} \, .
\end{equation}
Restricting these polynomials to $S_{n-1}$ gives the so-called \emph{spherical harmonics}.  We denote a basis for these as $\{ Y_{\ell,j} \mid 1 \le j \le d_\ell \}$.  They are eigenfunctions of the Laplace-Beltrami operator:
\begin{equation}
\label{eq:eigen-spherical}
\Delta_{S_{n-1}} Y_{\ell,j} = -\ell (n+\ell-2) Y_{\ell,j} \, .
\end{equation}
Any function $g$ in $L_2(S_{n-1})$ can be expanded in terms of spherical harmonics,
\begin{equation}
\label{eq:hatf-spherical}
g = \sum_{\ell \ge 0} g_\ell = \sum_{\ell \ge 0} \sum_{j=1}^{d_\ell} \widehat{g}(\ell,j) \,Y_{\ell,j} \, .
\end{equation}
This is analogous to the expansion into Fourier series over $[0,1)
\subset \mathbb{R}$, where small $\ell$ corresponds to smooth, low-frequency
variations, and larger $\ell$ corresponds to higher frequencies.

\section{Noise sensitivity and the heat equation on the cube and the sphere}

The standard notion of noise sensitivity for a Boolean function $f:\mathbb{Z}_2^n \to \{\pm 1\}$ is as follows~\cite{odonnell-book}.  Define a linear operator $K_\varepsilon$ on the space of probability distributions over $\mathbb{Z}_2^n$ that independently flips each bit with probability $\varepsilon$.  That is, if $d(x,y)$ denotes the Hamming distance,
\[
K_\varepsilon(x,y) = \varepsilon^{d(x,y)} (1-\varepsilon)^{n-d(x,y)} \, .
\]
Then if $\delta_x$ is the Kronecker delta function where $\delta_x(z) = 1$ if $z=x$ and $0$ otherwise,
\[
\mathbb{NS}_\varepsilon(f) = \Exp_{x,y} \Pr[f(y) \ne f(x)] \, ,
\]
where $x$ is uniform in $\{\pm 1\}^n$ and $y$ is chosen from the distribution $K_\varepsilon \delta_x$.  Equivalently, if we define the inner product on the cube as
\[
\langle f, g \rangle = \Exp_{x \in \mathbb{Z}_2^n} f(x)^* g(x) \, ,
\]
then
\[
\mathbb{NS}_\varepsilon(f) = \frac{1}{2} \left( 1 - \langle f, K_\varepsilon f \rangle \right) \, .
\]

We write $f$ in the Fourier basis, expanding in terms of the characters $\chi_k$,
\[
f(x) = \sum_{k \in \mathbb{Z}_2^n} \widehat{f}(k) \,\chi_k(x)
\quad \text{where} \quad
\chi_k(x) = (-1)^{k \cdot x}
\quad \text{and} \quad
\widehat{f}(k) = \langle f, \chi_k \rangle \, .
\]
Then we can use the fact that $\chi_k$ is an eigenvector of $K_\varepsilon$,
\begin{equation}
\label{eq:phi-eigen}
K_\varepsilon \chi_k = (1-2\varepsilon)^{|k|} \chi_k \, ,
\end{equation}
where $|k|$ denotes the Hamming weight of the frequency vector $k$.  Then we obtain
\begin{align}
\mathbb{NS}_\varepsilon(f)
&= \frac{1}{2} \left( 1 - \sum_{k \in \mathbb{Z}_2^n} \abs{\widehat{f}(k)}^2 (1-2\varepsilon)^{|k|} \right) \nonumber \\
&= \frac{1}{2} \sum_{k \in \mathbb{Z}_2^n} \abs{\widehat{f}(k)}^2 \left( 1-(1-2\varepsilon)^{|k|} \right) \, .
\label{eq:ns}
\end{align}
Here we used the fact that $\langle f, f \rangle = \sum_k \abs{\widehat{f}(k)}^2 = 1$ since $\abs{f(x)}^2=1$ for all $x$.  However, we can also take~\eqref{eq:ns} as the definition of $\mathbb{NS}_\varepsilon(f)$, in which case it can be applied to any function $f:\mathbb{Z}_2^n \to \mathbb{C}$.

We can also write the noise sensitivity in terms of a continuous-time (but discrete-space) heat equation on the hypercube.  Let $A$ be the adjacency matrix of the hypercube, and let
\[
L = A - n \mathds{1}
\]
be the graph Laplacian.  If we apply the heat equation
\[
\frac{\partial f}{\partial t} = Lf
\]
for time $\varepsilon'$ with initial condition $f(0)=f$, we have
\[
f(\varepsilon')
= \mathrm{e}^{\varepsilon' L} f \, .
\]
The characters $\chi_k$ are eigenfunctions of the Laplacian,
\[
L \chi_k = -2|k| \chi_k
\quad \text{and} \quad
\mathrm{e}^{\varepsilon' L} \chi_k = \mathrm{e}^{-2|k|\varepsilon'} \chi_k \, .
\]
Matching these eigenvalues with~\eqref{eq:phi-eigen}, we see that if
\begin{equation}
\label{eq:eps-prime}
\varepsilon' = \frac{1}{2} \log \frac{1}{1-2\varepsilon} = \varepsilon + O(\varepsilon^2) \, ,
\end{equation}
then
\[
K_\varepsilon = \mathrm{e}^{\varepsilon' L} \, ,
\]
and
\begin{align}
\mathbb{NS}_\varepsilon
&= \frac{1}{2} \left( 1-\langle f, f(\varepsilon') \rangle \right) \nonumber \\
&= \frac{1}{2} \left( 1-\langle f, \mathrm{e}^{\varepsilon' L} f \rangle \right) \nonumber \\
&= \frac{1}{2} \left( 1 - \sum_{k \in \mathbb{Z}_2^n} \abs{\widehat{f}(k)}^2 \,\mathrm{e}^{-2 \varepsilon' |k|} \right) \nonumber \\
&= \frac{1}{2} \sum_{k \in \mathbb{Z}_2^n} \abs{\widehat{f}(k)}^2 \left( 1 - \mathrm{e}^{-2 \varepsilon' |k|} \right) \, .
\label{eq:cube-heat}
\end{align}

In analogy with this heat-equation picture of the Boolean noise sensitivity, we define the \emph{spherical sensitivity} $\SS_t(g)$ of a function $g:S_{n-1} \to \{ \pm 1 \}$ as follows.  First define the inner product of two functions $f, g:S_{n-1} \to \mathbb{C}$ as
\begin{equation}
\label{eq:spherical-inner}
\langle f, g \rangle_{S}
= \Exp_{x \in S_{n-1}} f(x)^* \,g(x)
= \frac{1}{\Omega_{n-1}} \int_{S_{n-1}} \mathrm{d}x \,f(x)^* g(x) \, ,
\end{equation}
where $\Omega_{n-1}$ denotes the surface area of $S_{n-1}$,
\begin{equation}
\label{eq:omega}
\Omega_{n-1} = \frac{2 \pi^{n/2}}{\Gamma(n/2)} \, .
\end{equation}
The heat equation on $S_{n-1}$ is
\[
\frac{\partial g}{\partial t} = \Delta_{S_{n-1}} g \, ,
\]
and applying it for time $t$ with the initial condition $g(0)=g$ gives
\[
g(t) = \mathrm{e}^{\Delta_{S_{n-1}} t} g \, .
\]
Then we define $\SS_t(g)$ as
\begin{align}
\SS_t(g)
&= \frac{1}{2} \left( 1-\langle g, g(t) \rangle_{S} \right) \nonumber \\
&= \frac{1}{2} \left( 1-\langle g, \mathrm{e}^{t \Delta_{S_{n-1}}} g \rangle_{S} \right) \nonumber \\
&= \frac{1}{2} \sum_{\ell,j} \abs{\widehat{g}(\ell,j)}^2 \left( 1-\mathrm{e}^{-t \ell (n+\ell-2)} \right) \, ,
\label{eq:ss}
\end{align}
where we used~\eqref{eq:eigen-spherical} and the expansion~\eqref{eq:hatf-spherical}.

Here we assumed that $g$ takes values in $\{ \pm 1 \}$.  In that case, $\SS_t(g)$ is the probabiility that $g(x) \ne g(y)$ if $x$ is uniformly random and $y$ is the position of a particle that starts at $x$ and unergoes Brownian motion for time $t$.  However, as with~\eqref{eq:ns}, we will take~\eqref{eq:ss} as the definition of $\SS_t(g)$, thus extending the notion of sensitivity to arbitrary functions $g: S_{n-1} \to \mathbb{C}$.

Comparing~\eqref{eq:cube-heat} and~\eqref{eq:ss}, we see that flipping bits with probability $\varepsilon$ is roughly analogous to running the heat equation on the sphere for time $t = O(\varepsilon/n)$.  We will tighten this analogy in Theorem~\ref{thm:transfer} below.

\section{Zonal harmonics and Gegenbauer polynomials}
\label{sec:zonal}

In this section and the next, we continue our review of spherical harmonics and their associated orthogonal polynomials~\cite{szego,vilenkin,frye-efthimiou,gallier}.   We include somewhat more machinery than is strictly necessary to prove our main result.  However, in many cases this machinery gives us a more explicit picture of what is going on, and may be useful in proving more detailed results.

Let $\eta = (1,0,\ldots,0)$ be the north pole.  If $f(Rx) = f(x)$ for all rotation matrices that fix $\eta$, then $f(x)$ depends only on $z = \eta \cdot x$.  Such functions are called \emph{zonal}.  The inner product of two such functions can be written as a weighted inner product over the interval $-1 \le z \le 1$,
\begin{equation}
\label{eq:weighted-inner}
\langle f, g \rangle_{S}
= \frac{\Omega_{n-2}}{\Omega_{n-1}} \int_{-1}^1 \mathrm{d}z \,w_\alpha(z) \,f(z)^* g(z) \, ,
\end{equation}
where we wantonly abuse notation by identifying $f(z)$ with $f(x)$, and the weight
\begin{equation}
\label{eq:weight}
w_\alpha(z) = (1-z^2)^{\alpha-\frac{1}{2}}
\end{equation}
and the constant $\Omega_{n-2}$ account for the volume of the annulus between height $z$ and $z+\mathrm{d}z$.

There is a unique harmonic polynomial of each degree $\ell$.  A nice orthogonal family of such polynomials, called the \emph{zonal spherical harmonics} or the \emph{ultraspherical} or \emph{Gegenbauer} polynomials, are as follows.  For historical reasons, we parametrize them with a half-integer $\alpha$ rather than the integer dimension $n$:
\[
\alpha = \frac{n}{2} - 1 \, ,
\]
in which case the dimension~\eqref{eq:dim} becomes
\begin{equation}
\label{eq:dim-alpha}
d_\ell = \frac{\alpha+\ell}{\alpha} {2\alpha + \ell - 1 \choose \ell} \, .
\end{equation}
For each value of $\alpha$, we have a family of polynomials $\{ \gamma_\ell \mid \ell \in \mathbb{N} \}$ of degree $\ell$,
\begin{align}
\label{eq:gegen}
\gamma_\ell(z)
= \frac{1}{\sqrt{N^{(\ell)}}} \,G^{(\ell)}(z)
\quad \text{where} \quad
G^{(\ell)}(z)
&= \sum_{k=0}^{\lfloor \ell/2 \rfloor} (-1)^k \frac{(\ell-k+\alpha-1)!}{(\alpha-1)! \,k! \,(\ell-2k)!} (2z)^{\ell-2k} \\
\text{and} \quad
N^{(\ell)}
& = \left\langle G^{(\ell)}, G^{(\ell)} \right\rangle \nonumber \\
&= \frac{\Omega_{n-2}}{\Omega_{n-1}} \times \frac{\pi \,2^{1-2\alpha} \,\Gamma(\ell+2\alpha)}{(\ell+\alpha) \,\Gamma(\ell+1) \,\Gamma(\alpha)^2} \nonumber \\
&= \frac{\alpha}{\alpha+\ell}
{2\alpha + \ell - 1 \choose \ell} \, .
\end{align}
Note that $G^{(\ell)}$ is the $\ell$th Legendre polynomial when $\alpha = 1/2$ (i.e., when $n=3$).  Then for each $\alpha$, the $\gamma_\ell$ are orthonormal with respect to the inner product~\eqref{eq:weighted-inner}:
\[
\left\langle \gamma_\ell, \gamma^{(m)} \right\rangle_{S}
= \delta_{\ell,m} \, .
\]
Thus, given a zonal function $f(z)$, we can write
\[
f(z) = \sum_{\ell \ge 0} \widehat{f}^{(\ell)} \gamma_\ell(z)
\quad \text{where} \quad
\widehat{f}^{(\ell)}
= \left\langle f, \gamma_\ell \right\rangle_{S} \, .
\]
This transform is unitary, so inner products are preserved:
\[
\langle f, g \rangle_{S} = \sum_{\ell \ge 0} \widehat{f}^{(\ell)*} \,\widehat{g}^{(\ell)} \, .
\]

The Gegenbauer polynomials have deep roots in the representation
theory of Lie groups.  Let $\mathsf{SO}_n$ be the group of orthogonal
rotations of $\mathbb{R}^n$; then the harmonic polynomials of degree $\ell$
form an irreducible representation $\rho_\ell$ of $\mathsf{SO}_n$ with
dimension $d_\ell$.  We can think of $L_2(S_{n-1})$ as the subspace of
$L_2(\mathsf{SO}_n)$ consisting of functions that are right-invariant under
the subgroup $\mathsf{SO}_{n-1}$ that fixes the north pole $\eta$: that is,
functions $f(R)$ that only depend on $R\eta$.  The Gegenbauer
polynomials span the subspace of $L_2(\mathsf{SO}_n)$ consisting of functions
which are left- and right-invariant under $\mathsf{SO}_{n-1}$; that is,
functions that are zonal, depending only on $z = \langle \eta \cdot R\eta \rangle$, or equivalently on the latitude of $R\eta$.  The fact that $\gamma_\ell$ is the unique zonal polynomial of degree $\ell$ corresponds to the fact that $\mathsf{SO}_n$ and $\mathsf{SO}_{n-1}$ form a Gel'fand pair, i.e., this subspace is one-dimensional.

\section{Schur's lemma and evaluation maps}
\label{sec:schur}

Of course, $\eta$ is is an arbitrary choice for the north pole.
For any $w \in S_{n-1}$, there is a unique polynomial of degree $\ell$ which is zonal around $w$, i.e., which is fixed under the copy of $\mathsf{SO}_{n-1}$ that preserves $w$.  Given a function $f$ and $R \in \mathsf{SO}_n$, define $Rf$ as the function
\[
Rf(x) = f(R^{-1} x) \, .
\]
Now let $R \in \mathsf{SO}_n$ be any rotation such that $R\eta = w$.  Then if we write $\gamma_\ell^{(\eta)}(x) = \gamma_\ell(\eta \cdot x)$, we can define $\gamma_\ell^{(w)} = R \gamma_\ell^{(\eta)}$, so that
\[
\gamma_\ell^{(w)}(x) = \gamma_\ell^{(\eta)}(R^{-1} x) = \gamma_\ell(w \cdot x) \, .
\]
The inner products of these functions are again given by a Gegenbauer polynomial: for any $w, y \in S_{n-1}$,
\begin{equation}
\label{eq:gegen-inner}
\left\langle \gamma_\ell^{(w)}, \gamma_\ell^{(y)} \right\rangle_{S}
= \frac{1}{\Omega_{n-1}} \int_{S_{n-1}} \mathrm{d}x \,\gamma_\ell(w \cdot x) \,\gamma_\ell(y \cdot x)
= \frac{1}{\sqrt{d_\ell}} \,\gamma_\ell(w \cdot y) \, .
\end{equation}
Taking $w=y$ tells us how large $\gamma_\ell$ gets at the poles:
\begin{equation}
\label{eq:gegen-max}
\gamma_\ell(1)
= \abs{\gamma_\ell(-1)}
= \sqrt{d_\ell} \, .
\end{equation}
In addition, if we fix $w$ and take the expectation over $y$ of the inner product squared, we get
\begin{equation}
\Exp_y \babs{ \left\langle \gamma_\ell^{(w)}, \gamma_\ell^{(y)} \right\rangle_{S} }^2
= \frac{1}{d_\ell} \babs{ \left\langle \gamma_\ell, \gamma_\ell \right\rangle_{S} }^2
= \frac{1}{d_\ell} \, .
\end{equation}
Since averaging over $y$ is the same as averaging over $R$ according to the Haar measure on $\mathsf{SO}_n$, this is equivalent to
\begin{equation}
\label{eq:schur}
\Exp_R \babs{ \left\langle \gamma_\ell, R \gamma_\ell \right\rangle_{S} }^2
= \frac{1}{d_\ell} \, .
\end{equation}
This holds more generally for any spherical harmonic $f_\ell$ of degree $\ell$,
\begin{equation}
\label{eq:schur}
\Exp_R \babs{ \left\langle f_\ell, R f_\ell \right\rangle_{S} }^2
= \frac{\snorm{f_\ell}^2}{d_\ell} \, .
\end{equation}
where $\snorm{f_\ell}^2 = \langle f_\ell, f_\ell \rangle_{S} = \Exp_{x \in S_{n-1}} |f_\ell(x)|^2$.  This is a form of Schur's lemma: for any vector $v$ belonging to an irreducible representation $\rho$ of a group $G$, we have $\Exp_g \abs{ \langle v, gv \rangle }^2 = |v|^2/d_\rho$.

More generally, let $M$ be a linear operator on $L_2(S_{n-1})$.  Conjugating it with a random rotation yields an operator which commutes with all rotations.  By Schur's lemma any such operator is block diagonal, where each block is a scalar matrix operating on the degree-$\ell$ spherical harmonics.  Thus
\[
\Exp_R R^{-1} M R = \bigoplus_{\ell \ge 0} M_\ell
\quad \text{where} \quad
M_\ell = \frac{\tr M_\ell}{d_\ell} \,\mathds{1}_\ell \, ,
\]
where $\mathds{1}_\ell$ is the projection operator onto the space of degree-$\ell$ harmonics.  Thus if $f = \sum_{\ell \ge 0} f_\ell$ where each $f_\ell$ is a spherical harmonic of degree $\ell$,
\begin{align}
\Exp_R \,\langle Rf, M \cdot Rf \rangle_{S}
&= \left\langle f, \left( \Exp_R R^{-1} M R \right) f \right\rangle_{S} \nonumber \\
&= \sum_{\ell \ge 0} \langle f_\ell, M_\ell f_\ell \rangle_{S} \nonumber \\
&= \sum_{\ell \ge 0} \,\Exp_R \,\langle Rf_\ell, M \cdot Rf_\ell \rangle_{S} \, .
\label{eq:schur-separable}
\end{align}
In particular, if $f = \sum_\ell f_\ell$ then the expected outer product of $Rf$ with itself is
\begin{equation}
\label{eq:schur-outer}
\Exp_R \ket{Rf} \bra{Rf}
= \sum_{\ell \ge 0} \frac{\snorm{f_\ell}^2}{d_\ell} \,\mathds{1}_\ell \, .
\end{equation}
Thus if $g = \sum_\ell g_\ell$ and $h = \sum_\ell h_\ell$,
\begin{equation}
\label{eq:schur-outer-inner}
\Exp_R \,\langle g, Rf \rangle_{S} \langle Rf, h \rangle_{S}
= \sum_{\ell \ge 0} \frac{\snorm{f}^2}{d_\ell} \langle g_\ell, h_\ell \rangle_{S} \, .
\end{equation}

Another consequence of the irreducibility of $\rho_\ell$ is that any linear operator from $\rho_\ell$ to $\mathbb{C}$ can be written as $\langle \phi, \cdot \rangle_{S}$ where $\phi = \sum_{y \in S_{n-1}} c_y \gamma_\ell^{(y)}$ for some finite set of nonzero coefficients $c_y$.  In particular, for any $y \in S_{n-1}$ and any $\ell \ge 0$ there is an \emph{evaluation map} $\mathcal{E}^{(y)}_\ell: \rho_\ell \to \mathbb{C}$ such that, for any spherical harmonic $f$ of degree $\ell$, we have $\mathcal{E}^{(y)}_\ell(f) = f(y)$.  We can express it as
\begin{equation}
\label{eq:eval}
\mathcal{E}^{(y)}_\ell(f) = f(y) = \sqrt{d_\ell} \,\left\langle \gamma_\ell^{(y)} , f \right\rangle_{S} \, .
\end{equation}
To see this, think of $y$ as the north pole, and consider an orthonormal basis that includes $\gamma_\ell^{(y)}$.  Since $\gamma_\ell^{(y)}$ is the unique harmonic of degree $\ell$ that is zonal around $y$, all other basis functions are zero at $y$; otherwise they would have a nonzero projection onto $\gamma_\ell^{(y)}$ if we average over the subgroup $\mathsf{SO}_{n-1}$ of rotations that preserve $y$.  The normalization $\sqrt{d_\ell}$ then follows from~\eqref{eq:gegen-max}.

By summing over all $\ell$, we can similarly express the evaluation map for all integrable functions $f \in L_2(S_{n-1})$.  That is, we can define the evaluation map
\[
\mathcal{E}^{(y)} = \sum_{\ell \ge 0} \mathcal{E}^{(y)}_\ell \, .
\]
Then
\begin{equation}
\label{eq:eval-all}
\mathcal{E}^{(y)}(f)
= f(y)
= \sum_{\ell \ge 0} \sqrt{d_\ell} \,\left\langle \gamma_\ell^{(y)} , f \right\rangle_{S} \, .
\end{equation}
To put it differently, we can express the Dirac delta function as
\begin{equation}
\label{eq:dirac}
\delta(x-y)
= \sum_{\ell \ge 0} \sqrt{d_\ell} \,\gamma_\ell^{(y)}(x)
= \sum_{\ell \ge 0} \sqrt{d_\ell} \,\gamma_\ell(y \cdot x) \, .
\end{equation}

\section{Relating noise sensitivity and spherical sensitivity for randomly rotated functions}

In this section we will prove our main transfer theorem, bounding the expected noise sensitivity of a randomly rotated function in terms of its spherical sensitivity.

We identify the hypercube $\mathbb{Z}_2^n$ with the set $H = \{\pm 1/\sqrt{n} \}^n$ lying on the unit sphere.  If we restrict the inner product to this set, we obtain a cubical inner product, which we write
\[
\langle f, g \rangle_{H}
= \Exp_{x \in H} f(x)^* g(x) \, .
\]
In particular, for each frequency vector $k \in \mathbb{Z}_2^n$, if we extend the character $\chi_k$ to the sphere as a multilinear function of degree $|k|$,
\[
\chi_k(x) = \prod_{i: k_i=1} \sqrt{n} x_i \, ,
\]
then the $\chi_k$ are orthonormal with respect to the cubical inner product,
\[
\langle \chi_k, \chi_{k'} \rangle_{H} = \delta_{k,k'} \, .
\]
If we define a Boolean function $f|_H$ by restricting a function $f$ to the hypercube, its Fourier coefficients are
\[
\widehat{f}|_H(k) = \langle f, \chi_k \rangle_{H} \, .
\]
The \emph{energy} of $f$ at the character $k$ is $|\widehat{f}(k)|^2$.

In order to bound the noise sensitivity, we need to compute the expected energy of $Rf$ where $R$ is uniformly random, i.e., chosen according to the Haar measure in $\mathsf{SO}_n$; or equivalently, the expected energy of $f$'s restriction to a randomly rotated hypercube.  One simple observation is the following.  Since uniformly rotating any point on the cube yields a uniformly random point on the sphere, cubical inner products are equal to spherical inner products in expectation,
\begin{equation}
\label{eq:cube-sphere-exp}
\Exp_R \langle Rf, Rg \rangle_{H} = \langle f, g \rangle_{S} \, .
\end{equation}

In Appendix~\ref{app:kravchuk} we give a precise expression for the expected energy of a randomly rotated spherical harmonic.  However, here we just need a few facts.  First,  if we decompose a function into spherical harmonics, then its expected energy at each $k$ is the sum of the expected energies of its harmonics:
\begin{lemma}
\label{lem:separable}
Let $f \in L_2(S_{n-1})$, and write $f = \sum_{\ell \ge 0} f_\ell$ where each $f_\ell$ is a spherical harmonic of degree $\ell$.  Let $R \in \mathsf{SO}_n$ be uniform in the Haar measure.  Then for any $k \in \mathbb{Z}_2^n$,
\[
\Exp_R \babs{ \widehat{Rf}|_H(k) }^2
= \sum_{\ell \ge 0} \,\Exp_R \,\babs{ \widehat{Rf_\ell}|_H(k) }^2 \, .
\]
\end{lemma}

\begin{proof}
First we use the evaluation maps of Section~\ref{sec:schur} to write $\widehat{f}(k)$ as a spherical inner product rather than a cubical one.  Using~\eqref{eq:eval-all} we have
\[
\widehat{f}(k)
= \langle \chi_k, f \rangle_{H}
= \Exp_{x \in H} \chi_k(x) \,\mathcal{E}^{(x)}(f)
= \left\langle \psi_k, f \right\rangle_{S} \, ,
\]
where
\[
\psi_k = \Exp_{x \in H} \chi_k(x) \sum_{\ell \ge 0} \sqrt{d_\ell} \gamma_\ell^{(x)} \, .
\]
Then applying Schur's lemma~\eqref{eq:schur-separable} to the linear operator $M = \ket{\psi_k} \bra{\psi_k}$ gives
\begin{align*}
\Exp_R \babs{ \widehat{Rf}|_H(k) }^2
&= \Exp_R \,\langle Rf, \psi_k \rangle_{S} \langle \psi_k, Rf \rangle_{S} \\
&= \sum_{\ell \ge 0} \,\Exp_R \,\langle Rf_\ell, \psi_k \rangle_{S} \langle \psi_k, Rf_\ell \rangle_{S} \\
&= \sum_{\ell \ge 0} \,\Exp_R \,\babs{ \widehat{Rf_\ell}|_H(k) }^2 \, ,
\end{align*}
completing the proof.
\end{proof}

Secondly, restricting a spherical harmonic of degree $\ell$ to the hypercube can only give it nonzero energy at characters of Hamming weight $\ell$ or less:

\begin{lemma}
\label{lem:lower}
Let $f$ be a spherical harmonic of degree $\ell$.  If $|k| \ge \ell$, then $\widehat{f}|_H(k) = 0$.
\end{lemma}

\begin{proof}
Restricting any polynomial $f$ of degree $\ell$ to $H = \{ \pm 1/\sqrt{n} \}^n$ imposes the relations $x_i^2 = 1/n$ for all $i$, so there is a multilinear polynomial $f'$ of degree at most $\ell$ such that $f|_H = f'|_H$.  Each multilinear monomial is proportional to a character $\chi_k$ with $|k| \le \ell$, and these are orthogonal to all $\chi_k$ with $|k| > \ell$.
\end{proof}

Thirdly, for any integrable function $f$, the expected energy of $Rf$ summed over all characters of $\mathbb{Z}_2^n$ equals its norm on the sphere:
\begin{lemma}
\label{lem:total}
Let $f \in L_2(S_{n-1})$, and let $R \in \mathsf{SO}_n$ be uniform in the Haar measure.  Then
\[
\Exp_R \sum_{k \in \mathbb{Z}_2^n} \babs{ \widehat{Rf}|_H(k) }^2 = \snorm{f}^2 \, .
\]
\end{lemma}

\begin{proof}
Since the $\chi_k$ are orthonormal with respect to the inner product on the hypercube, for any $f$ we have
\[
\sum_{k \in \mathbb{Z}_2^n} \babs{ \widehat{f}|_H(k) }^2
= \sum_k \langle f, \chi_k \rangle_{H} \langle \chi_k, f \rangle_{H}
= \langle f, f \rangle_{H}
\]
Then~\eqref{eq:cube-sphere-exp} gives
\[
\Exp_R \sum_{k \in \mathbb{Z}_2^n} \babs{ \widehat{Rf}|_H(k) }^2
= \Exp_R \langle Rf, Rf \rangle_{H}
= \langle f, f \rangle_{S}
= \snorm{f}^2 \, .\qedhere
\]
\end{proof}

We are now ready to prove our transfer theorem, which bounds the expected noise sensitivity and average sensitivity in terms of the spherical sensitivity.
\begin{theorem}
\label{thm:transfer}
Let $f \in L_2(S_{n-1})$, and let $R \in \mathsf{SO}_n$ be uniform in the Haar measure.  Then
\begin{equation}
\label{eq:transfer-ns}
\Exp_R \mathbb{NS}_\varepsilon(Rf|_H)
\le \SS_t(f) \, ,
\end{equation}
where
\begin{equation}
\label{eq:t}
t
= \frac{1}{n} \log \frac{1}{1-2\varepsilon}
= \frac{2\varepsilon}{n} \big(1+O(\varepsilon)\big) \, .
\end{equation}
\end{theorem}

\begin{proof}
Starting with the Fourier-theoretic expression for the noise sensitivity~\eqref{eq:ns}, we have
\begin{align}
\Exp_R \mathbb{NS}_\varepsilon(Rf|_H)
&= \frac{1}{2} \Exp_R \,\sum_{k \in \mathbb{Z}_2^n} \babs{\widehat{Rf}|_H(\chi)}^2 \left(1 - (1 - 2\varepsilon)^{|k|}\right)
\nonumber \\
&= \frac{1}{2} \Exp_R \,\sum_k \sum_{\ell \ge 0} \babs{\widehat{Rf_\ell}|_H(k)}^2 \left(1 - (1 - 2\varepsilon)^{|k|} \right)
\label{eq:separable} \\
&= \frac{1}{2} \Exp_R \,\sum_\ell \sum_{k: |k| \le \ell} \babs{\widehat{Rf_\ell}|_H(k)}^2 \left(1 - (1 - 2\varepsilon)^{|k|} \right)
\label{eq:lower} \\
&\le \frac{1}{2} \Exp_R \,\sum_\ell \sum_k \babs{\widehat{Rf_\ell}|_H(k)}^2 \left(1 - (1 - 2\varepsilon)^{\ell} \right)
\nonumber \\
&\le \frac{1}{2} \sum_\ell \snorm{f_\ell}^2 \left(1 - (1 - 2\varepsilon)^{\ell} \right) \, .
\label{eq:total}
\end{align}
Here we used Lemma~\ref{lem:separable} in~\eqref{eq:separable}, Lemma~\ref{lem:lower} in~\eqref{eq:lower}, and Lemma~\ref{lem:total} in~\eqref{eq:total}.

On the other hand, using $\snorm{f_\ell}^2 = \sum_j \abs{ \widehat{f}(\ell,j) }^2$ in~\eqref{eq:ss} gives
\begin{align}
\SS_t(f)
&= \frac{1}{2} \sum_{\ell} \snorm{f_\ell}^2 \left( 1-\mathrm{e}^{-t \ell (n+\ell-2)} \right)
\nonumber \\
&\ge \frac{1}{2} \sum_{\ell} \snorm{f_\ell}^2 \left( 1-\mathrm{e}^{-t \ell n} \right) \, .
\label{eq:ss-lower}
\end{align}
Setting $1-2\varepsilon = \mathrm{e}^{-tn}$ completes the proof of~\eqref{eq:transfer-ns}.
\end{proof}

Although we don't need it below, we record an analogous theorem regarding the expected average sensitivity.

\begin{theorem}
\label{thm:transfer-as}
Let $f \in L_2(S_{n-1})$, and let $R \in \mathsf{SO}_n$ be uniform in the Haar measure.  Then for any $\alpha > 0$,
\begin{equation}
\label{eq:transfer-as}
\Exp_R \mathbb{AS}(Rf|_H)
\le \frac{2n}{1-\mathrm{e}^{-\alpha}} \,\SS_{\alpha/n^2}(f) \, .
\end{equation}
\end{theorem}

\begin{proof}
Recall the Fourier-theoretic expression for the average sensitivity,
\begin{align}
\mathbb{AS}(f)
&= - \frac{1}{2} \langle f, Lf \rangle \nonumber \\
&= \sum_{k \in \mathbb{Z}_2^n} \abs{\widehat{f}(k)}^2 |k| \, .
\label{eq:as}
\end{align}

Applying the same lemmas to~\eqref{eq:as} as we did to~\eqref{eq:ns} in the proof of the previous lemma, and noting that $|k| \le n$, gives
\[
\Exp_R \mathbb{AS}(Rf|_H)
\le \sum_\ell \snorm{f_\ell}^2 \max(\ell,n) \, .
\]
Setting $t=\alpha/n^2$ in~\eqref{eq:ss-lower}, we have for all $0 \le \ell \le n$
\[
\frac{1-\mathrm{e}^{-\alpha \ell/n}}{1-\mathrm{e}^{-\alpha}} \ge \frac{\ell}{n} \, ,
\]
since $1-\mathrm{e}^{-x}$ is concave.  This completes the proof of~\eqref{eq:transfer-as}.
\end{proof}

\section{Application: the Gotsman-Linial conjecture on average}

In this section we bound the spherical sensitivity of polynomial threshold functions, and apply Theorem~\ref{thm:transfer} to bound their expected noise sensitivity.

Our strategy for bounding the spherical sensitivity is similar to that of Kane~\cite{kane-surface-area}, who proved a similar bound on their Gaussian sensitivity.  He used the fact that adding Gaussian noise to a point $u$ can be thought of as choosing a random line through $u$, and then moving along that line by a distance $r$ chosen from the chi-squared distribution: then $p(x)$ can only change sign if it has a root on the intervening line segment.  Similarly, we add noise on the sphere by choosing a random great circle that passes through $u$, and then moving an angle $r$ along that circle where $r$ is chosen from a distribution derived from the heat equation on $S_{n-1}$.  The sensitivity is then at most the probability that the polynomial has a root on the resulting segment of the great circle.

First we need to bound the distribution of the angle, or equivalently the geodesic distance, that we travel on the sphere in time $t$.  Given two points $u, v \in S_{n-1}$ and a time $t \ge 0$, let $K_t(u,v)$ denote the \emph{heat kernel} on $S_{n-1}$ after time $t$.  That is, for any fixed $t$ and $u$, $K_t(u,v)$ is the probability distribution of the position $v$ of a particle that starts at $u$ and undergoes Brownian motion for time $t$.  As a linear operator, it is the solution to the partial differential equation
\[
\frac{\partial K_t}{\partial t} = \Delta_{S_{n-1}} K_t
\]
with the initial condition $K_0 = \mathds{1}$, so
\[
K_t(u,v) = \mathrm{e}^{t\Delta_{S_{n-1}}} \, .
\]
By comparing the heat kernel on $S_{n-1}$ to that on $\mathbb{R}^{n-1}$, we prove the following:

\begin{lemma}
\label{lem:sphere-flat}
Fix $u \in S_{n-1}$, and suppose that $v \in S_{n-1}$ is chosen with probability distribution $K_t(u,v)$.  Let $r$ denote the angle between $u$ and $v$.  Then
\[
\Exp[r] \le \sqrt{2(n-1)t} \, .
\]
\end{lemma}

\begin{proof}
In Appendix~\ref{app:sphere-flat} we offer an elementary calculus proof that $\Exp[r^2] \le 2(n-1)t$, which implies the lemma.  However, we can prove something much stronger: namely, that $r$ is stochastically dominated by the corresponding process on the flat tangent space $\mathbb{R}^{n-1}$.  The heat equation on $\mathbb{R}^{n-1}$ is driven by the Laplacian
\begin{equation}
\label{eq:heat-flat}
\Delta_{\mathbb{R}^{n-1}}
= \sum_{i=1}^{n-1} \frac{\partial^2}{\partial x_i^2} \, .
\end{equation}
Place $u$ at the origin, and let $r$ denote the distance from the origin.  Since $K_t(u,v) = f(r)$ is spherically symmetric, transforming to polar coordinates gives
\[
\frac{\partial f}{\partial t}
= \Delta_{\mathbb{R}^{n-1}} f
= \frac{\partial^2 \!f}{\partial r^2} + (n-2) \frac{1}{r} \frac{\partial f}{\partial r} \, .
\]
Similarly, for $S_{n-1}$, place $u$ at the north pole and let $r$ denote the angle between $u$ and $v$.  Then $K_t(u,v) = f(r)$ is a zonally symmetric function, and applying the Laplace-Beltrami operator gives
\begin{equation}
\label{eq:heat-sphere}
\frac{\partial f}{\partial t}
= \Delta_{S_{n-1}} f
= \frac{\partial^2 \!f}{\partial r^2} + (n-2) \frac{\cos r}{\sin r} \frac{\partial f}{\partial r} \, .
\end{equation}

We can view~\eqref{eq:heat-flat} and~\eqref{eq:heat-sphere} as governing the probability distributions of two stochastic processes on $r$.  These are well known in the theory of Brownian motion, and are referred to as Bessel and Jacobi processes respectively.  Since $\cos r / \sin r \le 1/r$, the comparison theorem of stochastic differential equations~\cite{ikeda-watanabe} implies that the distribution of $r$ on $S_{n-1}$ is stochastically dominated by its distribution on $\mathbb{R}^{n-1}$.  In particular, its second moment is at most the variance of $n-1$ independent variables $x_1,\ldots,x_{n-1} \in \mathbb{R}$ of variance $2t$, giving
\[
\Exp[r^2] \le 2(n-1)t \, .
\]
Noting that $\Exp[r] \le \sqrt{\Exp[r^2]}$ completes the proof.
\end{proof}

We remark that for small $n$, we get a small improvement by computing $\Exp[r]$ exactly on $\mathbb{R}^{n-1}$.  The fact that $r^2/2$ follows a chi-squared distribution with $n-1$ degrees of freedom implies
\[
\Exp[r]
\le \frac{2 \Gamma(n/2)}{\Gamma((n-1)/2)} \sqrt{t}
= \big( 1-O(1/n) \big) \sqrt{2(n-1)t} \, .
\]

\begin{lemma}
\label{lem:great-circle}
Let $p \in \mathbb{R}[x_1, \ldots, x_n]$ be a polynomial of degree $d$, and let $G$ a great circle on $S_{n-1}$.  If $p$ is not identically zero on $G$, then $p$ has no more than $2d$ roots on $G$.
\end{lemma}

\begin{proof}
Since applying a linear transformation to $x_1, \ldots, x_n$ doesn't change $p$'s degree, without loss of generality we can assume that $G$ is the unit circle in the plane spanned by the $x_1$ and $x_2$ axes: that is, the variety, or set of roots, of the polynomial
\[
q(x_1,x_2) = x_1^2 + x_2^2 - 1 \, .
\]
Restricting to this plane, i.e., setting $x_i=0$ for all $i > 2$, yields a polynomial $r(x_1,x_2)$ of degree $d_r \le d$.
B\'{e}zout's theorem~\cite{Shafarevich:1994,Schmid:1995} states that two polynomials $q, r$ of degree $d_q$ and $d_r$ can share at most $d_q d_r$ roots unless they share a common factor.  It is easy to check that $q$ is irreducible over $\mathbb{R}$ (and even over $\mathbb{C}$), since if it had a linear factor then $G$ would consist of the union of two lines.  Therefore, $q$ and $r$ share a common factor only if $q$ divides $r$, in which case $p$ is identically zero on $G$.  If they do not, they share at most $2d$ roots.
\end{proof}

Putting these lemmas together gives us a bound on the spherical sensitivity of a polynomial threshold function.
\begin{theorem}
\label{thm:poly-ss}
Let $p: \mathbb{R}^n \to \mathbb{R}$ be a polynomial of degree $d$ and let $f(x) = \mathop{\mathrm{sgn}}(p(x))$ where $\mathop{\mathrm{sgn}} z = +1$ for $z \ge 0$ and $\mathop{\mathrm{sgn}} z = -1$ for $z < 0$.  Then
\[
\SS_t(f) \le \frac{d}{\pi} \sqrt{2nt}  \, .
\]
\end{theorem}

\begin{proof}
Recall that $\SS_t(f)$ is the probability that $f(u) \ne f(v)$ where $u$ is chosen uniformly and $v$ is chosen from $K_t(u,v)$.  Equivalently, we can choosen $u$ uniformly, and then arrive at $v$ by choosing a uniformly random great circle $G$ passing through $u$ (by choosing a tangent vector from the Haar measure on $S_{n-2}$), choosing $r$ according to the heat kernel, and moving an angle $r$ along $G$.

If $p$ is identically zero on $S_{n-1}$, then $f$ is identically zero as well, in which case $\SS_t(f) = 0$.  Otherwise, with probability $1$ we have $f(u) \ne 0$, in which case $p$ is not identically zero on $G$.  By Lemma~\ref{lem:great-circle}, there are at most $2d$ roots of $p$ on $G$.  The probability that $p(u)$ and $p(v)$ have different signs is then at most the expected number of roots of $p$ on $G$ between $u$ and $v$.  Since $u$'s position on $G$ is uniformly random, this is simply $(2d/2\pi) \Exp[r]$, which by Lemma~\ref{lem:sphere-flat} is at most $(d/\pi) \sqrt{2nt}$.
\end{proof}

Theorems~\ref{thm:transfer} and~\ref{thm:poly-ss} immediately imply our bound on the expected noise sensitivity:

\begin{theorem}
\label{thm:poly-ns}
Let $p$ be a polynomial of degree $d$ and let $f(x) = \mathop{\mathrm{sgn}}(p(x))$.  Let $R \in \mathsf{SO}_n$ be uniform in the Haar measure.  Then
\begin{equation}
\label{eq:poly-ns}
\Exp_R \mathbb{NS}_\varepsilon(Rf|_H) \le \big( 1+O(\varepsilon) \big) \frac{2}{\pi} \,d \sqrt{\varepsilon} \, .
\end{equation}
\end{theorem}

Similar but simpler reasoning implies a bound on the expected average sensitivity:

\begin{theorem}
\label{thm:poly-as}
Let $p$ be a polynomial of degree $d$ and let $f(x) = \mathop{\mathrm{sgn}}(p(x))$.  Let $R \in \mathsf{SO}_n$ be uniform in the Haar measure.  Then
\begin{equation}
\label{eq:poly-as}
\Exp_R \mathbb{AS}(Rf|_H)
\le \big( 1+O(1/n) \big) \frac{2}{\pi} \, d \sqrt{n} \, .
\end{equation}
\end{theorem}

\begin{proof}
The angle between two adjacent corners of the hypercube is
\[
r = \cos^{-1} (1-2/n) = \frac{2}{\sqrt{n}} \,\big(1+O(1/n) \big) \, ,
\]
and $\Exp_R \mathbb{AS}(Rf|_H)$ is $n$ times the probability that $p$ changes sign between a uniformly random pair of points $r$ apart.  Using Lemma~\ref{lem:great-circle} as in the proof of Theorem~\ref{thm:poly-ns}, this probability is at most $(d/\pi)r$.
\end{proof}

\begin{remark}
For small $\varepsilon$, we can also prove Theorem~\ref{thm:poly-ns} by noting that the angle $r$ between two points $x, y$ on the hypercube where we have flipped each bit independently with probability $\varepsilon$ obeys $\Exp[\cos r] = 1-2\varepsilon$.
\end{remark}

\begin{remark}
The leading constant in~\eqref{eq:poly-as} is better than we would obtain from Theorem~\ref{thm:transfer-as}, even after optimizing the parameter $\alpha$.
\end{remark}

\begin{remark}
Theorems~\ref{thm:poly-ss} and~\ref{thm:poly-ns} immediately generalize from polynomials to any class of functions with a bound on the number of roots lying on a great circle.  If there are at most $b$ such roots, the expected noise sensitivity of the corresponding threshold function is $O(b \sqrt{\varepsilon})$ and its expected average sensitivity is $O(b \sqrt{n})$.
\end{remark}

\paragraph{Acknowledgments}  We are grateful to Fabrice Baudoin, Laura De Carli, Costas Efthimiou, Veit Elser, Josh Grochow, Ilia Krasikov, Ryan O'Donnell, Thomas H.\ Parker, Dan Rockmore, and James Stokes for helpful conversations.  This work was supported by NSF grants CCF-1117426 and CCF-1219117.