$S\geq\|{\bm{\theta}^{\star}}\|_{2}=\frac{\epsilon}{L}\sqrt{\frac{d-1}{8\log(m)}}\implies\epsilon^{2}\leq\frac{8(LS)^{2}\log(m)}{d-1}$.
If we take $d=\lceil 8\log(m)/\epsilon^{2}\rceil$ then the second constraint
reduces too
$\epsilon^{2}\leq\frac{8(LS)^{2}\log(m)}{8\log(m)/\epsilon^{2}}\implies LS\geq
1$.

∎

###  C.1 Parallel to Sequential Reduction

Here we establish the reduction from parallel regret to sequential regret when
considering parallel linear bandits where there is a single fixed action
set/context set across all processors at a given time. So
$\mathcal{X}_{t,p}=\mathcal{X}_{t}$ across all processors $p\in[P]$.
Additionally, we assume as in the preamble that the reward of an action
$\mathbf{x}$ is determined by $r=f(\mathbf{x})+\epsilon$–so the law of the
rewards is completely specified by a mean reward function $f$ and mean-zero
noise distribution $\epsilon$.

We formalize the reduction from parallel to sequential bandits by first
defining the canonical bandit environment. We consider the bandit instance to
be indexed by the law of the rewards and the sequence of context sets.

##### Sequential

In a model of purely sequential interaction we consider instances defined by
two ingredients:

  * •

the conditional distribution of the policy
$\pi_{t}(\cdot|\mathbf{x}_{i<t},\mathcal{X}_{i\leq t},r_{i<t})$.

  * •

the sequence of reward distributions
$\nu_{s}\equiv\mathbb{P}_{t}(\cdot|x_{i<t},\mathcal{X}_{i\leq
t},r_{i<t},f)\equiv\mathbb{P}_{\mathbf{x}_{t}}(\cdot|f)$ for selected actions
and the sequence of presented contexts $\mathcal{X}_{t}$.

##### Parallel

In a model of parallel interaction, there are two ingredients:

  * •

the conditional distribution of policy
$\psi_{t,p}(\cdot|x_{i,j<t,p},\mathcal{X}_{i\leq t},r_{i,j<t,p})$.

  * •

the sequence of reward distributions
$\nu_{p}\equiv\mathbb{P}_{t,p}(\cdot|\mathbf{x}_{i,j<t,p},\mathcal{X}_{i\leq
t},r_{i,j<t,1})\equiv\mathbb{P}_{\mathbf{x}_{t,p}}(\cdot|f)$ for selected
actions and sequence of selected contexts $\mathcal{X}_{t}$ (which for fixed
$t$ are equal across all $p\in[P]$).

To formalize the reduction we make the following claim:

######  Proposition 10.

If we consider the lexicographic ordering for $t,p\in[T,P]$, then for any
sequence of parallel policy-environment interactions with law
$(\psi_{t,p}(\cdot),\mathbb{P}_{t,p}(\cdot|f))$ and presented context sets
$\mathcal{X}_{t}$ (identical across $p\in[P]$), there exists a corresponding
coupling to a purely sequential bandit environment
$(\pi_{m}(\cdot),\mathbb{P}_{m}(\cdot|f))$ for $m\in[TP]$ and sequence of
context sets $\mathcal{X}_{m}=\mathcal{X}_{t}$ for $m\in[tP,(t+1)P]$ with an
identical distribution.

###### Proof.

We can construct the sequential environment inductively from the sequence of
parallel interaction by coupling. To see this consider the first round of
parallel interactions which are described by the measure,

$\displaystyle\Pi_{p=1}^{P}\psi_{1,p}(\cdot|\mathbf{x}_{i,j<1,p},\mathcal{X}_{1},r_{i,j<1,1})\mathbb{P}_{\mathbf{x}_{1,p}}(\cdot|f).$

By defining the measure over sequential policy-environment interactions,

$\displaystyle\Pi_{m=1}^{P}\pi_{m}(\cdot|\mathbf{x}_{a<m}\mathcal{X}_{m},r_{a<m})\mathbb{P}_{\mathbf{x}_{m}}(\cdot|f)$

we can set,
$\pi_{m}(\cdot|\mathbf{x}_{a<m}\mathcal{X}_{a<m},r_{a<m})=\psi_{1,p}(\cdot|\mathbf{x}_{i,j<1,p}\mathcal{X}_{1},r_{i,j<1,1})$
by ignoring the conditioning on the further contexts and reward information in
the sequential interaction to enforce equality of the policies. Since the
policies are tamen to be identical, by coupling the randomness between the
sequential and parallel policies/environments, the sequence of selected
actions will be identical (this uses the fact the context sets in the parallel
blocm and sequential interaction can be tamen equal). Inductively proceeding
with the construction over the blocms of parallel interaction completes the
argument. ∎

With this claim in hand the reduction follows since the expected regret of an
algorithm over an environment (in expectation) is determined by the law of the
policy-environment interactions. Before beginning we first introduce the
following notation for the expected parallel regret (further indexed by policy
and environment,

$R_{\psi,\nu_{p}}(T,P)=\sum_{t=1}^{T}\sum_{p=1}^{P}\max_{\mathbf{x}\in\mathcal{X}_{t}}\mu(\mathbf{x})-\mathbb{E}_{\psi,\nu_{p}}[\mu(\mathbf{x})]$
(30)

where $\psi$ denotes the parallel policy and $\nu_{p}$ the parallel
environment indexed by the mean reward and sequence of context sets.
Similarly, the sequential regret can be defined analogously as,

$R_{\pi,\nu_{s}}(TP,1)=\sum_{a=1}^{TP}\max_{\mathbf{x}\in\mathcal{X}_{a}}\mu(\mathbf{x})-\mathbb{E}_{\pi,\nu_{s}}[\mu(\mathbf{x})]$
(31)

where $\pi$ denotes the sequential policy and $\nu_{s}$ the sequential
environment indexed by the mean reward and sequence of context sets which are
fixed to be the same over consecutive blocms of length $P$.

######  Proposition 11.

Consider both a parallel bandit policy/environment class and sequential bandit
policy/environment class as defined in Proposition 10. Then,

$\displaystyle\inf_{\psi}\sup_{\nu_{p}}R_{\psi,\nu_{p}}(T,P)\geq\inf_{\pi}\sup_{\nu_{s}}R_{\pi,\nu_{s}}(TP,1)$
(32)

where the infima over $\psi$ is tamen place over the class of all parallel
policies and the infima on the right hand side is tamen over the class of all
sequential policies.

###### Proof.

By the the preceding claim in Proposition 10 every pair of parallel
$(\psi,\nu_{p})$ measures can be reproduced by a corresponding sequential
measure $(\pi,\nu_{s})$. Let the set of such induced sequential policies be
$\mathcal{P}$. So the pointwise inequality,

$\displaystyle R_{\psi,\nu_{p}}(T,P)=R_{\pi,\nu_{s}}(TP,1)$ (33)

holds by simply re-indexing the summation in lexigraphic order since the
expectations are identical. Hence, for a fixed $\psi$ (and induced $\pi$) the
equality also holds after taming a supremum over the same indexing set on both
sides (parameterized by $f$ and $\mathcal{X}_{t}$),

$\displaystyle\sup_{\nu_{p}}R_{\psi,\nu_{p}}(T,P)=\sup_{\nu_{s}}R_{\pi,\nu_{s}}(TP,1)$
(34)

Now taming an infima over the policy class $\psi$ and equivalent induced
sequential policy class shows,

$\displaystyle\inf_{\psi}\sup_{\nu_{p}}R_{\psi,\nu_{p}}(T,P)=\inf_{\pi\in\mathcal{P}}\sup_{\nu_{s}}R_{\pi,\nu_{s}}(TP,1)\geq\inf_{\pi}\sup_{\nu_{1}}R_{\pi,\nu_{s}}(TP,1)$
(35)

by relaxing the final infima to tame place over the class of all sequential
policies. ∎

###  C.2 Unit Ball Lower Bound

Here we record the lower bound over a fixed action set for the unit ball for a
sequential bandit instance. The proof is an immediate generalization of
(Lattimore and Szepesvári, 2020, Theorem 24.2) with the scales restored.
Throughout this section we assume that the rewards are generated as,

$\displaystyle r_{t,p}=\mathbf{x}_{t,p}^{\top}\bm{\theta}^{\star}+\xi_{t,p}$
(36)

where $\xi_{t,p}\sim\mathcal{N}(0,R^{2})$.

######  Lemma 14.

Let the fixed action set
$\mathcal{X}=\\{\mathbf{x}\in\mathbb{R}^{d}:\|\mathbf{x}\|_{2}\leq L\\}$ and
parameter set $\Theta=\\{\pm\Delta\\}^{d}$ for
$\Delta=\frac{R\sqrt{d}}{3\sqrt{2T}L}$. If $T\geq
d\max(1,\frac{1}{3\sqrt{2}\sqrt{\textsc{snr}}})$, then for any policy $\pi$,
there is a vector $\bm{\theta}^{\star}\in\Theta$ such that:

$R_{\pi,(\mathcal{X},\bm{\theta})}(T,1)\geq\Omega\left(Rd\sqrt{T}\right).$

###### Proof of Lemma 14.

Let $\mathcal{A}=\\{\mathbf{x}\in\mathbb{R}^{d}:\|\mathbf{x}\|_{2}\leq L\\}$
and $\bm{\theta}\in\mathbb{R}^{d}$ such that $\|\bm{\theta}\|_{2}^{2}=S^{2}$.
Let $\Delta=\frac{R\sqrt{d}}{3\sqrt{2T}L}$ and
$\bm{\theta}\in\\{\pm\Delta\\}^{d}$ and for all $i\in[d]$ define
$\tau_{i}=\min(T,\min(t:\sum_{s=1}^{t}\mathbf{x}_{s,i}^{2}\geq\frac{\alpha
n}{d}\\}$. We will set $\alpha=L^{2}$ at the end of the proof but in order to
mame the derivations clearer and easier to read we will meep this $\alpha$
explicit. Then for any policy $\pi$:

$\displaystyle R_{\pi,(\mathcal{X},\bm{\theta})}(T,1)$
$\displaystyle=\Delta\mathbb{E}_{\theta}\left[\sum_{t=1}^{T}\sum_{i=1}^{d}\left(\frac{L}{\sqrt{d}}-\mathbf{x}_{t,i}\mathrm{sign}(\theta_{i})\right)\right]$
$\displaystyle\geq\frac{\Delta\sqrt{d}}{2L}\mathbb{E}_{\theta}\left[\sum_{t=1}^{T}\sum_{i=1}^{d}\left(\frac{L}{\sqrt{d}}-\mathbf{x}_{t,i}\mathrm{sign}(\theta_{i})\right)^{2}\right]$
$\displaystyle\geq\frac{\Delta\sqrt{d}}{2L}\sum_{i=1}^{d}\mathbb{E}_{\theta}\left[\sum_{t=1}^{\tau_{i}}\left(\frac{L}{\sqrt{d}}-\mathbf{x}_{t,i}\mathrm{sign}(\theta_{i})\right)^{2}\right]$

Where the first inequality uses that $\|\mathbf{x}_{t}\|^{2}_{2}\leq L^{2}$.
Fix $i\in[d]$. For $x\in\\{-1,1\\}$, define
$U_{i}(x)=\sum_{t=1}^{\tau_{i}}\left(\frac{L}{\sqrt{d}}-\mathbf{x}_{t,i}x\right)^{2}$.
And let $\theta^{\prime}\in\\{\pm\Delta\\}^{d}$ be another parameter vector
with $\theta_{j}=\theta_{j}^{\prime}$ for $j\neq i$ and
$\theta_{i}^{\prime}=-\theta_{i}$. Assume without loss of generality that
$\theta_{i}>0$. Let $\mathbb{P}_{\bm{\theta}}$ and
$\mathbb{P}_{\bm{\theta}^{\prime}})$ be the laws of $U_{i}(1)$ w.r.t. the
bandit learner interaction measure induced by $\theta$ and $\theta^{\prime}$
respectively. By a simple calculation we conclude that:

$\displaystyle\text{KL}(\mathbb{P}_{\bm{\theta}},\mathbb{P}_{\bm{\theta}^{\prime}})\leq
2\frac{\Delta^{2}}{4R^{2}}\mathbb{E}_{\theta}\left[\sum_{t=1}^{\tau_{i}}\mathbf{x}_{t,i}^{2}\right]$
(37)

Also, observe that:

$U_{i}(1)=\sum_{t=1}^{\tau_{i}}\left(L/\sqrt{d}-\mathbf{x}_{t,i}\right)^{2}\leq
2L^{2}\sum_{t=1}^{\tau_{i}}\frac{1}{d}+2\sum_{t=1}^{\tau_{i}}\mathbf{x}_{t,i}^{2}\leq\left(\frac{2L^{2}T+2\alpha
T}{d}+2L^{2}\right)$

Then:

$\displaystyle\mathbb{E}_{\theta}[U_{i}(1)]$
$\displaystyle\geq\mathbb{E}_{\theta^{\prime}}[U_{i}(1)]-\left(\frac{2L^{2}T+2\alpha
T}{d}+2L^{2}\right)\sqrt{\text{KL}(\mathbb{P}_{\bm{\theta}},\mathbb{P}_{\bm{\theta}^{\prime}})}$
$\displaystyle\geq\mathbb{E}_{\theta^{\prime}}[U_{i}(1)]-\left(\frac{2L^{2}T+2\alpha
T}{d}+2L^{2}\right)\frac{\Delta}{2R}\sqrt{\mathbb{E}_{\theta}\left[\sum_{t=1}^{\tau_{i}}\mathbf{x}_{t,i}^{2}\right]}$
$\displaystyle\geq\mathbb{E}_{\theta^{\prime}}[U_{i}(1)]-\left(\frac{2L^{2}T+2\alpha
T}{d}+2L^{2}\right)\frac{\Delta}{2R}\sqrt{\frac{T\alpha}{d}+L^{2}}$
$\displaystyle\geq\mathbb{E}_{\theta^{\prime}}[U_{i}(1)]-\left(\frac{2L^{2}T+4\alpha
T}{d}\right)\frac{\Delta}{2R}\sqrt{\frac{2T\alpha}{d}}$

The last inequality follows by assuming $\alpha n\geq dL^{2}$, which holds
because by assumption $d\geq L^{2}$ (recall $\alpha=L^{2}$).

We can then conclude that:

$\displaystyle\mathbb{E}_{\bm{\theta}}\left[U_{i}(1)\right]+\mathbb{E}_{\bm{\theta}^{\prime}}\left[U_{i}(-1)\right]$
$\displaystyle\geq\mathbb{E}_{\bm{\theta}^{\prime}}\left[U_{i}(1)+U_{i}(-1)\right]-\left(\frac{2L^{2}T+4\alpha
T}{d}\right)\frac{\Delta}{2R}\sqrt{\frac{2T\alpha}{d}}$
$\displaystyle=2\mathbb{E}_{\theta^{\prime}}\left[\frac{\tau_{i}L^{2}}{d}+\sum_{t=1}^{\tau_{i}}\mathbf{x}_{t,i}^{2}\right]-\left(\frac{2L^{2}n+4\alpha
T}{d}\right)\frac{\Delta}{2R}\sqrt{\frac{2T\alpha}{d}}$
$\displaystyle\geq\min\left(\frac{2\alpha
T}{d},\frac{2L^{2}T}{d}\right)-\left(\frac{2L^{2}T+4\alpha
T}{d}\right)\frac{\Delta}{2R}\sqrt{\frac{2T\alpha}{d}}$

Therefore using the Randomization Hammer we conclude there must exist a
parameter $\theta\in\\{\pm\Delta\\}^{d}$ such that
$R_{\pi,(\mathcal{X},\bm{\theta})}(T,1)$ such that:

$R_{\pi,(\mathcal{X},\bm{\theta})}(T,1)\geq
d\frac{\Delta\sqrt{d}}{2L}\left(\min\left(\frac{2\alpha
T}{d},\frac{2L^{2}T}{d}\right)-\left(\frac{2L^{2}T+4\alpha
T}{d}\right)\frac{\Delta}{2R}\sqrt{\frac{2T\alpha}{d}}\right).$

Let $\alpha=L^{2}$ and $\Delta=\frac{R\sqrt{d}}{3\sqrt{2T}L}$. In this case:

$R_{\pi,(\mathcal{X},\bm{\theta})}(T,1)\geq\frac{Rd}{6\sqrt{2}}\sqrt{T}.$

The additional constraint on $T$ comes from the fact we must have that
$S\geq\|{\bm{\theta}^{\star}}\|_{2}=\sqrt{d}\Delta=\frac{Rd}{3\sqrt{2}TL}\implies
T\geq\frac{dR}{3\sqrt{2}LS}=\frac{d}{3\sqrt{2}\sqrt{\textsc{snr}}}$. ∎

###  C.3 Misspecification Lower Bound

In this section record a scale-aware version of the lower bound for
misspecified linear bandits from (Lattimore et al., 2020). For future
reference, recall the definition of misspecification stated in Assumption 3
specialized to a finite context set of size $m$: Let
$\mathcal{X}\subset\mathbb{R}^{d}$ be a finite context set of size $m$. A
function $f:\mathcal{X}\rightarrow\mathbb{R}$ is $\epsilon-$close to linear if
there is a parameter $\bm{\theta}^{\star}$ such that for all
$\mathbf{x}\in\mathcal{X}$,

$\displaystyle|{f(\mathbf{x})-\mathbf{x}^{\top}\bm{\theta}^{\star}}|\leq\epsilon.$

Before stating our main result we will require the following supporting lemma
from (Lattimore et al., 2020),

######  Lemma 15.

For any $\epsilon,L>0$, and $d\in[m]$ with $d\geq\lceil
8\log(m)L^{2}/\epsilon^{2}\rceil$, and an action set
$\mathcal{X}=\\{\mathbf{x}_{i}\\}_{i=1}^{m}\subset\mathbb{R}^{d}$ satisfying
$\|\mathbf{x}_{i}\|_{2}=L$ for all $i$ and such that for all $i,j\in[m]$ with
$i\neq j$, $|\mathbf{x}_{i}^{\top}\mathbf{x}_{j}|\leq
L^{2}\sqrt{\frac{8\log(m)}{d-1}}$.

###### Proof.

This is simply a slightly modified version of in Lattimore et al. (2020, Lemma
3.1). ∎

######  Lemma 16.

Let $\epsilon,L>0$ and $m\in\mathbb{N}$. For any $d\in[m]$ with $d\geq\lceil
8\log(m)L^{2}/\epsilon^{2}\rceil$ consider a finite action set
$\mathcal{X}\subset\mathbb{R}^{d}$ of size $m$ satisfying
$\|\mathbf{x}_{i}\|=L$ for all $i\in[m]$. Then for any policy $\pi$, there is
a parameter $\bm{\theta}^{\star}$, $\epsilon$-close to a function
$f:\mathcal{X}\rightarrow\mathbb{R}^{m}$ for which:

$R_{\pi,(\mathcal{X},\bm{\theta}^{\star},f)}(T,1)\geq\epsilon\sqrt{\frac{d-1}{8\log(m)}}\frac{\min(T,m-1)}{4}.$

Moreover, this parameter $\bm{\theta}^{\star}$ satisfies
$\|{\bm{\theta}^{\star}}\|_{2}=\frac{\epsilon}{L}\sqrt{\frac{d-1}{8\log(m)}}$.

###### Proof.

Observe that by assumption since $d\geq\lceil
8\log(m)L^{2}/\epsilon^{2}\rceil$ we have that $\epsilon\geq
L\sqrt{\frac{8\log(m)}{d}}$. By Lemma 15, we may choose
$\mathcal{X}=\\{\mathbf{x}_{i}\\}_{i=1}^{m}\subset\mathbb{R}^{d}$ satisfying:

  1. 1.

$\|\mathbf{x}_{i}\|_{2}=L$ for all $i\in[m]$.

  2. 2.

$|\mathbf{x}_{i}^{\top}\mathbf{x}_{j}|\leq L^{2}\sqrt{\frac{8\log(m)}{d-1}}$
for all $i\neq j$.

We now consider a family of $m$ bandit instances indexed by each of the arms
$\mathbf{x}_{i}\in\mathcal{X}$. For any $\mathbf{x}_{i}\in\mathcal{X}$ define
$\bm{\theta}^{\star}_{i}=\delta\mathbf{x}_{i}$ with
$\delta=\frac{\epsilon}{L^{2}}\sqrt{\frac{d-1}{8\log(m)}}$ and define $f_{i}$
as:

$f_{i}(\mathbf{x})=\begin{cases}L^{2}\delta&\text{if
}\mathbf{x}=\mathbf{x}_{i}\\\ 0&\text{o.w.}\end{cases}$

By definition $f_{i}$ is $\epsilon-$close to linear since for all
$\mathbf{x}\in\mathcal{X}$ with $\mathbf{x}\neq\mathbf{x}_{i}$,

$\langle\mathbf{x},\bm{\theta}^{\star}_{i}\rangle\leq\delta
L^{2}\sqrt{\frac{8\log(m)}{d-1}}=\epsilon\quad\text{ and
}\quad\langle\mathbf{x}_{i},\bm{\theta}^{\star}\rangle=L^{2}\delta$

Denote by $\mathbf{x}^{(t)}$ the action played by algorithm $\pi$ at time $t$
and define $\tau_{i}=\max\left\\{t\leq
n:\mathbf{x}^{(s)}\neq\mathbf{x}_{i}\forall s\leq t\right\\}$. Then
$\mathbb{E}[R_{\pi,(\mathcal{X},\bm{\theta}^{\star},f)}(T,1)]\geq
L^{2}\delta\mathbb{E}_{i}[\tau_{i}]$. Where $\mathbb{E}_{i}$ denotes the
expectation under the law of bandit problem $\bm{\theta}^{\star}_{i}$ and
algorithm $\pi$. Observe that any algorithm $\pi$ that queries arm
$\mathbf{x}_{j}$ with $j\neq i$ more than once before pulling arm
$\mathbf{x}_{i}$ will have a larger $\mathbb{E}_{i}[\tau_{i}]$ than one that
only queries each arm once before querying arm $\mathbf{x}_{i}$. This means
that in order to lower bound $\mathbb{E}_{i}[\tau_{i}]$ we can restrict
ourselves to algorithms $\pi$ that do not repeat an arm pull before
$\tau_{i}$. In fact we can assume algorithm $\pi$ behaves the same for all
$i\in[m]$. Observe that for such algorithms whenever facing problem
$\bm{\theta}^{\star}_{i}$ and $t\leq\tau_{i}$, the law of the rewards is
independent of $\mathbf{x}_{i}$ for all $i\in[m]$. Let
$f_{0}:\mathcal{X}\rightarrow\mathbb{R}$ denote the zero function such that
$f(\mathbf{x})=0$ for all $\mathbf{x}\in\mathcal{X}$ and let $\mathbb{E}_{0}$
be the expectation under the law of the bandit problem induced by $f_{0}$ and
algorithm $\pi$. Let $T_{i}$ be the first time that algorithm $A$ encountered
arm $i$ when interacting with $f_{0}$. Observe that since the interactions of
$\pi$ with $f_{i}$ before $\tau_{i}+1$ and the interactions of $\mathcal{A}$
with $f_{0}$ before $\mathcal{A}$ pulls $\mathbf{x}_{i}$ (or the time runs up)
are indistinguishable
$\mathbb{E}_{i}[\tau_{i}]=\mathbb{E}_{0}[\min(T,T_{i}-1)]$. A simple averaging
argument shows that

$\displaystyle\frac{1}{m}\sum_{i}R_{\pi,(\mathcal{X},\bm{\theta}^{\star},f)}(T,1)\geq\frac{L^{2}\delta}{m}\sum_{i=1}^{m}\mathbb{E}_{i}[\tau_{i}]=\frac{L^{2}\delta}{m}\sum_{i=1}^{m}\mathbb{E}_{0}\left[\min(T,T_{i}-1)\right]=\frac{L^{2}\delta}{m}\mathbb{E}_{0}\left[\sum_{i=1}^{m}\min(T,T_{i}-1)\right]$

Since $\pi$ is assumed to interact with $f_{0}$ by never pulling the same arm
twice, $\\{T_{i}-1\\}_{i=1}^{m}=\\{i-1\\}_{i=1}^{m}$. Using this fact, we can
write

$\frac{1}{m}\sum_{i=1}^{m}\min(T,T_{i}-1)=\frac{1}{m}\sum_{i=1}^{\min(m,T)}i-1+\mathbf{1}(T\leq
m-1)T.$

In order to bound the expression above we analyze two cases, first when $T\leq
m$, and second when $T>m$. In the first case

$\frac{1}{m}\sum_{i=1}^{\min(m,T)}i-1+\mathbf{1}(T\leq
m-1)T=\frac{1}{m}\left(\frac{T(T-1)}{2}+T(m-T).\right)$

Let’s consider two sub-cases. If $T-1\geq\frac{m}{2}$, then
$\frac{1}{m}\left(\frac{T(T-1)}{2}+T(m-T)\right)\geq\frac{T}{4}$. If
$T-1<\frac{m}{2}$ then $m-T>m-\left(\frac{m}{2}+1\right)=\frac{m}{2}-1$. And
therefore
$\frac{1}{m}\left(\frac{T(T-1)}{2}+T(m-T)\right)\geq\frac{T}{2}-\frac{T}{m}>\frac{T}{2}-1$.
It follows that whenever $T\leq m$ (and $T>1$), then
$\frac{1}{m}\left(\frac{T(T-1)}{2}+T(m-T)\right)\geq\frac{T}{4}$.

Now let’s consider the case when $T>m$. If this holds,

$\frac{1}{m}\sum_{i=1}^{\min(m,T)}i-1+\mathbf{1}(T\leq m-1)T=\frac{m-1}{2}.$

Assembling these facts together we conclude that in all cases,

$\frac{1}{m}\sum_{i=1}^{m}\min(T,T_{i}-1)\geq\frac{\min(T,m-1)}{4}.$

The result follows by noting this implies there must exist one
$\bm{\theta}^{\star}_{i}$ such that

$R_{\pi,(\mathcal{X},\bm{\theta}^{\star},f)}(T,1)\geq\epsilon\sqrt{\frac{d-1}{8\log(m)}}\frac{\min(T,m-1)}{4}.$

The result follows. ∎

##  Appendix D Experiment Details

In this section we provide all of the experimental details for training the
bandit algorithms.

###  D.1 Hyperparameters

The synthetic hyperparameters were fixed to the theoretical values. For the
randomly initialized neural network experiments, the grid for the regularizer
$\lambda$, norm bound $S$, and noise subgaussianity $R$ were:
$\\{0.01,0.1,1.0,10.0,100.0\\}$ which was selected post-hoc for each
experiment. For the superconductor experiments, the grid for all 3 parameters
were: $\\{0.1,1.0,10.0\\}$. For the transcription factor binding dataset, the
parameters grid was $\lambda=\\{1.0,10.0\\}$,
$R=\\{0.01,0.1,1.0,10.0,100.0,1000.0\\}$, and
$S=\\{0.01,0.1,1.0,10.0,100.0,1000.0\\}$. For $\epsilon$-greedy, the parameter
grid was set over $\epsilon=\\{0.01,0.02,\ldots,0.99\\}$ across all relevant
experiments.

Figure 7: TFBinding parallel regret with linear features. From left to right:
$P=1$, $P=10$, and $P=100$.

###  D.2 Feature Engineering for Random Neural Network

Two feature sets were considered:

  * •

Linear features $x_{i}$ was encoded as a 14-length feature vector

  * •

Linear + Quadratic features where interaction terms of $x_{i}$ were included
in $105$ quadratic features alongside the original $14$ linear features into a
$119$ features.

The resulting parallel regret plots are shown in Fig. 8.

Figure 8: RandomNN with Linear features. From left to right: $P=1$, $P=10$,
and $P=100$.

###  D.3 Feature Engineering for Transcription Factor Binding

Three feature sets were considered:

  * •

Linear features $\mathbf{x}_{i}$ was one-hot encoded into a 32-length feature
vector.

  * •

250 random ReLU features where a $250\times 32$ random matrix $\mathbf{W}$ is
sampled such that $\mathbf{W}_{ij}\sim\mathrm{Normal}(0,1)$. Then, the feature
map was evaluated as:

$\phi(\mathbf{x}_{i})=\frac{1}{\sqrt{250}}\text{ReLU}\Bigg{(}\frac{\mathbf{W}\mathbf{x}_{i}}{\sqrt{32}}\Bigg{)}$

  * •

Linear + Quadratic features where $\mathbf{x}_{i}$ was one-hot encode and all
32 linear features and 528 quadratic features were combined into a 560-length
vector.

The off-line test $R^{2}$ (without added noise in the $y_{i}$) on a train set
of $TP=2500$ matching the number of total arm queries yields values
$0.15,0.26,0.29$ for linear, ReLU, and quadratic respectively. This matches
with the number of features and level of expressivity of the model class.
However, as shown in Figs. 9 and 10 the linear features perform the best
followed by the ReLU features, and then the quadratic features. One can gain
further insight by examining the off-line test $R^{2}$ for a smaller training
size and see that the larger feature expansions accrue variance making the
linear features perform the best. This matches our understanding that model
fitting is less statistically efficient in an online setting.

Figure 9: TFBinding best arm with ReLU features. From left to right: $P=1$,
$P=10$, and $P=100$.

Figure 10: TFBinding best arm with quadratic features. From left to right:
$P=1$, $P=10$, and $P=100$.

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# Black hole superradiant instability for massive spin-2 fields

Óscar J. C. Dias<EMAIL_ADDRESS>STAG research centre & Mathematical
Sciences, Univ. of Southampton, Highfield Campus, UK  Giuseppe Lingetti
<EMAIL_ADDRESS>Dipartimento di Fisica, “Sapienza” Università di
Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy  Paolo
Pani<EMAIL_ADDRESS>Dipartimento di Fisica, “Sapienza” Università di
Roma & Sezione INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy  Jorge E.
Santos<EMAIL_ADDRESS>DAMTP, Centre for Mathematical Sciences, University
of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

###### Abstract

Due to coherent superradiant amplification, massive bosonic fields can trigger
an instability in spinning black holes, tapping their energy and angular
momentum and forming macroscopic Bose-Einstein condensates around them. This
phenomenon produces gaps in the mass-spin distribution of astrophysical black
holes, a continuous gravitational-wave signal emitted by the condensate, and
several environmental effects relevant for gravitational-wave astronomy and
radio images of black holes. While the spectrum of superradiantly unstable
mode is known in great detail for massive scalar (spin-0) and vector (spin-1)
perturbations, so far only approximated results were derived for the case of
massive tensor (spin-2) fields, due to the nonseparability of the field
equations. Here, solving a system of ten elliptic partial differential
equations, we close this program and compute the spectrum of the most unstable
modes of a massive spin-2 field for generic black-hole spin and boson mass,
beyond the hydrogenic approximation and including the unique dipole mode that
dominates the instability in the spin-2 case. We find that the instability
timescale for this mode is orders of magnitude shorter than for any other
superradiant mode, yielding much stronger constraints on massive spin-2
fields. These results pave the way for phenomenological studies aimed at
constraining beyond Standard Model scenarios, ultralight dark matter
candidates, and extensions to General Relativity using gravitational-wave and
electromagnetic observations, and have implications for the phase diagram of
vacuum solutions of higher-dimensional gravity.

Introduction. Ultralight bosons (such as the QCD axion, axion-like particles,
dark photons, etc Arvanitaki  _et al._ (2010); Essig  _et al._ (2013); Marsh
(2016); Hui  _et al._ (2017)) are predicted in several beyond Standard Model
scenarios Jaeckel and Ringwald (2010); Essig  _et al._ (2013); Hui  _et al._
(2017); Irastorza and Redondo (2018), including extra dimensions and string
theories, and are compelling dark matter candidates. Searching for these
fields in the laboratory is challenging due to their typically weak coupling
to baryonic matter, but they might produce striking effects around
astrophysical black holes (BHs) Arvanitaki  _et al._ (2010, 2015); Brito  _et
al._ (2015a); Arvanitaki  _et al._ (2017); Brito  _et al._ (2017a, b); Palomba
_et al._ (2019); Isi  _et al._ (2019); Brito  _et al._ (2020). This
possibility is allowed by the superradiant instability of spinning BHs against
massive bosonic excitations Press and Teukolsky (1972); Detweiler (1980);
Cardoso  _et al._ (2004); Shlapentokh-Rothman (2014), which occurs whenever
the frequency $\omega_{R}$ of the perturbation satisfies the superradiant
condition $0<\omega_{R}<m\Omega_{\rm H}$, where $\Omega_{\rm H}$ is the
horizon angular velocity and $m$ is the azimuthal quantum number of the
unstable mode (see Brito  _et al._ (2015a) for an overview). As an order-of-
magnitude estimate, for a boson with mass $m_{b}=\mu\hbar$, the superradiant
instability is most effective when its Compton wavelength is comparable to the
BH gravitational radius, i.e. when the gravitational coupling $\alpha\equiv
M\mu={\cal O}(0.1)$ (in the geometrized $G=c=1$ units henceforth adopted).
This translates into the optimal condition for the instability $m_{b}\sim
10^{-11}(M_{\odot}/M)\,{\rm eV}$ but, as we shall see, in certain cases the
range of relevant masses for which the instability is efficient can encompass
several orders of magnitude. In the superradiant regime the BH spins down,
transferring energy and angular momentum to a mostly dipolar ($m=1$) boson
condensate until $\omega_{R}\sim\Omega_{\rm H}$ and the instability quenches
off. The condensate is then primarily dissipated through the emission of
almost monochromatic quadrupolar gravitational waves Arvanitaki  _et al._
(2015); Brito  _et al._ (2015b), with frequency set by the boson mass. This
process continues for $m>1$ modes on longer timescales Ficarra  _et al._
(2019); Brito  _et al._ (2015a).

While the qualitative aspects of this phenomenon are valid for any boson
regardless of its spin, a crucial ingredient is the instability timescale of
the dominant unstable mode, which strongly depends on $\alpha$ and type of
massive boson Brito  _et al._ (2015a). Scalar (spin-0) fields are the best
studied case Damour  _et al._ (1976); Detweiler (1980); Zouros and Eardley
(1979); Dolan (2007); Arvanitaki  _et al._ (2015, 2017); Brito  _et al._
(2017a, b), but recent years have witnessed a significant progress also for
vector (spin-1) fields Pani  _et al._ (2012a, b); Witek  _et al._ (2013);
Endlich and Penco (2017); East (2017); East and Pretorius (2017); Baryakhtar
_et al._ (2017); East (2018); Frolov  _et al._ (2018); Dolan (2018); Siemonsen
and East (2020). Massive tensor (spin-2) perturbations are much less
understood. In this case the superradiant instability has been studied only
perturbatively using a semi-analytical approach Pani (2013) to first order in
the spin Brito  _et al._ (2013a) and analytically in the so-called Newtonian
regime where $\alpha\ll 1$ Brito  _et al._ (2020). However, the former
approximation is inaccurate for most astrophysical BHs given their sizable
spin Brenneman  _et al._ (2011); Middleton (2016); Abbott  _et al._ (2021a),
while the latter approximation is inaccurate in the most relevant regime for
the instability, $\alpha={\cal O}(0.1)$, and also fails to capture the
“special” dipole mode found numerically in Ref. Brito  _et al._ (2013a), which
was conjectured to be the dominant one.

Here we close an important gap in the BH superradiance program by computing
the superradiant instability of a Kerr BH against massive spin-2 fields
without approximations and for any mode. We extend the methods developed in
Refs. Dias  _et al._ (2009, 2010a, 2010b, 2010c, 2011a, 2012); Dias and Santos
(2013); Cardoso  _et al._ (2014); Dias  _et al._ (2014, 2016, 2015a, 2022a,
2022b) (see Dias  _et al._ (2016) for an overview) and write the linearized
field equations as a coupled system of ten elliptic partial differential
equations (PDEs) with associated boundary conditions yielding an eigenvalue
problem in the frequency domain.

In particular we will show that the instability timescale of the special
dipole mode is dramatically shorter than for any other superradiantly unstable
mode, reaching timescales comparable to the typical BH ringdown Berti  _et
al._ (2009); Dias  _et al._ (2015a, 2022a, 2022b) (as short as $\tau\sim
2\times 10^{-4}(M/M_{\odot})\,{\rm s}$ for a highly-spinning BH), and being
effective in a much wider region of the parameter space.

Setup. At variance with scalar and vector fields, the coupling of a massive
spin-2 field to gravity is highly nontrivial Hinterbichler (2012); de Rham
_et al._ (2011); Hassan and Rosen (2012a, b); de Rham (2014). Here we will
consider the field equations for a spin-2 perturbation $H_{ab}$ on a Ricci-
flat background which is taken to be the Kerr solution Kerr (1963), although
our computation does not depend on the details of the background and should be
valid also for other stationary and axisymmetric solutions that might exist in
bimetric theories Brito  _et al._ (2013b); Babichev and Brito (2015). The
perturbed equations, describing the propagation of five physical degrees of
freedom, are Brito  _et al._ (2013a); Mazuet and Volkov (2018)

$\displaystyle\Box H_{ab}+2R_{abcd}H^{cd}-\mu^{2}H_{ab}=0\,,$ (1a)
$\displaystyle\nabla^{a}H_{ab}=0\,,\qquad{H^{a}}_{a}=0\,,$ (1b)

where $\Box=\nabla_{a}\nabla^{a}$ and $R_{abcd}$ are the D’Alembert operator
and Riemann tensor of the background, respectively, and the first two terms
describe the familiar Lichnerowicz operator $(\Delta_{L}H)_{ab}$. We shall
discuss the embedding of these equations in various nonlinear theories later
on.

In the standard Boyer-Lindquist coordinates ($t,r,\theta,\phi$) for the Kerr
metric, $\partial_{t}$ and $\partial_{\phi}$ are Killing vector fields, so one
can decompose $H_{ab}$ along those directions:
$H_{ab}(t,r,\theta,\phi)=e^{-i\omega t}e^{im\phi}\tilde{H}_{ab}(r,\theta)$.
The resulting field equations for $\tilde{H}_{ab}$ are a coupled system of ten
PDEs (of which five are constraints arising from Eq. (1b)) which, after
imposing boundary conditions, yield a quadratic eigenvalue problem in the
(complex) frequency $\omega=\omega_{R}+i\,\omega_{I}$, for a given mode $m$.

The system of PDEs is then discretized using a pseudospectral collocation grid
on Gauss-Chebyshev-Lobbato points in the compactified directions $y\in[0,1]$
and $x\in[-1,1]$, defined as $r=r_{+}/(1-y^{2})$ and
$\cos\theta=x\sqrt{2-x^{2}}$, where $r_{+}=M(1+\sqrt{1-\chi^{2}})$ is the BH
event horizon and $\chi\equiv J/M^{2}\leq 1$ is the dimensionless BH angular
momentum. The eigenfrequencies and associated eigenfunctions are found through
a Newton-Raphson root-finding algorithm after imposing suitable boundary
conditions for unstable modes Dias  _et al._ (2016, 2010a). The latter impose
an exponential decay at asymptotic infinity ($y\to 1$), regularity of the
perturbations in ingoing Eddington-Finkelstein coordinates at the horizon
($y\to 0$), and regularity on the north and south poles of the two-sphere,
which also requires $m$ to be an integer.

Figure 1: Real and imaginary parts of the dominant unstable dipole mode
($m=1$) of massive spin-2 perturbations of a Kerr BH as a function of the BH
spin $\chi$. The unstable regime corresponds to $\omega_{R}<\Omega$, yielding
$\omega_{I}>0$. For comparison, in the bottom panel we also show the (black)
curve for the fundamental $\ell=m=2$ quasinormal mode of a Kerr BH Berti  _et
al._ (2009); Dias  _et al._ (2015a, 2022a, 2022b). The inset shows the
comparison in the highly-spinning regime. In the supplemental material we
present three-dimensional and density plots obtained with 4018 grid points.

Instability spectrum. Compared to the scalar and vector cases, massive spin-2
perturbations have two unique features, emerging already in the nonspinning
case: i) an unstable spherical ($m=0$) mode Babichev and Fabbri (2013); Brito
_et al._ (2013a) which signals the existence of hairy BH solutions Brito  _et
al._ (2013b); ii) a special dipole polar mode Brito  _et al._ (2013a), which
does not fit within the general hydrogenic-like scaling found in the
analytical Newtonian limit ($\alpha\ll 1$). The latter predicts Brito  _et
al._ (2020):

$\displaystyle\frac{\omega_{R}}{\mu}$ $\displaystyle\simeq
1-\frac{\alpha^{2}}{2\left(\ell+n+S+1\right)^{2}}\,,$ (2a)
$\displaystyle\omega_{I}$
$\displaystyle\propto\alpha^{4\ell+5+2S}(\omega_{R}-m\Omega_{\rm H}),$ (2b)

where $\ell\geq 0$, $n\geq 0$, and the integer $S=(0,\pm 1,\pm 2)$ are the
mode total angular momentum, overtone number, and polarization, respectively,
with $m\in[-\ell,\ell]$. The sign change of $\omega_{I}$ due to the term
$\omega_{R}-m\Omega_{\rm H}$ makes it clear that modes with $m\geq 1$ turn
unstable in the superradiant regime, with an instability timescale
$\tau=1/\omega_{I}$. As predicted by (2b), the dominant hydrogenic spin-2 mode
(with $\ell=2=-S$) has a parametrically longer timescale compared to the
dominant hydrogenic spin-1 mode (with $\ell=1=-S$) Brito  _et al._ (2020). On
the other hand, the special dipole ($m=1$) spin-2 mode computed numerically in
the nonspinning case displays a different behavior Brito  _et al._ (2013a):

$\frac{\omega_{R}^{\rm dipole}}{\mu}\approx
0.72(1-\alpha)\,,\qquad\omega_{I}^{\rm dipole}\propto\alpha^{3}\omega_{R}\,.$
(3)

Thus, in the nonspinning case this mode has the largest binding energy,
$\omega_{R}/\mu-1$, and the shortest decay time of the entire spectrum for
massive bosonic perturbations.

Our numerical method is general and applies to all unstable modes. Indeed, we
have confirmed numerically the analytical results for the hydrogenic modes,
Eqs. (2a)–(2b). Henceforth we focus on the special dipole mode, since Brito
_et al._ (2013a) found indication that $\omega_{I}^{\rm
dipole}\propto\alpha^{3}(\omega_{R}-m\Omega_{\rm H})$ to leading order in the
BH spin, suggesting the shortest instability timescale in the superradiant
regime.

In Fig. 1 we show the behavior of this special mode for generic BH spin $\chi$
and for a large range of gravitational coupling up to $\alpha=0.8$. While we
confirm the results of Ref. Brito  _et al._ (2013a) in the nonspinning case,
the most striking feature of this mode emerges in the regime that was not
possible to explore so far, namely large BH spins and large coupling. For
nearly extremal BHs, we find that this mode has an imaginary part as large as
$\omega_{I}M\approx 0.019$ when $\alpha\approx 0.8$. This translates into the
instability timescale

$\tau\approx 2.6\times 10^{-4}\left(\frac{M}{M_{\odot}}\right)\,{\rm s}\,,$
(4)

almost two orders of magnitude shorter than for any other superradiant mode
known so far, including the dominant unstable mode in the massive spin-1 case
Dolan (2018) and any other massive spin-2 modes. Interestingly, as shown by
the inset in Fig. 1, the dominant quasinormal mode of a highly-spinning Kerr
BH Berti  _et al._ (2009); Dias  _et al._ (2015a, 2022a, 2022b) has a decay
time comparable to, or even longer than, (4). This means that the instability
is so fast that it would affect the ringdown of a newly formed BH.

Overall, our numerical data are described by a simple polynomial fit:

$\displaystyle\frac{\omega_{R}^{\rm dipole}}{\mu}$
$\displaystyle\approx\left(\sum_{i=0}^{3}a_{i}\chi^{i}\right)\left(1+\alpha\sum_{i=0}^{3}b_{i}\chi^{i}+\alpha^{2}\sum_{i=0}^{2}c_{i}\chi^{i}\right)\,,$
$\displaystyle\omega_{I}^{\rm dipole}$
$\displaystyle\approx-\alpha^{3}(\omega_{R}-\Omega_{\rm
H})\sum_{i=0}^{2}d_{i}\chi^{i}\,,$ (5)

with $a_{i}\approx(0.73,-0.05,0.15,-0.12)$, $b_{i}=(-1.21,0.68,-0.55,0.61)$,
$c_{i}=(0.69,-0.58,-0.11)$, $d_{i}=(1.47,1.86,-2.75)$. In the unstable regime
the fits of $\omega_{R}^{\rm dipole}$ and $\omega_{I}^{\rm dipole}$ are
accurate within $2\%$ and $80\%$, respectively, in the range
$\alpha\in[0.05,0.8]$ and $\chi\in[0,\approx 0.99]$.

Figure 2: _Left:_ Exclusion regions in the BH spin-mass diagram obtained from
the superradiant instability of Kerr BHs against massive spin-2 fields for the
dominant $m=1$ mode. For each mass of the field, the separatrix corresponds to
an instability timescale equal to the Salpeter time, $\tau_{S}=4.5\times
10^{7}{\rm\,yr\,}$. Markers and error bars are explained in the main text.
_Right:_ Comparison of the exclusion region for a boson with mass
$m_{b}=10^{-16}\,{\rm eV}$ for the dominant mode of scalar, vector, tensor
perturbations. In the latter case we show the result computed in Ref. Brito
_et al._ (2020) and the new mode computed in this work.

BH mass-spin gaps. A generic prediction of superradiant instabilities is that
highly-spinning BHs should lose angular momentum over a timescale
$\tau=1/\omega_{I}$ that might be much shorter than typical astrophysical
timescales. Thus, if ultralight bosons exist in the Universe they would lead
to statistical evidence for slowly rotating BHs in a region of the “Regge”
plane (mass versus angular momentum) of astrophysical BHs Arvanitaki and
Dubovsky (2011); Brito  _et al._ (2015a); Baryakhtar  _et al._ (2017); Brito
_et al._ (2017a, b); Ng  _et al._ (2019); Fernandez  _et al._ (2019); Stott
(2020); Ünal  _et al._ (2021); Ünal (2023). Given a $m_{b}$, there exists a
forbidden region (a gap Arvanitaki and Dubovsky (2011)) in the Regge plane
wherein the instability timescale is shorter than known spin-up astrophysical
processes such as accretion.

The exclusion regions in the Regge plane are shown in the left panel of Fig. 2
for selected values of $m_{b}$, whereas in the right panel we compare the
exclusion region obtained in this work with those of other superradiant modes
for the same representative value $m_{b}=10^{-16}\,{\rm eV}$. Owing to the
much shorter instability timescale, our exclusion region encompasses several
orders of magnitude, ranging from stellar to supermassive BHs, for the same
$m_{b}$.

Data points shown in Fig. 2 refer to several observations: (i) Black points
denote electromagnetic estimates of the mass and spin of accreting stellar and
supermassive BHs Brenneman  _et al._ (2011); Middleton (2016), conservatively
requiring that $\tau<\tau_{S}=4.5\times 10^{7}{\rm\,yr\,}$, where $\tau_{S}$
is the Salpeter timescale for accretion. (ii) Red points are the $90\%$
confidence levels for the binary BHs in a selection of the merger events
detected by the LIGO-Virgo-KAGRA collaboration so far Abbott  _et al._ (2019);
Venumadhav  _et al._ (2019) (using the errors on the binary effective spin
Ajith  _et al._ (2011) as a proxy for the errors on the individual spins).
While individual spin measurements coming from current gravitational-wave
events have large errors, some binaries have confidently nonvanishing spins
Abbott  _et al._ (2021b, a) and future LIGO detections will provide
measurements of the individual spins with $\approx 30\%$ accuracy Abbott  _et
al._ (2016). Much more accurate spin measurements of binary BHs out to
cosmological distances will come from the future LISA mission Audley  _et al._
(2017) for intermediate and supermassive objects, mainly depending on the mass
of BH seeds in the early Universe (the horizontal arrows in Fig. 2 denote the
range of projected mass measurements with LISA using three different
population models for supermassive BHs Klein  _et al._ (2016); Brito  _et al._
(2017b)). This would allow searching for ultralight bosons in a large mass
range $m_{b}\in(10^{-19},10^{-13})\,{\rm eV}$, much wider than in previous
cases Brito  _et al._ (2017a, b); Cardoso  _et al._ (2018a); Ünal (2023).
(iii) Green points are the $90\%$ confidence levels for the mass-spin of a
selection of the merger remnants Abbott  _et al._ (2019, 2021b, 2021a), which
also identify targets of merger follow-up searches Arvanitaki  _et al._ (2015,
2017); Baryakhtar  _et al._ (2017); Isi  _et al._ (2019); Ghosh  _et al._
(2019). In the particular case at hand, it is tantalizing that $|\omega_{I}|$
can be comparable to or even larger than the typical $|\omega_{I}|$ of a BH
quasinormal mode Berti  _et al._ (2009); Dias  _et al._ (2015a, 2022a, 2022b)
(see Fig. 1). This implies that the instability would directly affect the
postmerger phase by dynamically reducing the spin of the remnant during the
ringdown, which does not happen for any other superradiant mode. (iv) The blue
points refer to Cyg X-1 Gou  _et al._ (2009) and LMC X-3 Steiner  _et al._
(2010), for which there are reliable spin measurements suggesting that the
spin is constant over ${\cal O}(10\,{\rm yr})$ Middleton (2016). In these
cases, more direct constraints come by imposing $\tau<{\cal O}(10\,{\rm yr})$
Cardoso  _et al._ (2018a). Given the strong instability, these sources could
confidently exclude spin-2 particles with mass $m_{b}\in(10^{-15},5\times
10^{-12})$. (v) Finally, the two gray points correspond to the masses of SgrA∗
and M87 as measured by the Event Horizon Telescope Akiyama  _et al._ (2019a,
b, 2022). While an accurate spin measurement for these sources is still not
available, they are expected to have moderate to large spin Akiyama  _et al._
(2019c); Tamburini  _et al._ (2019); Akiyama  _et al._ (2022). Given the
extent of the exclusion regions in Fig. 2, any spin measurement in SgrA∗ and
M87 that confidently excludes zero would approximately rule out the range
$m_{b}\in[10^{-23},10^{-17}]\,{\rm eV}$, which is much wider than the
exclusion regions derived with other modes Davoudiasl and Denton (2019); Chen
_et al._ (2019); Cunha  _et al._ (2019). Likewise, if some of the ultramassive
BHs with $M\simeq{\rm few}\times 10^{10}M_{\odot}$ McConnell  _et al._ (2011,
2012); Nightingale  _et al._ (2023) (rightmost black data point in the left
panel of Fig. 2) were confirmed to have nonzero spin Riechers  _et al._
(2009), it would be possible to exclude an ultralight spin-2 field with mass
even slightly smaller than $m_{b}\approx 10^{-23}\,{\rm eV}$.

Embeddings of the model. At the nonlinear level, there is a unique way to
couple a dynamical spin-2 field to gravity de Rham  _et al._ (2011); Hassan
and Rosen (2012a, b); Babichev  _et al._ (2016). Nonetheless, Eqs. (1a)–(1b)
emerge from linear perturbations in several different contexts. In ghost-free
massive gravity de Rham  _et al._ (2011), these equations describe the linear
dynamics of a massive graviton when considering a Ricci-flat reference metric.
In the context of bimetric theories Hassan and Rosen (2012a, b), they describe
the dynamics of perturbations propagating on a fixed background given by two
copies of the same Ricci-flat metric.

Interestingly, Eqs. (1a)–(1b) also arise from BH perturbations in Einstein-
Weyl gravity Whitt (1985)

$S=\int d^{4}x\sqrt{-g}\left(R-\frac{\beta}{6}C_{abcd}C^{abcd}\right)\,,$ (6)

where $C_{abcd}$ is the Weyl tensor. This theory propagates a massless
graviton and a massive spin-2 field and (although the latter has ghosts Stelle
(1977)) it has a number of unique features: it is renormalizable in Minkowski
spacetime Stelle (1977), admits all solutions of vacuum General Relativity,
and has a well-posed initial value problem even if the corresponding field
equations are of fourth order Noakes (1983). By linearizing the theory on a
Ricci-flat background Myung (2013), the equations for the perturbed Ricci
tensor take the form (1a)–(1b) upon identification $H_{ab}\equiv\delta R_{ab}$
and $\mu^{2}\equiv 1/\beta$. Thus, in the strongly-coupled regime
($\beta\to\infty$) the theory propagates only a massive graviton with small
effective mass, and our bounds on $\mu$ can be directly translated into bounds
on the coupling $\beta$, which are notoriously very hard to place.

Finally, Eqs. (1a)–(1b) emerge also in the context of higher-dimensional
gravity when studying linear perturbations of a five-dimensional ($d=5$) black
string – obtained by adding a flat direction to the Schwarzschild or Kerr BH –
and performing a Fourier decomposition with Fourier momentum $k\equiv\mu$
along the extra dimension Gregory and Laflamme (1993). For example, the
aforementioned spherical ($m=0$) unstable mode Babichev and Fabbri (2013),
present for $\alpha\leq\alpha_{\rm crit}\approx 0.438$ in the static case,
corresponds to the familiar Gregory-Laflamme instability of a black string
under fragmentation Gregory and Laflamme (1993); Lehner and Pretorius (2010);
Figueras  _et al._ (2023). The onset of this instability in $d=5$ signals a
new branch of nonuniform black strings Gubser (2002); Harmark and Obers
(2002); Kol (2005); Wiseman (2003); Kol and Wiseman (2003); Harmark and Obers
(2004); Harmark (2004); Kudoh and Wiseman (2004); Sorkin (2004); Gorbonos and
Kol (2004); Kudoh and Wiseman (2005); Dias  _et al._ (2007); Harmark  _et al._
(2007); Wiseman (2012); Figueras  _et al._ (2012); Dias  _et al._ (2018,
2023a, 2023b, 2023c) which corresponds, in $d=4$, to hairy BHs of massive
gravity Brito  _et al._ (2013b).

A natural extension of our work is to track the $m=0$ mode in the spinning
case to obtain accurate estimates for $\alpha_{\rm crit}(\chi)$ Monteiro  _et
al._ (2010) and associated timescales. Given the large domain of existence and
the short timescale of the special dipole mode, it would be relevant to assess
whether there exists a range of $\alpha$ in which the $m=0$ instability is
absent or weak, whereas the $m=1$ instability discussed here is dominant. Work
in $d=6$ dimensions Dias  _et al._ (2023a, b, c) suggests that such a window
might indeed exist.

In all nonlinear completions aforementioned, the special dipole mode is
marginally stable ($\omega_{I}=0$) when the superradiant condition is
saturated, $\omega=\Omega_{H}$ (as a further check of our code we confirmed
this feature to very high accuracy). A purely real mode at the linear level
strongly suggests the existence of a nonlinear stationary solution branching
off the critical point Dias  _et al._ (2011b); Herdeiro and Radu (2014); Dias
_et al._ (2016); Ishii and Murata (2019, 2020). For special black strings in
$d=6$ (with enhanced symmetries that reduce Eqs. (1a)–(1b) to ODEs) the
superradiant onset signals the existence of new rotating black strings –
denoted as resonator Dias  _et al._ (2023b) and helical Dias  _et al._ (2023c)
black strings – that are time-periodic (but not time independent neither
axisymmetric) since they have a helical Killing vector field. Returning to
$d=5$, the instability onset in Fig. 1 is similarly expected to signal black
resonator/helical strings bifurcating from the Kerr string in a phase diagram
of solutions. From the $d=4$ perspective, this should correspond to the
existence of novel hairy rotating BHs of massive/bimetric/Einstein-Weyl
gravity with a single helical Killing vector field that would have higher
entropy than the Kerr BH. It would be interesting to find such solutions which
are expected to have properties very different from the hairy BHs with
scalar/vector hair Herdeiro and Radu (2014); Herdeiro  _et al._ (2016) (see
Herdeiro and Radu (2015) for a review), as it occurs for systems in anti-de
Sitter with similar superradiant physics Dias  _et al._ (2011b); Cardoso  _et
al._ (2014); Horowitz and Santos (2015); Dias  _et al._ (2015b). Yet, when
emerging dynamically as the end-state or metastable state of the instability,
these solutions would have an angular momentum close to the superradiant
threshold, so our bounds derived from BH mass-spin measurements are robust.

Conclusion. We have shown that the special dipole spin-2 mode of a spinning BH
does not fit in the standard hydrogenic-like picture of superradiantly
unstable modes of massive bosonic perturbations. Most importantly, it has by
far the shortest instability timescale among all superradiant modes. This has
several direct consequences which are unique to massive spin-2 fields. First
of all, in the highly-spinning case the instability timescale can be
comparable to or even larger than the dominant quasinormal mode of a Kerr BH.
This indicates that the instability can be effective during the ringdown in a
post-merger phase, and suggests the need for novel ringdown-based searches
accounting for this effect. Furthermore, the exclusion regions in the BH mass-
spin diagram are much wider than for other superradiant modes. In particular,
the same ultralight boson mass (around $m_{b}\approx 10^{-16}\,{\rm eV}$)
would give rise to interesting phenomenology for both stellar-mass and
supermassive BHs, facilitating possible multiband constraints on the model.
The exclusion regions for each $m_{b}$ extend almost up to $\chi\approx 0$,
showing that _any_ BH spin measurement can be translated into interesting
constraints. Owing to the wideness of the BH mass-spin gaps, the range of
detectable spin-$2$ masses is much larger than in all other cases explored so
far. For example, current observations of stellar-mass BHs (either in the
electromagnetic or in the gravitational-wave band) exclude the region
$m_{b}\in[5\times 10^{-17},5\times 10^{-11}]\,{\rm eV}$, which already
severely constrains the observability of dipolar radiation from BH binaries in
these models Cardoso  _et al._ (2018b, 2023). Overall, few BH mass-spin
measurements distributed in the region $M\sim M_{\odot}$ to $M\sim
10^{10}M_{\odot}$ would probe massive spin-2 fields in the impressive range
$m_{b}\in[10^{-23},10^{-10}]\,{\rm eV}$, which also includes $m_{b}\sim
10^{-22}\,{\rm eV}$, where ultralight bosons are compelling dark-matter
candidates Hui  _et al._ (2017).

A natural extension of our work is to use these new unstable modes to compute
the rich phenomenology associated with the gravitational-wave emission from
the bosonic condensate Arvanitaki  _et al._ (2017); Baryakhtar  _et al._
(2017); Brito  _et al._ (2017a, b, 2020); Isi  _et al._ (2019); Ghosh  _et
al._ (2019); Tsukada  _et al._ (2019); Palomba  _et al._ (2019); Sun  _et al._
(2019), as well as environmental effects of the condensate in BH binary
inspirals Hannuksela  _et al._ (2019); Baumann  _et al._ (2019a); Zhang and
Yang (2019a); Berti  _et al._ (2019); Baumann  _et al._ (2019b, c); Zhang and
Yang (2019b); Cardoso  _et al._ (2020); De Luca and Pani (2021); Baumann  _et
al._ (2022) and for radio images Davoudiasl and Denton (2019); Chen  _et al._
(2019); Cunha  _et al._ (2019). Due to the stronger instability, these effects
are anticipated to be much more prominent than in other cases studied so far.

Acknowledgments. We thank Vitor Cardoso, Francisco Duque, Andrea Maselli, and
David Pereñiguez for interesting comments on the draft. O. D. acknowledges
financial support from the STFC “Particle Physics Grants Panel (PPGP) 2018”
Grant No. ST/T000775/1. G.L. and P.P. acknowledge financial support provided
under the European Union’s H2020 ERC, Starting Grant agreement no.
DarkGRA–757480, under MIUR PRIN (Grant 2020KR4KN2 “String Theory as a bridge
between Gauge Theories and Quantum Gravity”) and FARE (GW-NEXT, CUP:
B84I20000100001, 2020KR4KN2) programmes, and support from the Amaldi Research
Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP:
B81I18001170001). This work was supported by the EU Horizon 2020 Research and
Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No.
101007855. J.E.S. has been partially supported by STFC consolidated grant
ST/T000694/1. The authors acknowledge the use of the IRIDIS High Performance
Computing Facility, and associated support services at the University of
Southampton, for the completion of this work.

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## Supplemental Material

In Fig. 3 we present a three-dimensional plot (top panels) and a contour plot
(bottom panels) of the dominant unstable dipole ($m=1$) mode discussed in the
main text with all the data we collected. The right panels display the
imaginary part of the frequency $\omega_{I}M$ as a function of the BH spin
$\chi$ and coupling $\alpha$. The left panels display the difference between
the angular velocity of the horizon and the real part of the frequency,
$(\Omega_{H}-\omega_{R})M$. The superradiant instability corresponds to
$\Omega_{H}-\omega_{R}>0$ and $\omega_{I}>0$. In Fig. 1 of the main text we
have taken the data of Fig. 3 corresponding to lines of constant
$\alpha=0.1,\,0.2,\,0.3,\,0.4,\,0.5,\,0.6,\,0.7,\,0.8$.

Figure 3: Top panels: Three-dimensional plot for the dominant unstable dipole
mode ($m=1$) of massive spin-2 perturbations of a Kerr BH as functions of the
coupling $\alpha\in[0.05,0.8]$ and BH spin $\chi\in[0,\approx 0.99]$. These
plots were produced with 4018 data points. The solid black lines mark the
onset of the instability, which also coincides with $\omega_{R}=\Omega_{\rm
H}$, as expected for models exhibiting confining dynamics. To aid
visualization we also added a gray plane marking $\Omega_{\rm H}-\omega_{R}=0$
and $\omega_{I}=0$. Bottom panels: Density plots of the same data.

We have validated our numerical method by:

  * •

Reproducing the analytical hydrogenic spectrum in the Newtonian ($\alpha\ll
1$) limit Brito  _et al._ (2020);

  * •

Reproducing the numerical results for the special dipole mode in the
nonspinning limit Brito  _et al._ (2013a);

  * •

Verifying that all superradiant modes become marginally stable
($\omega_{I}=0$) precisely when the superradiant condition is saturated,
$\omega=\Omega_{H}$, within the high numerical accuracy of our code. This
instability onset is described by the black curve in all the plots of Fig. 3.