which implies (D.1) in light of Definition 4.1.

###  D.2. Remaining proof for Lemma 3.2

It remains to prove Claim 4.3. In light of how the $M_{I}$’s are defined in
(4.9), (4.11) precisely happens when, for any pair $I\subset[r]$ and
$\mathcal{J}\in\mathcal{P}_{I}$ such that
$(I,\mathcal{J})\neq([r],\\{[r]\\})$,

$\displaystyle x$
$\displaystyle\leq\mathcal{C}_{r}\mathcal{K}_{r}e\Bigg{[}\frac{M_{[r]}^{2}}{M_{I}^{1/(2|I^{c}|+|\mathcal{J}|/2)}}\Bigg{]}^{\frac{1}{2-(2|I^{c}|+|\mathcal{J}|/2)^{-1}}}$
(D.6)
$\displaystyle=\mathcal{C}_{r}\mathcal{K}_{r}e\cdot\sigma_{h}\cdot\frac{n^{\frac{(2r+\frac{1}{2})|I^{c}|+\frac{r}{2}|\mathcal{J}|-\frac{r}{2}}{4|I^{c}|+|\mathcal{J}|-1}}}{\Big{(}\log(2n^{|I^{c}|}+1)\Big{)}^{\frac{|I^{c}|}{4|I^{c}|+|\mathcal{J}|-1}}},$

where we have used the fact that

$\Big{(}n^{r-\frac{|I|}{2(2|I^{c}|+|\mathcal{J}|/2)}}\Big{)}^{\frac{1}{2-(2|I^{c}|+|\mathcal{J}|/2)^{-1}}}=n^{\frac{(2r+\frac{1}{2})|I^{c}|+\frac{r}{2}|\mathcal{J}|-\frac{r}{2}}{4|I^{c}|+|\mathcal{J}|-1}}.$

In particular, the right hand side on (D.6) is always equal to

(D.7) $\mathcal{C}_{r}\cdot\mathcal{K}_{r}\cdot e\cdot\sigma_{h}\cdot
n^{r/2}\text{ when }I=[r],$

because $|I^{c}|=0$ in this case. We claim that the right hand side on (D.6)
can be no smaller than

(D.8) $\mathcal{C}_{r}\cdot\mathcal{K}_{r}\cdot
e\cdot\sigma_{h}\cdot\frac{n^{\frac{(r+1)^{2}}{2(r+2)}}}{(\log(2n^{r}+1))^{\frac{r}{4r-1}}}\text{
for any }I\text{ other than }[r];$

if this claim is true, because
$n^{\frac{(r+1)^{2}}{2(r+2)}}/n^{\frac{r}{2}}=n^{\frac{1}{2(r+2)}}$ for all
$r\geq 1$, considering (D.6) and the upper limit for $x$ in (D.7) when
$I=[r]$, we will have proven Claim 4.3. Hence, it suffices to establish the
claimed lower bound in (D.8) for the right hand side of (D.6), when
$I\neq[r]$.

First we define the functions

$\mathtt{f}(|I^{c}|,|\mathcal{J}|)=\frac{(2r+\frac{1}{2})|I^{c}|+\frac{r}{2}|\mathcal{J}|-\frac{r}{2}}{4|I^{c}|+|\mathcal{J}|-1},$

and

$\mathtt{g}(|I^{c}|,|\mathcal{J}|)=\frac{|I^{c}|}{4|I^{c}|+|\mathcal{J}|-1},$

so that the general expression in (D.6) can be written as

(D.9)
$\mathcal{C}_{r}\mathcal{K}_{r}e\cdot\sigma_{h}\cdot\frac{n^{\mathtt{f}(|I^{c}|,|\mathcal{J}|)}}{(\log(2n^{|I^{c}|}+1))^{\mathtt{g}(|I^{c}|,|\mathcal{J}|)}}.$

Next we will respectively find the _minimum attainable value_ of
$\mathtt{f}(|I^{c}|,|\mathcal{J}|)$ and the _maximum attainable value_ of
$\mathtt{g}(|I^{c}|,|\mathcal{J}|)$ for $I\neq[r]$ and
$\mathcal{J}\in\mathcal{P}_{I}$:

  1. (i)

Minimum attainable value of $\mathtt{f}(|I^{c}|,|\mathcal{J}|)$: By taking
partial derivatives, we have

$\frac{\partial\mathtt{f}}{\partial|I^{c}|}=\frac{\frac{1}{2}(|\mathcal{J}|-1)}{(4|I^{c}|+|\mathcal{J}|-1)^{2}}\geq
0$

and

$\frac{\partial\mathtt{f}}{\partial|\mathcal{J}|}=\frac{-\frac{1}{2}|I^{c}|}{(4|I^{c}|+|\mathcal{J}|-1)^{2}}\leq
0.$

Hence, for $\mathtt{f}$ to attain its minimum value, $|I^{c}|$ should be as
small as possible at $|I^{c}|=1$ and $|\mathcal{J}|$ should be as large as
possible at $|\mathcal{J}|=r-1$, which gives

$\mathtt{f}(1,r-1)=\frac{2r+\frac{1}{2}+\frac{r}{2}(r-1)-\frac{r}{2}}{4+r-2}=\frac{\frac{r^{2}}{2}+r+\frac{1}{2}}{2+r}=\frac{(r+1)^{2}}{2(r+2)}.$

  2. (ii)

Maximum attainable value of $\mathtt{g}(|I^{c}|,|\mathcal{J}|)$: For any non-
empty $I^{c}\neq[r]$ (or equivalently, $I\neq\emptyset$), the largest
$\mathtt{g}(|I^{c}|,|\mathcal{J}|)$ can be is
$\frac{|I^{c}|}{4|I^{c}|}=\frac{1}{4}$ since the smallest $|\mathcal{J}|$ can
be is $1$. When $I^{c}=[r]$, then $|\mathcal{J}|=|\emptyset|$ must be $0$, in
which case we have

$\mathtt{g}(r,0)=\frac{r}{4r-1},$

which is larger than $1/4$ and hence the maximum attainable value of
$\mathtt{g}(|I^{c}|,|\mathcal{J}|)$.

In light of the expression in (D.9) and considering the minimum and maximum
attainable values for $\mathtt{f}$ and $\mathtt{g}$ respectively when
$I\neq[r]$ and $\mathcal{J}\in\mathcal{P}_{I}$ above, we have established the
lower bound in (D.8).

###  D.3. Remaining proof for Theorem 4.2

It remains to prove Claims 4.4, which shall use the following lemmas:

######  Lemma D.1 (Sub-Weibull order for $\mathbb{E}_{I,H}[(h^{(r)}_{\bf
i})^{2}]$).

For a given $I\subset[r]$ and ${\bf i}=(i_{1},\dots,i_{r})$ such that
$i_{1}\neq\dots\neq i_{r}$, as a variable $\mathbb{E}_{I,H}[(h^{(r)}_{\bf
i})^{2}]$ is

  1. (i)

a polynomial with maximum possible degree $4(r-|I|)$ in jointly normal random
variables, and

  2. (ii)

is sub-Weibull of maximum order $\alpha=\frac{1}{2(r-|I|)}$.

###### Proof of Lemma D.1.

For $(i)$, in light of the definition of a projection kernel in (3.4) and the
structure of $h$ defined by (2.2) and (2.3), $\pi_{r}(h)$ must be a polynomial
in the input ${\bf x}_{1},\dots,{\bf x}_{r}$ whose possible highest degree
terms stemming from $\delta_{{\bf x}_{1}}\times\cdots\times\delta_{{\bf
x}_{r}}\times P_{H}^{m-r}h$ are of the form

$\hat{\theta}_{u_{1}v_{1}}({\bf x}_{1})\dots\hat{\theta}_{u_{r}v_{r}}({\bf
x}_{r})$

when the coefficients are ignored. Hence, $(h^{(r)}_{\bf i})^{2}$ is also a
polynomial in the decoupled data ${\bf X}_{i_{1}}^{(1)},\dots,{\bf
X}_{i_{r}}^{(r)}$, with possible highest degree terms (modulo coefficient) of
the form

$\hat{\theta}_{u_{1}v_{1}}({\bf
X}_{i_{1}}^{(1)})\dots\hat{\theta}_{u_{r}v_{r}}({\bf
X}_{i_{r}}^{(r)})\hat{\theta}_{u_{r+1}v_{r+1}}({\bf
X}_{i_{1}}^{(1)})\dots\hat{\theta}_{u_{2r}v_{2r}}({\bf X}_{i_{r}}^{(r)}),$

which implies that $\mathbb{E}_{I,H}[(h^{(r)}_{\bf i})^{2}]$ is also a
polynomial with possible highest degree terms (modulo coefficient) of the form

$\prod_{l\in I^{c}}\hat{\theta}_{u_{l}v_{l}}({\bf
X}_{i_{l}}^{(l)})\hat{\theta}_{u_{l+r}v_{l+r}}({\bf X}_{i_{l}}^{(l)});$

the latter form implies $(i)$. $(ii)$ is a consequence of $(i)$ in light of
Lemma A.4. ∎

######  Lemma D.2 (Bound on the second moment of $h^{(r)}_{[r]}$ under $H$).

Under $H$, for a constant $C(r)>0$, the random quantity
$h^{(r)}_{[r]}=\frac{\pi_{r}(h)({\bf X}_{1}^{(1)},\dots,{\bf
X}_{r}^{(r)})}{r!}$ defined in (4.3) satisfy the second moment bound:
$\mathbb{E}_{H}[(h^{(r)}_{[r]})^{2}]\leq C(r)\sigma_{h}^{2}.$

###### Proof of Lemma D.2.

For each $k\in[m]$, let $\zeta_{k}\equiv\mathbb{E}_{H}[g_{k}^{2}]$ for
$g_{k}({\bf x}_{1},\dots,{\bf x}_{k})\equiv\mathbb{E}_{H}[h({\bf
x}_{1},\dots,{\bf x}_{k},{\bf X}_{k+1},\dots,{\bf X}_{m})]$. By classical
U-statistic theory (Korolyuk and Borovskich, 2013, p.31-32), it is known that

$\mathbb{E}_{H}\Big{[}\Big{(}\pi_{r}(h)({\bf X}_{1},\dots,{\bf
X}_{r})\Big{)}^{2}\Big{]}=\zeta_{r}-{r\choose 1}\zeta_{r-1}+{r\choose
2}\zeta_{r-2}+\cdots+(-1)^{r-1}{r\choose r-1}\zeta_{1},$

as well as
$0\leq\zeta_{1}\leq\zeta_{2}\leq\cdots\leq\zeta_{m}=\sigma_{h}^{2}$;
apparently, these two facts imply $\mathbb{E}_{H}[(h^{(r)}_{[r]})^{2}]\leq
C(r)\sigma_{h}^{2}$.

∎

We now begin to prove Claim 4.4. Note that the claim is trivial if $I=[r]$,
because in that case, $|I^{c}|=r-|I|=0$ and using the definition in (4.3) as
well as Lemma D.2,

$\bigg{\|}\max_{{\bf i}_{I^{c}}}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf
i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}\bigg{\|}_{p/2,H}=\bigg{\|}\mathbb{E}_{H}\bigg{[}\sum_{{\bf
i}}(h_{\bf
i}^{(r)})^{2}\bigg{]}\bigg{\|}_{p/2,H}=\mathbb{E}_{H}\bigg{[}\sum_{{\bf
i}}(h^{(r)}_{\bf i})^{2}\bigg{]}\\\ \leq
n^{r}\mathbb{E}_{H}[(h^{(r)}_{[r]})^{2}]\leq C(r)n^{r}\sigma_{h}^{2}.$

For any $I\subsetneq[r]$, i.e. $I$ is a proper subset of $[r]$, we have

$\displaystyle\bigg{\|}\max_{{\bf
i}_{I^{c}}}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}\bigg{\|}_{p/2,H}$
$\displaystyle\underset{(i)}{\leq}C(r)p^{2(r-|I|)}\bigg{\|}\max_{{\bf
i}_{I^{c}}}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{\psi_{\frac{1}{2(r-|I|)}},H}$
$\displaystyle\underset{(ii)}{\leq}C(r)p^{2(r-|I|)}\psi_{\frac{1}{2(r-|I|)}}^{-1}\Bigg{(}2n^{r-|I|}\Bigg{)}\max_{{\bf
i}_{I^{c}}}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{\psi_{\frac{1}{2(r-|I|)}},H}$ (D.10)
$\displaystyle\underset{(iii)}{=}C(r)p^{2(r-|I|)}\bigg{(}\log(2n^{r-|I|}+1)\bigg{)}^{2(r-|I|)}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf
i}_{I}}(h^{(r)}_{\bf i})^{2}\bigg{]}\Bigg{\|}_{\psi_{\frac{1}{2(r-|I|)}},H},$

where $\|\cdot\|_{\psi_{\alpha},H}$ means a sub-Weibull norm defined by an
expectation under $H$; the inequalities are explained as follows:

  1. (i)

Lemmas A.3 and D.1$(ii)$ imply $\max_{{\bf
i}_{I^{c}}}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}$ is sub-Weibull of order $\frac{1}{2(r-|I|)}$. The inequality
then follows from the first inequality in Lemma A.1 and that a constant
depending on $\frac{1}{2(r-|I|)}$ can be bounded by a further constant
depending only on $r$ because $0\leq|I|<r$;

  2. (ii)

The maximal inequality Lemma A.3;

  3. (iii)

The explicit expression
$\psi^{-1}_{\frac{1}{2(r-|I|)}}(\cdot)=(\log(\cdot+1))^{2(r-|I|)}$.

By the second inequality in Lemma A.1,

$\displaystyle\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{\psi_{\frac{1}{2(r-|I|)}},H}$ $\displaystyle\leq
C(r)\sup_{p\geq 1}p^{-2(r-|I|)}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf
i}_{I}}(h^{(r)}_{\bf i})^{2}\bigg{]}\Bigg{\|}_{p,H}$ $\displaystyle\leq
C(r)\Bigg{\\{}\sup_{p\geq 2}\Bigg{(}p^{-2(r-|I|)}\sum_{{\bf
i}_{I}}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{p,H}\Bigg{)}+\sup_{1\leq p\leq
2}\Bigg{(}p^{-2(r-|I|)}\sum_{{\bf
i}_{I}}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{p,H}\Bigg{)}\Bigg{\\}}$ (D.11) $\displaystyle\leq
C(r)\Bigg{\\{}\sup_{p\geq 2}\Bigg{(}p^{-2(r-|I|)}\sum_{{\bf
i}_{I}}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{p,H}\Bigg{)}+\sum_{{\bf
i}_{I}}\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{2,H}\Bigg{\\}}.$

Now for any $p\geq 2$, by Lemma 2.2 and Lemma D.1$(i)$,

$\displaystyle\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{p,H}$ $\displaystyle\leq
C(r)\times\big{(}p-1\big{)}^{2(r-|I|)}\times\bigg{\|}\mathbb{E}_{I,H}\big{[}(h^{(r)}_{\bf
i})^{2}\big{]}\bigg{\|}_{2,H}$
$\displaystyle=C(r)\times\big{(}p-1\big{)}^{2(r-|I|)}\times\sqrt{\mathbb{E}_{H}[(\mathbb{E}_{I,H}[(h^{(r)}_{\bf
i})^{2}])^{2}]}$ $\displaystyle\leq
C(r)p^{2(r-|I|)}\sqrt{\mathbb{E}_{H}[(h^{(r)}_{\bf i})^{4}]}\text{ by Jensen's
inequality}$ $\displaystyle=C(r)p^{2(r-|I|)}\|h^{(r)}_{\bf i}\|_{4,H}^{2}$
$\displaystyle\leq C(r)3^{2r}p^{2(r-|I|)}\|h^{(r)}_{\bf i}\|_{2,H}^{2}$ by
Lemma 2.2 ($h^{(r)}_{\bf i}$ is a degree-$2r$ polynomial in normal random
variables) (D.12)
$\displaystyle=C(r)p^{2(r-|I|)}\mathbb{E}_{H}\big{[}(h^{(r)}_{\bf
i})^{2}\big{]}.$

Moreover,

$\displaystyle\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{2,H}$
$\displaystyle\leq(\mathbb{E}_{H}[(h^{(r)}_{\bf i})^{4}])^{1/2}=\|h^{(r)}_{\bf
i}\|_{4}^{2}\text{ by Jensen's inequality }$ (D.13) $\displaystyle\ \leq
C(r)\mathbb{E}[(h^{(r)}_{\bf i})^{2}]\text{ by
Lemma~{}\ref{lem:hypercontraction}}.$

Combining (D.11) with (D.12) and (D.13), we get from Lemma D.2 that,

(D.14) $\Bigg{\|}\mathbb{E}_{I,H}\bigg{[}\sum_{{\bf i}_{I}}(h^{(r)}_{\bf
i})^{2}\bigg{]}\Bigg{\|}_{\psi_{\frac{1}{2(r-|I|)}},H}\leq C(r)\sum_{{\bf
i}_{I}}\mathbb{E}_{H}\big{[}(h^{(r)}_{\bf i})^{2}\big{]}\leq
C(r)n^{|I|}\sigma_{h}^{2},$

which implies Claim 4.4 for $I\subsetneq[r]$ from (D.10).

## References

  * Adamczak (2006) Radosł aw Adamczak.  Moment inequalities for $U$-statistics.  _Ann. Probab._ , 34(6):2288–2314, 2006.  ISSN 0091-1798,2168-894X.
  * Bakhshizadeh (2023) Milad Bakhshizadeh.  Exponential tail bounds and large deviation principle for heavy-tailed u-statistics.  _arXiv preprint arXiv:2301.11563_ , 2023.
  * Benedetti and Risler (1990) Riccardo Benedetti and Jean-Jacques Risler.  _Real algebraic and semi-algebraic sets_.  Actualités Mathématiques. [Current Mathematical Topics]. Hermann, Paris, 1990.  ISBN 2-7056-6144-1.
  * Berry (1941) Andrew C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates.  _Trans. Amer. Math. Soc._ , 49:122–136, 1941.  ISSN 0002-9947,1088-6850.
  * Bollen and Ting (2000) Kenneth A Bollen and Kwok-fai Ting.  A tetrad test for causal indicators.  _Psychological methods_ , 5(1):3, 2000.
  * Carbery and Wright (2001) Anthony Carbery and James Wright.  Distributional and $l^{q}$ norm inequalities for polynomials over convex bodies in $\mathbb{R}^{n}$.  _Mathematical research letters_ , 8(3):233–248, 2001.
  * Chakrabortty and Kuchibhotla (2018) Abhishek Chakrabortty and Arun Kumar Kuchibhotla.  Tail bounds for canonical u-statistics and u-processes with unbounded kernels.  2018\.
  * Chen et al. (2014) Bryant Chen, Jin Tian, and Judea Pearl.  Testable implications of linear structural equation models.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 28, 2014.
  * Chen et al. (2017) Bryant Chen, Daniel Kumor, and Elias Bareinboim.  Identification and model testing in linear structural equation models using auxiliary variables.  In _International Conference on Machine Learning_ , pages 757–766. PMLR, 2017.
  * Chen et al. (2011) Louis H. Y. Chen, Larry Goldstein, and Qi-Man Shao.  _Normal approximation by Stein’s method_.  Probability and its Applications (New York). Springer, Heidelberg, 2011\.  ISBN 978-3-642-15006-7.
  * Chen and Kato (2019) Xiaohui Chen and Kengo Kato.  Randomized incomplete $u$-statistics in high dimensions.  _The Annals of Statistics_ , 47(6):3127–3156, 2019.
  * Chernozhukov et al. (2013) Victor Chernozhukov, Denis Chetverikov, and Kengo Kato.  Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors.  _Ann. Statist._ , 41(6):2786–2819, 2013.  ISSN 0090-5364.
  * Chernozhukov et al. (2017) Victor Chernozhukov, Denis Chetverikov, and Kengo Kato.  Central limit theorems and bootstrap in high dimensions.  _Ann. Probab._ , 45(4):2309–2352, 2017.  ISSN 0091-1798,2168-894X.  doi: 10.1214/16-AOP1113.  URL https://doi.org/10.1214/16-AOP1113.
  * Chernozhukov et al. (2023) Victor Chernozhukov, Denis Chetverikov, and Yuta Koike.  Nearly optimal central limit theorem and bootstrap approximations in high dimensions.  _Ann. Appl. Probab._ , 33(3):2374–2425, 2023\.  ISSN 1050-5164,2168-8737.  doi: 10.1214/22-aap1870.  URL https://doi.org/10.1214/22-aap1870.
  * Chernozhuokov et al. (2022) Victor Chernozhuokov, Denis Chetverikov, Kengo Kato, and Yuta Koike.  Improved central limit theorem and bootstrap approximations in high dimensions.  _Ann. Statist._ , 50(5):2562–2586, 2022.  ISSN 0090-5364,2168-8966.  doi: 10.1214/22-aos2193.  URL https://doi.org/10.1214/22-aos2193.
  * Cox et al. (2015) David A. Cox, John Little, and Donal O’Shea.  _Ideals, varieties, and algorithms_.  Undergraduate Texts in Mathematics. Springer, Cham, fourth edition, 2015\.  ISBN 978-3-319-16720-6; 978-3-319-16721-3.  doi: 10.1007/978-3-319-16721-3.  URL https://doi.org/10.1007/978-3-319-16721-3.  An introduction to computational algebraic geometry and commutative algebra.
  * de la Peña et al. (2009) Victor H. de la Peña, Tze Leung Lai, and Qi-Man Shao.  _Self-normalized processes_.  Probability and its Applications (New York). Springer-Verlag, Berlin, 2009\.  ISBN 978-3-540-85635-1.  doi: 10.1007/978-3-540-85636-8.  URL https://doi.org/10.1007/978-3-540-85636-8.  Limit theory and statistical applications.
  * De la Pena and Giné (1999) Victor De la Pena and Evarist Giné.  _Decoupling: from dependence to independence_.  Springer Science & Business Media, 1999.
  * de Loera et al. (1995) Jesús A. de Loera, Bernd Sturmfels, and Rekha R. Thomas.  Gröbner bases and triangulations of the second hypersimplex.  _Combinatorica_ , 15(3):409–424, 1995.  ISSN 0209-9683.  doi: 10.1007/BF01299745.  URL https://doi.org/10.1007/BF01299745.
  * Drton and Xiao (2016) Mathias Drton and Han Xiao.  Wald tests of singular hypotheses.  2016\.
  * Drton et al. (2007) Mathias Drton, Bernd Sturmfels, and Seth Sullivant.  Algebraic factor analysis: tetrads, pentads and beyond.  _Probab. Theory Related Fields_ , 138(3-4):463–493, 2007.  ISSN 0178-8051,1432-2064.  doi: 10.1007/s00440-006-0033-2.  URL https://doi.org/10.1007/s00440-006-0033-2.
  * Drton et al. (2008) Mathias Drton, Hélène Massam, and Ingram Olkin.  Moments of minors of wishart matrices.  2008\.
  * Dufour et al. (2017) Jean-Marie Dufour, Eric Renault, and Victoria Zinde-Walsh.  Wald tests when restrictions are locally singular.  _Unpublished manuscript_ , 2017.  URL https://jeanmariedufour.research.mcgill.ca/Dufour_Renault_ZindeWalsh_2012_WaldTestsLocallySingularRestrictions_W.pdf.
  * Eaton (2007) Morris L. Eaton.  _Multivariate statistics_ , volume 53 of _Institute of Mathematical Statistics Lecture Notes—Monograph Series_.  Institute of Mathematical Statistics, Beachwood, OH, 2007.  ISBN 978-0-940600-69-0; 0-940600-69-2.  A vector space approach, Reprint of the 1983 original [MR0716321].
  * Esseen (1942) Carl-Gustav Esseen.  On the liapunoff limit of error in the theory of probability.  _Ark. Mat. Astron. Fys._ , A28(9):1–19, 1942\.
  * Fang and Koike (2021) Xiao Fang and Yuta Koike.  High-dimensional central limit theorems by Stein’s method.  _Ann. Appl. Probab._ , 31(4):1660–1686, 2021\.  ISSN 1050-5164,2168-8737.  doi: 10.1214/20-aap1629.  URL https://doi.org/10.1214/20-aap1629.
  * Gaffke (2002) Norbert Gaffke.  On the asymptotic null-distribution of the wald statistic at singular parameter points.  _Statistics & Risk Modeling_, 20(1-4):379–398, 2002.
  * Gaffke et al. (1999) Norbert Gaffke, Rolf Steyer, and Alina A von Davier.  On the asymptotic null-distribution of the wald statistic at singular parameter points.  _Statistics & Risk Modeling_, 17(4):339–358, 1999.
  * Giné and Nickl (2016) Evarist Giné and Richard Nickl.  _Mathematical foundations of infinite-dimensional statistical models_ , volume [40] of _Cambridge Series in Statistical and Probabilistic Mathematics_.  Cambridge University Press, New York, 2016.  ISBN 978-1-107-04316-9.
  * Giné et al. (2000) Evarist Giné, Rafał Latała, and Joel Zinn.  Exponential and moment inequalities for u-statistics.  In _High Dimensional Probability II_ , pages 13–38. Springer, 2000\.
  * Isserlis (1918) Leon Isserlis.  On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables.  _Biometrika_ , 12(1/2):134–139, 1918.
  * Janson (1984) Svante Janson.  The asymptotic distributions of incomplete u-statistics.  _Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete_ , 66(4):495–505, 1984.
  * Janson (1997) Svante Janson.  _Gaussian Hilbert spaces_ , volume 129 of _Cambridge Tracts in Mathematics_.  Cambridge University Press, Cambridge, 1997.  ISBN 0-521-56128-0.
  * Kelley (1935) Truman L Kelley.  Essential traits of mental life.  _(No Title)_ , 1935.
  * Koroljuk and Borovskich (1994) V. S. Koroljuk and Yu. V. Borovskich.  _Theory of $U$-statistics_, volume 273 of _Mathematics and its Applications_.  Kluwer Academic Publishers Group, Dordrecht, 1994.  ISBN 0-7923-2608-3.  Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors.
  * Korolyuk and Borovskich (2013) Vladimir S Korolyuk and Yu V Borovskich.  _Theory of U-statistics_ , volume 273.  Springer Science & Business Media, 2013.
  * Kuchibhotla and Chakrabortty (2022) Arun Kumar Kuchibhotla and Abhishek Chakrabortty.  Moving beyond sub-gaussianity in high-dimensional statistics: Applications in covariance estimation and linear regression.  _Information and Inference: A Journal of the IMA_ , 11(4):1389–1456, 2022.
  * Lehmann and Romano (2022) E. L. Lehmann and Joseph P. Romano.  _Testing statistical hypotheses_.  Springer Texts in Statistics. Springer, Cham, fourth edition, 2022.  ISBN 978-3-030-70577-0; 978-3-030-70578-7.
  * Leung and Drton (2018) Dennis Leung and Mathias Drton.  Algebraic tests of general gaussian latent tree models.  _Advances in Neural Information Processing Systems_ , 31, 2018.
  * Leung and Shao (2023a) Dennis Leung and Qi-Man Shao.  Another look at stein’s method for studentized nonlinear statistics with an application to u-statistics.  _arXiv preprint arXiv:2301.02098_ , 2023a.
  * Leung and Shao (2023b) Dennis Leung and Qi-Man Shao.  Nonuniform berry-esseen bounds for studentized u-statistics.  _Bernoulli, accepted_ , 2023b.
  * Lieb and Loss (2001) Elliott H. Lieb and Michael Loss.  _Analysis_ , volume 14 of _Graduate Studies in Mathematics_.  American Mathematical Society, Providence, RI, second edition, 2001.  ISBN 0-8218-2783-9.
  * Lopes (2022) Miles E. Lopes.  Central limit theorem and bootstrap approximation in high dimensions: near $1/\sqrt{n}$ rates via implicit smoothing.  _Ann. Statist._ , 50(5):2492–2513, 2022.  ISSN 0090-5364,2168-8966.  doi: 10.1214/22-aos2184.  URL https://doi.org/10.1214/22-aos2184.
  * Mitchell et al. (2019) Jonathan D Mitchell, Elizabeth S Allman, and John A Rhodes.  Hypothesis testing near singularities and boundaries.  _Electronic journal of statistics_ , 13(1):2150, 2019.
  * Peng et al. (2022) Wei Peng, Tim Coleman, and Lucas Mentch.  Rates of convergence for random forests via generalized u-statistics.  _Electronic Journal of Statistics_ , 16(1):232–292, 2022.
  * Pillai and Meng (2016) Natesh S. Pillai and Xiao-Li Meng.  An unexpected encounter with Cauchy and Lévy.  _Ann. Statist._ , 44(5):2089–2097, 2016.  ISSN 0090-5364,2168-8966.
  * Serfling (1980) Robert J. Serfling.  _Approximation theorems of mathematical statistics_.  Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1980.  ISBN 0-471-02403-1.
  * Shafer et al. (1993) Glenn Shafer, Alexander Kogan, and Peter Spirtes.  _Generalization of the tetrad representation theorem_.  Rutgers University. Rutgers Center for Operations Research [RUTCOR], 1993\.
  * Shao (2010) Qi-Man Shao.  Stein’s method, self-normalized limit theory and applications.  In _Proceedings of the International Congress of Mathematicians. Volume IV_ , pages 2325–2350. Hindustan Book Agency, New Delhi, 2010.  ISBN 978-81-85931-08-3; 978-981-4324-34-2; 981-4324-34-5.
  * Shiers et al. (2016) N. Shiers, P. Zwiernik, J. A. D. Aston, and J. Q. Smith.  The correlation space of Gaussian latent tree models and model selection without fitting.  _Biometrika_ , 103(3):531–545, 2016.  ISSN 0006-3444,1464-3510.
  * Silva et al. (2006) Ricardo Silva, Richard Scheines, Clark Glymour, Peter Spirtes, and David Maxwell Chickering.  Learning the structure of linear latent variable models.  _Journal of Machine Learning Research_ , 7(2), 2006.
  * Song et al. (2019) Yanglei Song, Xiaohui Chen, and Kengo Kato.  Approximating high-dimensional infinite-order $U$-statistics: statistical and computational guarantees.  _Electron. J. Stat._ , 13(2):4794–4848, 2019\.  ISSN 1935-7524.  doi: 10.1214/19-EJS1643.  URL https://doi.org/10.1214/19-EJS1643.
  * Spearman (1927) C. Spearman.  _The Abilities of Man: Their Nature and Measurement_.  Macmillan, 1927.
  * Spirtes et al. (2000) Peter Spirtes, Clark N Glymour, and Richard Scheines.  _Causation, prediction, and search_.  MIT press, 2000.
  * Sturma et al. (2022) Nils Sturma, Mathias Drton, and Dennis Leung.  Testing many and possibly singular polynomial constraints.  _arXiv preprint arXiv:2208.11756_ , 2022.
  * Sullivant et al. (2010) Seth Sullivant, Kelli Talaska, and Jan Draisma.  Trek separation for Gaussian graphical models.  _Ann. Statist._ , 38(3):1665–1685, 2010.  ISSN 0090-5364,2168-8966.  doi: 10.1214/09-AOS760.  URL https://doi.org/10.1214/09-AOS760.
  * Vaart and Wellner (1996) Aad W Vaart and Jon A Wellner.  Weak convergence.  In _Weak convergence and empirical processes_ , pages 16–28. Springer, 1996.
  * Vladimirova et al. (2020) Mariia Vladimirova, Stéphane Girard, Hien Nguyen, and Julyan Arbel.  Sub-weibull distributions: Generalizing sub-gaussian and sub-exponential properties to heavier tailed distributions.  _Stat_ , 9(1):e318, 2020.
  * Wick (1950) G. C. Wick.  The evaluation of the collision matrix.  _Phys. Rev. (2)_ , 80:268–272, 1950.  ISSN 0031-899X,1536-6065.
  * Wishart (1928) John Wishart.  Sampling errors in the theory of two factors.  _British Journal of Psychology_ , 19(2):180, 1928\.
  * Wong et al. (2020) Kam Chung Wong, Zifan Li, and Ambuj Tewari.  Lasso guarantees for $\beta$-mixing heavy-tailed time series.  _Ann. Statist._ , 48(2):1124–1142, 2020.  ISSN 0090-5364,2168-8966.
  * Zhang and Wei (2022) Huiming Zhang and Haoyu Wei.  Sharper sub-weibull concentrations.  _Mathematics_ , 10(13):2252, 2022.


* [3] F. Bernis and A. Friedman.  Higher order nonlinear degenerate parabolic equations.  Journal of Differential Equations, 83(1):179–206, 1990.
  * [4] J. Billingham.  On a model for the motion of a contact line on a smooth solid surface.  European Journal of Applied Mathematics, 17(3):347–382, 2006.
  * [5] T. D. Blake and J. M. Haynes.  Kinetics of liquidliquid displacement.  Journal of Colloid and Interface Science, 30(3):421–423, 1969.
  * [6] D. Bothe and J. Prüss.  On the interface formation model for dynamic triple lines.  In Mathematical fluid dynamics, present and future, volume 183 of Springer Proceedings in Mathematics & Statistics, pages 25–47. Springer, Tokyo, 2016.
  * [7] P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, and S. Zhou.  Droplet breakup in a model of the Hele-Shaw cell.  Physical Review E (3), 47(6):4169–4181, 1993.
  * [8] R. G. Cox.  The dynamics of the spreading of liquids on a solid surface. part 1. viscous flow.  Journal of Fluid Mechanics, 168:169–194, 1986.
  * [9] S. R. de Groot and P. Mazur.  Non-equilibrium thermodynamics.  Series in Physics. North-Holland Publishing Co., Amsterdam; Interscience Publishers Inc., New York, 1962.
  * [10] E. B. Dussan.  On the spreading of liquids on solid surfaces: Static and dynamic contact lines.  Annual Review of Fluid Mechanics, 11(1):371–400, 1979.
  * [11] E. B. Dussan V. and S. H. Davis.  On the motion of a fluid-fluid interface along a solid surface.  Journal of Fluid Mechanics, 65(1):71–95, 1974.
  * [12] T. Erneux and S. H. Davis.  Nonlinear rupture of free films.  Physics of Fluids A: Fluid Dynamics, 5(5):1117–1122, 1993.
  * [13] H. Garcke and S. Wieland.  Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system.  SIAM Journal on Mathematical Analysis, 37(6):2025–2048, 2006.
  * [14] L. Giacomelli, M. V. Gnann, and F. Otto.  Rigorous asymptotics of traveling-wave solutions to the thin-film equation and Tanner’s law.  Nonlinearity, 29(9):2497–2536, 2016.
  * [15] L. Giacomelli and F. Otto.  Rigorous lubrication approximation.  Interfaces and Free Boundaries, 5(4):483–529, 2003.
  * [16] H. P. Greenspan.  On the motion of a small viscous droplet that wets a surface.  Journal of Fluid Mechanics, 84(1):125–143, 1978.
  * [17] Y. Guo and I. Tice.  Stability of contact lines in fluids: 2D Navier-Stokes flow.  https://arxiv.org/pdf/2010.15713.pdf.
  * [18] L. M. Hocking.  A moving fluid interface. part 2. the removal of the force singularity by a slip flow.  Journal of Fluid Mechanics, 79(2):209–229, 1977.
  * [19] C. Huh and S. G. Mason.  The steady movement of a liquid meniscus in a capillary tube.  Journal of Fluid Mechanics, 81(3):401–419, 1977.
  * [20] C. Huh and L.E. Scriven.  Hydrodynamic model of steady movement of a solid/liquid/fluid contact line.  Journal of Colloid and Interface Science, 35(1):85 – 101, 1971\.
  * [21] G. Kitavtsev, A. Münch, and B. Wagner.  Thin-film models for an active gel.  Proceedings A, 474(2220):20170828, 20, 2018.
  * [22] A. Münch, B. Wagner, and T. P. Witelski.  Lubrication models with small to large slip lengths.  Journal of Engineering Mathematics, 53(3-4):359–383, 2005.
  * [23] W. Ren and W. E.  Derivation of continuum models for the moving contact line problem based on thermodynamic principles.  Communications in Mathematical Sciences, 9, 06 2011.
  * [24] Y. D. Shikhmurzaev.  The moving contact line on a smooth solid surface.  International Journal of Multiphase Flow, 19(4):589 – 610, 1993\.
  * [25] Y. D. Shikhmurzaev.  Moving contact lines in liquid/liquid/solid systems.  Journal of Fluid Mechanics, 334:211–249, 1997.
  * [26] Y. D. Shikhmurzaev.  Macroscopic mechanism of rupture of free liquid films.  Comptes Rendus Mécanique, 333(3):205–210, 2005.
  * [27] Y. D. Shikhmurzaev.  Capillary flows with forming interfaces.  Chapman & Hall/CRC, Boca Raton, FL, 2008.
  * [28] Y. D. Shikhmurzaev.  Moving contact lines and dynamic contact angles: a ‘litmus test’ for mathematical models, accomplishments and new challenges.  The European Physical Journal Special Topics, 229(10):1945–1977, 2020.
  * [29] T. P. Witelski.  Nonlinear dynamics of dewetting thin films.  AIMS Mathematics, 5(5):4229–4259, 2020.


# What Could be the Observational Signature of Dark Matter in Globular
Clusters?

Elaine C. F. S. Fortes  Oswaldo D. Miranda  Floyd W. Stecker  Carlos A.
Wuensche

###### Abstract

Here we investigate the possibility that some globular clusters (GCs) harbor
intermediate mass black holes (BH) in their centers and are also embedded in a
low-mass dark matter (DM) halo.

Up to date, there is no evidence on whether or not GCs have DM in their
constitution. For standard cold DM cosmology, it is expected that GCs form
with their own DM halos. Other studies investigate the possibility that GCs
were initially embedded in massive DM halos that evolved during the cluster
lifetime.

An additional intriguing question is related to the existence of intermediate
mass black holes (IMBH) in the of GCs. The determination of whether GCs hold
IMBHs would be able to answer important questions about GCs formation and the
circumstances that gave rise to the IMBHs. DM & IMBH in the context of GCs are
interesting subjects to be studied and we will perform such studies here,
assuming the coexistence of both of them in some GCs.

Having such information, we perform the study possible DM signals from GCs.
One important subject to be studied is the DM density profile. In the inner
regions of GCs, the DM density profile is still an open question of
fundamental importance to DM studies, specially for the study of radio and
$\gamma$-ray emission from DM annihilation in such regions. Here we consider
the case of IMBHs in the inner regions of three GCs: 47 Tuc, NGC 6266 and
$\omega$ Cen. The existence of a black hole (BH) directly affects the matter
distribution in its vicinity. These effects are significant for IMBHs as well
as supermassive black holes (SMBHs) and can lead to large DM overdensities
called spikes. In this paper, we direct our studies to the radio synchrotron
emission from DM annihilation with spike profiles around possible IMBHs in
GCs. We present our results for synchrotron emission produced by DM
annihilation via $b\bar{b}$ channel. We use the best-fit DM mass of 34 GeV and
annihilation channel in $b\bar{b}$ used to explain the Galactic center
“excess” and perform our analysis for the estimated radio flux. We direct our
attention specially to the $\gamma$-ray emission from 47 Tuc. When considering
a scenario with includes a BH and the combined analysis of multi-wavelength
studies, we conclude that some parameter choice as the annihilation cross
section $\langle\sigma v\rangle$ used to fit the $\gamma$-ray excess differs
by many orders of magnitude from the one necessary to fit the radio
observations for this globular cluster if we consider the BH scenario.

##  1 Introduction

Despite great advances in Particle Physics, Astrophysics and Cosmology, we
still have no knowledge about the nature or composition of dark matter (DM).
For particle DM, we have little knowledge of range of parameter space of the
model to which it might belong. Together with these uncertainties, there exist
questions related to DM density profiles. The structure of DM halos are the
subject of ongoing numerical and analytical studies.

N-body simulations have been used to study halo formation. These studies have
shown that, although the spherically-averaged density profiles of the
simulated halos are very similar, their profiles are significantly different
from the single power laws predicted by theory. This conclusion holds even
considering different halo masses or different cosmological models [1]. The
N-body profiles are characterized by a decline obeying an $r^{-3}$
proportionality at large radii and a cusp profile of the form $r^{-\gamma}$,
the so called $\gamma$-models [2, 3, 4].

Analytical studies of DM halos predict that the density profile of a
virialized halo should obey the power-law $\rho\propto r^{-9/4}$. The
$r^{-9/4}$ refers to an approach done semi-analytical models, which included a
variety of physical ingredients to describe the collapse and virialization of
halos that are spherically symmetric, have suffered no major mergers and have
undergone quiescent accretion. A great part of the analytical work is based on
the secondary infall model (SIM) introduced by Gunn & Gott (1972), Gott (1975)
and Gunn (1977) [5, 6, 7].

Similar results were also obtained when studying density profiles around
density peaks and considering an isotropic velocity dispersion in the core of
collisionless halos [1].

The most studied predictions for cold DM establish that DM halos should
exhibit steep central cusps, with their density proportional to $r^{-\gamma}$
. Semi-analytical approaches indicate the existence of a cusp having
$1<\gamma<2$ [8, 9, 10, 11]. However, other simulations had find values of
$\gamma$ ranging from 0.3, 1 and 1.5 [12, 13, 14, 15].

There are five common numerical profiles very well studied in the literature,
namely Navarro-Frenk-White (NFW), Einasto (EIN), Isothermal (ISO), Burkert
(BUR) and Moore (MOO) [15, 4, 16, 17, 18]. All of them find a cusp, assuming
spherical symmetry. Their forms as functions of the galactocentric coordinate
r can be found at [19]. An environment that could enhance and change such
distribution profiles is the surroundings of compact objects such as black
holes (BH).

The presence of intermediate mass black holes (IMBH) and supermassive black
holes (SMBH) modifies the distribution of dark matter in its vicinity, making
such regions, very promising locations to detect DM indirectly due to the
constant evolution of BH, such as growth, mass accretion, etc. DM annihilation
or decay products could be searched for in the vicinities of BH since these
massive objects could induce overdensities, called “spikes” or lighter
overdensities called “mini-spikes”. Indirect searches could also utilize the
detection of DM via the detection of fluxes of secondary particles such as
$\gamma$-rays, electrons, positrons and neutrinos in such regions.

In the literature there are many theoretical studies of the distribution of DM
in the surroundings of a BH. Gondolo and Silk first studied DM spike density
profile in the Galactic center, where a supermassive black hole $(2-4)\times
10^{6}M_{\odot}$ dominates the gravitational potential [20, 21, 22].

One important recent analysis invoking the “spiky” nature of DM where
performed at Ref. [23], where it was shown the huge enhancement to the
$\gamma$-ray flux from the Galactic center assuming such spike and considering
that the $1-3$ GeV excess observed in Fermi data could be due to DM
annihilations. More recent analysis trying to prove the existence of a DM
density spike allied to the radial dependence at Galactic center where
performed by Lacroix et al. [24], who also presented a solution for the cosmic
ray diffusion on very small scales, assuming a radial dependence based on NFW
$+$ spike profile.

Following the studies of Gondolo, in 2000s, Silk and Zhao analysed possible
lighter overdensities around IMBHs remnants of Pop III stars and presented the
first estimative of mini-spikes for the IMBH in the Milky Way [25]. A more
recent study used the idea of these lighter overdensities due to DM
annihilation in the surroundings of IMBH to account for the $\gamma$-ray
excess in the Galactic center [26].

In fact, the DM density profile around IMBHs is still unknown, despite many
suggestions of strong enhancements for it. On one hand, some authors claim
that effects of dynamical relaxation by stellar interactions, merger between
two halos, and the loss of DM falling into the IMBH may smooth the already
produced spike [27, 28]. On the other hand, all the range of dynamical effects
of IMBH-DM interactions, such as black hole growth time scale, core
relaxation, time from stellar dynamical heating and adiabatic response of DM
has not been fully explored.

In this paper we will revisit the already studied spike profile used to
explain the $\gamma$-ray emission from the globular cluster 47 Tuc [29].

We perform our simulations with this profile considering the synchrotron
emission from DM annihilations in three globular clusters, namely, 47 Tuc,
$\omega$ Cen and NGC 6266. We will study 34 GeV DM particles which annihilates
in $b\bar{b}$ channel, motivated by best-fit of the $\gamma-$ray “excess” for
the Galactic center (See, e.g., Ref. [29])111Although the millisecond pulsar
hypothesis for the excess $\gamma$-rays from the Galactic center has been
favored, the DM hypothesis has recently been revived [30].. We then compare
our results with the existing constraints from radio signatures for globular
clusters considering the scenarios with and without BHs. We also compare the
parameter results with the ones used to explain the $\gamma$-ray emission from
the globular cluster 47 Tuc considering the possible existence of an IMBH
[29].

The paper is organized as follows: in section 2 we present the globular
clusters to be studied, its characteristics and relevance for our analysis. In
section 3 we illustrate the DM density spike profile and its parameters used
in our simulations. In section 4, we review the techniques for calculation of
synchrotron emission from DM annihilations. In section 5 we present our
results of the synchrotron flux for the chosen globular clusters and in the
final section 6 we summarize and highlight our conclusions.

##  2 Globular Clusters

Globular clusters (GCs) consist of dense spherical concentrations of stars,
with typical half-light radii of 1-10 pc and belong to the list of the oldest
astrophysical objects with typical ages of 11.5 Gyr to 12.5 Gyr. The
conditions under which they are formed and their various possible formation
mechanisms, both inside and outside of galaxies, are open questions. These
questions involve the various formation, assembly, and evolution processes of
the larger galaxies [31].

When we talk about DM in GCs, up to date, there is no evidence on whether or
not GCs are embedded in a DM halo. In 1968, Peebles & Dicke first proposed a
primordial scenario of GCs within individual DM halos [32]. But, until today,
the existence of DM halo in GC formation has been debated [33, 34]. The
existence of DM in GCs could be inferred from the kinematics of GCs stars as
well. For standard cold DM cosmology, it is expected that GCs form with their
own DM halos. Other scenarios include the possibility that GCs were initially
embedded in massive DM halos that evolved during the cluster lifetime.

Another polemic question involving GCs is related to the IMBH possibly host by
them. One of the natural places to produce IMBH would be the centers of GCs
[35]. During the past years, several methods have been taken into
consideration for proving the existence of IMBH. The main efforts trying to
explore the presence of IMBHs in GCs, constraining their mass utilize two main
approaches which investigate dynamical (kinematic measurements such as radial
velocities and proper motions) and accretion signatures [36]. Some attempts
try to associate the ultraluminous X-ray sources with IMBH [37] and other ones
try to find faint IMBHs in the radio band [38], but until today, there is no
conclusive evidence for their existence.

But there are also some claims of observational evidence for these IMBHs in
GCs [39]. The prediction that IMBH (compact objects which mass is in the range
$100M_{\odot}\lesssim M_{BH}\lesssim 10^{5}M_{\odot}$) might exist in low-mass
stellar systems such as GCs was made in the 1970’s by Wyller [40], Silk &
Arons [41] and Frank & Rees [42]. Silk & Arons studied X-ray sources in a
large sample of GCs concluding that the observed flux of X-rays could be
explained considering a mass accretion of $10^{2}-10^{3}M_{\odot}$ of a
central BH . This discovery promoted the black-hole hunting in GCs using
mainly X-ray and radio emission [43]. The GCs have also been studied several
times across the electromagnetic spectrum and led to the discovery of a large
number of millisecond pulsars (MSPs).

The existence of a central BH in a globular could be inferred from several
properties of the cluster, such as the total mass, total luminosity and
velocity dispersion profile. Using these properties, it is possible to
establish correlations such as the well known $M-\sigma_{*}$, $M_{*}-M_{tot}$,
$M_{*}-L_{tot}$, which are satisfied by galaxies and which can also be
satisfied by IMBH, where some of them ($M-\sigma_{*}$) are stronger than
others [43].

The $M-\sigma_{*}$ relation can be obtained from theoretical N-body
simulations in combination with dynamical methods [44]. Recent radio continuum
observations of some GCs have put an upper limit on central IMBHs. However, we
emphasize that the existence of IMBH in GCs is still a subject of debate by
many groups and it’s a substantially controversial topic.

Stronger evidence concerning the presence of these massive objects in GCs may
be unveiled by future experiments such as LISA [45], which will be the first
dedicated space-based gravitational wave detector. More theoretical work will
also be very useful in investigating the existence of central IMBH in GCs.

Ref. [46] explored a model which unified the cosmic star formation rate with
the local galactic star formation rate via a hierarchical structure formation
scenario. As a consequence of the model, it was predicted that dark matter
halos of large mass could contain a number of halos of much smaller mass,
being able to form structures very similar to GCs. In particular, [47]
estimate the fraction and distribution of dark matter in the innermost regions
of NGC 6218 (M12) and NGC 288. They concluded that there is a large mass
fraction of non-luminous matter in these objects.

In this article we concentrate our efforts studying three GCs, namely, 47 Tuc
also known as NGC 104, $\omega$ Cen, also known as NGC 5139 and M62, also
known as NGC 6266. 47 Tuc and $\omega$ Cen are the two most concentrated and
massive GCs in our galaxy. The observations of Fermi-LAT satellite very early
discovered that 47 Tuc and $\omega$ Cen were $\gamma$-ray bright [48] 222On
the other hand, some atypical properties of $\omega$ Cen suggest that it might
be the remnant core of a dSph. If this statement is confirmed, $\omega$ Cen is
considered to be the best place to search for DM annihilation due to its
proximity to Earth and since it contains high density of DM as compact dwarf
galaxies and also emits $\gamma$-rays [49].. NGC 6266 is among the ten most
massive and luminous GCs in the galaxy. Simulations suggest that these
clusters may have an IMBH on their core.

These three GCs were observed by the Hubble Space Telescope (HST). The results
of these observations placed limits on the mass a possible IMBH [44]. X-ray
and radio observations were used to settle limits on the mass of a central
accreting black hole for 47 Tuc. [50].

NGC 6266 was found to be the most suitable object to search by HST which
measured this cluster’s internal proper motion dispersion profile and them
compared the results to those ones produced by N-body simulations of NGC 6266
with/without an IMBH [51].

In Table 1, we present the main characteristics for these GCs.

Globular Clusters
---
|  47 Tuc |  $\omega$ Cen |  NGC 6266
l (Degrees) |  305.9 |  309.1 |  353.6
b (Degrees) |  -44.9 |  +15 |  +7.3
d (kpc) |  4.59 |  5.21 |  7.05
$t_{BH}$(Gyr) |  11.75 |  11.52 |  11.78
$M_{BH}(M_{\odot})$ |  2.3$\times 10^{3}$ |  5$\times 10^{4}$ |  2$\times 10^{3}$
$\sigma_{*}$ (km/s) |  10 |  10 |  15
Table 1: Data from 47 Tuc, $\omega$ Cen and NGC 6266. The parameter l denotes
the Galactic latitude of the GC, b denotes Galactic longitude, d is the
distance of the GC from Earth, taken from Refs. [44, 52, 53, 54, 55]. The
parameter $t_{BH}$ denotes the age of the IMBH which the GC might hold,
$M_{BH}$ denotes the mass of the IMBH and $\sigma_{*}$ is the stellar velocity
dispersion. The numerical values for these parameters can be found at Refs
[56, 57, 58]. We emphasize that the BH masses used in this table are models
and are not conclusively confirmed.

In the next sections, we will study the impact of the existence of IMBH in
these GCs, considering a scenario of DM annihilation in the vicinity of the
IMBH. For the three GCs we will use the data of Table 1. We will study the
implication of the DM spike enhanced distribution in the synchrotron signal of
these GCs.

##  3 DM Spike Density Profile in Globular Clusters

We will use here the DM spike profile of Ref. [29] in order to study the
effect of DM on halos surrounding an IMBH in a GC. We emphasize that the full
range of dynamical effects of a BH was not explored in considering the spike.
Here, we just focus on a well motivated dense inner spike profile and try to
evaluate its effects on the synchrotron emission of DM in the GCs. We then
compare our results with the one obtained by Brown et al. ([29]) to fit the
$\gamma$-ray excess.

The spike profile considered here presents a sharply peaked radial dependence,
which accounts for both the presence of IMBH and dynamical processes in the
GC. The spike structure of this profile was motivated by the strong evidence
of DM mass with 47 Tuc. Brown et al. analysed the full 9-year Fermi-LAT data
in attempt to find a possible explanation of the $\gamma$-ray excess, not only
considering DM, but also millisecond pulsars (MSP). Approximations like this
were already used to study a DM population clustered in the vicinities of a
SMBH at the center of Centaurus A [29, 59].

The existence of a spike enhances the $\gamma$-ray signal from DM
annihilation. It can be represented by

$\hbox{$\rho$}(r)=\left\\{\begin{array}[]{lll}0&\hbox{}&r<2R_{s}\\\
\dfrac{\rho_{sp}(r)\rho_{sat}}{\rho_{sp}(r)+\rho_{sat}}&\hbox{}&2R_{s}\leqslant
r<R_{sp}\\\ \rho_{0}\left(\dfrac{r}{R_{sp}}\right)^{-5}&\hbox{}&r\geqslant
R_{sp}\end{array}\right.$ (3.1)

where $R_{s}=2GM_{BH}/c^{2}$ denotes the Scharzschild radius,
$R_{sp}=GM_{BH}/\sigma_{*}^{2}$ [60] denotes the radius of the spike, $G$
denotes the Newton’s constant of gravity, $M_{BH}$ is the mass of the black
hole, $\sigma_{*}$ is the stellar velocity dispersion,
$\rho_{sp}(r)=\rho_{0}(r/R_{sp})^{-3/2}$ and
$\rho_{0}\approx(3-\gamma_{sp})M_{BH}/(4\pi R_{sp}^{3})$. The parameter
$\gamma_{sp}=(9-2\gamma)/(4-\gamma)$ is expected to be between 2.25 and 2.5,
taking into account that $0<\gamma<2$. The parameter $\rho_{sat}$ is the
saturation density established by DM annihilations. In this case, this
corresponds to make $\rho_{sat}=\rho_{sat}^{ann}$, where

$\rho_{sat}^{ann}=\dfrac{m_{DM}}{\langle\sigma v\rangle t_{i}},$ (3.2)

basically establishing $\rho_{sat}=\rho_{sat}^{ann}$ corresponds to the
equality of characteristic annihilation time and infall time ($t_{i}$) of DM
towards the IMBH. Here, we will assume conservatively that $t_{i}=t_{BH}$,
where $t_{BH}$ denotes the age of the black hole, $m_{DM}$ is the mass of DM
candidate and $\langle\sigma v\rangle$ is the annihilation cross section.

For $r\geqslant R_{sp}$ Brown et al. [29] assumed a radial dependence $r^{-5}$
(from tidal stripping). This profile is normalized by requiring that the mass
inside the spike $M_{sp}$ be of order of the BH mass, resulting that the total
DM mass in the cluster is $\sim 1\%$ of the mass of 47 Tuc. In the end, the
authors fit 47 Tuc’s $\gamma$-ray spectrum considering MSP+DM spike. Their
best-fit solution had a DM candidate with mass 34 GeV and $\langle\sigma
v\rangle\sim 6\times 10^{-30}$ cm3/s.

There are some comments and replies to comments related to MSP & DM
interpretation in Brown’s papers [61, 62]. In the reply to comments they
conclude that such discussion motivates a deeper radio study of 47 Tuc. In
this paper, we present a first analysis of the radio signature from dark
matter annihilation, studying the same channel proposed by [29] to explain the
gamma excess in 47 Tuc. We will also present the theoretical radio flux for
other two GCs, namely, $\omega$ Cen and NGC 6266.

##  4 Synchrotron/Radio Flux From Dark Matter Annihilation

In this section we revisit the techniques to calculate the radio flux
originated from the propagation of electrons/positrons resulting from DM
annihilation processes. Then we compute the resulting synchrotron emission for
three GCs. The radio flux emission depends mainly on the magnetic field
strenght ($B$), on the DM mass ($m_{DM}$), annihilation channels and density
profile, $\rho(r)$, and on the density of the ionized gas ($n$).

The integrated synchrotron flux density produced by a generic distribution
reads:

$S_{\nu}(\nu)\approx\dfrac{1}{D^{2}}\int_{0}^{R_{a}}drr^{2}j_{\nu}(\nu,r),$
(4.1)

where $R_{a}$ is the radial extent of the region of interest [63]. The
parameter $D$ is the distance from the GC to Earth. The result is usually
given in Janskys. In eq. (4.1), $j_{\nu}$ denotes the synchrotron emissivity,
which can be expressed as

$j_{\nu}(\nu,r)=2\int_{m_{e}c^{2}}^{m_{DM}}dE\dfrac{dn_{e}}{dE}(E,r)P_{\nu}(\nu,E)$
(4.2)

where the factor 2 takes into account the electrons and positrons,
$\dfrac{dn_{e}}{dE}(E,r)$ is the electron equilibrium spectrum and
$P_{\nu}(\nu,E)$ denotes the synchrotron power for a certain frequency $\nu$.
Considering the isotropy in the distribution of relativistic electrons with
energy $E$ and uniform magnetic field, we develop the steps necessary to
obtain $P_{\nu}(\nu,E)$. We consider the analysis presented in Ref. [19].
First we have that:

$P_{\nu}(\nu,E,\alpha)=\dfrac{\sqrt{3}e^{3}B\sin\alpha F(x)}{m_{e}c^{2}},$
(4.3)

with $x=\nu/\nu^{\prime}$, $\nu^{\prime}=\nu_{c}\sin\alpha/2$ and
$\nu_{c}=3eB\gamma^{2}/(2\pi m_{e}c)$, $e$ denotes the electric charge, $B$ is
the magnetic field, $c$ is the velocity of light, $\gamma$ is the Lorentz
factor, related to the energy of a single electron by $E=\gamma m_{e}c^{2}$
and $F(x)$ is given by 333 We note that reference [63] considers a series
approach for $F(x)$. In their approximation, $s=x/\alpha$ and $F(s)\approx
1.25s^{1/3}\exp(-s)[648+s^{2}]^{1/12}$.

$F(x)=x\int_{x}^{\infty}K_{5/3}(x^{\prime})dx^{\prime},$ (4.4)

where $K_{n}$ the modified Bessel function of the second kind and order $n$.
As the next step we need to average the randomly oriented magnetic field over
the pitch angle $\alpha$ and after, we can express $P_{\nu}(\nu,E)$ as:

$P_{\nu}(\nu,E)=\dfrac{1}{2}\int_{0}^{\pi}d\alpha\sin\alpha
P\nu(\nu,E,\alpha),$ (4.5)

and finally we have

$P_{\nu}(\nu,E)=2\sqrt{3}\dfrac{e^{3}B}{m_{e}c^{2}}y^{2}\left[K_{4/3}(y)K_{1/3}(y)-\dfrac{3}{5}y\left(K_{4/3}^{2}(y)-K_{1/3}^{2}(y)\right)\right],$
(4.6)

where $y=\nu/\nu_{c}$. The term $\dfrac{dn_{e}}{dE}(E,r)$ in the eq. (4.2) is
the electron equilibrium spectrum.

In an astrophysical medium, the diffusion and energy losses modify the
injection spectrum from DM annihilation or decay. In the end, all of these
mechanisms should be taken into account and the electron equilibrium spectrum
is obtained from the diffusion equation

$\dfrac{\partial}{\partial
t}\dfrac{dn_{e}}{dE}=\triangledown\left[D(E,r)\triangledown\dfrac{dn_{e}}{dE}\right]\\\
+\dfrac{\partial}{\partial
E}\left[b_{loss}(E,r)\dfrac{dn_{e}}{dE}\right]+Q(E,r),$ (4.7)

where $Q(E,r)$ is the source term, $D(E,r)$ is the coefficient for spatial
diffusion and $b_{loss}$ is the loss term. If we neglect the diffusion
coefficient term, we have

$\dfrac{dn_{e}}{dE}(E,r)=\dfrac{\langle\sigma
v\rangle\rho(\,{\rm\textbf{r}})^{2}}{2m_{DM}^{2}b_{loss}(E,r)}\int_{E}^{m_{DM}}dE^{\prime}\dfrac{dN}{dE^{\prime}_{inj}},$
(4.8)

where $\langle\sigma v\rangle$ denotes de annihilation cross section,
$\rho$(r) is the spatial distribution of DM, $m_{DM}$ is the mass of DM
candidate, $b_{loss}$ is the energy loss term and $dN/dE^{\prime}_{inj}$ is
part of the source term which relates the electron injection spectrum from DM
annihilations. We took $dN/dE^{\prime}_{inj}$ from ref. [19], it can be also
obtained from packages presented in refs. [64, 65]. The source term is
expressed by:

$Q(E,r)=\dfrac{\langle\sigma
v\rangle\rho(\,{\rm\textbf{r}})^{2}}{2m_{DM}^{2}}\dfrac{dN}{dE_{inj}},$ (4.9)

The energy loss term is given by

$b_{loss}(E,r)=b_{syn}+b_{IC}+b_{brem}+b_{coul}$ (4.10)

where the terms $b_{syn}$, $b_{IC}$, $b_{brem}$ and $b_{coul}$ denote the loss
by synchrotron radiation, inverse Compton scattering, bremsstrahlung and
Coulomb interactions, respectively.

Since we are working with GCs in the galaxy, in energies below $\sim$ 5 GeV,
ionization and bremsstrahlung dominate. At higher energies Compton and
synchrotron processes dominate [66].

The energy loss due to all these processes can be expressed by [63].

$\displaystyle b_{loss}(E,r)$ $\displaystyle\approx
0.0254\left(\dfrac{E}{1GeV}\right)^{2}\left(\dfrac{B(r)}{1\mu
G}\right)^{2}+0.25\left(\dfrac{E}{GeV}\right)^{2}$
$\displaystyle+1.51n(0.36+log(\gamma/n))+6.13n(1+log(\gamma/n)/75.0),$

where this loss term has units of $10^{-16}$ GeV/s, $E=\gamma m_{e}c^{2}$, $B$
denotes the magnetic field, which we consider as 10 $\mu$G in all of our
analysis. In our simulations $n$ denotes the number of free electrons which
corresponds to the density of ionized gas. Here we have considered $n=1\times
10^{-3}$ cm-3. For GCs, some authors also consider the gas density expressed
as an exponential profile $n=n_{0}\exp(-\,{\rm\textbf{r}}/r_{D})$, where
$n_{0}$ is the initial gas density [67].

##  5 Numerical Results

In this section we present the numerical results for the synchrotron emission
theoretically predicted by our calculations for the GCs 47 Tuc, NGC 6266 and
$\omega$ Cen, assuming a DM candidate with $m_{DM}=34$ GeV annihilating into a
$b\bar{b}$ channel. When solving the diffusion-loss differential equation, we
assume that the cooling time scale of high energy electrons and positrons is
much smaller than their diffusion scale, so that the diffusion term in Eq.
(4.7) can be neglected.

For illustration, in Figure 1 we show the behavior of DM spike density profile
for 47 Tuc considering two situations. Spike-1 denotes the parameters we used
to fit the radio spectrum of this GC using the existing experimental limits
and upper bounds on 47 Tuc and also assuming the existence of a IMBH. Spike-2
denotes a best-fit scenario of $\gamma$-rays presented by [29] assuming the
existence of a IMBH, where $\langle\sigma v\rangle=6\times 10^{-30}$ cm3/s.
The spikes are formed with the typical radius of influence of the BH,
$R_{sp}=GM_{BH}/\sigma_{*}^{2}$. Out of the region of influence, the density
goes as $\rho\propto r^{-5}$.

Taking the set of parameters described in Table 1 together with those
described in Section 4, in order to fit the radio spectrum with a DM candidate
annihilating in $b\bar{b}$ with $m_{DM}=34$ GeV, we find that $\langle\sigma
v\rangle$ must be $\sim 5\times 10^{-37}$ cm3/s. This number was obtained
considering the presence of an IMBH in 47 Tuc.

In the same figure we also present the canonical NFW profile for the GCs 47
Tuc, $\omega$ Cen and NGC 6266. In this case, we consider that these GCs do
not have black holes in their centers. For calculating the NFW density profile
we had used the relation
$\frac{\rho(r)}{\rho_{crit}}=\frac{\delta_{c}}{(r/r_{s})(1+r/r_{s})^{2}}$,
where $r_{s}$ is the scale radius and $\delta_{c}$ is the characteristic
density. The parameter $\rho_{s}$ can be expressed as
$\rho_{s}=\rho_{crit}\cdot\delta_{c}$, where $\rho_{crit}=8.54\times 10^{-30}$
g/cm3.

Another step in the calculation of the DM density profile is related to the
virial radius and the concentration parameter $c$. This parameter is expressed
as $c=r_{200}/r_{s}$, where $r_{200}$ denotes the virial radius. The
characteristic density $\delta_{c}$ is given by [68]

$\delta_{c}=\frac{200}{3}\frac{c^{3}}{[\ln(1+c)-c/(1+c)]}$ (5.1)

One of the definitions of the parameter $c$ establishes that
$c=\log(r_{t}/r_{c})$, where $r_{c}$ denotes the core radius and $r_{t}$
denotes the tidal radius of the globular cluster [69]. For 47 Tuc,
$r_{c}=0.37$ pc and $r_{t}=56$ pc [34, 70]. So that, $c=2.17$. The calculation
of $\delta_{c}$ is immediate as we know the value of $c$ and in this case,
$\delta_{c}=1451.91$. It follows that $\rho_{s}=2.21\times 10^{-4}$ GeV/cm3.

As the last step, we have that the half light radius ($r_{h}$) of 47 Tuc is
$9.6\pm 0.3$ pc. Considering the that $r_{h}$ is around 2% of the virial
radius $r_{200}$, it results that $r_{200}=480$ pc, so that
$r_{s}=r_{200}/c\sim 0.22$ Kpc.

So the NFW DM density profile can be expressed as

$\rho(r)=\rho_{s}\frac{r_{s}}{r}\left(1+\frac{r}{r_{s}}\right)^{-2},$ (5.2)

with $\rho_{s}=2.21\times 10^{-4}$ GeV/cm3 and $r_{s}=0.22$ Kpc for 47 Tuc.

In a similar way For $\omega$ Cen, $r_{t}=65$ pc, $r_{s}=1.63\times 10^{-3}$
kpc and $r_{c}=3.9$ pc according to references [49, 71] which results in a
$\rho_{s}=7.45\times 10^{-5}$ GeV/cm3.

For NGC 6266, $r_{t}=18$ pc, $r_{h}=2.47$ pc and $r_{c}=1.70$ according to
references [72, 73]. Considering the that $r_{h}$ is around 2% of the virial
radius $r_{200}$, it results that $r_{s}=7.26\times 10^{-2}$ kpc and
$\rho_{s}=1.37\times 10^{-4}$ GeV/cm3. We had used all those information
provided above to plot the graphs related to the three NFW DM density profile.

---
Figure 1: The figure shows the spike density profile created around the IMBH
for the GC 47 Tuc. We present here two scenarios, shown in blue (Spike-1) and
dashed-pink (Spike-2). The spikes are formed within the radius of influence of
the BH ($R_{sp}$) of the globular cluster 47 Tuc. We take $m_{DM}=34$ GeV,
$\langle\sigma v\rangle=5\times 10^{-37}$ cm3/s for Spike-1 and $\langle\sigma
v\rangle=6\times 10^{-30}$ cm3/s for Spike-2 scenario. For effects of
comparison, we present in the same graph the canonical NFW DM density profile
for 47 Tuc (dashed-brown), $\omega$ Cen (dashed-double dot yellow) and NGC
6266 (dashed-dot green). In this case, we consider that there are no BHs in
the centers of these GCs.

In Table 2, we give the numerical values for the three GCs. We had obtained
these values with the set of parameters described in Table 1 and in Section 4,
considering the presence of IMBH in the GCs. In order to fit the radio
spectrum with a DM candidate annihilating in $b\bar{b}$ with mass $m_{DM}=34$
GeV, for $\omega$ Cen, we find that $\langle\sigma v\rangle$ must be $\sim
6\times 10^{-36}$ cm3/s; for NGC 6266 $\langle\sigma v\rangle$ must be $\sim
8\times 10^{-38}$ cm3/s.

Globular Clusters
---
|  47 Tuc |  $\omega$ Cen |  NGC 6266
$R_{a}$ (pc) |  56.7 |  55.1 |  30.7
$R_{s}$ (pc) |  2.19$\times 10^{-10}$ |  4.76$\times 10^{-9}$ |  1.90$\times 10^{-10}$
$R_{sp}$ (pc) |  9.84$\times 10^{-2}$ |  2.14 |  3.79$\times 10^{-2}$
$\rho_{sat}$ (GeV/cm3) |  1.83$\times 10^{20}$ |  1.55$\times 10^{19}$ |  1.14$\times 10^{21}$
$\langle\sigma v\rangle$ (cm3/s) |  $5\times 10^{-37}$ |  $6\times 10^{-36}$ |  $8\times 10^{-38}$
$\rho_{s}$ (GeV/cm3) |  $2.21\times 10^{-4}$ |  $7.45\times 10^{-5}$ |  $1.37\times 10^{-4}$
$r_{s}$ (kpc) |  0.22 |  $1.63\times 10^{-3}$ |  $7.26\times 10^{-2}$
Table 2:  The parameter $R_{a}$ denotes the radial extension of the globular
cluster used in the integration of Eq. 4.1, $R_{s}$ denotes the Schwarzschild
radius, $R_{sp}$ is radius of influence of the BH, $\rho_{sat}$ denotes
saturation density for the BH in the globular clusters, and $\langle\sigma
v\rangle$ represents the annihilation cross section. In the scenario with
IMBH, the $\langle\sigma v\rangle$ values are determined so that the radio
flux predicted by the model does not exceed the observational limits inferred
for each GC. We also present the parameters $\rho_{s}$ and $r_{s}$ used to fit
the NFW DM density profile for the three globular clusters.
---
Figure 2: The expected flux from DM as a function of frequency for the GCs.
Our DM candidate has mass $m_{DM}=34$ GeV and annihilates in $b\bar{b}$. We
had considered a scenario with an IMBH and $\langle\sigma v\rangle\sim 5\times
10^{-37}$ cm3/s for 47 Tuc, $\langle\sigma v\rangle\sim 6\times 10^{-36}$
cm3/s for $\omega$ Cen and $\langle\sigma v\rangle\sim 8\times 10^{-38}$ cm3/s
for NGC 6266. The GCs have observational upper limits taken from radio
continuum observations on the frequency 5 GHz (see [44]). The straigh line on
5 GHz indicates the 3$\sigma$-upper limits whose values are: 40 $\mu$Jy for 47
Tuc, 20 $\mu$Jy for $\omega$ Cen and 36 $\mu$Jy for NGC 6266. For comparison
only, we have included the measured flux densities of 47 Tuc X9. This source
is near to the center of 47 Tuc and the data are available at 5.5 GHz and 9
GHz [74]. For completeness, we had included the scenario with no IMBH
considering the NFW density profile for 47 Tuc.

In order to constrain the radio fluxes, we need to theoretically calculate
their signal and estimate the sensitivity of the existing radio surveys to
diffuse emission from these GCs. For the experimental values of radio flux, we
had considered the data of Ref. [57, 44]. In Figure 2 we give the expected
flux from DM annihilation as a function of frequency for 47 Tuc, $\omega$ Cen
and NGC 6266 considering the presence of an IMBH. Radio surveys of GCs have
produced only upper limits for the flux density, the latest data being mainly
derived by [44] through observations supported by the Australia Telescope
Compact Array (ATCA). The upper limit flux densities at 5 GHz are presented in
Fig. 2 and the values for $\langle\sigma v\rangle$ were chosen to respect
these constraints.

For comparison, we present the recent measurements for the 47 Tuc X9 source
[74]. This source possesses a double-peaked C IV emission line in its
ultraviolet spectrum, an indicative of matter accretion. Although it was
initially classified as a cataclysmic variable, the results of [74] show that
the spectral density is proportional to $\nu^{-0.4\pm 0.4}$ and may indicate
that X9 is a black hole accreting matter. The measurements for 47 Tuc X9 at
5.5 GHz and 9 GHz are just above our estimated radio spectrum for 47 Tuc.

In examining the individual radio flux densities, we see that the dark matter
annihilation cross section is much smaller than the one inferred from 47 Tuc
$\gamma$-ray flux. Through the observed upper limits, $\langle\sigma v\rangle$
must be within the range $\sim 10^{-35}-10^{-37}$cm3/s for the $b\bar{b}$
channel with $m_{DM}=34\,{\rm GeV}$. We normalized the spike profile similarly
to the one derived by [29], so that an increase in the cross section to
approximate the gamma–radio signatures should be accompanied by a decrease in
the amount of dark matter in the spike’s influence region.

All of these results can change in order of magnitude if we consider for
instance, the diffusion term in the diffusion-loss differential equation. In
this case, the radio flux can be reduced and the soft cutoff of the signal
decreases faster for lower frequencies when compared to the situation where
diffusion term was neglected. Different magnetic field configurations, gas
densities and annihilation cross sections, as well as the dark matter mass can
also change our results. However, even exploring the parameter space defined
by these variables, it is not possible to obtain signatures that are
consistent with the observational radio limits when we use, e.g.,
$\langle\sigma v\rangle\sim 10^{-30}$ cm3/s for 47 Tuc when we consider the
presence of an IMBH.

In Figure 2 we also present the expected flux from DM annihilation as a
function of frequency for the 47 Tuc considering a scenario with no BH and a
NFW DM density profile. For that flux we had considered the thermal
annihilation cross section $\langle\sigma v\rangle\sim 3\times 10^{-26}$
cm3/s. For $\omega$ Cen and NGC 6266, the radio flux are smaller that the ones
produced by 47 Tuc. To allow a better visualization of the figure, we didn’t
include the fluxes of $\omega$ Cen and NGC 6266 due to the NFW density profile
in the graph.

Considering the GC $\omega$ Cen, another limit can be obtained from reference
[49]. On it, they presented a result obtained from $\gamma$-ray emission for
this GC, concluding that if DM annihilation is responsible for $\omega$ Cen’s
$\gamma$-ray emission, they could infer DM particle properties from the
$\gamma$-ray spectral energy distribution. In this case, the observed spectrum
is very well fitted considering a model of DM annihilation in $b\bar{b}$ with
prompt photon emission and the best-fit mass is $m_{DM}=31\pm 4$ GeV for
$\langle\sigma v\rangle\sim 10^{-28^{+0.6}_{-1.2}}$ GeV/cm3 with 68%
confidence limits, assuming a NFW distribution of DM plus a black hole.

##  6 Final Remarks

In this work, we present the results obtained for the radio flux of three GCs,
namely, 47 Tuc, $\omega$ Cen and NGC 6266 and the constraints for these
signals obtained from radio observations considering the presence or not of an
IMBH. First, to study the fluxes we had considered an enhanced DM spike
density profile due to the presence of an IMBH, modeled for explaining the
observed $\gamma$-ray flux of 47 Tuc [29] and further extended our analyses
for other two GCs. Considering that the GC holds an IMBH, we find that the
annihilation cross section $\langle\sigma v\rangle$ used to explain the
observed $\gamma$-ray spectrum of 47 Tuc is too high to agree with the
continuum radio observations of 47 Tuc. For this GC, our simulations indicate
that $\langle\sigma v\rangle\sim 5\times 10^{-37}$ cm3/s would be in agreement
with the radio surveys. We have here assumed that the DM population is
dominant over the MSP population in producing the $\gamma$-ray and radio
emission. A viable particle physics model is presented at [75], in which
$\langle\sigma v\rangle$ is very suppressed (being much smaller than the
required thermal annihilation cross section), and that could be used for DM
interpretation of the$\gamma$-ray excess in 47 Tuc.

In considering a spike profile motivated to fit $\gamma$-ray flux of 47 Tuc,
we have assumed the popular scenario of a 34 GeV DM candidate annihilating
into a $b\bar{b}$ channel. We have also investigated the possibility of
probing DM in the inner part of two other GCs containing IMBHs, exploring the
ranges of most important parameters used in determining synchrotron flux.

Although we have performed a simplified analysis, neglecting the diffusion
terms in our calculations and considering a fixed magnetic configuration, we
have shown that the spike profile studied here can leave strong signatures for
the synchrotron flux from GCs.

For completeness, we had also presented the predicted radio fluxes for 47 Tuc,
$\omega$ Cen and NGC 6266 with no IMBH considering NFW DM density profile. The
fluxes of $\omega$ Cen and NGC 6266 are below the ones presented by 47 Tuc. In
the calculation of all of the flux due to NFW density profile, we had used the
thermal annihilation cross sections.

In the future it would be interesting to investigate other channels for dark
matter annihilation or decay, exploring different values for $m_{DM}$ and
simultaneously, taking into account the inferred $\gamma$-ray and radio
constraints.

## Acknowledgments

E. C. F. S. Fortes would like to thank Marco Cirelli and Javier Montaño
Domínguez for very useful discussion. E. C. F. S. Fortes and O. D. Miranda
thanks FAPESP for financial support under contract 2018/21532-4.

## References

  * [1] A. Del Popolo and P. Kroupa, Density profiles of dark matter haloes on Galactic and Cluster scales, Astron. Astrophys. 502 (2009) 733 [arXiv:0906.1146].
  * [2] L. M. Widrow, “Distribution Functions for Cuspy Dark Matter Density Profiles,´´ The Astrophysical J. 131, 39-46 (2000) doi:10.1086/317367
  * [3] L. M. Widrow, “Semi-analytic models for dark matter halos,” [arXiv:astro-ph/0003302 [astro-ph]].
  * [4] A. W. Graham, D. Merritt, B. Moore, J. Diemand and B. Terzic, “Empirical models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models,” Astron. J. 132, 2685-2700 (2006) doi:10.1086/508988 [arXiv:astro-ph/0509417 [astro-ph]].
  * [5] Gunn, J.R., Gott, J.R, “On the Infall of Matter Into Clusters of Galaxies and Some Effects on Their Evolution, ´´ ApJ, 176, 1, 1972.
  * [6] Gott, J.R, “On the Formation of Elliptical Galaxies,´´ ApJ, 201, 296, 1975.
  * [7] Gunn, J. E, “Massive galactic halos. I. Formation and evolution,´´ ApJ, 218, 592, 1977.
  * [8] Navarro, J.F., Frenk, C.S., & White, S.D.M., “Simulations of X-ray clusters,´´ MNRAS, 275, 270, 1995.
  * [9] Navarro, J.F., Frenk, C.S., & White, S.D.M., “The Structure of Cold Dark Matter Halos,´´ ApJ, 462, 563, 1996.
  * [10] Zhao, H.S., “Analytical Models for Galactic Nuclei,´´ MNRAS, 278, 488, 1996.
  * [11] S. Cole and C. G. Lacey, “The Structure of dark matter halos in hierarchical clustering models,” Mon. Not. Roy. Astron. Soc. 281, 716 (1996) doi:10.1093/mnras/281.2.716 [arXiv:astro-ph/9510147 [astro-ph]].
  * [12] D. Syer and S. D. M. White, Dark halo mergers and the formation of a universal profile, Mon. Not. Roy. Astron. Soc. 293 (1998) 337 [arXiv:astro-ph/9611065].
  * [13] Y. Hoffman and J. Shaham, Local density maxima: Progenitors of structure, Astrophys. J. 297 (1985) 16.
  * [14] A. V. Kravtsov, A. A. Klypin, J. S. Bullock and J. R. Primack, The Cores of dark matter-dominated galaxies: Theory versus observations, Astrophys. J. 502 (1998) 48 [arXiv:astro-ph/9708176].
  * [15] J. F. Navarro, C. S. Frenk and S. D. White, “The Structure of cold dark matter halos,” Astrophys. J. 462, 563-575 (1996) doi:10.1086/177173 [arXiv:astro-ph/9508025 [astro-ph]].
  * [16] K. Begeman, A. Broeils and R. Sanders, “Extended rotation curves of spiral galaxies: Dark haloes and modified dynamics,” Mon. Not. Roy. Astron. Soc. 249, 523 (1991)
  * [17] J. N. Bahcall and R. Soneira, “The Universe at faint magnetidues. 2. Models for the predicted star counts,” Astrophys. J. Suppl. 44, 73-110 (1980) doi:10.1086/190685
  * [18] A. Burkert, “The Structure of dark matter halos in dwarf galaxies,” IAU Symp. 171, 175 (1996) doi:10.1086/309560 [arXiv:astro-ph/9504041 [astro-ph]].
  * [19] J. Buch, M. Cirelli, G. Giesen and M. Taoso, PPPC 4 DM secondary: A Poor Particle Physicist Cookbook for secondary radiation from Dark Matter, JCAP 1509 (2015) 037 [arXiv:1505.01049].
  * [20] P. Gondolo and J. Silk, Dark matter at the galactic center, Nucl. Phys. Proc. Suppl. 87 (2000) 87 [arXiv:hep-ph/0001070].
  * [21] P. Gondolo and J. Silk, Dark matter annihilation at the galactic center, Phys. Rev. Lett. 83 (1999) 1719 [arXiv:astro-ph/9906391].
  * [22] P. Gondolo, Either neutralino dark matter or cuspy dark halos, Phys. Lett. B 494 (2000) 181 [arXiv:hep-ph/0002226].
  * [23] B. D. Fields, S. L. Shapiro and J. Shelton, Galactic Center Gamma-Ray Excess from Dark Matter Annihilation: Is There A Black Hole Spike?, Phys. Rev. Lett. 113 (2014) 151302 [arXiv:1406.4856].
  * [24] T. Lacroix, C. Boehm and J. Silk, Probing a dark matter density spike at the Galactic Center, Phys. Rev. D 89 (2014) 063534 [arXiv:1311.0139].
  * [25] H. S. Zhao and J. Silk, Dark minihalos with intermediate mass black holes, Phys. Rev. Lett. 95 (2005) 011301 [arXiv:astro-ph/0501625].
  * [26] T. Lacroix and J. Silk, Intermediate mass black holes and dark matter at the Galactic center, Astrophys. J. 853, (2018) L16 [arXiv:1712.00452].
  * [27] E. Vasiliev and M. Zelnikov, Dark matter dynamics in Galactic center, Phys. Rev. D 78 (2008) 083506 [arXiv:0803.0002].
  * [28] P. Ullio, H. Zhao and M. Kamionkowski, Dark-matter spike at the galactic center?, Phys. Rev. D 64 (2001) 043504 [arXiv:astro-ph/0101481].
  * [29] A. M. Brown, T. Lacroix, S. Lloyd, C. Boehm and P. Chadwick, Understanding the $\gamma$-ray emission from the globular cluster 47 Tuc: evidence for dark matter?, Phys. Rev. D 98, (2018) 041301 [arXiv:1806.01866].
  * [30] R. K. Leane and T. R. Slatyer, Revival of the Dark Matter Hypothesis for the Galactic Center Gamma Ray Excess, Phys. Rev. Letters 123, 241101 (2019).
  * [31] D. A. Forbes, N. Bastian, M. Gieles and R. A. Crain et al., Globular cluster formation and evolution in the context of cosmological galaxy assembly: open questions, Proc. Royal. Astron. Soc. 474 (2018) 2210 [arXiv:1801.05818].
  * [32] P. Peebles and R. Dicke, “Origin of the Globular Star Clusters,” Astrophys. J. 154, 891 (1968) doi:10.1086/149811
  * [33] D. C. Heggie and P. Hut, “Dark matter in globular clusters,” IAU Symp. 174, 303 (1996) [arXiv:astro-ph/9511115 [astro-ph]].
  * [34] R. Ibata, C. Nipoti, A. Sollima, M. Bellazzini, S. Chapman and E. Dalessandro, “Do globular clusters possess Dark Matter halos? A case study in NGC 2419,” Mon. Not. Roy. Astron. Soc. 428, 3648 (2013) doi:10.1093/mnras/sts302 [arXiv:1210.7787 [astro-ph.CO]].
  * [35] M. Miller and D. P. Hamilton, “Production of intermediate-mass black holes in globular clusters,” Mon. Not. Roy. Astron. Soc. 330, 232 (2002) doi:10.1046/j.1365-8711.2002.05112.x [arXiv:astro-ph/0106188 [astro-ph]].
  * [36] J. Anderson and R. P. van der Marel, “New Limits on an Intermediate Mass Black Hole in Omega Centauri: I. Hubble Space Telescope Photometry and Proper Motions,” Astrophys. J. 710, 1032-1062 (2010) doi:10.1088/0004-637X/710/2/1032 [arXiv:0905.0627 [astro-ph.GA]].
  * [37] J. M. Miller, A. Fabian and M. Miller, “A Comparison of intermediate mass black hole candidate ULXs and stellar-mass black holes,” Astrophys. J. 614, L117-L120 (2004) doi:10.1086/425316 [arXiv:astro-ph/0406656 [astro-ph]].
  * [38] T. J. Maccarone, R. Fender and A. K. Tzioumis, “Finding faint intermediate-mass black holes in the radio band,” Astrophysics 300, 239 (2005) doi:10.1007/s10509-005-1188-5 [arXiv:astro-ph/0412014 [astro-ph]].
  * [39] K. Gebhardt, R. Rich and L. C. Ho, “An Intermediate-mass black hole in the globular cluster G1: Improved significance from new Keck and Hubble Space Telescope observations,” Astrophys. J. 634, 1093-1102 (2005) doi:10.1086/497023 [arXiv:astro-ph/0508251 [astro-ph]].
  * [40] A. A. Wyller, Observational Aspects of Black Holes in Globular Cluster, Astroph. J. 160 (1970) 443.
  * [41] J. Silk and J. Aarons, On the nature of the globular cluster X-ray sources, Astrop. J. 200 (1975) L131.
  * [42] J. Frank and M. J. Rees, Effects of massive black holes on dense stellar systems, Mon. Not. Roy. Astron. Soc. 176 (1976) 633.
  * [43] N. Lützgendorf et al., The $M_{BH}-\sigma$ relation for intermediate-mass black holes in globular clusters, Astron. Astrophys. 555 (2013) A26 [arXiv:1304.7156].
  * [44] T. N. Lu and A. K. H. Kong, Radio Continuum Observations of 47 Tucanae and $\omega$ Centauri: Hints for Intermediate-mass Black Holes?, Astrop. J. 729 (2011) L25 [arXiv:1102.1668].
  * [45] LISA – Laser Interferometer Space Antenna.
  * [46] C. Gribel, O. D. Miranda and J. W. Vilas-Boas, Connecting the Cosmic Star Formation Rate with the Local Star Formation, Astrophys. J. 849 (2017) 108 [arXiv:1711.02481].
  * [47] A. Sollima, F. R. Ferraro, L. Lovisi, F. Contenta, E. Vesperini, L. Origlia, E. Lapenna, B. Lanzoni, A. Mucciarelli, E. Dalessandro and C. Pallanca, Searching in the dark: the dark mass content of the Milky Way globular clusters NGC288 and NGC6218, Mon. Not. Roy. Astron. Soc. 462 (2016) 1937 [arXiv:1607.05612].
  * [48] Fermi–LAT Collaboration, A population of gamma-ray emitting globular clusters seen with the Fermi Large Area Telescope, Astron. Astrophys. 524 (2010) A75 [arXiv:1003.3588].
  * [49] A. M. Brown, R. Massey, T. Lacroix, L. E. Strigari, A. Fattahi and C. Boehm, The glow of annihilating dark matter in Omega Centauri, arXiv:1907.08564.
  * [50] J. E. Grindlay, C. Heinke, P. D. Edmonds and S. S. Murray, High resolution X-ray imaging of a globular cluster core: compact binaries in 47tuc, Science 292 (2001) 2290 [arXiv:astro-ph/0105528].
  * [51] B. J. McNamara, T. E. Harrison, H. Baumgardt and P. Khalaj, A search for an intermediate-mass black hole in the core of the globular cluster NGC 6266, Astrophys. J. 745 (2012) 175.
  * [52] S. Chen, H. Richer, I. Caiazzo and J. Heyl, Distances to the Globular Clusters 47 Tucanae and NGC 362 Using Gaia DR2 Parallaxes, Astrophys. J. 867 (2018) 132 [arXiv:1807.07089].
  * [53] I. McDonald, J. Th. Van Loon, L. Decin, M.L. Boyer, A.K. Dupree, A. Evans, R. D. Gehrz and C. E. Woodward, Giants in the globular cluster $\omega$ Centauri: dust production, mass-loss and distance, Mon. Not. Roy. Astron. Soc. 394 (2009) 831 [arXiv:0812.0326].
  * [54] B. J. McNamara and J. McKeever, The Dynamical Distance, RR Lyrae Absolute Magnitude, and Age of the Globular Cluster NGC 6266 , Astron. J 142 (2011) 163.
  * [55] B. J. McNamara and J. McKeever, Erratum: The Dynamical Distance, RR Lyrae Absolute Magnitude, and Age of the Globular Cluster NGC 6266 , Astron. J. 143 (2012) 54.
  * [56] B. Kızıltan, H. Baumgardt, A. Loeb, An intermediate-mass black hole in the centre of the globular cluster 47 Tucanae, Nature 542 (2017) 203 [arXiv:1702.02149].
  * [57] N. Lützgendorf et al., Limits on intermediate-mass black holes in six Galactic globular clusters with integral-field spectroscopy, Astron. Astrophys. 552 (2013) A49 [arXiv:1212.3475].
  * [58] E. Noyola, K. Gebhardt and M. Bergmann, Gemini and Hubble Space Telescope Evidence for an Intermediate Mass Black Hole in $\omega$ Centauri, Astrophys. J. 676 (2008) 1008 [arXiv:0801.2782].
  * [59] A. M. Brown, C. Bœhm, J. Graham, T. Lacroix, P. Chadwick and J. Silk, Discovery of a new extragalactic population of energetic particles, Phys. Rev. D 95 (2017) 063018 [arXiv:1603.05469].
  * [60] P. J. E. Peebles, Star Distribution Near a Collapsed Object, Astrophys. J. 178 (1972) 371.
  * [61] R. Bartels and T. Edwards, Comment on “Understanding the $\gamma$-ray emission from the globular cluster 47 Tuc: Evidence for dark matter?”, Phys. Rev. D 100 (2019) 068301 [arXiv:1807.08800].
  * [62] A. M. Brown, T. Lacroix, S. Lloyd, C. Boehm and P. Chadwick, Reply to Comment on “Understanding the $\gamma$-ray emission from the globular cluster 47 Tuc: Evidence for dark matter?”, Phys. Rev. D 100 (2019) 068302 [arXiv:1909.01773].
  * [63] E. Storm, T. E. Jeltema, M. Splettstoesser and S. Profumo, Synchrotron Emission from Dark Matter Annihilation: Predictions for Constraints from Non-detections of Galaxy Clusters with New Radio Surveys, Astrophys. J. 839 (2017) 33 [arXiv:1607.01049].
  * [64] T. Bringmann, J. Edsjö, P. Gondolo, P. Ullio and L. Bergström, DarkSUSY 6: An Advanced Tool to Compute Dark Matter Properties Numerically, JCAP 1807 (2018) 033 [arXiv:1802.03399].
  * [65] G. Bélanger, F. Boudjema, A. Pukhov and A. Semenov, micrOMEGAs3: A program for calculating dark matter observables, Comput. Phys. Commun. 185 (2014) 960 [arXiv:1305.0237].
  * [66] F. W. Stecker, Equilibrium Spectra of Secondary Cosmic-Ray Positrons in the Galaxy and the Spectrum of Cosmic Gamma-Rays resulting from Their Annihilation, Astrophys. Space Sci. 3 (1969) 579.
  * [67] G. Beck, An excess of excesses examined via dark matter radio emissions from galaxies, JCAP 08 (2019) 019 [arXiv:1905.05599].
  * [68] J. F. Navarro, C. S. Frenk and S. D. White, “A Universal density profile from hierarchical clustering,” Astrophys. J. 490, 493-508 (1997) doi:$10.1086/304888$ [arXiv:astro-ph/9611107 [astro-ph]].
  * [69] G. Meylan, “The internal dynamics of globular clusters,” doi:10.1142/9789812793621${}_{0}004$ [arXiv:astro-ph/9912495 [astro-ph]].
  * [70] R. R. Lane, A. H. Kupper and D. C. Heggie, “The Tidal Tails of 47 Tucanae,” Mon. Not. Roy. Astron. Soc. 423, 2845 (2012) doi:10.1111/j.1365-2966.2012.21093.x [arXiv:1204.2549 [astro-ph.GA]].
  * [71] S. Henleywillis, A. M. Cool, D. Haggard, C. Heinke, P. Callanan and Y. Zhao, “A Deep X-ray Survey of the Globular Cluster Omega Centauri,” Mon. Not. Roy. Astron. Soc. 479, no.2, 2834-2852 (2018) doi:10.1093/mnras/sty675 [arXiv:1803.04822 [astro-ph.HE]].
  * [72] A. Mackey and S. van den Bergh, “The Properties of Galactic globular cluster sub-systems,” Mon. Not. Roy. Astron. Soc. 360, 631-645 (2005) doi:10.1111/j.1365-2966.2005.09080.x [arXiv:astro-ph/0504142 [astro-ph]].
  * [73] Mihwa Han, Sang Hyun Chun, Samyaday Choudhury, Howoo Chiang, Sowon Lee and Young Jong Sohn “Spatial configuration of stars around three metal-poor globular clusters in the galatic bulge, NGC 6266, NGC 6273, and NGC 6681: Surface density map and radial density profile” Journal of Astronomy and Space Sciences, 34, 83-97 (2017).
  * [74] J. C. A. Miller-Jones et al., Deep radio imaging of 47 Tuc identifies the peculiar X-ray source X9 as a new black hole candidate, Mon. Not. Roy. Astron. Soc. 453 (2015) 3918 [arXiv:1509.02579].
  * [75] N. Okada and O. Seto, “Inelastic extra $U(1)$ charged scalar dark matter,” Phys. Rev. D 101, no.2, 023522 (2020) doi:10.1103/PhysRevD.101.023522 [arXiv:1908.09277 [hep-ph]].