††institutetext: Department of Physics, University of Alberta, Edmonton, AB
T6G 2E1 Canada

# Greybody factors for higher-dimensional non-commutative geometry inspired
black holes

Zachary Cox 2 and Douglas M. Gingrich111Also at TRIUMF, Vancouver, BC V6T 2A3
Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Greybody factors are computed for massless fields of spin 0, 1/2, 1, and 2
emitted from higher-dimensional non-commutative geometry inspired black holes.
Short-range potentials are used with path-ordered matrix exponentials to
numerically calculate transmission coefficients. The resulting absorption
cross sections and emission spectra are computed on the brane and compared
with the higher-dimensional Schwarzschild-Tangherlini black hole. A non-
commutative black hole at its maximum temperature in seven extra dimensions
will radiate a particle flux and power of 0.72-0.81 and 0.75-0.81,
respectively, times lower than a Schwarzschild-Tangherlini black hole of the
same temperature. A non-commutative black hole at its maximum temperature in
seven extra dimensions will radiate a particle flux and power of 0.64-0.72 and
0.60-0.64, respectively, times lower than a Schwarzschild-Tangherlini black
hole of the same mass.

##  1 Introduction

Non-commutative (NC) space-time geometry has allowed insights into the quantum
nature of gravity. Within the effective theory for NC black holes, the point-
like sources in the energy-momentum tensor, that are normally represented by
Dirac delta functions of position, are replaced by Gaussian smeared matter
distributions of width $\sqrt{2\theta}$. Effective theories for non-
commutativity have enable calculations of black hole properties distinctly
different from those of classical gravity. The NC black hole has a finite
maximum temperature. A minimum mass and horizon radius exist at which the
temperature is zero and the heat capacity vanishes which may terminate Hawking
evaporation. These properties are in clear contradistinction to the classical
black hole with temperature becoming infinite as it approaches zero mass and
horizon radius.

Many studies have elucidated the nature of NC geometry inspired black holes
but little attention has been devoted to Hawking emission. Models for NC
geometry inspired chargeless, non-rotating black holes were developed by
Nicolini, Smailagic and Spallucci Nicolini:2005vd . The model was extended to
the case of charge in four dimensions Ansoldi:2006vg , generalized to higher
dimensions by Rizzo Rizzo:2006zb , and then to charge in higher dimensions
Spallucci:2008ez . A review of the developments can be found in Ref.
Nicolini:2008aj . Phenomenological considerations for searches with the Large
Hadron Collider (LHC) experiments appeared in Ref. Gingrich:2010ed . Graybody
factor calculations for massless scalar emission were presented in Ref.
Nicolini:2011nz .

The aim of this paper is to present the emission spectra from non-rotating NC
inspired black holes in higher dimensions for massless fields of spin 0, 1/2,
1, and 2. The results are compared with the higher-dimensional Schwarzschild-
Tangherlini (ST) black hole. One of our goals is to separate the temperature
characteristics from the transmission factors in the comparison of emission by
performing calculations of the graybody factors for different black hole
masses.

Throughout, we will work in units of $\hbar=c=k=1$, and use the PDG PDG
definition for the higher-dimensional ADD Planck scale $M_{D}$. Note that
$M_{*}^{n+2}=8\pi/(2\pi)^{n}M_{D}^{n+2}$ is often used in the literature. When
$M_{*}\sim 1$ TeV is taken, the value of $M_{D}$ will be more than four times
beyond the current experimental lower bounds on $M_{D}$ for $n\geq 3$, and
ruled out for $n=1$ and 2 extra dimensions.

##  2 Non-commutative geometry inspired black holes

A nice review of NC geometry inspired black holes already exists
Nicolini:2008aj . We will only present the mathematical results used here. The
$g_{00}$ component of the NC inspired metric is

$h(r)=1-\frac{1}{k_{n}}\frac{M}{M_{D}}\frac{1}{(M_{D}r)^{n+1}}P\left(\frac{n+3}{2},\frac{r^{2}}{4\theta}\right)\,,$
(1)

where

$k_{n}=\frac{n+2}{2^{n}\pi^{(n-3)/2}\Gamma\left(\frac{n+3}{2}\right)}$ (2)

and $P$ is the normalized lower-incomplete gamma function

$P\left(\frac{n+3}{2},\frac{r^{2}}{4\theta}\right)=\frac{1}{\Gamma\left(\frac{n+3}{2}\right)}\gamma\left(\frac{n+3}{2},\frac{r^{2}}{4\theta}\right)=\frac{1}{\Gamma\left(\frac{n+3}{2}\right)}\int_{0}^{r^{2}/(4\theta)}dte^{-t}t^{(n+3)/2-1}\,.$
(3)

The symbols are $M$ for the mass of the black hole and $r$ for the radial
distance from the centre of the black hole.

For the effective potentials to be discussed soon, we will also require the
derivative of the metric function with respect to $r$:

$h(r)^{\prime}=\frac{M}{k_{n}\Gamma\left(\frac{n+3}{2}\right)}\frac{1}{(M_{D}r)^{n+2}}\left[(n+1)\gamma\left(\frac{n+3}{2},\frac{r^{2}}{4\theta}\right)-2\left(\frac{r^{2}}{4\theta}\right)^{\frac{n+3}{2}}e^{\frac{-r^{2}}{4\theta}}\right]\,.$
(4)

These expressions reduce to the higher-dimensional commutative forms in the
limit $\theta\to 0$ or $P\to 1$.

The horizon radius for non-commutative inspired black holes is given by

$\frac{M}{M_{D}}=\frac{k_{n}}{P\left(\frac{n+3}{2},\frac{r_{\mathrm{h}}^{2}}{4\theta}\right)}(M_{D}r_{\mathrm{h}})^{n+1}\,.$
(5)

We are unaware of a closed form solution to this equation and so have solved
it numerically for $r_{\mathrm{h}}$. In the commutative limit, the horizon
reduces to the usual higher-dimensional ST case, which can be written as

$r_{S}=\frac{1}{M_{D}}\left(\frac{1}{k_{n}}\frac{M}{M_{D}}\right)^{\frac{1}{n+1}}\,.$
(6)

Depending on the context, $r_{\mathrm{h}}$ will be used to refer to both NC
and ST black hole horizon radii.

For given $M$, $n$, $M_{D}$, and $\theta$, there can be one, two, or no
horizon. For reasons related to positive temperature given later, only the
outer horizon radius is relevant to the work presented here. Figure 1 shows
$h(r)$ for the case of a single event horizon for $n=7$, $M=20.4M_{D}$, and
$M_{D}=\sqrt{\theta}=1$, and a comparison with the ST black hole.

Figure 1: Metric function $h(r)$ for non-commutative (solid line) and
commutative (dashed line) black holes with $n=7$, $M=20.4M_{D}$, and
$M_{D}=\sqrt{\theta}=1$.

Because there is a minimum mass, there are masses below which the black hole
will not form, and above the minimum mass the horizon radius is double valued.
As the mass increases, the inner horizon radius shrinks to zero, while the
outer horizon radius approaches the commutative value. The situation thus
depicted in Figure 1 represents the minimum mass case in which the values of
the horizon radii for NC and ST black holes are maximally different. This will
be a useful condition when considering maximum differences in the transmission
coefficients between NC and ST black holes.

##  3 Transmission coefficients

Hawking radiation from black holes is typically studied by examining the
response to perturbations. Hence, understanding modifications of Hawking
radiation due to NC geometry requires one to solve the equations of motion for
various spin perturbations on the NC inspired black hole metric.

The Teukolsky equation describes spin 0, 1/2, 1, and 2 field perturbations in
the background metric due to the black hole Teukolsky:1973ha ;
teukolsky1973part2 ; Teukolsky:1974yv . The partial differential equations can
be separated. The angular equation can be numerically solved to obtain the
energy eigenvalues or separation constant. The energy eigenvalues are then
used in the radial equation. The radial equation can be solve to find
transmission coefficients for fields emitted from the black hole horizon. The
transmission coefficients describe the probability that a particle, generated
by quantum fluctuations at the horizon of a black hole, escapes to spatial
infinity.

The Teukolsky radial equation can be numerically solved directly, but the
convergence of the solution at the integration boundaries is not clear. The
difficulty with the radial equation in the context of a scattering problem is
that the first order radial derivatives create complex $is\omega$ terms which
have a $1/r$ behaviour at infinity. This can eventually lead to problems in
the numerical computations.

Another approach uses a Chandrasekhar transformation Chandrasekhar  to cast
the radial equation into an effective Schrödinger equation with a short-range
barrier potential different for each spin field, allowing a more careful
numerical treatment. The equation takes the form

$\left[\frac{d^{2}}{dr^{2}_{*}}+\omega^{2}-V_{s}(r)\right]\psi_{s}=0\,,$ (7)

where $s$ is the spin of the field and $r_{*}$ a generalized tortoise
coordinate. We are now faced with a potential-barrier problem.

The purpose of this transformation is that the potentials $V_{s}(r)$ are now
short-ranged. They vanish faster than $1/r$ which is advantageous for
numerical computations. It should be noted that these potentials contain a
dependence on $\omega$ through the connection coefficient to the angular
equation. Working with real-valued potentials has benefits.

Arbey et al. Arbey:2021jif  have derived general potentials $V_{s}$ for
spherical symmetric metrics for different spins $s$ of massless fields. The
effective potentials seen by $s=0,1/2,1$, and 2 massless fields can be written
as

$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle
h\left[\frac{\nu_{0}}{r^{2}}+\frac{h^{\prime}}{r}\right]\,,$ (8)
$\displaystyle V_{1/2}$ $\displaystyle=$
$\displaystyle\nu_{1/2}\frac{h}{r^{2}}\pm\sqrt{\nu_{1/2}h}\left[\frac{h^{\prime}}{2r^{2}}-\frac{h}{r^{2}}\right]\,,$
(9) $\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\nu_{1}\frac{h}{r^{2}}\,,$ (10) $\displaystyle V_{2}$
$\displaystyle=$ $\displaystyle
h\left[\frac{\nu_{2}+2h}{r^{2}}-\frac{h^{\prime}}{r}\right]\,,$ (11)

where $\nu_{0}=\nu_{1}=\ell(\ell+1)$, $\nu_{1/2}=\ell(\ell+1)+1/4$,
$\nu_{2}=\ell(\ell+1)-2$, and $\ell$ is the angular momentum quantum number.

For spin 1/2, one must take the positive sign in Eq. (9) to get a positive
potential for all $r$. For the spin-2 case, for high $n$ and $\ell=2$, it is
possible to get a negative potential. We will discuss this further below when
describing absorption cross section results. These potentials differ from
those in Ref. Arbey:2021jif  in that they are missing a $1/\sqrt{2}$ factor in
the second term of Eq. (5.1d), and Eq. (5.9c) should have a first constant of
4 rather than 2 in the second term and a constant of 3 rather than 1 in the
last term.

Figure 2 shows two representative cases for the NC black hole potentials with
$M_{D}=\sqrt{\theta}=1$ for spin 0, 1/2, 1, and 2; the case of $l=s$ is shown.

Figure 2: Effective potentials for non-commutative black holes for spin 0,
1/2, 1, and 2 massless fields with $M_{D}=\sqrt{\theta}=1$: (left)
$n=0,M=60M_{D}$ and (right) $n=7,M=360M_{D}$.

The Schrödinger Eq. (7) is in the tortoise coordinate $r_{*}$ while the
potential is in $r$; or one can view $r=r(r_{*})$. The relationship between
the coordinates is defined by

$\frac{dr_{*}}{dr}=\frac{1}{h(r)}\,.$ (12)

We note that at $r\to+\infty$, $r_{*}\to+\infty$, and as $r\to
r_{\mathrm{h}}$, $r_{*}\to-\infty$.

One thus needs to solve differential Eq. (12) for $r_{*}$ and also invert it
to obtain $r$. While the result is well known for the four-dimensional
Schwarzschild metric it is less apparent for others. Analytic expressions
exist for $r_{*}$ for the higher dimensional ST case but we are unaware of
such analytic expressions for the NC case. We have thus numerically integrated
Eq. (12). For the initial condition, we take $r=r_{*}$ for a very large value
(approximating $+\infty$), and integrate backwards. The procedure has been
validated in the ST case by comparing the numerical integrations with the
analytic results. The analytic formulae are taken from Ref. Arbey:2021yke , to
which we have added the $n=7$ case:

$r_{*}=r+\frac{r_{S}}{16}\left[\sqrt{2}\ln\frac{x^{2}-\sqrt{2}x+1}{x^{2}+\sqrt{2}x+1}+2\ln\frac{x-1}{x+1}+4\tan^{-1}\frac{1}{x}-2\sqrt{2}\tan^{-1}\frac{1}{\sqrt{2}x-1}\right.\\\
\left.+2\sqrt{2}\tan^{-1}\frac{1}{\sqrt{2}x+1}\right]\,,$ (13)

where $x=r/r_{S}$. We note that Eq. (A5) in Ref. Arbey:2021yke  is missing an
essential negative sign under the square-root for the first
$\sqrt{\varphi_{-}}$.

We integrate the tortoise equation using a variable step size over the range
$r/r_{\mathrm{h}}=[\epsilon,350]$, where $\epsilon\approx 10^{-16}$. To obtain
the inverse relation for $r$ in terms of $r_{*}$, we have integrated the
tortoise equation and numerically inverted it using linear interpolation.

Gray and Visser Gray:2015xig  showed that the Bogoliubov coefficients relating
incoming and outgoing waves on a potential barrier can be directly obtained
from the following path-ordered exponential

$\left[\begin{array}[]{cc}\alpha&\beta^{*}\\\
\beta&\alpha^{*}\end{array}\right]=\mathcal{P}\exp\left(-\frac{i}{2\omega}\int_{-\infty}^{+\infty}V_{s}(r_{*})\left[\begin{array}[]{cc}1&e^{-2i\omega
r_{*}}\\\ -e^{2i\omega r_{*}}&-1\end{array}\right]dr_{*}\right)\,,$ (14)

where $\mathcal{P}$ is a path-ordering operator. Using the product calculus
definition of path-ordered integrals, they compute the Bogoliubov coefficients
via the product integrals

$\left[\begin{array}[]{cc}\alpha&\beta^{*}\\\
\beta&\alpha^{*}\end{array}\right]=\prod_{+\infty}^{-\infty}\left[I+A(r_{*})dr_{*}\right]\,,$
(15)

where $I$ is the identity matrix and $A(r_{*})$ is the transfer matrix given
by

$A(r_{*})=-\frac{i}{2\omega}V_{s}(r_{*})\left[\begin{array}[]{cc}1&e^{-2i\omega
r_{*}}\\\ -e^{2i\omega r_{*}}&-1\end{array}\right]\,.$ (16)

The product integral can be approximated numerically by

$\left[\begin{array}[]{cc}\alpha&\beta^{*}\\\
\beta&\alpha^{*}\end{array}\right]=\lim_{N\to\infty}\left[(I+A((r_{*})_{N-1})h)\ldots(I+A((r_{*})_{1})h)\right]\,,$
(17)

where $(r_{*})_{i}>(r_{*})_{i-1}$ and $h$ is the step size. We have taken
$N=10^{4}$.

The transmission probabilities $\Gamma(\omega)$ are related to the Bogoliubov
coefficients by

$\Gamma(\omega)=\frac{1}{|\alpha(\omega)|^{2}}\,.$ (18)

By using this procedure, one does not actually solve numerically a
differential equation. The problem becomes one of performing a single
numerical integral, Eq. (17).

##  4 Hawking emission

In studying Hawking emission from NC geometry inspired black holes, we
consider the following. The absorption cross section is an observable
representing an effective area embodying the likelihood of a particle to be
scattered by the black hole:

$\sigma_{s}(\omega)=\frac{\pi}{\omega^{2}}\sum_{\ell\geq
s}(2\ell+1)\Gamma_{s,l}(\omega)\,.$ (19)

where $\Gamma_{s,\ell}(\omega)$ are transmission coefficients, greybody
factors, for angular momentum mode $l$, and $2\ell+1$ is a multiplicity factor
for the azimuthal modes $m$ in spherical geometry.

One of the most interesting aspects of NC inspired black holes is their
temperature properties. The temperature in terms of the horizon radius is
given by

$T=\frac{n+1}{4\pi
r_{\mathrm{h}}}\left[1-\frac{2}{n+1}\left(\frac{r_{\mathrm{h}}}{2\sqrt{\theta}}\right)^{n+3}\frac{e^{-r_{\mathrm{h}}/(4\theta)}}{\gamma\left(\frac{n+3}{2},\frac{r_{\mathrm{h}}^{2}}{4\theta}\right)}\right]\,.$
(20)

The quantity in square brackets modifies the usual higher-dimensional
commutative form. In addition, since $r_{\mathrm{h}}$ has been modified, it
also leads to temperature differences. Since the inner horizon radius
corresponds to negative temperature, we only considered the outer radius.
Figure 3 shows the temperature versus horizon radius for NC black holes with
$M_{D}=\sqrt{\theta}=1$. The temperature vanishes at the minimum mass and
there is a maximum temperature. Also shown for comparison are the usual
temperatures for ST black holes.

Figure 3: Black hole temperature versus horizon radius for different number of
extra dimensions with $M_{D}=\sqrt{\theta}=1$. The solid lines are for non-
commutative inspired black holes and the dashed lines for Schwarzschild-
Tangherlini black holes. The number of extra dimensions increases from 0
(bottom) to 7 (top).

Of interest to us will be the NC black hole maximum temperature
$T_{\mathrm{max}}$ and mass $M_{\mathrm{e}M}$ at which the maximum temperature
occurs. In addition, we will use values of the ST black hole mass
$M_{\mathrm{eT}}$ at which the temperature is the same as the NC black hole
maximum temperature. These values are shown in Table 1.

$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
$T_{\mathrm{max}}$ | 0.015 | 0.030 | 0.043 | 0.056 | 0.067 | 0.078 | 0.089 | 0.098
$M_{\mathrm{e}M}$ | 60.47 | 108.31 | 157.48 | 204.89 | 249.06 | 289.35 | 325.56 | 357.74
$M_{\mathrm{e}T}$ | 66.95 | 135.96 | 224.84 | 332.05 | 456.43 | 597.05 | 753.05 | 923.67
Table 1: Non-commutative inspired black hole maximum temperature
$T_{\mathrm{max}}$, mass at which the maximum temperature occurs
$M_{\mathrm{e}M}$, and mass $M_{\mathrm{{e}T}}$ of a Schwarzschild-Tangherlini
black hole that has temperature $T_{\mathrm{max}}$, for different number of
extra dimensions $n$ with $M_{D}=\sqrt{\theta}=1$.

We now introduce the main physical variables that can be formulated using the
transmission coefficients. The number of particles emitted per unit time and
per unit frequency is

$\frac{d^{2}N}{dtd\omega}=\frac{1}{2\pi}\frac{1}{\exp(\omega/T)-(-1)^{2s}}\sum_{\ell\geq
s}(2\ell+1)\Gamma_{s,\ell}(\omega)\,.$ (21)

The energy emitted per unit time (or power) and per unit frequency is

$\frac{d^{2}E}{dtd\omega}=\frac{1}{2\pi}\frac{\omega}{\exp(\omega/T)-(-1)^{2s}}\sum_{\ell\geq
s}(2\ell+1)\Gamma_{s,\ell}(\omega)\,.$ (22)

We acknowledge a damping factor $\exp(-\theta\omega^{2}/2)$ that was developed
in Ref.Nicolini:2011nz  that multiplies Eq. (21) and Eq. (22). Including this
factor gives a sub-dominate effect Nicolini:2011nz  which is not particularly
relevant to our discussion and will be ignored.

The black body radiation leaving the horizon sees an effective potential
barrier due to the geometry of the space-time surrounding the black hole. The
potential barrier attenuates the radiation such that an observer at spatial
infinity away from the black hole will measure a different emission spectrum
than one at the horizon by a factor $\Gamma_{s,l}(\omega)$ called the greybody
factor. Thus the grey body factor $\Gamma_{s,l}(\omega)$ represents the
fraction of the black body emission which penetrates through the potential
barrier and escapes to spatial infinity. The expressions for absorption cross
section Eq. (19) and emission rates Eq. (21) and Eq. (22) are only applicable
to radiation on the brane.

##  5 Results

We present calculations of transmission coefficients, absorption cross
sections, and spectra for spin 0, 1/2, 1, and 2 fields. We have validated the
method by comparing with well known results for four dimensional Schwarzschild
and Kerr black holes, as well as the ST black holes Harris:2003eg ;
Park:2005vw  that we use for comparison with the NC results.

The model of NC geometry inspired black holes in higher dimensions has three
unknown parameters $n$, $M_{D}$, and $\sqrt{\theta}$. Typically we present
results for each extra dimension $n$. Usually it is necessary to fix the other
two parameters. One possibility for fixing the parameters is to be guided by
experimental constraints. Updating the approach taken in Ref. Gingrich:2010ed
(see Appendix A) restricts the values of $\sqrt{\theta}M_{D}$ that can be
probed from 0.25 to 0.98, different for each number of extra dimensions. The
allowed range in $\sqrt{\theta}M_{D}$ for any given number of extra dimensions
is severely restricted.

Calculations using values of $\sqrt{\theta}M_{D}<1$ begin to probe the details
of the matter smearing distribution and become model dependent. However, the
primary goal in this paper is to study the differences in Hawking emission
from NC and ST black holes so we choose the usual condition
$M_{D}=\sqrt{\theta}=1$. This implies that our phenomenological predictions
will not have particular consequence for the physics at the LHC.

###  5.1 Transmission coefficients

The fundamental calculated quantity is the transmission coefficient as a
function of frequency for different black hole masses, number of extra
dimensions, spin, and $\ell$ modes. Figure 4 shows transmission coefficients
for $n=7$ and $s=0,1/2,1,2$ as a function of frequency for different $\ell$
modes. The solid lines are for NC black holes and dashed lines for ST black
holes. The quantum number $\ell$ increases from $\ell=s$ going from left to
right. A black hole mass of $358M_{D}$ is used and corresponds to the NC black
hole maximum temperature. We observe that the NC and ST black hole
transmissions coefficients at this mass are very similar, differing slightly
for higher $l$. This is because they have a similar horizon radius at this
mass of 5.68 for NC black holes and 5.76 for ST black holes.

Figure 4: Transmission coefficients $\Gamma_{s,\ell}(\omega)$ for
$s=0,1/2,1,2$ as a function of frequency $\omega$. The quantum number $\ell$
increases from $\ell=s$ going from left to right. The solid lines are for non-
commutative black holes and dashed lines for Schwarzschild-Tangherlini black
holes. A black hole mass of $358M_{D}$ has been used and
$M_{D}=\sqrt{\theta}=1$ taken.

If the horizon radius difference between NC and ST black holes is
significantly different, the comparison changes. Figure 5 shows transmission
coefficients for $n=7$ and $s=0,1/2,1,2$ as a function of frequency for
different $\ell$ modes. A black hole mass of $358M_{D}$ and $924M_{D}$ are
used for the NC black hole and ST black hole, respectively, corresponding to
the NC black hole maximum temperature. In this case, significant differences
are observed for a horizon radius of 5.68 for NC black holes and 6.48 for ST
black holes.

Figure 5: Transmission coefficients $\Gamma_{s,\ell}(\omega)$ for
$s=0,1/2,1,2$ as a function of frequency $\omega$. The quantum number $\ell$
increases from $\ell=s$ going from left to right. The solid lines are for non-
commutative black holes $M=358M_{D}$ and dashed lines for Schwarzschild-
Tangherlini black holes $M=924M_{D}$. A black hole temperature of $0.098M_{D}$
has been used and $M_{D}=\sqrt{\theta}=1$ taken.

To examine more significant differences in transmission coefficients, a value
for the black holes mass at the minimum NC black hole mass can be chosen and
is shown in Figure 6. The horizon radius of the NC black hole is 2.32 and that
of the ST black hole 4.02. We observe significant differences between NC and
ST black hole transmission coefficients with increasing $\ell$. The NC black
hole transmission probabilities begin to rise at higher frequencies but rise
more steeply than the ST transmission probabilities. This behaviour was first
observed for spin 0 in Ref. Nicolini:2011nz .

Figure 6: Transmission coefficients $\Gamma_{s,\ell}(\omega)$ for
$s=0,1/2,1,2$ as a function of frequency $\omega$. The quantum number $\ell$
increases from $\ell=s$ going from left to right. The solid lines are for non-
commutative black holes and dashed lines for Schwarzschild-Tangherlini black
holes. A black hole mass of $20.4M_{D}$ has been used and
$M_{D}=\sqrt{\theta}=1$ taken.

The number of effective $\ell$ modes used in the calculations can vary. The
total number of $\ell$ modes considered are 15, 15, 14, 13 for $s=0,1/2,1,2$,
respectively. The number of effective $\ell$ modes giving a non-negligible
contribution in the frequency range $0<\omega\leq 1$ is different depending on
$M$, $n$, and $s$. Typically, $s=0$ and 1/2 have the same number of effective
modes, while $s=1$ has one less and $s=2$ has two less modes. The $s=1$ and 2
cases have transmission coefficients that turn-on at higher frequencies
relative to the $s=0$ and 1/2 cases, i.e. because of $\ell\geq s$, the higher
spins are missing the lower $\ell$ modes. As $n$ increases, the transmission
coefficients become more spread out, and thus less modes will contribute to
the given frequency range. The $n=0$ case has about four more modes than
$n=7$. The lowest masses we consider will have about three less effective
modes than the highest masses we consider. The number of effective modes that
will fit into the frequency range is largely determined by the spacing of the
transmission coefficients in frequency.

Another important characteristic of the transmission coefficients is how
steeply they rise with increasing frequency. In general, the turn-on steepness
is largely independent of spin except for the $\ell=0$ and 1 modes. The more
effective number of modes, the steeper the turn-on. Visually, the turn-on is
most steep for $s=1$ and less step for $s=2$.

The differences in transmission coefficients between NC and ST black holes
depend significantly on their relative horizon radii. Typically a bigger
horizon radius will give transmission coefficients that turn-on lower in
frequency; the difference becoming more pronounced as $\ell$ increases. In
addition, it is observed that at lower masses although the ST black hole
transmission coefficients turn-on sooner, the NC black hole coefficients rise
steeper and become higher before plateauing to unity, especially for $s=1/2$.

###  5.2 Absorption cross sections

The absorption cross section depends on the weighted sum over $\ell$ of
transmission coefficients and inversely as $1/\omega^{2}$. Figure 7 shows
cross sections versus frequency for $s=0,1/2,1,2$. The solid lines are for NC
black holes and dashed lines for ST black holes. Black hole masses
corresponding to the NC black hole maximum temperature have been used; equal
NC and ST black hole masses, $M_{\mathrm{e}M}$ in Table 1). Differences in
cross sections are observed at low frequencies. These differences are most
significant for $s=1/2$ and less pronounced for $s=2$.

Figure 7: Absorption cross sections versus frequency $\omega$ for
$s=0,1/2,1,2$. The solid lines are for non-commutative black holes and dashed
lines for Schwarzschild-Tangherlini black holes. $M_{D}=\sqrt{\theta}=1$ has
been taken. Black hole masses corresponding to the non-commutative black hole
maximum temperature have been used.

Hawking radiation for spin-2 fields in the ST metric was first discussed by
Park Park:2005vw . Direct comparison is not possible since we use a different
effective potential which is taken from Ref. Arbey:2021jif  but originally
comes from Ref. Moulin:2019ekf . The difference in the general form of the
potential appears significant but when substituting the particular ST metric,
the difference is replacing the $-1$ coefficient of the second term in Eq.
(11) by $-(n+1)$. Noteworthy in Ref. Park:2005vw  is the acknowledgement that
the spin-2 potential can become negative – potential well – for some masses
(or radii), number of extra dimensions, and $\ell$ modes. The potential well
can occur in the region $r_{*}\sim 0$. For the ST metric, the condition for
non-negative potential is $n\leq 3$. If $4\leq n\leq 7$, the $\ell=2$ mode
feel a potential well. The depth of the potential well, and height of the
barrier, increase with increasing number of extra dimensions. A trade-off can
occur between barrier suppression and well enhancement. For the ST case, this
causes the $n=5$–7 spin-2 cross sections to rise slightly faster at low-
frequency than the $n=0$–4 cross sections. The same observations are made for
the NC case. However, the effect is small and will not concern use for the
remainder of this paper.

The differences in Figure 7 at low frequency are predominately due to
differences in horizon area. It is enlightening to effectively remove these by
scaling the cross sections by $4\pi r_{\mathrm{h}}^{2}$ as shown in Figure 8.
The cross sections are now in better agreement for $\omega\lesssim 0.25$ but
have almost constant residual differences for $\omega\gtrsim 0.25$. These
differences are due to the universal nature of the cross section – to be
discussed later.

Figure 8: Normalized absorption cross sections versus frequency $\omega$ for
$s=0,1/2,1,2$. The solid lines are for non-commutative black holes and dashed
lines for Schwarzschild-Tangherlini black holes. $M_{D}=\sqrt{\theta}=1$ has
been taken. Black hole masses corresponding to the non-commutative black hole
maximum temperature have been used.

At the mass giving maximum NC black hole temperature, the horizon radius of
the NC and ST black holes are similar. To examine larger differences due to
the transmission coefficients, we take masses near the minimum NC back hole
horizon; values from Table 10 $+10M_{D}$ (for numerical stability). Figure 9
shows significant differences for all but the low $n$ cases.

Figure 9: Absorption cross sections versus frequency $\omega$ for
$s=0,1/2,1,2$. The solid lines are for non-commutative black holes and dashed
lines for Schwarzschild-Tangherlini black holes. $M_{D}=\sqrt{\theta}=1$ has
been taken. Black hole masses corresponding to the non-commutative black hole
minimum mass $+10M_{D}$ have been used.

For ST black holes, the absorption cross section results are the same as Ref.
Harris:2003eg . Although the absorption cross section results for NC black
holes and $s=0$ agree qualitatively with Ref. Nicolini:2011nz , they are
quantitatively different222We do not understand the normalization of Figure 7
in Ref. Nicolini:2011nz . Figure 3 Ref. Nicolini:2011nz  shows values of
$r_{\mathrm{h}}\sim 4.7-5.7$ at the maximum temperature. These values
correspond to horizon areas of about $4\pi r_{\mathrm{h}}^{2}\sim 180-400$
which are in contradiction to what is shown in Figure 7 Ref. Nicolini:2011nz
..

The transmission coefficients for spin 0 and 1/2 turn on immediately for tiny
frequencies, leading to finite absorption cross sections at zero frequency.
Transmission coefficient for spin 1 and 2 are essential zero at $\omega=0$,
leading to an absorption cross section of zero at zero frequency. The usual
oscillations are seen and the number of peaks correspond to the number of
$\ell$ modes. The oscillation are more predominate at low $n$ where the
transmission coefficients rise the steepest. After normalizing by the horizon
radius, the absorption cross sections for NC black holes are higher than the
ST back hole.

For spin 0, the low-frequency limit should correspond to the area of the black
hole for both ST and NC black holes: $\sigma_{0}^{(0)}=4\pi
r_{\mathrm{h}}^{2}$. Numerically, for $\omega=0.001$, we obtain the black hole
area to better than 0.9% for both ST and NC black holes for high and low mass
and for all number of dimensions and spins; except for $n=0$ for which it
agrees to 1.7% for ST black holes of mass $M_{\mathrm{e}T}$. For spin 1/2, the
low-frequency limit is given by Kanti:2002ge

$\sigma_{0}^{(1/2)}=2^{\frac{n-3}{n+1}}4\pi r_{\mathrm{S}}^{2}\,,$ (23)

which we are able to reproduce numerically to better than 0.4%, except for the
$n=0$ case in which we obtain 2% agreement. We also mention that we obtain
zero absorption cross section at $\omega=0.001$ for spin 1 and 2 fields from
NC and ST black holes to 0.01%.

In the high-frequency limit, it has been shown that the absorption cross
section approaches a universal geometrical optics limit of
$\sigma_{\infty}=\pi b_{c}^{2}$, where $b_{c}=r_{c}/\sqrt{h(r_{c})}$ and
$r_{c}$ is given by the solution to $r_{c}h^{\prime}(r_{c})-2h(r_{c})=0$
Decanini:2011xi . Using the ST metric, one obtains

$\sigma_{\infty}=\left(\frac{n+3}{2}\right)^{\frac{2}{n+1}}\frac{n+3}{n+1}\pi
r_{\mathrm{S}}^{2}\,,$ (24)

where $r_{\mathrm{S}}$ is the ST horizon radius. This result was first
obtained a long time ago Emparan:2000rs . In the case of the NC metric, we
have calculated $\sigma_{\infty}$ numerically. For $\omega=1$, we obtain the
optical cross section for ST and NC black holes to better than 3% for high
mass, all number dimensions, and all spins. For low mass, the NC accuracy
remains but the ST $n=7$ and $s=1$ case worsens by up to 5%. Visually, we
already approach the geometrical limit for $\omega\gtrsim 0.25$. Reproducing
these known analytical values is a good test of the numerical validity of our
calculations.

###  5.3 Particle spectra

The particle spectra have an additional dependence on temperature and the
dependence on frequency is only indirectly through the sum of transmission
coefficients and the statistical factor. This time it is not possible to take
the minimum NC black hole mass (zero temperature) as the spectra will vanish
due to the statistical factor. An interesting choice is to take the NC black
hole mass at its maximum temperature. For the ST black hole comparison,
logical choices are to take the same mass or the mass that gives the same
temperature. If the same temperature is taken, the statistical factor in the
particle spectra will be identical and the only difference will be the
transmission coefficient sum part of the formula. First, we consider the case
of equal mass which means the temperature of the ST black hole will be hotter,
and hence lead to significantly more particle flux. Figure 10 shows particle
spectra versus frequency for $s=0,1/2,1$, and 2. The solid lines are for NC
black holes and dashed lines for ST black holes. The number of extra
dimensions increases from 0 to 7 as the curves move from bottom to top. Black
hole masses $M_{\mathrm{e}M}$ corresponding to the NC black hole maximum
temperature have been used, as shown in Table 1.

Figure 10: Particle spectra versus frequency $\omega$ for $s=0,1/2,1,2$. The
solid lines are for non-commutative black holes and dashed lines for
Schwarzschild-Tangherlini black holes. The number of extra dimensions
increases from 0 to 7 as the curves move from bottom to top. Black hole masses
corresponding to the non-commutative black hole maximum temperature have been
used. $M_{D}=\sqrt{\theta}=1$ has been taken.

To remove the temperature dependence, different mass NC and ST black holes are
compared. Figure 11 shows particle spectra versus frequency. Black hole masses
$M_{\mathrm{e}M}$ for NC black holes and $M_{\mathrm{e}T}$ for ST black holes
corresponding to the NC black hole maximum temperature have been used, as
shown in Table 1.

Figure 11: Particle spectra versus frequency $\omega$ for $s=0,1/2,1,2$. The
solid lines are for non-commutative black holes and dashed lines for
Schwarzschild-Tangherlini black holes. The number of extra dimensions
increases from 0 to 7 as the curves move from bottom to top. Black hole masses
corresponding the same temperature as the non-commutative black hole maximum
temperature have been used. $M_{D}=\sqrt{\theta}=1$ has been taken.

###  5.4 Energy spectra

The energy spectra are similar to the particle spectra but include a
multiplicative frequency factor. Figure 12 shows energy spectra versus
frequency for $s=0,1/2,1$ and 2. The solid lines are for NC black holes and
dashed lines for ST black holes. The number of extra dimensions increases from
0 to 7 as the curves move from bottom to top. Black hole masses
$M_{\mathrm{e}M}$ corresponding to the NC black hole maximum temperature have
been used, as shown in Table 1.

Figure 12: Energy spectra versus frequency $\omega$ for $s=0,1/2,1,2$. The
solid lines are for non-commutative black holes and dashed lines for
Schwarzschild-Tangherlini black holes. The number of extra dimensions
increases from 0 to 7 as the curves move from bottom to top. Black hole masses
corresponding to the non-commutative black hole maximum temperature have been
used. $M_{D}=\sqrt{\theta}=1$ has been taken.

To remove the temperature dependence, different mass NC and ST black holes are
compared. Figure 13 shows energy spectra versus frequency. Black hole masses
$M_{\mathrm{e}M}$ for NC black holes and $M_{\mathrm{e}T}$ for ST black holes
corresponding to the NC black hole maximum temperature have been used, as
shown in Table 1.

Figure 13: Energy spectra versus frequency $\omega$ for $s=0,1/2,1,2$. The
solid lines are for non-commutative black holes and dashed lines for
Schwarzschild-Tangherlini black holes. The number of extra dimensions
increases from 0 to 7 as the curves move from bottom to top. Black hole masses
corresponding the same temperature as the non-commutative black hole maximum
temperature have been used. $M_{D}=\sqrt{\theta}=1$ has been taken.

###  5.5 Particle fluxes and total power

To make the comparison quantitative, we integrate the particle spectra and
energy spectra over frequency out to $\omega=1$ to obtain the particle flux
and power, respectively. Table 2 and Table 3 show the NC to ST particle flux
ratios for the cases of equal mass and equal temperature, respectively. We
observe the ratio of spin 0 and 1 fields are not very sensitive to number of
extra dimensions for $n>0$. The biggest change in particle flux ratio with
number of dimensions is for spin 2.

| $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 0.73 | 0.71 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.71
1/2 | 0.65 | 0.69 | 0.70 | 0.71 | 0.71 | 0.72 | 0.72 | 0.72
1 | 0.54 | 0.60 | 0.63 | 0.65 | 0.67 | 0.68 | 0.69 | 0.69
2 | 0.40 | 0.49 | 0.53 | 0.57 | 0.59 | 0.61 | 0.63 | 0.64
Table 2: Ratio of particle flux from non-commutative to Schwarzschild-Tangherlini black holes versus spin $s$ and number of extra dimensions $n$ at the mass of the non-commutative black hole of maximum temperature, $M_{\mathrm{e}M}$.  | $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 0.80 | 0.79 | 0.79 | 0.79 | 0.79 | 0.79 | 0.79 | 0.79
1/2 | 0.71 | 0.77 | 0.79 | 0.80 | 0.80 | 0.81 | 0.81 | 0.81
1 | 0.59 | 0.67 | 0.71 | 0.74 | 0.75 | 0.76 | 0.77 | 0.78
2 | 0.44 | 0.54 | 0.60 | 0.64 | 0.67 | 0.69 | 0.71 | 0.72
Table 3: Ratio of particle flux from non-commutative to Schwarzschild-
Tangherlini black holes versus spin $s$ and number of extra dimensions $n$ at
masses corresponding to the non-commutative black hole maximum temperature:
$M_{\mathrm{e}M}$ for non-commutative and $M_{\mathrm{e}T}$ for Schwarzschild-
Tangherlini black holes.

Table 4 and Table 5 show the NC to ST power ratios for the cases of equal mass
and equal temperature, respectively. The same observations can be made as for
the particle fluxes.

| $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 0.67 | 0.64 | 0.62 | 0.62 | 0.62 | 0.62 | 0.63 | 0.63
1/2 | 0.61 | 0.63 | 0.63 | 0.63 | 0.63 | 0.63 | 0.64 | 0.64
1 | 0.51 | 0.57 | 0.59 | 0.60 | 0.61 | 0.62 | 0.63 | 0.63
2 | 0.39 | 0.46 | 0.50 | 0.53 | 0.55 | 0.57 | 0.59 | 0.60
Table 4: Ratio of power emitted from non-commutative to Schwarzschild-Tangherlini black holes versus spin $s$ and number of extra dimensions $n$ at the mass of the non-commutative black hole of maximum temperature, $M_{\mathrm{e}M}$.  | $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 0.82 | 0.80 | 0.79 | 0.79 | 0.79 | 0.79 | 0.79 | 0.80
1/2 | 0.74 | 0.79 | 0.80 | 0.80 | 0.80 | 0.80 | 0.81 | 0.81
1 | 0.63 | 0.71 | 0.75 | 0.77 | 0.78 | 0.79 | 0.79 | 0.80
2 | 0.47 | 0.58 | 0.64 | 0.68 | 0.70 | 0.72 | 0.74 | 0.75
Table 5: Ratio of power emitted from non-commutative to Schwarzschild-
Tangherlini black holes versus spin $s$ and number of extra dimensions $n$ at
masses corresponding to the non-commutative black hole maximum temperature:
$M_{\mathrm{e}M}$ for non-commutative and $M_{\mathrm{e}T}$ for Schwarzschild-
Tangherlini black holes.

Concentrating on NC geometry inspired black holes, we calculate the particle
flux and total power for each number of extra dimensions and compare it to the
$n=0$ case shown in Table 6 and Table 7, respectively. Direct comparison with
Ref. Harris:2003eg  of the results for ST black holes (not shown) is not
possible since the black hole mass used is not stated. The results are
however, similar.

| $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 1 | 5 | 12 | 25 | 44 | 69 | 101 | 141
1/2 | 1 | 10 | 29 | 61 | 104 | 160 | 230 | 312
1 | 1 | 22 | 93 | 235 | 459 | 772 | 1179 | 1685
2 | 1 | 59 | 406 | 1354 | 3189 | 6149 | 10412 | 16131
Table 6: Particle flux ratios for different number of extra dimensions $n$ relative to $n=0$ versus spin $s$ for non-commutative black holes with the maximum temperature.  | $n$
---|---
$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
0 | 1 | 9 | 32 | 84 | 174 | 316 | 519 | 794
1/2 | 1 | 15 | 60 | 153 | 311 | 551 | 887 | 1334
1 | 1 | 30 | 160 | 469 | 1031 | 1916 | 3187 | 4899
2 | 1 | 82 | 668 | 2512 | 6494 | 13504 | 24379 | 39872
Table 7: Power emission ratios for different number of extra dimensions $n$
relative to $n=0$ versus spin $s$ for non-commutative black holes with the
maximum temperature.

Shown in Table 8 and Table 9 are the case of particle flux and power for each
spin compared to the spin-0 case. Direct comparison with Ref. Harris:2003eg
of the results for ST black holes (not shown) is not possible since the black
hole mass used is not stated. However, the results are the same as Ref.
Harris:2003eg  for most cases, except for a difference of 1% for some $n=7$
spins.

$n$ | $s=0$ | $s=1/2$ | $s=1$ | $s=2$
---|---|---|---|---
0 | 1 | 0.33 | 0.08 | 0.005
1 | 1 | 0.68 | 0.38 | 0.06
2 | 1 | 0.78 | 0.62 | 0.15
3 | 1 | 0.79 | 0.77 | 0.25
4 | 1 | 0.78 | 0.87 | 0.33
5 | 1 | 0.76 | 0.93 | 0.41
6 | 1 | 0.74 | 0.97 | 0.47
7 | 1 | 0.73 | 0.99 | 0.53
Table 8: Particle flux ratios for different spin $s$ relative to $s=0$ versus number of extra dimensions $n$ for non-commutative black holes with the maximum temperature.  $n$ | $s=0$ | $s=1/2$ | $s=1$ | $s=2$
---|---|---|---|---
0 | 1 | 0.50 | 0.17 | 0.01
1 | 1 | 0.86 | 0.61 | 0.14
2 | 1 | 0.92 | 0.86 | 0.30
3 | 1 | 0.91 | 0.97 | 0.44
4 | 1 | 0.89 | 1.03 | 0.55
5 | 1 | 0.87 | 1.05 | 0.63
6 | 1 | 0.85 | 1.06 | 0.69
7 | 1 | 0.84 | 1.07 | 0.74
Table 9: Power emission ratios for different spin $s$ relative to $s=0$ versus
number of extra dimensions $n$ for non-commutative black holes with the
maximum temperature.

##  6 Discussion

Transmission coefficients for spin 0, 1/2, 1, and 2 fields from NC geometry
inspired black holes of extra dimension from 0 to 7 have been calculated. The
NC black hole transmission coefficients are similar to the ST black hole
transmission coefficients when their horizon radius are similar. However,
there are major differences when the black hole masses are similar but the
horizon radius are significantly different. The major difference in
transmission coefficients is that the NC black hole transmission coefficients
turn-on at slightly higher frequency. The differences between NC black hole
and ST black hole transmission coefficients are about the same for all spins.

The absorption cross section of different spin fields from NC geometry
inspired black holes of different number extra dimensions have been
calculated. Significant differences in NC black hole and ST black hole
absorption cross sections occur at low frequencies while the cross sections at
high frequencies approach the geometrical optics limits. For masses near the
minimum NC black hole mass, the differences are more apparent, particularly
for higher dimensions.

The particle and energy spectra on the brane of different spin fields from NC
geometry inspired black holes of different number of extra dimensions have
been calculated. For equal masses, the NC black hole spectra are significantly
lower than for ST black holes, mainly due to the lower temperature. However,
at equal temperature the NC black hole spectra are still significantly lower
than for ST black hole spectra.

When integrating the particle spectra over frequency the particle flux from NC
black holes is significantly less than that from ST black holes. For spin-0
fields, the reduction can be from 0.70-0.80 depending on if the black holes
have equal mass (lower number) or equal temperature (higher number). The
dependence on the number of dimensions is small. For spin-1/2 fields, the
ratio is from 0.65-0.81, also not too dependent on number of extra dimensions.
For spin-1 fields, the ratio is 0.54-0.78. But for spin-2 fields, the
reduction is most significant from 0.40-0.72, increasing with increasing
number of dimensions. The general trends in the power are similar to the
trends in the particle flux.

Considering the NC geometry inspired black hole emission on its own, it is
common to compare the fluxes to the spin-0 case or the $n=0$ case. Increases
in particle flux and power relative to the $n=0$ case are observed for
increasing number of dimensions. The increase is most prominent for spin 2 and
smallest for spin 0. The particle flux and power of spin-1/2 fields relative
to spin-0 fields is less, with $n=0$ being the lowest and $n=3$ the highest
for the particle flux and $n=2$ for power. The particle flux of spin-1 fields
relative to spin-0 fields is less and decreases significantly with decreasing
number of dimensions. The power of spin-1 fields relative to spin-0 fields is
less or greater, depending on the number of dimensions; being significantly
less for $n=0$ and slightly greater for $n=7$. The spin-2 fields particle
fluxes and power are always significantly less than for spin-0 fields, being
1% for $n=0$.

We have presented greybody factors, absorption cross sections, and particle
and energy spectra for all spin fields from higher-dimensional non-commutative
geometry inspired black holes for the first time. The calculations are
numerical and thus valid over the entire frequency range. The emission of
higher spin fields is useful for relating possible experimental observation of
black hole radiation to theory. The reduction in emission due to the greybody
factors, not temperature, that we observe are hopefully model independent.
This work represents another step towards possibly elucidating some aspects of
quantum gravity.

##  Appendix A Experimental constraints

The experimental lower bounds on $M_{D}$ and the maximum energy of the LHC
will restrict the values of $\sqrt{\theta}$ that can be probed by experiments
at the LHC. We do not expect black holes to form for masses much less than
$M_{D}$. This give a lower bound on $M$. We will consider only the hard limits
on the Planck scale set by accelerator experiments D0:2008ayi ; ATLAS:2021kxv
: $M_{D}>11.2$ TeV for $n=2$, $M_{D}>8.5$ TeV for $n=3$, $M_{D}>7.1$ TeV for
$n=4$, $M_{D}>6.4$ TeV for $n=5$, $M_{D}>5.9$ TeV for $n=6$, and $M_{D}>0.8$
TeV for $n=7$. The maximum mass of the black hole is likely to be limited by
the statistics of the maximum parton energies in a proton-proton interaction
but in no case can it be larger than the proton-proton centre-of-mass energy.
Thus, we will only be interested in the case where the minimum black hole mass
is below the LHC current maximum energy of 13.6 TeV and above the experimental
lower bound on the Planck scale.

We obtain a valid range of $\sqrt{\theta}M_{D}$ for each number of extra
dimensions by restricting the minimum black hole mass at the LHC to be in the
range $1<M_{\mathrm{min}}/M_{D}<13.6$ TeV/$M_{D}$, as discussed above. The
results are given in Table 10. We see that $\sqrt{\theta}$ is very restricted
and there is no single value of $\sqrt{\theta}$ that lies in the allowed range
for all number of extra dimensions.

$n$ | $(r_{\mathrm{h}})_{\mathrm{min}}/\sqrt{\theta}$ | $(M_{\mathrm{min}}/M_{D})(\sqrt{\theta}M_{D})^{-(n+1)}$ | $\sqrt{\theta_{\mathrm{min}}}M_{D}$ | $\sqrt{\theta_{\mathrm{max}}}M_{D}$
---|---|---|---|---
0 | 3.02 | 47.9 |  |
1 | 2.68 | 63.2 |  |
2 | 2.51 | 65.2 | 0.248 | 0.265
3 | 2.41 | 58.8 | 0.361 | 0.406
4 | 2.34 | 48.6 | 0.460 | 0.524
5 | 2.29 | 37.9 | 0.546 | 0.619
6 | 2.26 | 28.2 | 0.621 | 0.699
7 | 2.23 | 20.3 | 0.686 | 0.978
Table 10: Values of minimum horizon radius $(r_{\mathrm{h}})_{\mathrm{min}}$
in units of $\sqrt{\theta}$ and minimum mass $M_{\mathrm{min}}$ in units of
$M_{D}(\sqrt{\theta}M_{D})^{n+1}$. The last two columns show the range of
$\sqrt{\theta}$ in units of $1/M_{D}$ that can be probed at the Large Hadron
Collider.

To study the phenomenology of NC inspired black holes at the LHC experiments
one can take $M_{D}$ above the experimental limits and the following values
$\sqrt{\theta}=0.3$ for $n=2$, $\sqrt{\theta}=0.4$ for $n=3$,
$\sqrt{\theta}=0.5$ for $n=4$, $\sqrt{\theta}=0.6$ for $n=5$,
$\sqrt{\theta}=0.7$ for $n=6$, and $\sqrt{\theta}=0.8$ for $n=7$.

###### Acknowledgements.

We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC). Nous remercions le Conseil de recherches en
sciences naturelles et en génie du Canada (CRSNG) de son soutien.

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