# Characterizing Beam Errors for Radio Interferometric Observations of
Reionization

Ainulnabilah Nasirudin,1 David Prelogovic,1 Steven G. Murray,2 Andrei
Mesinger,1 and Gianni Bernardi3,4,5
1Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
2School of Earth and Space Exploration, Arizona State University, Tempe, AZ
3INAF-Istituto di Radioastronomia, via Gobetti 101, 40129 Bologna, Italy
4Department of Physics and Electronics, Rhodes University, PO Box 94,
Grahamstown, 6140, South Africa
5South African Radio Observatory (SARAO), 2 Fir Street, Observatory, Cape
Town, 7925, South Africa
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A limiting systematic effect in 21-cm interferometric experiments is the
chromaticity due to the coupling between the sky and the instrument. This
coupling is sourced by the instrument primary beam; therefore it is important
to know the beam to extremely high precision. Here we demonstrate how known
beam uncertainties can be characterized using databases of beam models. In
this introductory work, we focus on beam errors arising from physically offset
and/or broken antennas within a station. We use the public code OSKAR to
generate an “ideal” SKA beam formed from 256 antennas regularly-spaced in a
35-m circle, as well as a large database of “perturbed” beams sampling
distributions of broken/offset antennas. We decompose the beam errors (“ideal”
minus “perturbed”) using Principal Component Analysis (PCA) and Kernel PCA
(KPCA). Using 20 components, we find that PCA/KPCA can reduce the residual of
the beam in our datasets by $60-90\%$ compared with the assumption of an ideal
beam. Using a simulated observation of the cosmic signal plus foregrounds, we
find that assuming the ideal beam can result in $1\%$ error in the EoR window
and $10\%$ in the wedge of the 2D power spectrum. When PCA/KPCA is used to
characterize the beam uncertainties, the error in the power spectrum shrinks
to below $0.01\%$ in the EoR window and $\leq 1\%$ in the wedge. Our framework
can be used to characterize and then marginalize over uncertainties in the
beam for robust next-generation 21-cm parameter estimation.

###### keywords:

reionization – interferometric – statistical

††pubyear: 2021††pagerange: Characterizing Beam Errors for Radio
Interferometric Observations of Reionization–C.1

##  1 Introduction

Measuring the Epoch of Reionization (EoR) 21-cm signal is one of the key
science goals of current and upcoming low-frequency interferometers such as
the Murchison Widefield Array (MWA) (Tingay et al., 2013; Wayth et al., 2018),
the Hydrogen Epoch of Reionization Experiment (HERA) (DeBoer et al., 2017),
the Low Frequency Array (LOFAR) (van Haarlem et al., 2013), the Giant
Metrewave Radio Telescope (GMRT) (Swarup et al., 1991) and the upcoming Square
Kilometre Array (SKA) (Dewdney et al., 2009; Mellema et al., 2013). To date,
several upper limits of the EoR power spectrum have been published (see e.g.
Barry et al. (2019), Li et al. (2019), Trott et al. (2020), Mertens et al.
(2020), and Abdurashidova et al. (2021)). For an actual detection of the EoR
signal, an unprecedented level of precision is required. This is because
foregrounds and instrumental systematics dominate over the reionization signal
by several orders of magnitude.

Possibly the most complicated and pronounced instrumental systematic comes
from the uncertainty in the model of the primary beam (Jacobs et al., 2017;
Line et al., 2018; Sutinjo et al., 2015). For the wide field-of-view (FoV)
instruments common to 21-cm cosmology, this beam must be well-characterized
over essentially the entire sky, including in side-lobes close to the horizon.
The beam itself is a highly multi-dimensional quantity, changing over
direction, frequency, pointing and polarization. Furthermore, accurate
measurements in the far-field regime are incredibly difficult; individual
elements are far too large to be characterized with anechoic chambers. While
novel techniques such as mapping with pulsars (Newburgh et al., 2014) and
drones (Jacobs et al., 2017) show some promise, it is unclear if they will
achieve the necessary angular resolution and coverage required.

In principle, simulations of the beam via electromagnetic modelling are highly
accurate and very illuminating; however they can only be computed for an ideal
antenna/station and take significant resources. An SKA “station" simulation is
typically performed by combining electromagnetic simulations of isolated
antennas with the station layout. There are several ways that such a
simulation can be inaccurate, including that it does not take into account
full mutual coupling effects, potential in-situ small physical defects of
antennas, potential deviations of precise locations with respect to the
planned layout, and the potential for individual antennas to be offline for a
particular measurement.

For interferometers like the SKA, the primary beam pattern is dictated by the
dipole array structure in each station but there are several sources of
deviations from an ideal primary beam. One important source of deviation is
the electromagnetic coupling among closely packed elements (Fagnoni et al.,
2021; Sutinjo et al., 2020; Bolli et al., 2021). In addition, after the
antennas are physically put on site, many small things can happen to them that
would make them respond slightly differently from the ideal. For example,
although the layout is accurate when the antennas are first deployed, this may
not be true after a certain time has passed, since the physical position of
the antenna can be displaced or even be offline due to multiple reasons e.g
natural events and work maintenance. Indeed, Joseph et al. (2020) find that
the MWA has a maximum of $12.5\%$ broken antennas per station at any given
time during the observation period.

Accurate models of the beam are incredibly important for 21-cm calibration and
power spectrum estimation. Uncertainties in the beam model must be propagated
through to parameter estimation. However, this requires defining a suitably
compact basis in which to represent the uncertainties, since the beam itself
in angular/frequency/time/polarization space is far too complex to propagate.
Thus in this paper, we develop a method for describing the residuals of the
beam model with respect to an ‘ideal’ simulation using PCA-basis sets for
compression. Our general framework can be applied to any beam uncertainty that
can be modeled. As a demonstration here we use simplified, but realistic, beam
perturbations due to two sources of error: antenna displacement within a
station, and the possibility that some fraction of antennas are dead.

The paper is organized as follows. We first describe the beam simulation and
the basis set that we use in §2. In §3, we present the results of our analysis
that motivates our choice for the perturbation model and the impacts on the
reconstructed beam errors. In §4, we study the impact of using the ideal beam
and the reconstructed beam errors on the power spectrum of a mock sky
consisting of the 21-cm signal and foregrounds in an interferometric
framework. Finally, we discuss our findings and conclude the paper in §5.

##  2 Method for characterizing beam uncertainties

In this section, we explain the method used to generate the beam and the basis
set used to characterize the beam errors. We generate electromagnetic
simulations of an idealised beam, as well as thousands of realizations of non-
ideal beams. This training set is used in two PCA-based approaches to identify
the primary modes of variation amongst the perturbed beams, and to determine
the number of such modes that adequately reconstruct any perturbed beam. The
linear coefficients of these modes thus represent a compressed parameter space
which can be marginalized in parameter estimation. We first create a database
of perturbed beam realizations and sample the errors in §2.1. We then
introduce our PCA and kPCA methods for characterizing the beam errors in §2.2.

###  2.1 Database of beam models

We use OSKAR111https://github.com/OxfordSKA/OSKAR to simulate the primary beam
response of a station based on the antenna layout. The station has 256
antennas positioned within a circle of diameter 35m, motivated by the design
of upcoming SKA stations. We define the ideal station as one in which all the
antennas are regularly-spaced within the circle, with a spacing of $\sim 1.8$m
between each antenna222The actual SKA stations are expected to have pseudo-
randomly distributed antennas, but this assumption does not affect the goal of
this paper.. We generate the beam at 150, 170 and 190 MHz for a zenith-
pointing observation. The exact input parameters we use in OSKAR are presented
in Table 3 in Appendix B.

Our perturbation model is based on two scenarios:

  1. 1.

broken (i.e. offline) antennas, in which some antennas are excluded from the
beam synthesis process. The number of broken antennas in each realization is
sampled from a uniform distribution between $N_{\rm broken}=$ 1 – 12
(corresponding to $\lesssim 5\%$ of the total 256 antennas), with their
positions assigned randomly.

  2. 2.

offset antennas, in which all antennas are displaced from the ideal position
following a zero-mean normal distribution with $\sigma_{x}=\sigma_{y}=3\,$cm.

Figure 1:  The ideal antenna layout (top left panel), along with the same
layout but with broken (top right panel), offset (bottom left panel), and both
broken+offset antennas (bottom right panel).

We create three separate datasets, each comprised of 10,000 perturbed beam
realizations: (i) broken+offset, (ii) broken only, and (iii) offset only.
Figure 1 shows the ideal antenna layout, along with three examples of the same
layout but with broken-only, offset-only, and both broken+offset antennas,
respectively. The 10,000 realizations in each dataset are divided into 7,000
training and 3,000 test realizations.

Throughout, we refer to the simulated power beam of the “ideal" station as
$B_{\rm ideal}(\nu,\theta,\phi)$, where $\nu$ is the frequency, $\theta$ is
the zenith angle, and $\phi$ the angle around the zenith pole, and the the
simulated power beam of a particular realization of a perturbed station as
$B_{\rm true}(\nu,\theta,\phi)$. Instead of modeling the perturbed beams
themselves, we model their residuals, i.e:

$\Delta
B(\nu,\theta,\phi)=B_{\textrm{true}}(\nu,\theta,\phi)-B_{\textrm{ideal}}(\nu,\theta,\phi).$
(1)

A particular model of the residuals will be represented as $\widehat{\Delta
B}$, where we use subscripts to denote the basis used (see next subsection).
Note that the residual of residuals is equivalent to the residual in the
modeled beam, i.e. $\Delta B-\widehat{\Delta B}\equiv B_{\rm true}-(B_{\rm
ideal}+\widehat{\Delta B})$. For visualization purposes, in Figure 3 we show
the ideal beam, $B_{\textrm{ideal}}$ (left panel), an example realization of
$B_{\textrm{true}}$ (middle panel) from the broken+offset data, and the
corresponding $\Delta B$ (right panel) at $\nu=$150 MHz.

Figure 2:  The ideal beam, $B_{\textrm{ideal}}(\nu)$ (left panel), an example
realization of $B_{\textrm{true}}(\nu)$ (middle panel) broken+offset data, and
the consequent $\Delta B(\nu)$ (right panel) at $\nu=$150 MHz.

Figure 3:  Mean of the fractional residual, $\mu(X)$ (top panels), and their
standard deviations, $\sigma(X)$ (bottom panels), over the 10,000 realizations
for the broken+offset (left), offset-only (middle) and broken-only (right)
datasets at 150 MHz, where the maximum value for the colour-scale has been
limited to 100$\%$.

To quantify the beam errors we calculate the fractional residual, $X=|\Delta
B|/B_{\rm ideal}$ for each model in the datasets, and then compute the mean
and standard deviation of $X$ over all realizations at each frequency, $\nu$.
We present the sample mean (top panels) and sample standard deviation (bottom
panels) at 150 MHz for the three sets in Figure 3. For the broken+offset (left
panels) datasets, the mainlobe areas are mostly unaffected by the different
antenna configuration as shown by $\sigma(X)$ that is consistent with 0,
although the sidelobes can differ by as much as 100$\%$ near the nulls. In
contrast, the impact of the antenna offsets (middle panels) is very large,
with errors exceeding 100% around the nulls and in the side-lobes.
Interestingly, the residuals in the broken-only (right panels) dataset are
intermediate between the ones in the offset-only and broken+offset. This could
imply that the impact of "breaking" and "offsetting" can partially compensate
for one another. Indeed, one might consider that the positional offsets
partially act to “fill the gaps” left by the broken antennas, resulting in a
net lower error when both effects are at play.

###  2.2 Perturbation Basis Set

We utilize Principal Component Analysis and Kernel Principal Component
Analysis to model the beam residuals, $\Delta B$, arising from the different
antenna configurations. We describe each in turn. These are applied to the
7000 samples in each of our three beam error training sets, and then tested on
the remaining 3000 samples.

####  2.2.1 Principal Component Analysis

The goal of a traditional Principal Component Analysis (PCA) is to reduce the
dimensionality of a dataset by performing a linear change of basis and
determining the extent to which each eigenvector captures the variation within
the dataset. The most significant basis vectors are termed the “principal
components”. Typically, a dataset $Y=(y_{1},y_{2},...,y_{n-1},y_{n})$ is first
standardized with respect to its mean and standard deviation,

$Z=\frac{Y-\mu(Y)}{\sigma(Y)},$ (2)

and its covariance matrix, $C(Z)$, is computed. The eigenvectors, v, and
eigenvalues, $a$, are then calculated following

$C\textbf{v}=a\textbf{v}.$ (3)

Finally, $a$ are arranged in descending order, yielding principal components
in order of significance in which a feature vector can be formed with some
number of features or components, $N$, and the reconstructed beam,
$\widehat{\Delta B}_{N;\rm PCA}=\sum_{i}^{N}a_{i}\textbf{v}_{i}$. The
principal components have the limitation that they are linear transformations
of the input dataset; non-linear transformations that require fewer terms to
adequately describe the data may exist.

####  2.2.2 Kernel Principal Component Analysis

Kernel Principal Component Analysis (KPCA) extends the PCA method via non-
linear transformations of the dataset. The data is first mapped to an
arbitrary higher dimension, often referred to as the feature space, and then
linear PCA is performed on this feature space. The feature space, however,
does not need to be explicitly computed. Instead, it is sufficient to compute
the kernel,

$K(y_{i},y_{j})=\psi(y_{i})^{T}\psi(y_{j}),$ (4)

where $\psi(y_{i})$ is the non-linear transformation from real to feature
space (Schölkopf et al., 1997). One downside of KPCA is that a unique, one-to-
one inverse relation that transforms $\psi(y_{i})$ back to $y_{i}$ does non
exist. However, other methods such as ridge regression (Hoerl & Kennard,
1970a, b) can be used for this purpose, which is what is being used here. For
a simple introduction to KPCA, we refer the reader to Appendix C.

To model our dataset comprised of $\Delta B(\nu)$, we developed
SPax333https://github.com/dprelogo/SPax , an efficient PCA and KPCA code that
is GPU and CPU-optimized. The following kernels are available within SPax for
KPCA:

$\displaystyle K(y_{i},y_{j})$
$\displaystyle=y_{i}^{T}y_{j}\,\textrm{[linear]}$ (5)
$\displaystyle=\textrm{tanh}\left(\kappa
y_{i}^{T}y_{j}\right)\,\textrm{[tanh]}$ (6) $\displaystyle=\left(\kappa
y_{i}^{T}y_{j}+r\right)^{d}\,\textrm{[polynomial]}$ (7)
$\displaystyle=\exp\left(-\kappa[|y_{i}+y_{j}|]^{2}\right)\textrm{[radial
basis]}$ (8) $\displaystyle=y_{i}^{T}y_{j}/(y_{i}^{2}\cdot
y_{j}^{2})\,\textrm{[cosine]}.$ (9)

The kernels, $K$, regularization parameter, and hyper-parameters $\kappa,r,$
and $d$ are flexible; different kernels (and/or hyper-parameter values) can be
used for the transform and inverse transform respectively, in order to improve
the fit to the training dataset. We perform hyper-parameter optimization using
a simple, coarse grid search, selecting the parameter combination that
minimizes the Mean Square Error (MSE) between $\Delta B$ and the reconstructed
residual using $N$ features, $\hat{\Delta B}_{N}$ for all three frequencies.
We note that reconstructing the beam error/residual is obtained by finding the
best fit set of eigenvalues using the given eigenvector basis. Hyper-parameter
calibration is performed separately for all three datasets, and we allow the
inverse kernel to be different from the transform kernel. Our course grid
assumes integer values, with $N=10$, $d=2$, and $r=1$ where applicable to
reduce computation. The best set of parameters that gives the lowest MSE for
each dataset are presented in Table 1, in which the subscript “inv” refers to
the parameter for the inverse transform. We highlight that using more
sophisticated hyper-parameter Bayesian optimization should yield even better
results; we defer this to future work when we apply our method to mock data.

The relative performance of KPCA vs. (linear) PCA can depend strongly on the
processes which generate the data itself. In simplest terms, if the data
itself is a linear combination of effects, then PCA is optimal. However, if
the data is inherently a nonlinear transformation from a more compact basis,
then KPCA may be better in compressing the information content. In practice,
if the data is inherently most compact in a non-linear basis, we may expect
KPCA to outperform (i.e. have a smaller MSE) linear PCA when reconstructing
with a ‘small’ number of components. However, linear PCA is guaranteed to
achieve perfect reconstruction if using $N\rightarrow N_{\rm dim}$ (i.e. an
MSE of close to zero), whereas KPCA is not444KPCA essentially performs
standard linear PCA in a non-linearly transformed space. While it is
guaranteed to minimize the MSE in this space, it is not guaranteed to minimize
MSE in the space of the data. It is difficult to judge whether this is better
or worse without understanding the natural basis of the data., and therefore
there may be a crossover at some $N$.

| Broken + Offset | Broken Only | Offset Only
---|---|---|---
$\kappa$ | 54 | 2 | 79
$\kappa_{\rm inv}$ | 1 | 1 | 23
$K$ | tanh | tanh | tanh
$K_{\rm inv}$ | poly | rbf | poly
Table 1:  The best set of parameters and kernels that gives the lowest MSE for
each dataset based on a simple, coarse Monte-Carlo “grid-search”. The
subscript “inv” refers to the parameter and kernel for the inverse transform.
Figure 4:  The mean (lines) and standard deviation (shaded regions) of
$\mathcal{R}_{N}$ for the broken+offset dataset with varying $N$ (number of
components) using PCA (dash red line) and KPCA (solid blue line).

Figure 5:  $\Delta B$ (left-most panels), $\hat{\Delta B}$ (second and fourth
columns from the left), and $\hat{\Delta B}-\Delta B$ (middle and right-most
columns) from one sample realization of the broken+offset data for all three
frequencies.

Figure 6:  $\Delta B$ (left-most panels), $\hat{\Delta B}$ (second and fourth
columns from the left), and $\hat{\Delta B}-\Delta B$ (middle and right-most
columns) from one sample realization of the broken-only set for all three
frequencies.

##  3 Results: how well is the beam error recovered?

To decide how many components to include in the reconstruction of the beam
across all frequencies, we vary $N$ and evaluate the reconstruction error
$\mathcal{R}_{N}\equiv|\widehat{\Delta B}_{N}-\Delta B|$. We present the mean,
$\langle\mathcal{R}_{N}\rangle$, and standard deviation,
$\sigma(\mathcal{R}_{N})$ (lines and shaded regions, respectively) across the
3,000 realizations of broken+offset test data and all three frequencies for
PCA (red) and KPCA (blue) in Figure 4. Although PCA yields lower values of the
mean reconstruction error $\langle\mathcal{R}_{N}\rangle$ at $N\geq 50$, its
decrease with the number of components is slow. In contrast, with KPCA
$\langle\mathcal{R}_{N}\rangle$ decreases rapidly by $N=20$ and then plateaus
somewhat as $N$ increases. This relative performance is in qualitative
agreement with our expectations from the previous section.

Since we want to model the beam error with the least number of components
possible, below we limit ourselves to the first $N=20$ components for both PCA
and KPCA. Hereafter, $\widehat{\Delta B}_{N=20}$ is referred to simply as
$\widehat{\Delta B}$.

We illustrate the recovery of the beam error in Figures 6 – 7, for a randomly-
chosen sample from each of our test sets. From left to right, we show the
actual beam error ($\Delta B$), the PCA-reconstructed beam error
($\widehat{\Delta B}_{\rm PCA}$), the difference between the PCA-reconstructed
and actual error ($\widehat{\Delta B}_{\rm PCA}-\Delta B$), the KPCA-
reconstructed beam error ($\widehat{\Delta B}_{\rm KPCA}$), and the difference
between the KPCA-reconstructed and actual error ($\widehat{\Delta B}_{\rm
KPCA}-\Delta B$). Rows correspond to our three frequency bins.

With 20 features, both PCA and KPCA can effectively reconstruct $\Delta B$ of
our sample from the broken+offset example shown in Fig. 6, including the
frequency evolution of the features, the size of the mainlobe, and the
magnitude of the perturbation. All these result in average difference of
$\leq|10^{-4}|$. At 190 MHz, however, the model seems to be slightly less
sensitive to structures in the sidelobe region, as is apparent in the middle
and right-most panel on the bottom row, especially with PCA. For reference,
the distribution of the KPCA eigenvalues (which, as expected, follow a
Gaussian distribution), are presented in Figure 11 in the Appendix.

| Broken + Offset | Broken Only | Offset Only
---|---|---|---
$\langle|\Delta B|\rangle$ | 3.7$\times 10^{-4}$ | 8.4$\times 10^{-4}$ | 5.8$\times 10^{-3}$
$\sigma(|\Delta B|)$ | 4.5$\times 10^{-4}$ | 2.3$\times 10^{-3}$ | 8.5$\times 10^{-3}$
$\langle\mathcal{R}_{\textrm{PCA}}\rangle$ | 4.4 $\times 10^{-5}$ | 1.9 $\times 10^{-4}$ | 2.0$\times 10^{-3}$
$\sigma(\mathcal{R}_{\textrm{PCA}})$ | 5.7 $\times 10^{-5}$ | 1.9 $\times 10^{-4}$ | 1.8 $\times 10^{-3}$
$\langle\mathcal{R}_{\textrm{ KPCA}}\rangle$ | 3.9$\times 10^{-5}$ | 1.7$\times 10^{-4}$ | 2.2$\times 10^{-3}$
$\sigma(\mathcal{R}_{\textrm{KPCA}})$ | 4.9$\times 10^{-5}$ | 3.9$\times 10^{-4}$ | 2.5$\times 10^{-3}$
Table 2:  The MSE and standard deviation of MSE of $\Delta B$ and
|$\hat{\Delta B}-\Delta B$| with PCA and KPCA for the three test datasets.

For the broken-only sample in Fig. 6, both PCA and KPCA are able to capture
the overall details of $\Delta B$, including the evolution of the features and
the size of the mainlobe, as shown in the second column from the left and
fourth column from left in Figure 6. However, the error in the reconstruction
can be up to an order of magnitude higher with PCA and there are more small
scale features compared to the reconstruction with KPCA.

For the offset-only example shown in Figure 7, both PCA and KPCA perform worse
than seen in the previous two examples. The reconstructions (second and fourth
columns from left) somewhat resemble the large-scale structures of $\Delta B$,
but instead of having two large “half-ring” structures in the sidelobe, both
PCA and KPCA model them as multiple radial features. Moreover, the
reconstructed error can be up to two orders of magnitude higher than in the
previous examples, as is evident in the third and right-most columns.

To summarise, we present the mean and standard deviation of $|\Delta B|$ and
$\mathcal{R}_{20}$ over the 3,000 test realizations and all three frequencies
for the three test datasets with PCA and KPCA in Table 2. Using PCA/KPCA,
there is up to a factor of 10 reduction in beam error compared with the
assumption of an ideal beam.

Figure 7:  $\Delta B$ (left-most panels), $\hat{\Delta B}$ (second and fourth
columns from the left), and $\hat{\Delta B}-\Delta B$ (middle and right-most
columns) from one sample realization of the offset-only set for all three
frequencies.

##  4 Impact of Beam Reconstruction on the Power Spectrum

Although we have established that PCA and KPCA do a good job in capturing the
beam error from our datasets, the reconstruction is not perfect. Hence in this
section, we investigate the impact of these residual errors on the recovery of
the power spectrum, using a realistic sky composed of the EoR signal and
point-source foregrounds.

###  4.1 Foreground Model

Following Nasirudin et al. (2020), we simulate extra-galactic point-source
foregrounds with a flux-density source count distribution with the power-law
relation

$\frac{dN}{dS}(S,\nu)=\alpha
S_{\nu}^{-\beta}\left(\frac{\nu}{\nu_{0}}\right)^{-\gamma\beta}[{\rm
Jy}^{-1}{\rm sr}^{-1}],$ (10)

where $dN/dS$ is the source spatial density per unit flux density, $S_{\nu}$
is the flux at a specific frequency $\nu$, $\beta$ is the slope of the source-
count function, and $\gamma$ is the mean spectral-index of point sources.
Based on an observational result from Intema et al. (2011), we set $\alpha$ =
4100 Jy-1 sr-1, $\beta$ = 1.59, and $\gamma=0.8$ at $\nu_{0}$ = 150 MHz .
Having drawn source fluxes from the above distribution, we situate them
uniformly randomly across the sky. We sample the point sources between $S_{\rm
min}=50$mJy and $S_{\rm max}=50\mu$ Jy555Choudhuri et al. (2021) found that
the brightest sources to be particularly important in the presence of beam
variations/non-redundancy, but because we assume that brighter sources have
been perfectly peeled from the observation, hence we only model those below
the peeling threshold. at 150 MHz. The observation consists of 128 linearly-
spaced frequency channels between 150 to 165 MHz.

###  4.2 Reionization Model

The differential brightness temperature, $\delta T_{B}$, during the EoR can be
approximated as

$\displaystyle\delta T_{b}(z)\approx$ $\displaystyle 27x_{\rm
HI}(1+\delta_{\rm
nl})\left(\frac{H(z)}{\mathrm{d}v/\mathrm{d}r+H(z)}\right)\left(1-\frac{T_{\gamma}}{T_{s}}\right)$
$\displaystyle\times\left(\frac{1+z}{10}\frac{0.15}{\Omega_{m}h^{2}}\right)^{\frac{1}{2}}\left(\frac{\Omega_{b}h^{2}}{0.023}\right)\
[{\rm mK}],$ (11)

where $x_{\mathrm{HI}}$ is the neutral fraction, $\delta_{\mathrm{nl}}$ is the
evolved Eulerian overdensity, $H$ is the evolving Hubble constant, d$v$/d$r$
is the gradient of the line-of-sight velocity component, $T_{\gamma}$ is the
temperature of the CMB, $T_{\mathrm{s}}$ is the spin temperature of neutral
hydrogen (HI), $z$ is the redshift, $\Omega_{m}$ is the dimensionless matter
density parameter, $\Omega_{b}$ is the dimensionless baryonic density
parameter and $h$ is the normalized Hubble constant (Furlanetto et al., 2006).

We use the efficient semi-numerical EoR modelling tool, 21cmFASTv3 (Mesinger
et al., 2011; Park et al., 2019; Murray et al., 2020), to generate the
lightcone of $\delta T_{b}$ during the EoR. For a detailed description of the
code and astrophysical model, we refer readers to Mesinger et al. (2011), Park
et al. (2019) and Murray et al. (2020). In our research, we use the default
parameter values of 21cmFASTv3, which are shown to reproduce current high-$z$
observations (Park et al., 2019) and simulate the lightcone of a $512$ Mpc$/h$
box. This choice of parameters corresponds to a neutral fraction of 0.5 at
$z\sim 6.5$. Because the lightcone covers only $\sim 3.3^{\circ}$ at 150 MHz,
we tile it across the 20∘ mock sky and coarsen the grids to match with the
resolution of the beam.

###  4.3 Interferometric Framework

At wavelength, $\lambda$, the baseline displacement, u $=(u,v)$, is defined as
$\mathbf{u}=\mathbf{x}/\lambda$, where x is the physical displacement between
the stations in meters. The sky coordinate, $\mathbf{l}$, is defined as
$\mathbf{l}=(l,m)=(\sin\theta\cos\phi,\sin\theta\sin\phi$).

Figure 8:  From left to right, the PS of cosmic signal, foregrounds, and both
cosmic signal and foregrounds respectively, for a mock sky convolved with
$B_{\rm true}$. The black dash line shows the extent of the wedge which is
calculated following Equation 14.

Figure 9:  From left to right, the fractional difference between the PS of
both cosmic signal and foregrounds convolved with $B_{\rm ideal}$,
$\hat{B}_{\rm PCA}$, and $\hat{B}_{\rm KPCA}$ with respect to the PS of the
same sky convolved with $B_{\rm true}$.

Using the flat-sky approximation, the visibility at frequency $\nu$,
$V(\textbf{u}_{j},\nu)$, for each baseline $j$ is defined as

$V(\textbf{u}_{j},\nu)=\int S(\textbf{l},\nu)B(\textbf{l},\nu)\exp(-2\pi
i\textbf{u}_{j}\cdot\textbf{l})d\textbf{l}\ \ \ \mathrm{[Jy]},$ (12)

where $S(\textbf{l},\nu)$ and $B(\textbf{l},\nu)$ are the flux density of each
point-source and the beam attenuation at l and $\nu$. The observed
interferometric visibility is identical to the Fourier transform of the
product of signal and the beam model under the flat-sky approximation. Here,
we assume all stations have the exact same layout hence the same beam.

For computation purposes, we Fast Fourier Transform over the 2-D image to a
regular-spaced 2D grid $\mathbf{u}_{k}$, and then interpolate
$V(\textbf{u}_{k},\nu)$ from the regular 2D grid to the baselines
$\textbf{u}_{j}$. We then apply a Blackman-Harris taper $H(\nu)$ over the
frequency axis, and calculate the delay transform (i.e. Fourier transform of
un-gridded visibilities along the frequency axis),

$V(\textbf{u}_{j},\tau)=\int V(\textbf{u}_{j},\nu)H(\nu)\exp(-2\pi
i\tau\cdot\nu)d\nu\ \ \ [{\textrm{ Jy Hz}}].$ (13)

The delay power spectrum is then calculated by cylindrically averaging the
power of the visibilities within radial bin r $=\sqrt{u^{2}+v^{2}}$, which is
proportional to the angular mode, $k_{\perp}$. We approximate the delay power
spectrum as the power spectrum, in which $\tau$ is proportional to the line-
of-sight mode, $k_{\parallel}$. The conversion of the power spectrum, r and
$\tau$ to cosmological units are outlined in Appendix A. For reference, we
calculate the wedge region given by

$k_{\parallel}\leq k_{\perp}\frac{\textrm{sin}(\theta_{\rm
FoV})E(z)\int_{0}^{z}dz^{\prime}/E(z^{\prime})}{(1+z)}\ \ \ [h\
\mathrm{Mpc}^{-1}],$ (14)

where $\theta_{\rm FoV}$ is the angular radius of the FoV (Thyagarajan et al.,
2013; Dillon et al., 2014).

###  4.4 Impact of Different Beams on the Power Spectrum

To understand the impact of beam errors on the recovery of the cylindrical
power spectrum (hereafter PS), we convolve our sky described in §4.1 and 4.2
with the sample realization of $B_{\rm true}$ shown in Figure 6. Following the
steps outlined in §4.3, we then simulate the effects of interferometric
observation and calculate the respective PS. From left to right, the panels in
Figure 9 show the PS of a sky consisting of the cosmic signal, foregrounds,
and both cosmic signal plus foregrounds, respectively, that has been convolved
with $B_{\rm true}$. In the foreground PS (middle panel), the well known key
features shown in e.g. Dillon et al. (2014) and Barry et al. (2016) are
clearly visible, mainly the foreground dominated region in dark blue where
$k_{\parallel}\leq 0.11\,h$ Mpc-1, the yellow-blue region of the wedge at
$k_{\perp}\geq 0.1\,h$ Mpc-1, and the mostly red EoR window.

Finally in Figure 9 we present the fractional error in the PS of the beam-
convolved total signal (EoR + FG) with respect to the "true" PS from the third
panel of Figure 9, i.e. $P(B*[{\rm FG+EoR}])/P(B_{\rm true}*[{\rm FG+EoR}])$
for $B=[B_{\rm ideal},\widehat{B}_{\rm PCA},\widehat{B}_{\rm KPCA}]$ (left to
right panels respectively). From the left panel we see that not accounting for
beam errors mis-estimates the power spectrum throughout $k$-space, with errors
peaking at $\sim 10$% in the wedge region. Instead, modeling $\widehat{\Delta
B}$ using either PCA or KPCA reduces the error in the recovered power spectrum
by over a factor of a hundred in the EoR window and a factor of ten in the
wedge (compare middle and right to the left panel).

Because the visibilities in the wedge are highly correlated, any deficit or
surplus of beam attenuation with respect to $B_{\rm true}$ is reflected in the
entire wedge region. Indeed, as expected, beam errors affect foregrounds more
than the cosmic signal, even in the PS space.

Finally, we stress that this exercise is highly idealized, providing only a
mimimum estimate of the PS recovery error. In practice, we will not be able to
fit the PCA and KPCA coefficients to the true beam directly, as we have done
here. Instead, eigenvalues would need to be co-varied when performing
calibration and inference. We defer this analysis to future work.

##  5 Conclusions

Some of the most important systematics in radio interferometry arise from
imperfect knowledge of the telescope beam. In this work we demonstrate an
empirical approach to characterizing known sources of beam errors. Focusing on
offline and offset antennas for an SKA-like beam, we generate thousands of
realizations of beam errors. We use these realizations to define a beam error
basis using PCA and KPCA.

We demonstrate that both PCA and KPCA perform well in recovering beam errors
from offline and offset antennae. Compared with assuming an "ideal" beam,
using the top 20 components in either basis can reduce the MSE by
$\sim$tens–100% over our test datasets.

We demonstrate how this beam error characterization translates to improved
power spectrum recovery. We generate a mock sky comprised of point source
foregrounds and the cosmic signal, and recover the cylindrical power spectra
assuming different beam models. For a random realization of beam error, we
find that assuming the "ideal" beam results in PS errors that peak at $\sim
10$% around the wedge region. Instead if either PCA or KPCA is used to
characterize the true beam with 20 components, the PS error is reduced by a
factor of $\sim$10–100 throughout $k$-space.

We stress that we did not include additional errors from, e.g. calibration, in
this work. We expect that fractional errors in sky-based calibration will be
much more sensitive to errors in the assumed beam model, and these are further
squared when propagated to power spectrum space. Depending on the spectral
structure of these calibration errors, inaccuracies as small as $10^{-5}$ can
be crippling to a power spectrum estimation (Barry et al., 2016). Therefore,
we expect improved beam characterization to be even more important when
calibration is also included; we defer this to future work.

Our general framework of using an empirical basis to characterize systematics
should prove useful for an end-to-end inference pipeline for 21-cm
interferometry. The principal eigenvectors from PCA and KPCA can provide an
optimal basis for systematics, with the corresponding eigenvalues being co-
varied together with cosmological parameters when performing Bayesian
inference. We will demonstrate this in a follow-up work.

## Acknowledgements

We thank P. Bull for helpful comments on a draft version of this work. This
work was supported by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No
638809 – AIDA – PI: Mesinger). The results presented here reflect the authors’
views; the ERC is not responsible for their use. We gratefully acknowledge
computational resources of the Center for High Performance Computing (CHPC) at
Scuola Normale Superiore (SNS).

## Data Availability

The data from this study will be shared on reasonable request to the
corresponding author.

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##  Appendix A Unit Conversion

The conversion of $\delta T_{B}$ to $S(\nu)$ (and vice versa) follows the
Rayleigh-Jeans law,

$S(\nu)=\left(\frac{2k_{B}\nu^{2}\delta T_{B}}{c^{2}}\right)\times 10^{26}\ \
\ [{\rm Jy/sr}],$ (15)

where $k_{B}$ is the Boltzmann constant.

Under the assumption that $\tau$ is equivalent to the Fourier counterpart of
the line-of-sight mode, $\eta$, both $k_{\perp}$ and $k_{\parallel}$ are
converted from r and $\tau$ in Fourier dimensions following

$k_{\perp}=\frac{2\pi|\textbf{r}|}{D_{M}(z)}\ \ \ [{\rm Mpc^{-1}h}],$ (16)

and

$k_{\parallel}=\frac{2\pi H_{0}f_{21}E(z)}{c(1+z)^{2}}\tau\ \ \ [{\rm
Mpc^{-1}h}]$ (17)

from Morales & Wyithe (2010). Here, $z$ is the observation redshift,
$D_{M}(z)$ is the transverse comoving distance, $H_{0}$ is the Hubble
constant, $f_{21}$ is the rest frequency of the 21 cm hydrogen hyperfine
transition and $E(z)$ is defined as

$E(z)=\sqrt[]{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}},$ (18)

where $\Omega_{\Lambda}$, and $\Omega_{k}$ are the dimensionless density
parameters for dark energy and the curvature of space.

##  Appendix B Extra Materials

In this section, we present some extra materials concerning the research for
interested readers. The parameter input used for OSKAR is presented in Table 3
and the SKA-like station layout is shown in Figure 10. In addition, Figure 11
shows the Probability Density Functions (PDFs) of the first 20 components in
the higher dimension space from the KPCA, ordered from largest (top left
panel) to smallest (bottom right panel) variance.

Parameter | Values
---|---
FoV (∘) | 20
RA of Observation (∘) | 0
Dec of Observation (∘) | -27
Latitude of Telescope (∘) | -27
Longitude of Telescope (∘) | 117
Observation Time (UTC) | 17:00:00
Observation Date | 1 August 2020
Frequencies (MHz) | [150, 170, 190]
Table 3:  The OSKAR parameter input used in this research. Figure 10:  The
SKA-like station layout. Figure 11:  The Probability Density Functions (PDFs)
of the first 20 components in the higher dimension space from the KPCA,
ordered from largest (top left panel) to smallest (bottom right panel)
variance.

##  Appendix C Kernel PCA example

The purpose of this simple example is to qualitatively describe the main
ingredients of the KPCA algorithm - in particular: data space - $y$, feature
space - $\psi(y)$, and two kernels ($\kappa$ and $\widetilde{\kappa}$)
defining mappings from one to another. Moreover, we would like to show very
different roles the two kernels have in the process.

In Figure 12 we show a bi-modal distribution (with the two modes labeled ’1’
and ’2’). The horizontal axis represents data space. In this simple example,
we would like to use KPCA to accentuate the bi-modality of this distribution.

Figure 12: Initial distribution on which we would like to run a KPCA
algorithm.

We first pull samples from the distribution and use the kernel
$\kappa(y_{i},y_{j})=\psi(y_{i})\cdot\psi(y_{j})$ as a part of the KPCA
algorithm. Here $\psi(y)$ is in feature space, which is not directly
accessible and can be infinite dimensional. Performing KPCA in this space and
selecting the first $N$ components amounts to selecting basis vectors in the
feature space following the largest variance of the samples. After fitting the
data, this subset of a feature space $\boldsymbol{\psi}_{N}$ is accessible and
the mapping $\boldsymbol{\psi}_{N}(y)$ is known.

###  C.1 Mapping to and from feature space

For the example above, one can show the first component of the feature space
$\psi_{1}(y)$ (see Figure 13). Samples from the modes in the distribution are
distinguished by different colors, for better visualization. Starting from the
distribution on the top, we pass it through the (learned) transformation shown
in the middle, getting the distribution on the right. As expected, the two
modes (blue vs orange) are much better separated in the first component of the
feature space, $\psi_{1}$, than they were in data space, $y$.

Figure 13: Mapping from data space to the KPCA feature space
$\boldsymbol{\psi}_{N}$. Starting from the top distribution in data space,
points are transformed through the learned function in the middle into the
first component of the feature space, shown on the right. Samples from two
underlying distributions are denoted with different colors for better
visualization.

Contrary to linear PCA however, an exact inverse transform to return from
$\psi_{1}$ back to data space generally does not exist. Therefore, we define
the inverse transform using kernel ridge regression. In linear ridge
regression from $\boldsymbol{\psi}_{N}$ back to $y$, we would minimize the
mean square error over the data:

$\sum_{i}\left(y_{i}-\mathbf{w}^{T}\boldsymbol{\psi}_{N}\right)^{2}+\frac{\lambda}{2}||\mathbf{w}||^{2}\,,$
(19)

where $\mathbf{w}$ are the weights and second term is a standard $l_{2}$
regularization. However, as the inverse function is highly non-linear, we
firstly transform $\boldsymbol{\psi}_{N}$ into another (possibly infinite)
feature space $\phi$, and learn the transformation $\widetilde{\mathbf{w}}$
back to $y$. The minimization is then:

$\sum_{i}\left(y_{i}-\widetilde{\mathbf{w}}^{T}\phi\left(\boldsymbol{\psi}_{N}\right)\right)^{2}+\frac{\lambda}{2}||\widetilde{\mathbf{w}}||^{2}\,.$
(20)

One can prove that the feature space $\phi$ does not have to be accessed and
is only implicitly defined by the kernel
$\widetilde{\kappa}\left((\boldsymbol{\psi}_{N})_{i},(\boldsymbol{\psi}_{N})_{j}\right)=\phi\left((\boldsymbol{\psi}_{N})_{i}\right)\cdot\phi\left((\boldsymbol{\psi}_{N})_{j}\right)$.

In Figure 14, we show the results of such procedure. We can see that the
learned mapping is indeed non-linear and the initial distribution is well
preserved.

Figure 14: Mapping from the feature space
$\boldsymbol{\psi}_{N}=(\psi_{1},\psi_{2},...)$ back to the data space, $y$.
Kernel ridge regression feature space $\phi$ is never accessed and only
defined by the kernel $\widetilde{\kappa}$. Samples from the two underlying
modes are separated in color for better visualization.


††institutetext: 1 Institutionen för Fysik och Astronomi, Box 803, SE-751 08
Uppsala, Sweden
2 Instituut voor Theoretische Fysica, K.U.Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium

# Over-extremal brane shells from string theory?

Ulf Danielsson1, Vincent Van Hemelryck2, Thomas Van Riet2
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We demonstrate that, if the usual phenomenological compactifications of IIB
string theory with warped throats and anti-branes make sense, there must exist
spherical brane shells in 4d that are overcharged. They correspond to
classical over-extremal objects but without the usual naked singularities. The
objects are made from D3-particles that puff into spherical 5-branes that
stabilise at finite radii in 4d and whose inside corresponds to the
supersymmetric AdS vacuum. One can think of these shells as stabilised Brown-
Teitelboim bubbles. We find that these objects can be significantly larger
than the string scale depending on the details of the warped compactification.

UUITP-26/22

##  1 Introduction

String theory has shown to provide the rich structure needed to describe the
quantum nature of black holes, at least for black holes that are
supersymmetric or are close to being so. For charged non-rotating black holes
this means they are necessarily extremal. The main success of string theory
has been in counting the microstates Sen:1995in ; Strominger:1996sh , but a
clear understanding of the spacetime structure of the inside of a black hole
is yet to be found, especially for black holes that are more realistic in an
astrophysical sense and hence are not extremal. Many bottom-up ideas exist for
what could replace the interior of a black hole, but for actual top-down
constructions it is less clear. Aside the fuzzball proposal Mathur:2005zp ;
Bena:2022rna  there is another appealing possibility as first described in
Danielsson:2017riq : black holes could be spherical shells whose inside
correspond to a stable supersymmetric AdS space whereas the outside is a meta-
stable vacuum. When the meta-stable vacuum decays through bubble nucleation,
the bubbles tend to expand and overtake the universe, unless there is a
mechanism that stabilises them so they remain at finite radius. When they do
so, the objects behave in many ways as sensible black holes to an outside
observer Danielsson:2017riq ; Danielsson:2017pvl ; Danielsson:2021ruf ;
Danielsson:2021ykm . Although this scenario is highly “string-inspired”, an
actual embedding within string theory is lacking. In particular one needs to
find the mechanism that stabilises the radius of the bubble. It is the aim of
this paper to provide such an embedding, but we will see that our examples are
actually describing over-extremal objects. This by itself is very intriguing
since it is highly unclear whether over-extremal objects, like the charged
particles of the Standard Model, can be macroscopic in size as well. Although
our objects are over-extremal and so not of astrophysical interest, if the
working assumption in this paper is correct (ie phenomenological IIB
compactifications) then our results show that at least the principle of
stabilising bubbles with SUSY AdS vacua inside does occur within string
theory. Whether the size of our over-extremal balloon like objects can truly
become large (beyond particle physics length scales or the de Broglie
wavelength) seems to depend on how far flux numbers, warping effects and
spacing between vacuum energies can be tuned. The possibility of this tuning
is a crucial debate within the current Swampland paradigm Vafa:2005ui ;
Brennan:2017rbf ; Palti:2019pca  that has been discussed for instance in
Bena:2018fqc ; Gao:2020xqh ; Carta:2021lqg ; Plauschinn:2021hkp ;
Marchesano:2021gyv ; Bena:2020xrh ; Grana:2022dfw ; Broeckel:2021uty ;
Bastian:2021hpc ; Demirtas:2021nlu ; Demirtas:2021ote ; Crino:2022zjk .

It is well-known in string theory that brane bubbles can be stabilised at
finite radii, see e.g. the mechanism for brane-flux decay Myers:1999ps ;
Kachru:2002gs ; Gautason:2015tla  but in all such examples the bubbles form
inside the extra dimensions. To our knowledge we provide the first mechanism
where this happens inside the external dimensions. The ideas for our
construction originated from discussions in reference Montero:2021otb .

The rest of this paper is organised as follows. In section 2 we explain the
set-up of wrapped D3-particles in the context of uplifting scenario’s
involving space-filling anti-D3-branes and we explain how to compute the
potential energy as a function of the radius of the puffed-up D3-particles
into spherical 5-branes. In section 3 we then investigate in detail the
stabilised brane shells that emerge: their size, mass and charge. We end with
a discussion in section 4.

##  2 Puffed up and wrapped D3-branes

One way to construct massive point-like objects from 10-dimensional string
theory is to wrap D$p$ branes over $p$-dimensional cycles inside the extra
dimensions. If this can be done in a stable manner, one finds a particle in
4d. If we wrap many branes the particle can be more massive and even become
black-hole like.

Let us take the example of IIB string theory and wrap D3-branes over 3-cycles.
The resulting particles will be charged under a U(1) gauge field $A_{\mu}$
that comes from reducing the RR gauge potential $C_{4}$ over that 3-cycle with
volume tensor $\epsilon_{3}$:

$C_{4}=A_{\mu}dx^{\mu}\wedge\epsilon_{3}+\ldots$ (1)

where $x^{\mu}$ are 4d coordinates. Intuitively one can think of this charge
as the reason the particle is stable since it will be BPS once the cycles is
supersymmetrically calibrated. In other words, the charge guarantees the
stability by means of the BPS bound. However, for compactifications with less
than $\mathcal{N}=2$ supersymmetry, the situation is more involved. For
example, let us consider IIB reduced over Calabi-Yau orientifolds, the
starting point for the majority of string-phenomenological moduli
stabilisation scenarios. In this case the orientifold projections make it
impossible for D3-particle states to be stable EnriquezRojo:2020hzi ,
essentially because the gauge fields are projected out by the orientifolding.

In what follows we construct massive D3-states on orientifold-odd cycles, and
give them a charge by relying on the Freed-Witten effect and the presence of
anti-branes in uplifting scenarios, inspired from the discussions in
Montero:2021otb . In simple words: adding anti-D3-branes to uplift the KKLT
Kachru:2003aw  or the LVS scenario Balasubramanian:2005zx  AdS vacuum111See
Danielsson:2018ztv ; Gao:2020xqh ; Lust:2022lfc ; Junghans:2022exo ;
DallAgata:2022abm ; Gao:2022uop  for some recent works discussing the very
existence of these vacua. introduces exactly the U(1) gauge field to charge
the objects, which we will see is really necessary for our purposes.

The detailed set-up is the following: consider D3-branes wrapping the A-cycle
of the KS throat Klebanov:2000hb  embedded in some compact Calabi-Yau
geometry. Since there are $M$ units of $F_{3}$ piercing that cycle as well,
there are $M$ fundamental strings leaving from every D3-brane that attach to
the anti-D3-branes, by means of the Freed-Witten effect Freed:1999vc .

In what follows, we show that the corresponding D3-particles polarise into a
spherical NS5-brane shell. We first discuss this in the usual way with probe
actions and then use GR techniques to compute backreaction effects. We will
find that qualitatively there is not much difference, but the comparison shows
nicely how the more bottom-up ideas from Danielsson:2017riq , using junctions
conditions, are indeed realised in string theory.

###  2.1 Brane polarisation in 4d using probes

There are two ways to investigate whether D$p$-branes puff into spherical
branes D$(p+2)$ Myers:1999ps . Either using the non-Abelian worldvolume theory
on the D$p$-brane or using the Abelian D$(p+2)$ action. We will follow the
second approach. This means that we consider a NS5-brane of spherical shape in
4d that, when it pinches to a point, describes the worldvolume action of a
D3-particle charged under the space-filling anti-D3 worldvolume gauge field.
Then we compute the energy in terms of its radius and check whether it has a
meta-stable minimum at finite radius. There will be three contributions to the
energy of the brane shell: the DBI energy, the Coulomb energy and the pressure
energy. The latter is related to the fact that the inside of the bubble is
SUSY AdS and the outside is essentially Minkowski (de Sitter with small vacuum
energy). Let us compute these 3 separate ingredients:

The DBI energy: The DBI action of the NS5-brane, in string units, is

$S_{\text{NS5}}=-\frac{\mu_{5}}{g_{s}^{2}}\int\mathrm{d}x^{6}\sqrt{-\det(g_{MN}+2\pi
g_{s}\mathcal{F}_{MN})}\,,$ (2)

with $2\pi\mathcal{F}_{2}=2\pi F_{2}-C_{2}$. The WZ part of the NS5 action
will be specified later. The DBI action of a D3-brane instead goes like

$S_{\text{D3}}=-\frac{\mu_{3}}{g_{s}}\int\mathrm{d}x^{4}\sqrt{-\det(g_{ab}+\ldots)}$
(3)

where, in string units, $4\pi^{2}\mu_{5}=\mu_{3}$. In order to compute the
potential energy as a function of the radius we need to know the induced
metric, the worldvolume flux $F_{2}$ and the RR potential $C_{2}$. If we call
the radius of the NS5-brane $R$ then the string frame metric induced on the
worldvolume is:

$\displaystyle\frac{\mathrm{d}s_{6}^{2}}{\ell^{2}_{s}}=e^{2A_{0}}\left(-\mathrm{d}T^{2}+R^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2})\right)+b_{0}^{2}g_{s}M\left(\mathrm{d}\psi^{2}+\sin^{2}\psi(\mathrm{d}\chi^{2}+\sin^{2}\chi\mathrm{d}\xi^{2})\right)\,,$
(4)

where we used a standard expression for the metric at the tip of the Klebanov-
Strassler throat Kachru:2002gs  (using that the B-cycle pinches. The
warpfactor at the tip is denoted $e^{2A_{0}}$, $M$ is the $F_{3}$-flux quantum
through the A-cycle, $g_{s}$ is the string coupling and $b_{0}^{2}\approx
0.93$ an order one numerical constant. The coordinates $t,\theta$ and $\phi$
are in the external space and the coordinates $\psi,\chi,\zeta$ parametrise
the local A-cycle of the internal compact dimensions. From here onward we work
in string units where $\ell_{s}=1$.

In order for the spherical NS5 to correspond to $N_{\text{D3}}$ D3-branes when
$R=0$, the NS5 world volume flux should be:

$F_{2}=\frac{N_{\text{D3}}}{2}\text{vol}(S^{2})=\frac{N_{\text{D3}}}{2}\sin\theta\mathrm{d}\theta\wedge\mathrm{d}\phi\,,\qquad\frac{1}{2\pi}\int_{S^{2}}F_{2}=N_{\text{D3}}\,.$
(5)

To find the potential we integrate the DBI term over all spatial coordinates
and use that
$S_{\text{DBI}}(R,\dot{R}=0)=-\int\mathrm{d}T\>V_{\text{DBI}}(R)$. This will
be interpreted as the gravitational term responsible for the shell wanting to
collapse under its own weight. With the background expression of $C_{2}$ near
the tip,

$C_{2}=M\left(\psi-\frac{1}{2}\sin(2\psi)\right)\sin\chi\mathrm{d}\chi\wedge\mathrm{d}\xi\,,\qquad\int_{S^{3}}\mathrm{d}C_{2}=(2\pi)^{2}M\,,$
(6)

the DBI potential becomes

$V_{\text{DBI}}(R)=\mu_{3}\left(4\pi\tau e^{A_{0}}\sqrt{(\pi
g_{s}N_{\text{D3}})^{2}+(e^{A_{0}}R)^{4}}\right).$ (7)

The tension-like parameter $\tau$ is defined as

$\displaystyle\tau=2\frac{(g_{s}M)^{\frac{3}{2}}}{g_{s}^{2}}\mathcal{C}\,,$
(8)

with $\mathcal{C}$ the following numerical constant

$\mathcal{C}=b_{0}\int\mathrm{d}\psi\sqrt{4b_{0}^{4}\sin^{4}\psi+(-2\psi+\sin
2\psi)^{2}}=10.3591\,.$ (9)

The Coulomb energy: The shell must be charged under the gauge theory coming
from the anti-D3s due to the Freed-Witten effect. We implement this by adding
the following coupling.

$S_{\text{Coulomb}}=\alpha\mu_{5}\int_{\Sigma_{\text{NS5}}}F_{3}\wedge\tilde{A}_{1}\wedge
F_{2},$ (10)

where $\alpha$ is a constant and $\tilde{A}_{1}$ is the worldvolume gauge
field of the anti-D3. These terms we added by hand since there is no
supersymmetric brane action anymore to guide us. Nevertheless this coupling
seems quite unique and when integrating it over all spatial coordinates we get
the expected potential for a static charged shell:

$V_{\text{Coulomb}}=\alpha\mu_{3}\frac{g^{2}Q^{2}_{\text{shell}}}{2R}\,,$ (11)

Where $Q_{\text{shell}}=MN_{\text{D3}}$ is the charge of the shell and
$g^{2}=\pi g_{s}$ is the gauge coupling. This is to be compared with the
textbook expression for the energy of a charged shell

$V_{\text{Coulomb}}=\frac{1}{4\pi}\frac{g^{2}Q^{2}_{\text{shell}}}{2R}\,,$
(12)

Since we worked in string units where $\ell_{s}=1$ and we have
$\mu_{3}=(2\pi)/(2\pi\ell_{s})^{4}$, this fixes $\alpha$ to be
$\alpha=2\pi^{2}$.

The pressure energy: Since the D3-shell is understood as an NS5-brane, it is
expected that the $H$-flux on the B-cycle drops by one unit, $K\to K-1$, i.e.
it triggers a brane/flux decay, as one passes the shell. Therefore the vacuum
energy density inside the charged sphere will be lower than outside. If we
assume that the energy density difference between the uplifted and SUSY vacuum
is $2\mu_{3}e^{4A}$, we have a pressure energy contribution to the potential:

$V_{\text{pressure }}=-2g_{s}^{-1}\mu_{3}e^{4A}\frac{4\pi}{3}R^{3}\,.$ (13)

The complete potential: The total potential $V$ becomes
$V=V_{\text{DBI}}+V_{\text{Coulomb}}+V_{\text{pressure}}$:

$V=e^{A_{0}}\mu_{3}\left(4\pi\tau\sqrt{\left(e^{A_{0}}R\right)^{4}+N_{D3}^{2}\pi^{2}g_{s}^{2}}+2\pi^{3}g_{s}\frac{(MN_{\text{D3}})^{2}}{2(e^{A_{0}}R)}-2\frac{4\pi}{3}g_{s}^{-1}(e^{A_{0}}R)^{3}\right)\,.$
(14)

We see that the potential only depends on the combination $e^{A_{0}}R$. Hence
the potential stabilises $e^{A_{0}}R$ at a value that depends on the
coefficient in $V/e^{A_{0}}$, which are _warping independent_. So the size of
$e^{A_{0}}R$ depends on $M,N_{\text{D3}}$ and $g_{s}$. Note that the radius
measured by the 10d metric
$\mathrm{d}s^{2}_{10}=e^{2A_{0}}\mathrm{d}s_{4}^{2}+\ldots$ is shrunk by a
factor $e^{A_{0}}$ and so the physical size of the shell in string units is
$R_{\text{phys}}=Re^{A}$.

###  2.2 Brane polarisation beyond probe level using junction conditions

Using Smarr relations and blackfold techniques, brane polarisation has been
discussed beyond probe level in for instance Armas:2018rsy ; Armas:2019asf ;
Nguyen:2021srl ; Cohen-Maldonado:2015ssa ; Cohen-Maldonado:2016cjh . The more
pedestrian 4d GR-inspired method uses Israel junction conditions instead as in
the original proposal of Danielsson:2017riq . In here we merge the probe
results of the previous section with the junction conditions and find
corrections to the probe expressions, which however do not alter the final
results much.222In Danielsson:2022fhd , the relation between the brane action
and junction conditions was explored in a different context. This serves as a
very non-trivial consistency check of the brane potential and provides a
stringy interpretation to the bottom-up procedure of junction conditions.

We start out with a charged shell in 4d with a Reissner-Nordström metric on
the outside and an AdS metric on the inside. In principle the spacetime on the
outside should be asymptotically (A)dS but since the cosmological constant is
supposed to be tiny, it is safe to ignore it. We therefore work with the
ordinary RN-metric which is asymptotically Minkowski. Both the RN (+) and AdS
(-) metrics take the following form

$\mathrm{d}s_{\pm}^{2}=-f_{\pm}(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f_{\pm}(r)}+r^{2}\mathrm{d}\Omega_{2}^{2},$
(15)

where the $f_{\pm}(R)$ are

$\displaystyle f_{+}(r)=1-\frac{2G_{4}m}{r}+\frac{G_{4}(gQ)^{2}}{4\pi
r^{2}}\,,$ (16) $\displaystyle f_{-}(r)=1+k^{2}r^{2}\,.$ (17)

Note that now $r,t$ are now dimensionful, unlike $R,T$ before. The shell that
separates the RN exterior from the AdS interior is in our case a wrapped
NS5-brane. The junction condition across this shell is obtained by setting the
difference of the extrinsic curvature equal to the effective tension
$\mathcal{T}$ of the shell. The Israel junction condition then becomes
Israel:1966rt

$\mathcal{T}(r)=\frac{2}{8\pi
G_{4}}\frac{1}{r}\left(\sqrt{f_{-}(r)}-\sqrt{f_{+}(r)}\right)\,.$ (18)

The benefit of having a top down description is that we can fix the effective
tension from the previous subsection. The effective tension $\mathcal{T}(r)$
depends on the radius of the shells and can be found from the DBI part of the
NS5 action as follows

$S^{\text{DBI}}_{\text{NS5}}=\mathcal{T}(r)\int\sqrt{-h}\,\mathrm{d}^{3}\,x\,.$
(19)

with $h$ the induced metric on the 3d shell. To compare the junction condition
analysis with the previous section we need to compare coordinates. In order to
compare them note that the previous section ignored backreaction, which means
$f_{\pm}=1$. Hence we have

$t=\ell_{s}e^{A_{0}}T\,,\qquad r=\ell_{s}e^{A_{0}}R\,.$ (20)

To find the tension we use (from equating the on-shell actions) that

$4\pi\mathcal{T}r^{2}\mathrm{d}t=V^{\text{DBI}}_{\text{NS5}}\mathrm{d}T\,,$
(21)

which implies:

$\mathcal{T}(r)=\frac{\mu_{3}\ell_{s}\tau}{r^{2}}\sqrt{(\pi
g_{s}N_{\text{D3}})^{2}\ell_{s}^{4}+r^{4}}.$ (22)

Note that in the large $r$ regime the tension becomes $r$-independent which is
the usual assumption in the bottom-up literature. Combining everything
together, one can rewrite the junction condition as:

$\mu_{3}\ell_{s}\tau=\frac{2}{8\pi G_{4}}\frac{r}{\sqrt{(\pi
g_{s}N_{\text{D3}})^{2}\ell_{s}^{4}+r^{4}}}\left(\sqrt{1+k^{2}r^{2}}-\sqrt{1-\frac{2G_{4}m}{r}+\frac{G_{4}(gQ)^{2}/4\pi}{r^{2}}}\right)\,.$
(23)

The shell potential can then be obtained by multiplying the tension with the
square root determinant of the induced metric, which is done such that one
takes the averaged determinant roots. The other pieces in the potential are
argued in the usual way and coincide with the probe expressions. In the end we
obtain:

$\displaystyle V=$ $\displaystyle 4\pi
r^{2}\mathcal{T}(r)\frac{1}{2}\left(\sqrt{1+k^{2}r^{2}}+\sqrt{1-\frac{2G_{4}m}{r}+\frac{G_{4}(gQ)^{2}/4\pi}{r^{2}}}\right)$
$\displaystyle-\frac{1}{2G_{4}}\left(k^{2}r^{3}-\frac{G_{4}(gQ)^{2}/4\pi}{r}\right)\,.$
(24)

The consistency of this procedure can easily be verified by using the junction
condition to write the tension as in the right hand side of equation (18) such
that all terms drop against each other and we find the following on-shell
value of the potential:

$V=m\,,$ (25)

which indeed is the total energy (mass). The difference between the approach
based on the junction condition and the brane probes can then be understood as
follows; in the junction condition formalism one has an extra parameter in the
potential, namely $m$, but one also has an extra condition: namely $V$ at the
minimum has to equal this parameter $m$. Whereas in the probe approach one
simply minimises the potential and $m$ never appeared as an external
parameter. The junction condition formalism can be seen as corrected version
of the brane probe formalism, but for our purposes the two approaches give
qualitatively similar results.

To compare (24) with (14), we use333The absence of warping in the last
equation is due to a rescaling of coordinates.

$\displaystyle Q=MN_{\text{D3}}\,,\qquad g^{2}=\pi g_{s}\,,\qquad
3k^{2}M_{\textrm{Pl}}^{2}=\frac{2}{g_{s}}\mu_{3}\,.$ (26)

The only difference is that the junction conditions take into account
backreaction on the bulk metric such that $\mathcal{T}(r)r^{2}$ is now not
multiplied with a constant, but with an $r$-dependent function which becomes
$1$ in the probe approximation.

##  3 The stabilised brane shells

The shells have a radius that is determined by minimising the potential (24).
In order to proceed we will work in Planck-like units where $G_{4}=1$ and
ignore the cosmological constant assuming that the Hubble length is much
larger than the shell radius. The potential to minimise then becomes:

$\displaystyle V$
$\displaystyle=\frac{\ell_{s}^{-3}\tau}{(2\pi)^{2}}\sqrt{(\pi
g_{s}N_{\text{D3}})^{2}\ell_{s}^{4}+r^{4}}\left(1+\sqrt{1-\frac{2m}{r}+\frac{(gQ)^{2}/4\pi}{r^{2}}}\right)+\frac{(gQ)^{2}/4\pi}{2r}\,.$
(27)

Since the D3-brane tension is $\mu_{3}=2\pi/(2\pi\ell_{s})^{4}$, we need to
express the string length $\ell_{s}$ in Planck units, which should be a large
number.

In these units, the junction condition (18) becomes

$\frac{1}{\pi^{3}\ell_{s}^{3}}\tau=\frac{2r}{\sqrt{(\pi
g_{s}N_{\text{D3}})^{2}\ell_{s}^{4}+r^{4}}}\left(1-\sqrt{1-\frac{2m}{r}+\frac{(gQ)^{2}/4\pi}{r^{2}}}\right)\,.$
(28)

To highlight our approximation it is informative to make a plot of the
potential in (27) versus the potential of (24). In figure 1 we have plotted
both on top of each other, where the blue curve has the cosmological constant
included and the orange curve is without. The orange curve only has one
minimum corresponding to the brane shell whereas the blue curve also has a
maximum and then a runaway direction. In the limit of small cosmological
constant this runaway piece and the maximum will be send to very far
distances, corresponding to the fact that the nucleation probability toward
the true vacuum is highly suppressed. More accurately our assumptions are that
the length scale associated to the _difference_ in vacuum energies (inside and
outside the bubble) is parametrically larger than the radius of the bubble.
For jumps in the cosmological constant that are not small, one can verify that
it becomes more difficult to find a meta-stable brane shell.

Figure 1: A sketch of the potential $V$ as a function of $r$, with and without
the cosmological constant. The blue curve includes the cosmological
contribution and the orange is one without. For the sake of illustration we
have plotted the potential for a large cosmological constant.

From now onward we therefore continue with the approximation that the
cosmological constant can safely be ignored and use (27). In what follows we
will discuss the minima of this potential. Interestingly it will turn out that
minimising the potential at finite radius is never possible for extremal or
sub-extremal configurations.

In order to find minima of the potential we use that

$gQ/\sqrt{4\pi}=\frac{\sqrt{g_{s}}}{2}MN_{\text{D3}}>1\,.$ (29)

This can be argued from $g_{s}M\gg 1$ such that the A-cycle can be described
within supergravity. Alternatively this condition can be argued from the
stability of the conifold modulus Bena:2018fqc .

From equation (28) one can infer a maximal value of $\tau$. This works as
follows. The only negative contribution to $\tau$ comes from the square root,
which would vanish at the horizon radius. The overall term is largest for
minimal $r$ and a minimal horizon radius occurs for an extremal black hole.
Hence $\tau$ is maximised for extremal configurations where $m=gQ/\sqrt{4\pi}$
at the extremal radius $r=m$:

$8\pi\mu_{3}\tau<8\pi\mu_{3}\ell_{s}\tau_{\text{max}}=\frac{2gQ/\sqrt{4\pi}}{\sqrt{(\pi
g_{s}N_{\text{D3}})^{2}\ell_{s}^{4}+(gQ)^{4}/(4\pi)^{2}}}.$ (30)

This bound can be translated into a bound for our string parameters, and is
given by

$\displaystyle\frac{2\mathcal{C}}{(2\pi)^{2}}(\ell_{s}^{-2}M)^{\frac{1}{2}}(\ell_{s}^{2}N_{\text{D3}})^{-1}\sqrt{16\pi^{2}(\ell_{s}^{2}N_{\text{D3}})^{2}+(\ell_{s}^{-2}M)^{4}(\ell_{s}^{2}N_{\text{D3}})^{4}}<1.$
(31)

Recall that $\mathcal{C}$ is a numerical constant defined in (9) and
$\ell_{s}$ here is a number since it is the string length in Planck units.
Remarkably this bound does not depend on the string coupling and only involves
the combinations $\ell_{s}^{2}N_{\text{D3}}$ and $\ell_{s}^{-2}M$.

We are now ready to numerically determine the shell radius $r_{*}$ and mass
$m$ as a function of $M,g_{s},\ell_{s},N_{\text{D3}}$ from the equations:

$\partial_{r}V(r)|_{r=r_{*}}=0,\qquad V(r_{*})=m.$ (32)

To solve the system, it is useful to rewrite $r$ and $m$ as
$r\equiv\frac{gQ}{\sqrt{4\pi}}\,\delta$ and
$m\equiv\frac{gQ}{\sqrt{4\pi}}\,\epsilon$, hence over-extremality means that
$\epsilon<1$. The potential then depends trivially on the string coupling:

$\displaystyle V$
$\displaystyle=\sqrt{g_{s}}\left[\frac{\ell_{s}^{-3}M^{3/2}\mathcal{C}}{2(2\pi)^{2}}\sqrt{(4\pi
N_{\text{D3}})^{2}\ell_{s}^{4}+(MN_{\text{D3}})^{4}}\left(1+\sqrt{1-\frac{2\epsilon}{\delta}+\frac{1}{\delta^{2}}}\right)+\frac{MN_{\text{D3}}}{4\delta}\right]\,.$
(33)

We furthermore require that

$V(r_{*})=m=\frac{\sqrt{g_{s}}}{2}MN_{\text{D3}}\epsilon\,.$ (34)

The solutions of (32) for $\delta$ and $\epsilon$ are independent of $g_{s}$
and only depend on the combinations $\ell_{s}^{2}N_{\text{D3}}$ and
$\ell_{s}^{-2}M$. The physical radius, in Planck units, is then found by
multiplying $\delta$ with $gQ/\sqrt{4\pi}$. Solutions can be found numerically
and we plot them below in Figure 2.

Figure 2: All these plots have $g_{s}=0.1$ and show the mass $m$ versus the
radius $r$ for different values of the string length in Planck units and the
$F_{3}$-flux $M$. The dots represent different values of $N_{\text{D3}}$,
ranging from 10 from the left to 985 to the right in steps of 25 units.

Recall that the physical radius and mass our functions of
$N_{\text{D3}},M,g_{s},\ell_{s}$ with $\ell_{s}$ the string length in Planck
units. The last of these 3 values are ultimately determined by the
compactification whereas $N_{\text{D3}}$ is chosen in the sense that it
determines the state we investigate. Since that state is charged in 4d, there
is no tadpole constraint, unlike for the 3-form flux quantum $M$. However, if
$N_{\text{D3}}$ gets too large there are no more solutions because of the
bound (31). Our numerical plots below use $g_{s}=0.1$, which is close to the
boundary value of control, but lowering $g_{s}$ has a trivial effect on the
physical radius and mass, as we explained before since they will just scale
down as $\sqrt{g_{s}}$. We display for graphs with varying values for $M$ and
$\ell_{s}$. Each of these graphs shows a range for $N_{\text{D3}}$ values
between $10$ and $985$ with steps of $25$.

We observe the following general patterns

  * •

All our solutions correspond to over-extremal objects and no extremal or sub-
extremal configurations can be found. This can be understood intuitively since
what stabilises the shell is the Coulomb repulsion which needs to be large
enough. To quantitatively see why this exactly forbids extremal or sub-
extremal solutions we rely on our discussion about the maximal value of the
tension-like parameter $\mathcal{\tau}$ around equation (30).

  * •

The larger $N_{\text{D3}}$ the larger the mass and the radius, which is to be
expected.

  * •

The larger $\ell_{s}$ the larger the physical radius and the smaller the mass,
also as expected.

  * •

Less obvious are dependencies on $M$; the radius is not very sensitive to $M$,
but the mass strongly increases with increasing $M$.

Clearly all our solutions are in the regime where 10d supergravity, used to
derive the configurations, can be trusted. It is interesting to contemplate
how large these shells can actually become with reasonable compactification
parameters $M,g_{s},\ell_{s}$. Note that the relation between string length
$\ell_{s}$ and Planck length $\ell_{\text{Pl}}$ is, up to $\pi$ factors:
$\ell_{s}^{2}=g_{s}^{-2}V_{6}\ell_{\text{Pl}}^{2}$ with $V_{6}$ the
compactification volume in string units. Hence, our quoted $\ell_{s}$ values
are in the conventional ball park.

It is interesting to note that the radius we find is of the order of the
“classical radius” $r_{c}$ for a particle of mass $m$. The classical radius is
usually defined as

$r_{c}=\frac{g^{2}Q^{2}}{m}\,,$ (35)

in natural units and up to numerical factors of order one. From our
definitions above we have that

$rm=\frac{(gQ)^{2}}{4\pi}\epsilon\delta\,.$ (36)

Note that the product $\epsilon\delta$ is independent of the value of $g_{s}$
as can be seen from minimising the shell potential. Hence, if our numerical
solutions are such that the product $\epsilon\delta$ is order 1 typically we
indeed find the classical radius. In figure 3 we plot this product for the
top-left configuration of figure 2.

Figure 3: The product $\epsilon\delta$ for $\ell_{s}^{2}=10^{8},M=10$ and
various values of $N_{\text{D3}}$, ranging from 10 from the left to 985 to the
right in steps of 25 units.

We indeed notice that all values are order unity and one can further prove
analytically that the value cannot drop below $1/2$.444To see this one uses
that the positive tension implies that the combination
$1-\sqrt{1-\frac{2\epsilon}{\delta}+\frac{1}{\delta^{2}}}$ needs to be
positive.

Finally, it is useful to compare the radius with a corresponding black hole
radius. Of course for over-extremal black holes there is no black hole
horizon, so the best guess is to use the radius of the corresponding extremal
object, which is $r_{ex}\sim\frac{gQ}{M_{\textrm{Pl}}}$ and is a good measure
for the length scale associated to the gravitational backreaction. We then
find that

$\frac{r}{r_{ex}}\approx\frac{gQM_{\textrm{Pl}}}{m}>1\,,$ (37)

because our configurations are over-extremal. So one could argue that our
over-extremal fuzzball indeed corrects GR at the expected length scales.

It is interesting to compare the energy of excitations of the shell with the
mass of the shell itself. Instead of solving the Schrödinger equation for
perturbations, we simply use the uncertainty principle to make some estimates.
For a charged shell, we have, using the junction conditions

$\frac{c^{2}r\dot{r}^{2}}{2G_{4}}+V(r)=mc^{2},$ (38)

with $V(r)$ our brane potential. Using the conjugate momentum, we find:

$\frac{G_{4}}{2c^{2}}\frac{p^{2}}{r}+V(r)=mc^{2}.$ (39)

One can estimate the level spacing by approximating the potential near the
minimum as a quadratic function and then use the standard spacing formula for
the harmonic oscillator. A quick and dirty way to arrive at the same formula
involves Heisenberg’s uncertainty relation $pr\sim\hbar$, and compare the
excitation energy due the kinetic energy $\frac{G_{4}}{2c^{2}}\frac{p^{2}}{r}$
with $mc^{2}$. This relies on the fact that the quantum ground state energy is
of the order of the level spacing. By using that the terms in the potential
are of the same order and balancing the kinetic energy with the Coulomb energy
we find that $E\sim(gQ)^{3}M_{\textrm{Pl}}$. If we compare this with $mc^{2}$
we find that the spectrum of excitations become continuous in the limit where
$m<<(gQ)^{3}M_{\textrm{Pl}}$, while for $m>>(gQ)^{3}M_{\textrm{Pl}}$ the first
excitation is much heavier than the ground state. Our solutions belong to the
first category, as expected for classical configurations.

##  4 Discussion

Let us recapitulate what we have done. We found that in the “vanilla” flux
compactifications involving warped throats with SUSY-breaking anti-branes,
D3-particles puff into spherical shells. These shells can be regarded as
stabilised bubbles of vacuum decay, with the SUSY vacuum inside the bubble.
This realises explicitly the ideas of black holes as brane shells put forward
in Danielsson:2017riq ; Danielsson:2021ruf ; Danielsson:2021ykm , albeit for
over-extremal objects, so that the name black holes is somewhat a misnomer.
The shells can be significantly larger than the Planck scale, and can be
pushed towards the classical regime. From that perspective it is interesting
that we find over-extremal configurations since the naked singularity one
would expect, is not resolved in a quantum mechanical sense as for the
particles of the Standard Model (whose wavelength is vastly bigger than the
classical backreaction radius). However, to get to actual astrophysical length
scales one would have to lower the string scale to non-realistic values,
values which is furthermore in tension with several Swampland bounds since it
would mean the vacua exist at parametric weak coupling and large volume
Ooguri:2018wrx ; Lust:2019zwm . However, note that our construction does not
rely on de Sitter uplifts. Our computations are equally valid for uplifts of
AdS to another AdS, as long as the cosmological length scales are large and
the uplift energy small.

Since these objects resolve the naked singularity of over-extremal particles
they are somewhat exotic objects in the sense of describing over-extremal
fuzzballs. Yet, they seem a genuine prediction of theories based on
compactifications with warped throats. We also note, in line with suggestions
in Montero:2021otb , that these objects are the WGC particles Arkani-
Hamed:2006emk  for the U(1) gauge field living on anti-branes and in that
sense it is reassuring that we found that we can only sustain over-extremal
shells.

It remains an interesting open problem to find a string theory embedding for
black hole mimickers, in the style of Danielsson:2017riq ; Danielsson:2021ruf
; Danielsson:2021ykm , using shells that have realistic properties from an
astrophysics viewpoint.

### Acknowledgments

We like to thank Rob Tielemans and Miguel Montero for useful discussions. The
work of TVR is supported by the KU Leuven C1 grant ZKD1118C16/16/005. VVH is
supported by grant nr. 1185120N of the Research Foundation - Flanders (FWO).

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