# Jet-Inflated Cocoons in Dying Stars: New LIGO-Detectable Gravitational Wave
Sources

Ore Gottlieb,1∗ Hiroki Nagakura,2 Alexander Tchekhovskoy,1
Priyamvada Natarajan,3,4,5 Enrico Ramirez-Ruiz,6
Jonatan Jacquemin-Ide,1 Nick Kaaz,1 Vicky Kalogera1

1Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA),
Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA
2Division of Science, National Astronomical Observatory of Japan, 2-21-1
Osawa,
Mitaka, Tokyo 181-8588, Japan
3Department of Astronomy, Yale University, 52 Hillhouse Avenue,
New Haven, CT 06520, USA
4Department of Physics, Yale University, P.O. Box 208121,
New Haven, CT 06520, USA
5Black Hole Initiative, Harvard University, 20 Garden Street,
Cambridge MA 02138, USA
6Department of Astronomy and Astrophysics, University of California,
Santa Cruz, CA 95064, USA
∗Corresponding author. Email<EMAIL_ADDRESS>
> Long Gamma-Ray Bursts (LGRBs), the most powerful events in the Universe, are
> generated by jets that emerge from dying massive stars. Highly beamed
> geometry and immense energy make jets promising gravitational wave (GW)
> sources. However, their sub-Hertz GW emission is outside of ground based GW
> detectors’ frequency band. Using a 3D general-relativistic
> magnetohydrodynamic simulation of a dying star, we show that jets inflate a
> turbulent, energetic bubble-cocoon that emits strong quasi-spherical GW
> emission within the ground-based GW interferometer band, $100-600$ Hz, over
> the characteristic jet activity timescale, $\approx 10-100$ s. Our
> prediction for the source amplitude makes this the first non-inspiral GW
> source detectable by current interferometers out to hundreds of Mpc, with
> $\approx 0.1-3$ detectable events expected during LIGO/Virgo/Kagra’s
> observing run O4. These GWs are likely accompanied by detectable energetic
> core-collapse supernova and cocoon electromagnetic emission, making jetted
> stellar explosions promising multi-messenger sources.

Core-collapse supernovae (CCSNe) provide a unique opportunity to study the
last stages of stellar life-cycles, the synthesis of heavy elements, and the
birth of compact objects (?, ?, ?). However, the intervening opaque stellar
gas limits the prospects for learning about the underlying physics of the
explosion mechanism and the compact object environment from electromagnetic
signals. Fortuitously, CCSNe produce two extra messengers: neutrinos and
gravitational waves (GWs); both carry information from the stellar core to the
observer with negligible interference along the way (?, ?, ?). Numerical
studies (?, ?) showed that CCSNe can be highly asymmetric, giving rise to a
substantial time-dependent gravitational quadrupole moment, which generates GW
emission. However, with a small fraction ($E\approx 10^{46}~{}{\rm erg}$) of
the CCSN energy going in GW production, the ground-bound based interferometers
LIGO/Virgo/Kagra (LVK) (?) can detect only nearby ($\lesssim 1$ Mpc) events at
design sensitivity (?).

A special class of CCSNe - collapsars (?) is associated with long-duration
gamma-ray bursts (LGRBs), which originate in energetic jets powered by a
rapidly rotating newly-formed compact object, a black-hole (BH) or a neutron
star. Their enormous power makes LGRB jets promising GW sources. The GW memory
effect (?, ?) implies that the GW frequency is inversely proportional to the
timescale over which the metric is perturbed. For jets, this timescale is set
by the longer of the launching and acceleration timescales (?, ?, ?, ?, ?, ?,
?). Thus, the characteristic duration of LGRBs $\gtrsim 10$ s places the GW
emission from LGRB jets at the sub-Hz frequency band, too low for LVK, but
potentially detectable by the proposed space-based Decihertz Interferometer GW
Observatory (?).

As the jets drill their way out of the collapsing star, they shock the dense
stellar material and build a cocoon - a hot and turbulent structure that
envelops the jets (Fig. 1) (?, ?, ?, ?). The cocoon is generated as long as
parts of the jets are moving sub-relativistically inside the star. LGRB jets
break out from the star after $t_{b}\approx 10$ s (?) and spend a comparable
amount of time outside of the star before their engine turns off (?). After
breakout, jets are expected to stop depositing energy into the cocoon (?, ?),
unless they are intermittent or wobbly (?, ?). This implies that the cocoon
energy $E_{c}\approx 10^{51}-10^{52}$ erg (?) is at least comparable to that
of the jet (the energy retained by the jet after breakout), $E_{c}\gtrsim
E_{j}$.

Figure 1: Jets (light blue) inflate the energetic cocoon (dark blue) inside of
a dying massive star (pale yellow). The jets run into and shock against the
collapsing star, forming a backflow (yellow arrows). Infalling star runs into
and shocks against the backflow (red arrows). The shocked jet and shocked
stellar components form the cocoon and turbulently mix inside of it (white
arrows).

Here, we show that the cocoon is an attractive new GW source for current
detectors. First, the cocoon evolves over shorter timescales than the jets,
and, as we demonstrate below, its GW signal lies within LVK frequency band.
Second, relativistic jets are subject to the anti-beaming effect (?) that
inhibits the GW emission within their opening angle, whereas the cocoon can
emit GWs in all directions across the sky.

The GW strain can be approximated as

$h\approx\frac{2G}{Dc^{4}}\frac{d^{2}Q}{dt^{2}}~{},$ (1)

where $G$ is the gravitational constant, $c$ is the speed of light, $D$ is the
distance to the source, and $Q$ is the gravitational quadrupole. From Eq. 1,
cocoon-powered GW characteristic strain is (?)

$h_{\rm coc}\approx\frac{4G}{Dc^{4}}E_{c}\epsilon\simeq
10^{-23}~{}\frac{100~{}{\rm Mpc}}{D}\frac{E_{c}\epsilon}{10^{52}~{}{\rm
erg}}~{},$ (2)

where $\epsilon$ is the degree of asymmetry of the cocoon, which depends on
the viewing angle $\theta_{\rm obs}$ (see Fig. 1).

We carry out a high resolution 3D general-relativistic magnetohydrodynamic
(GRMHD) simulation of energetic LGRB jets in a collapsing star by repeating
the simulation from (?), but with an increased data output frequency, as
needed to resolve the time-variability of the gravitational quadrupole. This
simulation follows the jets from the BH for $\simeq 3.1$ s until they reach
distance $\simeq 1.5R_{\star}$ and energy $E_{j}\simeq 2\times 10^{52}$ erg,
where $R_{\star}$ is the stellar radius. Fig. 2 depicts the hourglass-shaped
cocoon upon breakout from the star (see the full animation and accompanying
sonification in https://oregottlieb.com/gw.html). For observers facing the jet
axis (Fig. 2a), the projected shape of the cocoon is close to circular
($\epsilon\ll 1$), significantly suppressing the quadrupole moment. Off-axis
observers (Fig. 2b,c), on the other hand, will see an asymmetric cocoon with
an order unity asymmetry $\epsilon\approx 1$, thus maximizing the observed GW
signal strain amplitude in Eq. 2.

Figure 2: Three-dimensional (3D) rendering of the cocoon upon breakout from
the star at different viewing angles. At $\theta_{\rm obs}=0^{\circ}$ (a) the
axisymmetric projection results in a weak on-axis signal as the cocoon appears
nearly circular, whereas at larger angles the asymmetric shape of the cocoon
enables strong GW emission (b,c). The colormap delineates ${\rm
log}(T^{00}r^{2})$ in c.g.s., which is an order of magnitude estimate to the
gravitational quadrupole density, where $T^{00}$ is the contravariant energy
density component of the stress-energy tensor. See details in supplementary
materials in (?).

Turbulent motions in the jet111Jets are also subject to instabilities (?, ?)
that may introduce high frequency GWs. Computing those requires going beyond
analytic and numerical modeling that assume continuous and steady axisymmetric
jets.-cocoon result in a stochastic GW signal, and lead to a broad spectrum of
GW frequencies that is challenging to evaluate analytically. However, we can
estimate the main features of the emerging GW spectrum frequency range as
follows. The smallest length-scales of the cocoon emerge over the thickness of
the shocked region $\Delta r_{\rm sh}\approx
10^{-2}R_{\star}\Gamma^{-2}\approx 10^{8}$ cm (?), where $\Gamma\approx 3$ is
the jet head Lorentz factor inside the star, and $R_{\star}\approx 10^{11}$
cm. These shocked regions evolve over $t_{\rm min}\approx\Delta r_{\rm
sh}/c_{s}\approx 2\times 10^{-3}$ s, where $c_{s}\approx c/\sqrt{3}$ is the
relativistic sound speed, implying the highest GW frequency, $f_{\rm
max}\approx t_{\rm min}^{-1}\approx 500$ Hz. The maximum timescale is the time
that the jet energizes the cocoon, $t_{\rm max}\gtrsim t_{b}\approx 10~{}$s,
thereby setting the minimum GW frequency, $f_{\rm min}\approx t_{\rm
max}^{-1}\lesssim 0.1$ Hz. The cocoon energy is distributed quasi-uniformly in
the logarithm of the proper-velocity, $10^{-2.5}\lesssim\Gamma\beta\lesssim 3$
(?, ?). Thus, although various cocoon components evolve on different
timescales, they carry comparable amounts of energy, and are expected to
result in a flat GW spectrum between $f_{\rm min}$ and $f_{\rm max}$.

Figure 3: 3D rendering of off-axis (a) and on-axis (b) projections of the
cocoon mass density after breakout from the star (star is in the center shown
in white), taken from (?). On- and off-axis strain amplitude in time (c) and
frequency (d) domains. Normalized amplitude (by maximal off-axis amplitude)
spectrograms for a sliding window of 50 ms for (average of $\phi_{\rm
obs}=0^{\circ}$ and $\phi_{\rm obs}=90^{\circ}$) off-axis (e) and on-axis (f)
observers. The off-axis amplitude is constant in frequency at $100~{}{\rm
Hz}\lesssim f\lesssim 600~{}{\rm Hz}$, and in time from $\approx 1$ s until
the jet shuts off and the cocoon relaxes.

To compute the GW signal, we post-process $\gtrsim$ petabyte of simulation
data output at a high cadence, enabling us to numerically calculate the second
derivatives of the gravitational quadrupole moment (see Supplementary
Materials in (?))222For comparison, in (?) we also calculate the weak GW
signal from non-jetted explosions, whose emission is dominated by an expanding
accretion shock.. Fig. 3a,b shows a 3D mass density rendering of the cocoon
after breakout from the star, taken from (?). Similar to its pre-breakout
shape in Fig. 2, the cocoon is asymmetric when observed off-axis (Fig. 3a),
and near-axisymmetric when observed on-axis (Fig. 3b). In fact, the strain
amplitudes in Fig. 3c,d show that on-axis emission is weaker by 1-2 orders of
magnitude than off-axis emission, whose spectrum is roughly flat at
$100~{}{\rm Hz}\lesssim f\lesssim 600~{}{\rm Hz}$, followed by an exponential
cutoff, $h(f)\propto{\rm exp}\left[-\left(f/600~{}{\rm Hz}\right)^{2}\right]$
(dotted line), consistent with our $f_{\rm max}$ estimate. The off-axis strain
amplitude spectrogram (Fig. 3e) shows that as the cocoon expands ($t\lesssim
1.2$ s in our simulation), the GW emission turns on, intensifies and shifts
only slightly toward lower frequencies, as the spectrum does not vary
considerably between different regions in the cocoon, owing to mixing. At
$\simeq 1.2$ s, the jets traverse about a third of the star, the cocoon is
fully formed, and the signal plateaus in amplitude. The GW emission is
expected to last until the jet engine shuts off at $t_{j}$ (longer than our
simulation), and the cocoon turbulent motions relax333GW travel time effects
may slightly prolong the signal duration for observers close to the jet axis,
see (?).. On the other hand, the on-axis GW signal becomes gradually stronger
as the jet breaks out from the star (Fig. 3f), owing to a stronger deviation
from axisymmetry in the absence of a cocoon confinement by the dense stellar
envelope after breakout. Both on- and off-axis signals are qualitatively
different from traditional GWs in CCSNe which have a peak frequency that rises
over a much shorter GW emission timescale ($\ll 1$ s) (?, ?).

The off-axis GW emission from the energetic cocoon
($E_{c}~{}\approx~{}2E_{j}~{}\simeq~{}4\times 10^{52}$ erg) is detectable out
to distances as far as $\approx 200$ Mpc (Fig. 3d), in agreement with Eq. 2
(to within a factor of two). For on-axis observers the GW detection horizon of
such cocoons is much smaller $\approx 10$ Mpc, owing to the low degree of non-
axisymmetry which renders a direct on-axis detection unlikely. We use our
numerical result to calibrate Eq. 2, assuming a linear scaling of the strain
amplitude with the cocoon energy444We neglect secondary effects on the cocoon-
powered GW emission, such as the density profile of the star and turbulent
mixing, that may affect the cocoon shape and its GW spectrum, and cause
deviations from the linear scaling shown in Eq. 2.,

$h_{\rm coc}\approx 10^{-23}~{}\frac{200~{}{\rm
Mpc}}{D}\frac{E_{c}\epsilon}{10^{52}~{}{\rm erg}}~{}.$ (3)

Approximating $E_{c}\approx E_{j}$ and $\epsilon(\theta_{\rm obs})\approx 1$,
we can express Eq. 3 via the BH spin $a$, mass $M_{\rm BH}$ and magnetic field
strength on the horizon $B_{\rm BH}$, assuming that the jet is powered by the
Blandford-Znajek mechanism (?, ?, ?),

$h_{\rm coc}\gtrsim 10^{-23}~{}\frac{200~{}{\rm
Mpc}}{D}\left(\frac{t_{j}}{10~{}{\rm s}}\right)\left(\frac{\,{M_{\rm
BH}}}{5\,{\rm M_{\odot}}}\frac{B_{\rm BH}}{10^{15}~{}{\rm
G}}\frac{a}{0.8}\right)^{2}~{}.$ (4)

Therefore, detections of cocoon-powered GWs can help us constrain the
properties of the central BH and its environment deep inside the dying star.

To estimate the expected number of detectable GW events in LVK observing run
O4, we connect $E_{c}$ to the observed jet energy via $E_{c}\approx E_{j}=\xi
E_{j,\rm obs}$, and consider two types of jets with opening angle
$\theta_{j}\approx 0.1^{+0.07}_{-0.03}$ rad (?): i) a traditional axisymmetric
jet, for which we adopt $\xi=1$ and conventional local LGRB rate
${\cal{R}_{\rm GRB}}\approx 100~{}{\rm Gpc^{-3}}~{}{\rm yr}^{-1}$ (?); and ii)
a jet wobbling by $\theta_{w}\approx 0.2$ rad as in our simulations, for which
the local GRB rate is an order of magnitude lower (?). When a jet wobbles by
$\theta_{w}\approx 2\theta_{j}$, it is observed on average only
$\left(\theta_{j}/\left[\theta_{w}+\theta_{j}\right]\right)^{2}\simeq 10\%$ of
the time, namely $\xi=10$. Assuming the $\gamma$-ray energy is $20\%$ of the
total jet energy (?), the isotropic equivalent $\gamma$-ray energy is
$\,{E_{\gamma,{\rm iso}}}\approx 0.4\theta_{j}^{-2}E_{j,\rm obs}$. The
distribution of LGRBs in $\,{E_{\gamma,{\rm iso}}}$ can be approximated by a
power-law (?), which we find to be $N_{\rm LGRBs}\propto\,{E_{\gamma,{\rm
iso}}}^{-1.5}$. This implies that the most energetic GRBs dominate cocoon-
powered GW detections, so that the $\,{E_{\gamma,{\rm iso}}}$ power-law
cutoff, $\,{E_{\gamma,{\rm iso,m}}}$ sets the expected number of detectable
events, see (?).

Assuming the LVK detection threshold of the characteristic strain at the
relevant frequencies, $|h_{\rm crit}|\approx 10^{-22}$ (Fig. 3d) and isotropic
GW emission $\left(\epsilon\left(\theta_{\rm obs}\right)\approx 1\right)$, we
use Eq. 3 to show in Fig. 4 the number of detectable cocoon-powered GW events
per year in the upcoming LVK observing run O4 for the above two types of jets
(?), where the shaded areas depict the standard deviation in $\theta_{j}$. If
jets propagate along a fixed axis, the detection probability (red line) is
$\simeq 10\%$. If jets wobble (blue line), then a jet (and the jet-powered
cocoon) has more energy for a given $\,{E_{\gamma,{\rm iso}}}$, since most of
its energy ($\approx 90\%$) is beamed away from our line of sight. This
dramatically increases the expected number of detectable GW events to a few
during observing run O4. Had the latter case been the true rate, then such
events might already exist in the data from previous LVK runs. However, this
signal was never explicitly searched for, and might be easily overlooked due
to its noisy nature compared to inspiral GWs. We emphasize that the large
uncertainties in $\theta_{j}$, the distribution of the most energetic GRBs,
and the prompt/afterglow energy ratio (?), in turn introduce large
uncertainties in the expected number of detectable GW events555Additionally,
in our estimate we ignore jets that are choked in the stellar envelope and do
not generate a GRB. If such phenomenon is common among massive stars (?), then
the predicted cocoon-powered GW detection rate is increased significantly, see
(?).. Interestingly, the detection rate of quasi-isotropic GWs from cocoons
with energies $E_{c}<E_{\rm max}$ constrains the fraction of SN Ib/c
progenitors that power LGRBs with $E_{c}<E_{\rm max}$, as shown on the right
vertical axis in Fig. 4, see (?).

Figure 4: The number of detectable cocoon-powered GW events in one year of LVK
observing run O4 as a function of the maximal isotropic equivalent
$\gamma$-ray energy $\,{E_{\gamma,{\rm iso,m}}}$ (the cutoff of the power-law
fit to $N_{\rm LGRBs}(E_{\gamma,{\rm iso}})$ distribution), assuming LVK
detection threshold of $|h_{\rm crit}|=10^{-22}$ and isotropic GW emission. If
the jets propagate along a fixed axis, then $\simeq 0.1$ events are expected
(red line), whereas if jets wobble (blue line), multiple GW events could be
detectable. Variations in jet opening angle (shaded areas) introduce
uncertainties into predicted GW detection rate. The detection rate of cocoons
with $E_{c}<E_{\rm max}$ also indicates the abundance of LGRBs with such
cocoons among SNe Ib/c (right vertical axis labels).

LGRBs are likely to be rich multi-messenger events. While a coincident LGRB-GW
detection is unlikely due to the weak on-axis GW emission, the GW signal is
likely accompanied by a wide range of electromagnetic counterparts powered by
the SN explosion and the cocoon: shock breakout in $\gamma$\- and X-rays
(seconds to minutes), cooling emission and radioactive decay in UV/optical/IR
(days to months), and broadband synchrotron (afterglow) emission (days to
years) (?, ?). The earliest radiative signal emerges when the cocoon or SN
shock wave breaks out from the star, producing a nearly coincident
electromagnetic counterpart to the cocoon-powered GWs. However, the shock
breakout signal originates in a thin layer, and primarily depends on the
breakout shell velocity and structure of the progenitor star, rather than the
total explosion energy (?, ?). Thus, although under favorable conditions of
viewing angle and progenitor structure it is possible to detect a shock
breakout in $\gamma$-rays, those signals will typically go unnoticed.

After releasing the shock breakout emission, the stellar shells and the cocoon
expand adiabatically and give rise to an optical cooling signal. SNe Ib/c
cooling emission lasts weeks and peaks at an absolute magnitude $M_{\rm
AB}\approx-18$ (?), whereas cocoons with energies of $E_{c}\gtrsim 10^{52}$
erg that emit strong GWs, will power even brighter ($M_{\rm AB}\lesssim-19$)
quasi-isotropic ($\theta_{\rm obs}\lesssim 1.0$ rad) UV/optical cooling
emission on timescales of days (?, ?). The optical emission of the cocoons and
SNe is sufficiently long and bright to be detected at all relevant distances
of a few hundred Mpc by Zwicky Transient Facility (ZTF) (?, ?) and upcoming
Rubin Observatory (?). Furthermore, cocoons with $E_{c}\gtrsim 10^{52}~{}{\rm
erg~{}s^{-1}}$ will explode the entire massive star and may power
superluminous supernovae (SLSNe) with $M_{\rm AB}\lesssim-21$ (?) over
$\approx$ months timescale.

Finally, the interaction of such energetic cocoons with the circumstellar
medium will produce a detectable broadband afterglow, assuming typical ambient
densities and standard equipartition parameters (?). The timescale over which
the afterglow emission emerges varies from days to years, as it depends on the
specific parameters of the system and the observer’s viewing angle, thereby
posing a challenge to its detection. However, the early cooling signal that
can be detected by a rapid search will enable an early localization of the
event and a targeted search for the later multi-band afterglow, which will
potentially alleviate the afterglow detection difficulty.

In fact, the ZTF online catalog (?) already contains $\approx 100$ CCSNe
Ib/c/SLSNe that were discovered during LVK observing run O3. 14 of these are
SNe Ic-BL: the only confirmed SN type to be associated with LGRBs (?). About
50 of these CCSNe (and 10 SNe Ic-BL) lie within 170 Mpc of Earth, implying
that if their progenitors harbored (off-axis) GRBs with conventional
$E_{c}\lesssim 10^{52}~{}{\rm erg}$ (?), GWs from their cocoon might
conceivably be detectable in LVK run O3 data. It will thus be intriguing to
perform a targeted search for GWs from these progenitors.

In this report, we propose and investigate the first non-inspiral GW source
that is detectable by LVK out to hundreds of Mpc. Future calculations of the
zoo of cocoon-powered GWs for different LGRB progenitors will enable the
characterization of amplitude/spectrum to source properties, and will be
addressed in a follow-up work. This will enable a more efficient search for
GWs from collapsar cocoons, enhance the wealth of information regarding the
physical properties of the source to be extracted from the GW signal, and aid
follow-up searches for electromagnetic counterparts of these multi-messenger
events, ultimately providing a better understanding of the relation between
CCSNe and LGRBs.

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## Acknowledgments

We are grateful to Kenta Hotokezaka for valuable discussion that improved the
paper. We thank David Radice, Fabio De Colle and Brian Metzger for helpful
comments. HN thanks Hirotada Okawa for useful comments and discussions. This
research used resources of the Oak Ridge Leadership Computing Facility, which
is a DOE Office of Science User Facility supported under Contract DE-
AC05-00OR22725. An award of computer time was provided by the ASCR Leadership
Computing Challenge (ALCC), Innovative and Novel Computational Impact on
Theory and Experiment (INCITE), and OLCF Director’s Discretionary Allocation
programs under award PHY129.

## Author Contributions

OG initiated the study, performed the simulations, conducted the analysis and
wrote the manuscript. OG, HN, AT, PN and ER-R discussed the results, their
interpretation and implications. OG, HN, AT and JJ discussed the numerical
methods. OG, AT, NK and VK discussed the observational prospects. All
coauthors provided comments on the manuscript.

## Funding

OG is supported by a CIERA Postdoctoral Fellowship. OG and AT acknowledge
support by Fermi Cycle 14 Guest Investigator program 80NSSC22K0031. AT was
supported by NSF grants AST-2107839, AST-1815304, AST-1911080, AST-2031997,
and NASA grant 80NSSC18K0565. PN gratefully acknowledges support at the Black
Hole Initiative (BHI) at Harvard as an external PI with grants from the Gordon
and Betty Moore Foundation and the John Templeton Foundation. ER-R thanks the
Heising-Simons Foundation and the NSF (AST-1911206, AST-1852393, and
AST-1615881) for support.

## Competing Interests

The authors declare no competing interests.

## Data and Materials availability

All numerical data used in this paper will be shared upon request to the
corresponding author.

## Supplementary Materials

Gravitational wave calculation
Numerical setup
GRB population
GWs from accretion shocks
Fig. S1
References (56-73)

### Gravitational wave calculation

When a fluid particle accelerates, it induces a perturbation in the metric,
which results in emission of GW radiation. Here, we assume the background
metric to be flat since all the GW energy is essentially released far away
from the BH. The monopole and dipole gravitational radiation are set to zero
by conservation of mass and momentum, respectively. Thus, the lowest non-
vanishing multipole of gravitational emission is the quadrupole component. The
GW strain can be approximated by the quadrupole formula for the transverse-
traceless (TT) gauge metric perturbation

$h\approx\frac{2G}{Dc^{4}}\frac{d^{2}Q}{dt^{2}}~{},$ (5)

where the gravitational quadrupole is

$Q_{ij}=\int{T^{00}\left(x_{i}x_{j}-\frac{1}{3}\delta_{ij}r^{2}\right)d^{3}x}~{},$
(6)

$T^{00}$ is the contravariant energy density component of the stress-energy
tensor, and the integration is over equal arrival time surfaces (radii with
retarded times that correspond to a given observer time). Eqs. 5 and 6 show
that axisymmetric modes and stationary sources do not emit GWs. It then
follows that asymmetric, time-evolved energetic objects such as the off-axis
cocoon, are most promising in generating detectable GWs.

For a single GW with angular frequency $\omega$, that without the loss of
generality propagates along the $\hat{z}$-axis, the plane wave solution has
two independent components

$h_{ij}(t,z)=\left(\begin{matrix}0&0&0&0\\\ 0&h_{+}&h_{\times}&0\\\
0&h_{\times}&-h_{+}&0\\\ 0&0&0&0\end{matrix}\right){\rm
cos}\left[\omega\left(t-\frac{z}{c}\right)\right]~{},$ (7)

where

$h_{+}(t)=\frac{G}{Dc^{4}}\left(\frac{d^{2}Q_{xx}}{dt^{2}}-\frac{d^{2}Q_{yy}}{dt^{2}}\right)\\\
;\\\ h_{\times}(t)=\frac{2G}{Dc^{4}}\frac{d^{2}Q_{xy}}{dt^{2}}~{}$ (8)

are the two independent polarizations of the plane wave moving along the
$\hat{z}$-axis. The GW detector response function to the planar GW is a linear
combination of the two polarizations (?) and is given by:

$h(t)=F_{+}h_{+}(t)+F_{\times}h_{\times}(t)~{},$ (9)

where $F_{+},F_{\times}$ are functions of the antenna-pattern of the detector
that depend on the sensitivity to each polarization. For an order of magnitude
estimate, the GW strain can be approximated as

$h(t)\approx\sqrt{h_{+}^{2}(t)+h_{\times}^{2}(t)}~{}.$ (10)

To characterize the sensitivity for detection at a given frequency, one
defines the characteristic strain as (?)

$h_{c}(f)=2f|\tilde{h}(f)|~{},$ (11)

where

$\tilde{h}(f)=\sqrt{\frac{\tilde{h}_{+}^{2}(f)+\tilde{h}_{\times}^{2}(f)}{2}}~{}$
(12)

is the Fourier transform of the interferometer response to the dimensionless
GW strain.

By using the quadrupole formula, we make the following approximations:

i) To separate the contribution of the cocoon from that of the jet, we
integrate the gravitational quadrupole only over plasma with asymptotic
proper-velocity smaller than the upper limit of the cocoon velocity,
$\Gamma\approx 3$ (?). At these velocities the anti-beaming effect, which
attenuates the GW emission of relativistic sources that move toward the
observer (?), is small and therefore the quadruple approximation
applies666However, at mildly-relativistic velocities, higher multipoles are no
longer negligible, and their contribution might be comparable to that of the
quadrupole. Nevertheless, since most of the cocoon energy is at sub-
relativistic velocities, our estimates are not expected to change
significantly.. To verify that the exclusion of the jet does not artificially
increase the quadrupole contribution at the expanse of lower moments, we also
calculate the quadrupole with the jet contribution, and find that the change
in the quadrupole is minimal since the jet is less energetic than the cocoon.

ii) The anti-beaming effect is also applicable to the mildly-relativistic part
of the cocoon777Recently, (?) estimated the GW signal from a GRB jet outflow
numerically, however their use of the GW memory expressions assumes a coherent
motion and thus is inapplicable to the cocoon. that forms before breakout and
expands to an angle of $\approx 1/\Gamma$. Since the energy in the cocoon is
distributed uniformly in log-scale of the velocity from $\beta\approx
10^{-2.5}$ to $\Gamma\approx 3$, only a small amount of energy undergoes anti-
beaming, and is attenuated for observers at $\lesssim 0.5$ rad. Thus, anti-
beaming in the cocoon may introduce an error of a few tens of percents at low
frequencies for observers close to the axis, where the projected quasi-
axisymmetric shape of the cocoon leads to weaker GW emission anyway.

iii) Eq. 6 includes an integration over surfaces with equal arrival time, i.e.
different fluid elements at the explosion frame (lab time) contribute to the
GW emission at different observer times. Our expensive numerical calculation
which processes $\gtrsim 1$ petabytes of data prevents us from performing an
accurate calculation with the GW travel time effect. Instead, the calculation
is performed in the explosion frame such that it assumes that the lab time is
identical to the observer time. Since we obtain a strain amplitude that is
roughly constant in space and time, this assumption is reasonable. That is,
different regions in the cocoon maintain similar spectrum at all times such
that the GW travel time is not anticipated to alter our result considerably.
This is especially true for off-axis observers, where the GW travel time
difference is on the order of the cocoon width divided by the speed of light,
which is much shorter than the GW signal duration. For on-axis observers, the
GW travel time difference can be a few seconds, that would moderately prolong
the GW emission duration, but should not affect its spectrum qualitatively.

### Numerical setup

We follow (?) who performed high resolution 3D GRMHD simulations of a
collapsing star with strong magnetic fields (maximal B-field at the time of
the BH collapse is $B\approx 10^{12.5}$ G), that lead to the formation of
powerful bipolar jets with $E_{j}\approx 10^{53}$ erg near a Kerr BH of mass
$\,{M_{\rm BH}}=4.2\,{\rm M_{\odot}}$ and dimensionless spin $a=0.8$. The
simulations followed the jet head for $18$ seconds, until it is well outside
the stellar envelope of mass $14\,{\rm M_{\odot}}$, at $10R_{\star}$, where
$R_{\star}=4\times 10^{10}$ cm is the stellar radius. Here, we use the code
h-amr (?) to rerun one of their simulations - the configuration with initial
jet magnetization of $\sigma_{0}=15$, until the jet head breaks out from the
collapsing star and reaches $\simeq 1.5$ stellar radii at $t\simeq 3.1$ s.

To properly follow the time evolution of the gravitational quadrupole in the
simulation, we output data files every $\Delta t=r_{g}/c\approx 0.02$ ms,
where $r_{g}$ is the BH gravitational radius. This timescale guarantees that
the physical processes in the disk are temporally resolved. We verify this by
calculating the second derivative with respect to time of the gravitational
quadrupole using $\Delta t=2r_{g}/c$, and find that the GW signal is
independent of the interval choice. We also verify that the quadrupole moment
does not change when multiplying the spatial resolution of each axis by 1.5.
The integration to $t\gtrsim 3~{}{\rm s}\approx 150,000~{}r_{g}/c$ implies
that we post-process 150,000 files with more than a petabyte of data.

## GRB population

To estimate the number of detectable events in LVK during observing run O4,
one needs to evaluate the LGRB distribution in $\,{E_{\gamma,{\rm iso}}}$. The
detected LGRB population can be modeled by a log-normal distribution, whereas
the total number of LGRBs can be fit by a power-law that coincides with the
log-normal distribution tail (figure 5 in (?)). We define the distribution
tail as LGRBs with $E_{\gamma,\rm iso}>10^{54}$ erg, since those are less
prone to detection bias, and are relevant for producing detectable GW
emission. Then, by fitting a power-law to the log-normal distribution tail, we
find $N_{\rm LGRBs}\propto\,{E_{\gamma,{\rm iso}}}^{-1.5}$. Assuming a LVK
detection threshold of $|h_{\rm crit}|\simeq 10^{-22}$ and using Eq. 3, the
number of detectable cocoon-powered GW events per year in LVK observing run O4
is (Fig. 4)

$N_{\rm O4}(\,{E_{\gamma,{\rm iso,m}}})=\int_{0}^{N(\,{E_{\gamma,{\rm
iso,m}}})}\left(\frac{\theta_{j}^{2}\xi\,{E_{\gamma,{\rm iso}}}}{8\times
10^{52}~{}{\rm erg}}\right)^{3}\frac{\cal{R}_{\rm GRB}}{\rm
Gpc^{-3}~{}yr^{-1}}f_{p}(\,{E_{\gamma,{\rm iso}}})dN(\,{E_{\gamma,{\rm
iso}}})~{},$ (13)

where we assume the $\gamma$-ray energy to be 20% of the total jet energy,
$f_{p}$ is the normalized density function of the $N_{\rm
LGRBs}(\,{E_{\gamma,{\rm iso}}})$ distribution, $\,{E_{\gamma,{\rm iso,m}}}$
is the distribution cutoff, and $\xi=1$ if jets propagate along one axis, and
$\xi=10$ if they wobble.

The detection rate of the quasi-isotropic GW signal can also be used to
estimate the abundance of LGRBs among their progenitors - SNe of type Ib/c,
whose rate is ${\cal{R}_{\rm SNe~{}Ib/c}}\approx 2.6\times 10^{4}~{}{\rm
Gpc^{-3}~{}yr^{-1}}$ (?). If the number of detected events with $E_{c}<E_{\rm
max}$ in LVK observing run O4 is $N_{\rm O4}(E_{c}<E_{\rm max})$, then using
Eq. 3, the fraction of SNe Ib/c progenitors that power LGRBs with
$E_{c}<E_{\rm max}$ can be estimated as (Fig. 4)

$\frac{N_{\rm LGRBs}(E_{c}<E_{\rm max})}{N_{\rm
SNe~{}Ib/c}}\approx\frac{N_{\rm O4}(E_{c}<E_{\rm max})}{\frac{{\cal{R}_{\rm
SNe~{}Ib/c}}}{{\rm Gpc^{-3}~{}yr^{-1}}}\left(\frac{E_{\rm max}}{5\times
10^{52}~{}{\rm erg}}\right)^{3}}~{}.$ (14)

Another noteworthy jet populations for production of detectable cocoon-powered
GWs are short GRB jets in binary neutron star (BNS) mergers, and jets that
fail to break out (and deposit all their energy in the cocoon) either from the
BNS ejecta (?) or from the star. (?) estimated that choked jets could be even
more abundant than those that successfully break out, based on the GRB
duration distribution. However, LGRB jets with a short central engine activity
are unlikely to drive the powerful cocoons required for producing a detectable
GW signal, as their cocoon structure will be quasi-spherical due to early
choking. Alternatively, if jets fail because of the progenitor structure, e.g.
it has an extended envelope such as in SN Ib progenitors (?, ?), their cocoon
could be very powerful, and even give rise to energetic explosions such as
fast blue optical transients (?). In this case, the cocoon-powered GW
detection rate can be significantly higher. Short GRBs are $3-30$ (depending
on whether jets wobble) more abundant than LGRBs (?), but are also about an
order of magnitude less energetic (?). Because the detectable GW events are
dominated by the most energetic GRBs, it might be too challenging for LVK to
detect cocoons that accompany short GRB jets in binary mergers.

## GWs from accretion shocks

We consider the case of a rapidly rotating star in which the magnetic field is
insufficient to power relativistic jets via the Blandford-Znajek (BZ)
mechanism. Instead, disk winds energize the expansion of an accretion shock.
(?) showed that reducing the magnetic field magnitude, such that the BZ power
falls below the one needed to overcome the ram pressure of the falling stellar
material, results in an expanding accretion shock. We follow this logic and
reduce the magnetic field strength in the star by two of orders of magnitude,
while keeping the stellar mass, radius, angular momentum and magnetic field
geometry identical to the jet simulation. Using a stellar density profile

$\rho(r)\propto r^{-\alpha}\left(1-\frac{r}{R_{\star}}\right)^{3}~{},$ (15)

we consider two cases of an expanding accretion shock in a stellar core:
$\alpha=0$ (constant core density) and $\alpha=2$ (steep core density).

We note that the expanding shock is similar to standing accretion shock
instabilities (SASI) in CCSNe (where the rotation can be slow), which are also
candidates for GW emission (?, ?, ?, ?, ?, ?, ?). However, there are two
important differences: i) while SASI is driven by the core bounce of the
collapse, the energy source of the expanding shock is disk winds; ii) the
stagnation of the shock in SASI occurs as the supersonic collapsing plasma
cannot be fully accreted onto the BH, forming a shock front, whereas in our
simulation the fast rotation forms a disk that keeps energizing the expanding
shock at all times.

Eq. 6 dictates that $Q\approx M_{\rm sh}r^{2}$, where $M_{\rm sh}$ is the mass
contained in the shock, which depends on the stellar core density profile. The
virial theorem for a gravitationally bound object at radius $r$ suggests that
$\ddot{Q}\approx Mv^{2}\approx GM^{2}/r$. In our simulation of a star with a
steep density profile, the accretion shock expands from the disk size at
$r_{\rm sh}\approx 100$ km to $r_{\rm sh}\gtrsim 10^{3.5}$ km, and the mass
contained in the shock is $M_{\rm sh}\approx\,{\rm M_{\odot}}$. It then
follows from Eq. 5 that the strain amplitude of the shock is

$h_{\rm sh}\approx\frac{r_{s}^{2}}{r_{\rm sh}D}\simeq 10^{-22}\left(\frac{{\rm
200~{}kpc}}{D}\right)\left(\frac{M_{\rm sh}}{\,{\rm
M_{\odot}}}\right)^{2}\frac{10^{3.5}~{}{\rm km}}{r_{\rm sh}}~{},$ (16)

where $r_{s}$ is the Schwarzschild radius of the shock. The GW frequency is
governed by the Keplerian orbit, such that as the shock expands over time, it
covers the following frequency range

$f_{\rm sh}\approx\sqrt{\frac{G\,{M_{\rm BH}}}{r_{\rm sh}^{3}}}\approx
30-10^{3}~{}{\rm Hz}~{}.$ (17)

Eqs. 16 and 17 show that the expansion velocity of the shock essentially
dictates the GW signal evolution. When $\alpha=0~{}(\alpha=2)$, the shock
gradually (rapidly) expands to larger radii, thus the GW spectrum evolves slow
(fast) toward lower frequencies.

Fig. S1 depicts the numerically calculated strain amplitudes during the first
$\approx 1$ s of the simulations, in time (a) and frequency (b). When
$\alpha=0$, the shock expands slowly to larger radii that contain more mass,
so the GW signal intensifies while gradually shifting toward lower
frequencies. When $\alpha=2$, the shock expands rapidly to reach longer orbits
at lower frequencies, but with most of the shock mass lies at small radii, the
strain amplitude remains largely unchanged. In both cases, the LVK detection
horizon is $\approx 100$ kpc, in agreement with Eq. 16 to an order of
magnitude, and the spectrum has an exponential cutoff at high frequencies,
$h(f)\propto{\rm exp}\left[-\left(f/1.3~{}{\rm kHz}\right)^{2}\right]$ (dotted
line), consistent with Eq. 17.

Fig. S1: The strain amplitude in time (a) and in frequency (b) shows that the
GW emission from an expanding accretion shock is weaker by $\approx 3$ orders
of magnitude than that from the cocoon. The lines for $\theta_{\rm
obs}=90^{\circ}$ are averages of observers at $\phi_{\rm obs}=0$ and
$\phi_{\rm obs}=90^{\circ}$.

We find that most of the symmetry breaking takes place in the disk at a small
radius. Thus, when most of the mass is concentrated at the shock front
($\alpha=0$), the axisymmetric spiral mode $m=1$ of the shock (see animation
in https://oregottlieb.com/transients.html) attenuates the on-axis GW signal.
By contrast, when most of the mass is around the disk ($\alpha=2$), then the
asymmetric mass distribution leads to a quasi-isotropic GW emission. The
magnitude and dependence on the viewing angle are in a rough agreement with
the results reported for SASI-powered GWs in CCSNe (?, ?, ?, ?, ?, ?, ?). Most
of the considered CCSN progenitors maintain a relatively steep density
profile, where SASI’s characteristic frequencies are in the range of 20-100 Hz
(?), and thus their results resemble those of our $\alpha=2$ model.


# Symplectic capacities
of disc cotangent bundles of flat tori

Gabriele Benedetti111Department of Mathematics, Vrije Universiteit Amsterdam,
De Boelelaan 1111, 1081 HV Amsterdam<EMAIL_ADDRESS>Johanna
Bimmermann222Fakultät für Mathematik, Ruhr-Universität Bochum,
Universitätsstraße 150, D-44801 Bochum, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>and Kai Zehmisch††footnotemark:

###### Abstract

We show that on the unit disc cotangent bundle of flat Riemannian tori, all
normalized capacities coincide with twice the systole. The same result holds
for flat, reversible Finsler tori and normalized capacities that are greater
than or equal to the Hofer–Zehnder capacity.

##  1 Introduction

Given $n\in\mathbb{N}$, symplectic capacities are invariants of symplectic
manifolds $(M,\omega)$ of dimension $2n$ used to study symplectic embeddings
between such manifolds. As introduced by Ekeland and Hofer in [7], a
(normalized) symplectic capacity is a function

$c\colon\big{\\{}\text{Symplectic manifolds of dimension
$2n$}\big{\\}}\to[0,\infty]$

satisfying the following properties:

Embedding

If $\varphi\colon(M_{1},\omega_{1})\to(M_{2},\omega_{2})$ is any symplectic
embedding, then the inequality $c(M_{1},\omega_{1})\leq c(M_{2},\omega_{2})$
holds.

Scaling

If $r>0$, then $c(M,r\omega)=rc(M,\omega)$.

Normalization

If $\omega_{0}$ is the standard symplectic form on $\mathbb{C}^{n}$, then
$c(B_{r})=\pi r^{2}=c(Z_{r})$, when $B_{r}$ is the open Euclidean ball of
radius $r$ and $Z_{r}=D_{r}\times\mathbb{C}^{n-1}$ is the infinite symplectic
cylinder having the open disc $D_{r}\subset\mathbb{C}$ of radius $r$ as base.

The first condition yields obstructions to the existence of symplectic
embeddings. In view of this, the third condition encodes the celebrated
Gromov’s non-squeezing theorem: $B_{r_{1}}$ can be symplectically embedded in
$Z_{r_{2}}$ if and only if $r_{1}\leq r_{2}$, see [10].

For this reason, finding a function that satisfies the axioms of a capacity is
a difficult problem. Nevertheless, a number of capacities have been discovered
in the last forty years, see [6]. For instance, the Hofer–Zehnder capacity
$c_{\mathrm{HZ}}$ is defined using compactly supported Hamiltonian systems on
$(M,\omega)$ and has a special dynamical significance: if
$c_{\mathrm{HZ}}(M,\omega)$ is finite then every proper Hamiltonian $H\colon
M\to\mathbb{R}$ admits periodic orbits on almost all energy levels. See [11,
Ch. 4, Thm. 1] and [13] for a proof of this fact, and [17] for examples of
closed manifolds having infinite Hofer–Zehnder capacity.

Despite these advances, it is a challenging task to compute capacities, even
for simple classes of manifolds, or to show that two capacities coincide on
these classes. For instance, the strong Viterbo conjecture asserts that all
capacities coincide on the class of convex domains in $\mathbb{C}^{n}$ [14,
Section 14.9, Problem 53]. Another remarkable class of examples is given by
canonical Finsler disc cotangent bundles $(D^{*}_{F}Q,\omega_{\mathrm{can}})$.
Here,

  * •

$Q$ is a closed manifold and

$D^{*}_{F}Q=\\{(q,p)\in T^{*}Q\mid F^{*}(q,p)<1\\},$

where $F\colon TQ\to\mathbb{R}$ is a smooth Finsler norm on $Q$ and
$F^{*}\colon T^{*}Q\to\mathbb{R}$ denotes the dual norm

$F^{*}(q,p):=\sup_{\begin{subarray}{c}v\in T_{q}Q\\\
F(q,v)<1\end{subarray}}p\cdot v,\qquad\forall(q,p)\in T^{*}Q$

  * •

$\displaystyle\omega_{\mathrm{can}}=\sum_{i=1}^{n}\mathrm{d}p_{i}\wedge\mathrm{d}q_{i}$
is the canonical symplectic form.

Characteristics $(q,p)\colon\mathbb{R}\to\partial(D^{*}_{F}Q)$ project to
Finsler geodesics $q\colon\mathbb{R}\to Q$, that is, extremals of the Finsler
length

$\ell_{F}(q):=\int_{a}^{b}F(q(t),\dot{q}(t))\mathrm{d}t$

at fixed endpoints. We denote by $\mathrm{sys}(F)$ the Finsler length of the
shortest non-contractible closed geodesic on $Q$. We call a Finsler norm
reversible if $F(q,v)=F(q,-v)$ for all $(q,v)\in TQ$. The most fundamental
examples of reversible Finsler norms are induced by Riemannian metrics $g$,
that is, $F(q,v):=\sqrt{g_{q}(v,v)}$ for all $(q,v)\in TQ$.

Capacities of disc cotangent bundles are known only in some symmetric
examples. The ball-capacity (also known as Gromov-width) of codisc bundles of
$S^{2}$ and $\mathbb{R}\mathrm{P}^{2}$ with respect to the round metric was
computed by Ferreira and Ramos [8]. These results were also generalized to
spheres of revolution in [9]. Furthermore, the Hofer–Zehnder capacity of disc
cotangent bundles of $\mathbb{C}\mathrm{P}^{n}$ with respect to the
Fubini–Study metric and of $\mathbb{R}\mathrm{P}^{n}$ with respect to the
round metric was computed by the second author in [3]. Finally, in [12] Jiang
computed the value of any normalized capacity for the codisc bundle of the
$\ell^{1}$-norm on the $n$-torus. It is this last result that we aim to
generalize in the present paper.

Indeed, our main theorem computes the capacity of disc cotangent bundles of
flat, reversible Finsler norms on the torus
$\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$. We recall that a Finsler norm
is flat if, in the global trivialization
$T\mathbb{T}^{n}=\mathbb{T}^{n}\times\mathbb{R}^{n}$ induced by the quotient
$\mathbb{R}^{n}\to\mathbb{T}^{n}$, there is
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ with $F(q,v)=f(v)$ for all
$(q,v)\in\mathbb{T}^{n}\times\mathbb{R}^{n}$. In this case
$D^{*}_{F}\mathbb{T}^{n}=\mathbb{T}^{n}\times K^{*}$, where
$K^{*}=\\{p\in(\mathbb{R}^{n})^{*}\mid f^{*}(p)<1\\}$.

######  Theorem 1.1.

If $F$ is a flat, reversible Finsler norm on $\mathbb{T}^{n}$ and $c$ is a
(normalized) symplectic capacity, then

$c(D_{F}^{*}\mathbb{T}^{n},\omega_{\mathrm{can}})\leq 2\mathrm{sys}(F).$

If, in addition,

  1. 1.

there is a Finsler norm $G$ induced by a flat Riemannian metric with $F\geq G$
and $\mathrm{sys}(F)=\mathrm{sys}(G)$, or

  2. 2.

$c\geq c_{\mathrm{HZ}}$, where $c_{\mathrm{HZ}}$ denotes the Hofer–Zehnder
capacity,

then

$c(D_{F}^{*}\mathbb{T}^{n},\omega_{\mathrm{can}})=2\mathrm{sys}(F).$

######  Remark 1.2.

In particular,
$c(D_{F}^{*}\mathbb{T}^{n},\omega_{\mathrm{can}})=2\mathrm{sys}(F)$ for all
normalized capacities, if $F$ is induced by a Riemannian metric or if $F$ is
an $\ell^{p}$-norm with $p\leq 2$. As recalled above, the case $p=1$ was
proved in [12].

Before ending this introduction, let us mention that it would be interesting
to extend the computation of capacities to the larger class of twisted disc
cotangent bundles $(D^{*}_{F}Q,\omega_{\mathrm{can}}+\pi^{*}\sigma)$, where
$\pi\colon D^{*}_{F}Q\to Q$ is the footpoint projection, and $\sigma$ is a
closed two-form on $Q$. In this case, characteristics on
$\partial(D^{*}_{F}Q)$ correspond to magnetic geodesics which describe the
motion of a particle in the stationary magnetic field $\sigma$. We refer the
reader to [4, 3, 2] for computations of the Hofer–Zehnder capacity for twisted
disc cotangent bundles of hermitian symmetric spaces, where the norm and the
magnetic field are induced by the associated canonical Kähler structure.

Acknowledgments. G.B. gratefully acknowledges support from the Simons Center
for Geometry and Physics, Stony Brook University at which some of the research
for this paper was performed during the program Mathematical Billiards: at the
Crossroads of Dynamics, Geometry, Analysis, and Mathematical Physics. G.B. is
partially supported by the DFG under Germany’s Excellence Strategy EXC2181/1 -
390900948 (the Heidelberg STRUCTURES Excellence Cluster). J.B. and K.Z. are
partially supported by the DFG under the Collaborative Research Center SFB/TRR
191 - 281071066 (Symplectic Structures in Geometry, Algebra and Dynamics). We
are grateful to Alberto Abbondandolo, Hansjörg Geiges, Felix Schlenk and
Stefan Suhr for inspiring conversations and their valuable suggestions on this
work.

##  2 Proof of Theorem 1.1

Let us identify $\mathbb{T}^{n}$ with the quotient of $\mathbb{R}^{n}$ by the
standard lattice $\mathbb{Z}^{n}$. Geodesics of a flat Finsler norm $F$ are
straight lines and therefore

$s:=\mathrm{sys}(F)=\min\\{f(v)\mid v\in\mathbb{Z}^{n},\ v\neq 0\\},$ (1)

where $F(q,v)=f(v)$ for all $(q,v)\in\mathbb{T}^{n}\times\mathbb{R}^{n}$. We
write $K=\\{v\in\mathbb{R}^{n}\mid f(v)<1\\}$ and
$K^{*}=\\{p\in(\mathbb{R}^{n})^{*}\mid f^{*}(p)<1\\}$, where $f^{*}$ is the
dual Finsler norm, which, by reversibility, has the formula
$f^{*}(p)=\sup_{v\in K}|p\cdot v|$.

###  2.1 Upper bound

Let $e_{1},\ldots,e_{n}$ be the standard basis of $\mathbb{Z}^{n}$.

Every $p\in(\mathbb{R}^{n})^{*}$ has coordinates $p_{1}:=p\cdot
e_{1},\ldots,p_{n}:=p\cdot e_{n}$. Since $(f^{*})^{*}=f$, we have for all
$k=1,\ldots,n$

$s_{k}:=f(e_{k})=\sup_{p\in K^{*}}|p\cdot e_{k}|=\sup_{p\in K^{*}}|p_{k}|.$

Therefore,

$D^{*}_{F}\mathbb{T}^{n}\subset\big{(}\mathbb{T}\times(-s_{1},s_{1})\big{)}\times\ldots\times\big{(}\mathbb{T}\times(-s_{n},s_{n})\big{)}.$
(2)

For all $k=1,\ldots,n$, let $r_{k}:=\sqrt{\frac{2}{\pi}s_{k}}$ and observe
that the maps

$\displaystyle\phi_{k}\colon\big{(}\mathbb{T}\times(-s_{k},s_{k}),\omega_{\mathrm{can}}\big{)}\rightarrow(D_{r_{k}},\omega_{0}),\quad\phi_{k}(q,p)=\sqrt{\frac{p+s_{k}}{\pi}}e^{2\pi
iq}$

are symplectic embeddings [11, Ch. 4, Prop. 4]. By (2), we get a symplectic
embedding

$\phi_{1}\times\ldots\times\phi_{n}\colon(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\to(Z_{r_{1}},\omega_{0}).$

Hence,

$c(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\leq\pi r^{2}_{1}=2s_{1}.$
(3)

Finally, let $u=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}$ be such that
$f(u)=\min\\{f(v)\mid v\in\mathbb{Z}^{n},\ v\neq 0\\}=s$. In particular, the
numbers $k_{1},\ldots,k_{n}$ are coprime and, therefore, up to the action of
$GL(n,\mathbb{Z})$, we can assume that $u=e_{1}$. In this case, $s_{1}=s$ and
we conclude from (3) that
$c(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\leq 2s$.

###  2.2 Lower bound

Denote by $\pi\colon\mathbb{R}^{n}\to\mathbb{T}^{n}$ the quotient map. We
claim that $\pi$ is injective on $\tfrac{s}{2}K$. Indeed, if
$u_{1},u_{2}\in\tfrac{s}{2}K$ and $\pi(u_{1})=\pi(u_{2})$, then there is
$v\in\mathbb{Z}^{n}$ such that $u_{2}-u_{1}=v$. Now $-u_{1}\in\tfrac{s}{2}K$
by reversibility and, thus,
$\frac{u_{2}+(-u_{1})}{2}=\tfrac{1}{2}v\in\tfrac{s}{2}K$ by convexity. Thus,
$v\in sK\cap\mathbb{Z}^{n}$, which implies that $v=0$ by definition (1), and
the claim is proved. By the claim, we see that $(\tfrac{s}{2}K)\times
K^{*}\subset D^{*}_{F}\mathbb{T}^{n}$. We get

$c(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\geq
c\big{(}(\tfrac{s}{2}K)\times K^{*},\omega_{\mathrm{can}}\big{)}=c(K\times
K^{*},\tfrac{s}{2}\omega_{\mathrm{can}})=\tfrac{s}{2}c(K\times
K^{*},\omega_{\mathrm{can}}).$ (4)

#### Case 1

Suppose that there is a Finsler norm $G$ induced by a flat Riemannian metric
such that $F\geq G$ and $\mathrm{sys}(G)=\mathrm{sys}(F)=s$. In particular,
$D^{*}_{G}\mathbb{T}^{n}\subset D^{*}_{F}\mathbb{T}^{n}$ and, therefore,
$c(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\geq
c(D^{*}_{G}\mathbb{T}^{n},\omega_{\mathrm{can}})$. Thus, it is enough to prove
the lower bound for $G$, or in other words, we can assume without loss of
generality that $F$ is induced by a flat Riemannian metric. In this case, $K$
is an ellipsoid and there is a linear map
$A\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $K=A(B_{1})$. The map
$A\times(A^{-1})^{*}$ is a symplectomorphism of
$(\mathbb{R}^{n}\times(\mathbb{R}^{n})^{*},\omega_{\mathrm{can}})$ and
therefore $c(K\times K^{*},\omega_{\mathrm{can}})=c(B_{1}\times
B^{*}_{1},\omega_{\mathrm{can}})$. By [5, Lemma 3.7], there is a symplectic
embedding

$\Phi\colon(B_{r},\omega_{0})\to(B_{1}\times
B_{1}^{*},\omega_{\mathrm{can}}),\qquad r:=\sqrt{\tfrac{4}{\pi}}.$

The existence of $\Phi$ also follows from an unpublished argument by Yaron
Ostrover and Felix Schlenk [16, Theorem 12.14(i)], and, for $n=2$, by [15]. We
conclude that

$c(B_{1}\times B^{*}_{1},\omega_{\mathrm{can}})\geq\pi r^{2}=4.$

Actually, $c(B_{1}\times B^{*}_{1},\omega_{\mathrm{can}})\leq 4$ is also true
by [1, Remark 4.2], although we don’t need this here. Using (4), we arrive at
the desired lower bound
$c(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\geq\tfrac{s}{2}4=2s$.

#### Case 2

Suppose that $c\geq c_{\mathrm{HZ}}$. We have

$c(K\times K^{*},\omega_{\mathrm{can}})\geq c_{\mathrm{HZ}}(K\times
K^{*},\omega_{\mathrm{can}})=4$

by [1, Theorem 1.7]. Actually, $c(K\times K^{*},\omega_{\mathrm{can}})\leq 4$
is also true by [1, Remark 4.2], although we don’t need this here. Using (4),
we get $c(D^{*}_{F}\mathbb{T}^{n},\omega_{\mathrm{can}})\geq\tfrac{s}{2}4=2s$.

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