# Development of a new Front End electronics in Silicon and Silicon-Germanium
technology for the Resistive Plate Chamber detector for high rate experiments

L. Pizzimento  R. Cardarelli  G. Aielli  E. Alunno Camelia  S. Bruno  A.
Caltabiano  P. Camarri  A. Di Ciaccio  B. Liberti  L. Massa  A. Rocchi

###### Abstract

The upgrade of the Resistive Plate Chamber (RPC) detector, in order to
increase the detector rate capability and to be able to work efficiently in
high rate environment, consists in the reduction of the operating voltage
along with the detection of signals which are few hundred µV small. The
approach chosen by this project to achieve this objective is to develop a new
kind of Front End electronics which, thanks to a mixed technology in Silicon
and Silicon-Germanium, enhance the detector performances increasing its rate
capability. The Front End developed is composed by a preamplifier in Silicon
BJT technology with a very low inner noise (1000 $e^{-}$ rms) and an
amplification factor of $0.3-0.4$ $mV/fC$ and a new kind of discriminator in
SiGe HJT technology which allows a minimum threshold of the order of $0.5$
$mV$. The performances of this kind of Front End will be shown. The results
are obtained by using the CERN H8 beamline with a full-size RPC chamber of $1$
$mm$ gas gap and $1.2$ $mm$ thickness of electrodes equipped with this kind of
Front End electronics.

##  1 RPC rate capability and Front End electronics upgrade

The RPC rate capability is mainly limited by the current that can be driven by
the high resistivity electrodes. It can be improved by modifying the highly
interconnected parameters which define the voltage drop on the electrodes.
These parameters may be derived by applying the Ohm’s law:

$V_{gas}=V_{a}-R\cdot IV_{gas}=V_{a}\rho\cdot\dfrac{d}{S}\cdot\braket{Q}\cdot
S\cdot\Phi_{particles}=V_{a}-\rho\cdot d\cdot\braket{Q}\cdot\Phi_{particles}$
(1.1)

This equation shows several ways to increase the detectable particle flux:

  * •

Decrease the electrode resistivity; This approach requires a large
technological effort in order to find out a different resistive material
suitable for the electrodes. Moreover, this variation could cause an
increasing of the detector operating current, leading to a possible ageing
problems due to the more current driven.

  * •

Reduce the electrode thickness;

  * •

Reduce the average charge per count Q. The reduction of this parameter is the
only one which permits to increase the rate capability while operating the
detector at fixed current (increasing of a factor 10 the flux of incident
particles while reducing of the same factor the average charge per count will
give the net effect of keeping the current driven by the electrodes costant).
The drawback of this approach is that reducing the average charge per count
implies a reduction of the injected charge inside the Front End, hence the
signal which needs to be detected will be much more smaller.

For this reason, a very sensitive Front End electronics is mandatory in order
to detect such small signal. Moreover an high suppression of the noise induced
inside the detector by the electronics and by external sources is required,
since the Front End developed for this approach is much more influenced by
these effects. Also the chamber structure, as a Faraday cage, requires a very
careful optimization in order to minimize the noise induced by external
sources.

This project aims at increasing the rate capability of the RPC by developing a
very sensitive Front End electronics which is able to detect very small
signals, of the order of hundred of $\mu V$, allowing a huge reduction of the
average charge per count.

Figure 1: Sketch of a Resistive Plate Chamber (RPC) detector

##  2 The new Front End Electronics

The developed Front End electronics boards is composed as follows:

  * •

eight channels of a new preamplifier,

  * •

two full-custom ASIC Discriminators with four channels each

  * •

PullUp system and LVDS transmitters integrated inside the board

The overall features of the preamplifier and of the ASIC discriminator are
reported in the Table below;

Amplifier Properties
---
Si standard component |
Amplification factor | 0.2-0.4 mV/fC
Power Consumption | 3-5 V 0.5–1 mA
Bandwidth | 100 MHz

Discriminator Properties
---
SiGe full custom |
Discrimination Threshold | 0.5 mV
Power Consumption | 2-3 V 4-5 mA
Bandwidth | 100 MHz

Table 1: Overall properties of preamplifier and full-custom ASIC discriminator
Figure 2: Sketch of the Front End board layout

###  2.1 Discrete components preamplifier

The preamplifier developed in the INFN laboratory of Rome Tor Vergata for the
Resistive Plate Chamber detector is made in Silicon Bipolar Junction
Transistor (BJT) technology. The main feature of this new kind of preamplifier
is a fast charge integration with the possibility to match the input impedance
to a transmission line. The working principle of this amplifier is shown in
Figure 3.

Figure 3: Working principle of the preamplifier

These pictures show how the injected signal is integrated. The output of the
preamplifier is a signal which has the rise time and the maximum amplitude
directly proportional to the duration of the injected signal and its charge.
The fall time is an intrinsic parameter of the preamplifier and it is not
related with the physical injected signal but depends only on the constants of
the preamplifier. The detailed performance of the silicon BJT preamplifier are
shown in the following Table.

Amplifier Properties
---
Voltage supply | 3-5 Volt
Sensitivity | 0.2-0.4 mV/fC
Noise (independent from detector) | 1000 $e^{-}$RMS
Imput impedance | 100-50 Ohm
Bandwidth | 10 - 100 MHz
Power Consumption | 5 mW/ch
Rise time $\delta(t)$ input | 300-600 ps
Radiation hardness | 1 Mrad, $10^{13}n$ $cm^{-2}$
Table 2: Detailed properties of the preamplifier

Thanks to its outstanding signal to noise ratio and its sensitivity this
preamplifier allows the anticipation of the efficiency curve of the RPC
detector of around 400 V with respect to the one previously achieved with the
preamplifier actually in use in ATLAS RPCs.

###  2.2 Full-custom ASIC Discriminator

The new full-custom Discriminator circuit dedicated to the RPC detector for
high rate environment is developed by using the Silicon-Germanium Hetero
Junction bipolar Transistor (HJT) technology. The principle of SiGe
heterojunction bipolar transistor is to introduce a Silicon-Germanium impurity
in the base of the transistor. The advantage of this device is that the band
structure introduces a drift field for electrons into the base of the
transistor, thus producing a ballistic effect that reduces the base transit
time of the carriers injected in the collector. The net effect is to improve
the transition frequency and to introduce a directionality in the charge
transport allowing a much lower value of B-C capacitance; hence a much higher
charge amplification can be achieved. The main idea behind this new
discriminator is the limit amplifier. If the signal surpasses the threshold,
it will be amplified until saturation giving as output a square wave (see
Figure 4). Moreover, the Full Width Half Maximum of the discriminator output
is proportional to the time that the input signal stays over the threshold,
allowing the time over threshold measurement directly with the discriminator.
The SiGe HJT technology is particularly suitable for this kind of working
principle, since the performances of this discriminator are mainly defined by
the switching frequency of the transistor and its $\beta$.

Figure 4: Working principle of the discriminator

The main features of this kind of discriminator are:

  * •

optimal characteristic function for the RPC; the characteristic function of
the discriminator is reported in the Figure 5. It is shown also a very small
logic state transition of around $300$ $\mu V$, which becomes practically
negligible when this discriminator is operated within the RPC detector and its
charge distribution is taken into account. Moreover the threshold can be
easily regulated with a minimum value of few $mV$.

  * •

Time-over-threshold measurement directly with the discriminator; In the Figure
6 the calibration curve of this measurement is reported.

  * •

Minimum pulse width of $3$ $ns$; this is one of the most important parameter
for a discriminator since it defines the shortest signal which can be injected
inside the discriminator keeping the working regime linear (see Figure 7).

Figure 5: Characteristic function of the discriminator and its transition
region Figure 6: Calibration curve of the Time-Over-Threshold measurement
realized by the discriminator Figure 7: Minimum injectable signal duration in
order to have a linear working regime

##  3 RPC and Front End electronics test at H8 beamline

This test was realized inside the North Area of CERN, precisely at the H8
beamline. The H8 beam is a secondary particle beam that provides hadrons,
electrons or muons of energies between $10$ and $400$ $GeV/c$, as well as
$450$ $GeV/c$ (primary) protons and up to $400$ $GeV/c$ per charge primary Pb
ions. Only the muon beam was used for this test. In this test two measurements
were performed:

  * •

RPC overall test and performances; the aim of this test was to ensure that the
whole system was working correctly and on the other side to study the
performances of the RPC detector equipped with this new Front End electronics.

  * •

Parameters "phase space" scans; the phase space defined by the Front End
electronics parameters was studied. The aim of these scans was to find out all
the possible Front End electronics working points and their properties.

The RPC detector tested is composed by 3 mono-gap equipped with the new Front-
End electronics. The gas gap is 1 mm thick and the electrode thickness is 1.2
mm. The mixture used was $95\%$ TFE, $4.7\%$ I-C4H10, $0.3\%$ SF6. The trigger
system was composed by two scintillators of around 10cm x 10cm, and the tested
chamber was put between them. The data acquisition was realized by using a
CAEN TDC with 100 ps of time resolution.

The low voltage system was realized in such a way to have the possibility to
regulate each parameter individually, allowing a deep parameters regulation
for each Front-End board. The Front-End board low voltages studied were:

  * •

Pull Up and Discriminator voltages; those two voltages regulate and define the
stability of the system.

  * •

Threshold and Amplifier voltages; these two voltages define the effective
threshold which is applied on the induced signal.

The main results obtained during the testbeam are reported in the following
figures.

Figure 8: Efficiency curves for each gap of the RPC detector tested, obtained
by doing the logic "AND" between the $\eta$ and $\phi$ readout panels Figure
9: Efficiency curve obtained by doing the majority 2 on 3 of the 3 gaps Figure
10: Average number of strips that fired in a single event Figure 11: On the
left: time resolution achieved with the Time Of Flight method without any
correction; On the right: time resolution achieved by applying the time walk
correction

##  4 Conclusion

The newly developed Front-End electronics achieved a minimum threshold value
of $1$ $mV$ of the amplified signal, the possibility to realize directly
inside the Front End board the Time Over Threshold measurement and , along
with $1$ $mm$ RPC gas gap, a time resolution of around $350$ $ps$. With these
performances, the developed Front-End electronics, composed by the full-custom
ASIC Discriminator in SiGe technology and the new preamplifier in Si
technology, is able to detect signals of few $fC$ allowing the use of the RPC
detector in high rate experiments.

## References

  * [1] R. Santonico, and R. Cardarelli. _Nucl. Instr. And Meth._ 187 (1981) 377. R.
  * [2] Cardarelli, and R. Santonico. _Nucl. Instr. And Meth._ A263 (1988) 20-25.
  * [3] R. Cardarelli et al., _RPC performance vs. front-end electronics, Nucl. Instr. and Meth._ A 661 (2012), Supplement 1, S198-S200. Doi: 10.1016/j.nima.2010.09.136
  * [4] R. Cardarelli et al., _Performance of RPCs and diamond detectors using a new very fast low noise preamplier_ , Journal of Instrumentation Vol. 8, Issue 1 (2013) P01003.
  * [5] D. J. Paul, _Si/SiGe heterostructures: From material and physics to devices and circuits_ , Semiconductor Science and Technology 19, (2004) R75. doi:10.1088/0268- 1242/19/10/R02
  * [6] John D Cressler, _Sige hbt technology: A new contender for si-based rf and microwave circuit applications. IEEE Transactions on Microwave Theory and techniques_ , 46(5):572589, 1998.
  * [7] Lawrence E Larson, _Sige hbt bicmos technology as an enabler for next generation communications systems_ , Publisher (2004).
  * [8] Author, _IHP_ , _innovations for high performance microelectronics_ https://www.ihpmicroelectronics.com/en/start.html (04/05/2018).
  * [9] M. Ullan et al, _Radiation hardness evaluation of sige hbt technologies for the front-end electronics of the atlas upgrade. Nuclear Instrument s and Methods in Physics Research_ , 2007.
  * [10] M. Ullan et al,Evaluation of two SiGe HBT technologies for the ATLAS sLHC upgrade, 2008.


# Exploring the $H_{0}$ tension and the evidence of dark sector interaction
from 2D BAO measurements

Armando Bernui<EMAIL_ADDRESS>Observatório Nacional, Rua General José Cristino
77, São Cristóvão, 20921-400, Rio de Janeiro, RJ, Brazil  Eleonora Di
Valentino<EMAIL_ADDRESS>School of Mathematics and Statistics,
University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom
William Giarè<EMAIL_ADDRESS>School of Mathematics and Statistics,
University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom
Suresh Kumar<EMAIL_ADDRESS>Department of Mathematics, Indira Gandhi
University, Meerpur, Haryana 122502, India  Rafael C. Nunes
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio Grande
do Sul, 91501-970 Porto Alegre RS, Brazil  Divisão de Astrofísica, Instituto
Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos
Campos, 12227-010, São Paulo, Brazil

###### Abstract

We explore observational constraints on a cosmological model with an
interaction between dark energy (DE) and dark matter (DM), using a compilation
of 15 measurements of the 2D BAO (i.e., transversal) scale in combination with
Planck-CMB data, to explore the parametric space of a class of interacting DE
models. We find that 2D BAO measurements can generate different observational
constraints compared to the traditional approach of studying the matter
clustering in the 3D BAO measurements. Contrary to the observations for the
$\Lambda$CDM and IDE models when analyzed with Planck-CMB + 3D BAO data, we
note that Planck-CMB + 2D BAO data favor high values of the Hubble constant
$H_{0}$. From the joint analysis with Planck-CMB + 2D BAO + Gaussian prior on
$H_{0}$, we find $H_{0}=73.4\pm 0.88$ km/s/Mpc. We conclude that the $H_{0}$
tension is solved in the IDE model with strong statistical evidence (more than
3$\sigma$) for the IDE cosmologies.

##  I Introduction

The standard cosmological model (the $\Lambda$CDM-cosmology framework) has
consolidated as the best scenario able to fit the astronomical observations
carried out over the past two decades, but recently some tensions and
anomalies have turned out to be statistically significant while analyzing
different data sets, placing the $\Lambda$CDM cosmology in a crossroad (see
Abdalla  _et al._ (2022) for a review). The most statistically significant
tension in the literature is in the determination of the Hubble constant,
$H_{0}$, between the Cosmic Microwave Background (CMB) and the direct local
distance ladder measurements. Assuming the $\Lambda$CDM scenario and the
Planck-CMB data Aghanim  _et al._ (2020a), these observations are at more than
5$\sigma$ tension with the SH0ES measurement Riess  _et al._ (2021, 2022).
And, in general, while the early time estimates prefer a lower value for
$H_{0}$, the late time measurements are in agreement with a higher value (see
Abdalla  _et al._ (2022) for a review). The lower value of $H_{0}$ inferred
from the Planck-CMB data is actually in very good agreement with Baryon
Acoustic Oscillations (BAO) + Big Bang Nucleosynthesis (BBN) constraints Cuceu
_et al._ (2019); Schöneberg  _et al._ (2019, 2022), and the other CMB
experiments like ACTPolDR4 and SPT-3G Aiola  _et al._ (2020); Dutcher  _et
al._ (2021). Motivated by several current observational discrepancies,
unlikely to disappear completely by introducing multiple and unrelated
systematic errors, it has been widely discussed in the literature whether new
physics beyond the $\Lambda$CDM can solve these tensions, in particular the
$H_{0}$ tension Di Valentino  _et al._ (2021a); Perivolaropoulos and Skara
(2022); Schöneberg  _et al._ (2021). One of the most popular dark energy (DE)
models in literature, designated by interacting DE (IDE) models Wang  _et al._
(2016), where a non-gravitational interaction between the dark matter (DM) and
DE is postulated to exist, have been intensively investigated as a possibility
to solve the current cosmological $H_{0}$ tension Yang  _et al._ (2021a); Gao
_et al._ (2021); Yao and Meng (2021); Martinelli  _et al._ (2019); Lucca and
Hooper (2020); Yang  _et al._ (2020a); Di Valentino  _et al._ (2020a); Pan
_et al._ (2019a); Kumar  _et al._ (2019); Yang  _et al._ (2018); Kumar and
Nunes (2016); Murgia  _et al._ (2016); Kumar and Nunes (2017); Di Valentino
_et al._ (2017); Yang  _et al._ (2019a, b); Pan  _et al._ (2019b); Di
Valentino  _et al._ (2020b); Di Valentino and Mena (2020); Di Valentino  _et
al._ (2021b); Rezazadeh  _et al._ (2022); Gómez-Valent  _et al._ (2020); Yang
_et al._ (2020b); Yao and Meng (2020); Pan  _et al._ (2020); Di Valentino
(2020); Kumar (2021); Yang  _et al._ (2021b); Lucca (2021a); Halder and Pandey
(2021); Gariazzo  _et al._ (2022); Nunes and Di Valentino (2021); Akarsu  _et
al._ (2021, 2022); Yang  _et al._ (2022).111It has also been investigated
where the IDE can solve/alleviate the $S_{8}$ tension Pourtsidou and Tram
(2016); Lucca (2021b); Poulin  _et al._ (2022); Naidoo  _et al._ (2022). On
the other hand, it is also discussed where in fact is the ability of this
class of models in actually solving the $H_{0}$ tension using robust external
probes other than Type Ia supernovae (SN) and BAO sample Nunes  _et al._
(2022), or even the non-ability for late times modification in solving the
$H_{0}$ problem Camarena and Marra (2021); Efstathiou (2021); Lemos  _et al._
(2019); Cai  _et al._ (2022).

The state-of-the-art constraints on IDE cosmologies arise primarily from CMB
data in combination with measurements of the late-time background expansion
history from BAO and SN. On the other hand, the traditional approach to study
the matter clustering in the three-dimensional (3D) BAO phenomena is to assume
a fiducial cosmology to calculate the 3D comoving distances, then perform the
2-point correlation function from which one estimates the sound horizon scale
at the end of the baryon drag epoch, $r_{s}$, the spherically averaged
distance $D_{V}$, and as sub-products the Hubble parameter $H$, and the
angular diameter distance $D_{A}$. This means that 3D BAO analyses are model
dependent and their results –to some extent– too; not to mention that usually,
the assumed fiducial cosmology is flat-$\Lambda$CDM or type-$\Lambda$CDM. The
way to investigate the impact of this assumption is to consider other fiducial
cosmologies, but these consistency analyses do not include, e.g., non-flat
geometries or non-$\Lambda$CDM models (e.g., extended DE cosmologies like IDE,
which may differ significantly from the $\Lambda$CDM model).

One can study the BAO phenomena without assuming a fiducial cosmology, this
can be done with the transversal BAO, termed 2D BAO. An additional advantage
is that the examination of 2D BAO phenomena does not assume a 3D geometry
because it works with data on spherical shells, with redshift thickness
$\Delta z$, and consider only their angular distribution. The redshift shell,
where data analysis is displayed, cannot be too thick (because the projection
effect smooths and shifts the BAO signal222The projection effect, present in
any shell with $\Delta z\neq 0$, introduces a shift in the 2D BAO signal that
is corrected using cosmological model; at the end one verifies that the shift
correction is a small fraction of the error measurement –for a large set of
models/parameters (Sánchez  _et al._ , 2011)– and for this the measurement is
weakly dependent on a fiducial cosmological model. Also, smaller is $\Delta
z$, smaller is the shift.) neither too thin (because the number density of
data should be enough to obtain a good BAO signal to noise). Given a deep
astronomical survey with large surveyed area, one divides the volume data in
several disjoint (to avoid correlation between contiguous shells) redshift
shells. The analyses of these shells provide the angular scale of the 2D BAO
signal $\theta_{\text{BAO}}(z)$ at redshift $z$, or equivalently give
measurements of $D_{A}(z)$ with $r_{s}$ as a parameter. Given the weakly
model-dependent features of 2D BAO measurements, it seems natural to use
$\\{\theta_{\text{BAO}}(z)\\}$ data to test non-$\Lambda$CDM models, and to
perform comparisons between analyses with diverse data sets.

Distinctive features of 2D BAO measurements that make them potentially useful
for studying non-$\Lambda$CDM models include:

  1. 1.

Important systematic effects in 3D BAO, like Redshift Space Distortions, are
absent in thin redshift bins 2D BAO measurements;

  2. 2.

While 3D BAO measures $D_{V}(z)$, which is a combination of $H$ and $D_{\\!A}$
distances, additional datasets are required to finally obtain $H(z)$ or
$D_{\\!A}(z)$;

  3. 3.

3D BAO requires a passive cosmic tracer (i.e., cosmic objects that do not
evolve in the large 3D volume in analysis), and in practice, a few
measurements of $\\{D_{V}(z_{i})\\}$ are expected along the universe history.
On the other hand, 2D BAO can be done with diverse cosmic tracers, in many
thin redshift bins.

  4. 4.

Although 3D BAO have smaller error bars than the 2D case, this is a
consequence of the use of a fiducial cosmology, and ultimately, one has to
decide whether to use data with model-dependent $\sim 1\%$ errors, or
measurements with weakly model-dependent $\sim 10\%$ errors. In this scenario,
it is advisable to use caution and prefer the use of 2D BAO data to examine
non-$\Lambda$CDM models such as IDE.

Overall, these factors lead us to argue that 2D BAO data are suitable to
provide remarkable observational constraints in non-$\Lambda$CDM models.

The aim of this Letter is to investigate the impact of the
$\theta_{\text{BAO}}(z)$ measurements on a class of IDE models well studied in
the literature, where in a homogeneous and isotropic universe, the dark
interaction is quantified by

$\displaystyle\nabla_{\mu}T_{i}^{\mu\nu}=Q_{i}^{\nu}\,,\quad\sum\limits_{i}{Q_{i}^{\mu}}=0~{}.$
(1)

Here the index $i$ runs over DM and DE, and the four-vector $Q_{i}^{\mu}$
governs the interaction.

In the present work, we consider a very well known parametric form of the
interaction function $Q$, namely, $Q=\mathcal{H}\,\xi\,\rho_{x}$, where $\xi$
is the coupling parameter between the dark components. All model formalism,
the background cosmic evolution, and the DM and DE density perturbation mode
evolution are well described in Gavela  _et al._ (2010). Also, as well
described in previous works, in order to avoid early times instabilities, in
developing the results of this work, we fix the equation of state of DE to
$w=-0.9999$, and impose the coupling parameter to be negative, that is,
$\xi<0$. This condition corresponds to the energy flow from DM to DE.

This work is structured as follows. In Section II we present the data sets and
methodology used in this work. In Section III we discuss the main results of
our analysis. In Section IV we outline our final considerations and
perspectives.

##  II Data sets and Methodology

We describe below the observational data sets and the statistical methods we
use to explore our parameter space.

  * •

CMB: Measurements of CMB temperature and polarization power spectra, as well
as their cross-spectrum, from the Planck 2018 legacy data release. We consider
the high-$\ell$ Plik likelihood for TT (in the multipole range $30\leq\ell\leq
2508$) as well as TE and EE (in the multipole range $30\leq\ell\leq 1996$), in
combination with the low-$\ell$ TT-only ($2\leq\ell\leq 29$) likelihood based
on the Commander component-separation algorithm in pixel space, as well as the
low-$\ell$ EE-only ($2\leq\ell\leq 29$) SimAll likelihood Aghanim  _et al._
(2020b). We refer to this data set as Planck.

  * •

CMB lensing: Measurements of the CMB lensing power spectrum as reconstructed
from the temperature 4-point correlation function Aghanim  _et al._ (2020c).
We refer to this data set as lensing.

  * •

BAO: Baryon Acoustic Oscillation (BAO) distance and expansion rate
measurements from the final measurements of the SDSS collaboration covering
eight distinct redshift intervals, obtained and improved over the past 20
years Alam  _et al._ (2021). These consists of isotropic BAO measurements of
$D_{V}(z)/r_{s}$ (with $D_{V}(z)$ and $r_{s}$ the spherically averaged volume
distance, and sound horizon at baryon drag respectively) and anisotropic BAO
measurements of $D_{M}(z)/r_{s}$ and $D_{H}(z)/r_{s}$ (with $D_{M}(z)$ the
comoving angular diameter distance and $D_{H}(z)=c/H(z)$ the Hubble distance).
All these measurements are compiled in Table 3 in Alam  _et al._ (2021),
regarding BAO-only data. We refer to this data set as BAO.

  * •

Transversal BAO: Measurements of BAO 2D, $\theta_{\text{BAO}}(z)$, obtained in
a weakly model-dependent approach, compiled in table I in Nunes  _et al._
(2020); de Carvalho  _et al._ (2021). These measurements were obtained using
public data releases (DR) of the Sloan Digital Sky Survey (SDSS), namely: DR7,
DR10, DR11, DR12, DR12Q (quasars), and following the same methodology. It is
important to notice that due to the cosmological-independent methodology,
these transversal BAO measurements have their errors larger than the errors
obtained using a fiducial cosmology. The reason for this fact is that, while
in the former methodology the error is given by the measure of how large is
the BAO bump, in the latter approach the model-dependent best-fit of the BAO
signal quantifies a smaller error. Typically, in the former methodology the
error can be of the order of $\sim 10\%$, but in some cases it can arrive to
$\sim 18\%$, and in the later approach it is of the order of few percent
Sánchez  _et al._ (2011). Another important feature of these 2D BAO data is
that the measurements at different $z$ are not correlated. In fact, the
methodology adopted to perform these measurements excluded the possibility for
covariance between measurements because the analyses of the 2-point angular
correlation function were done with cosmic objects belonging to disjoint
redshift shells to avoid correlation between contiguous data bins. Notice also
that the production of the random sets, necessary to the 2PACF analyses, is
also done in a model-independent way: consider a data sample of cosmic objects
in a thin redshift bin, then one shifts randomly the angular coordinates of
each object obtaining a set with the same number of cosmic objects as the data
sample with randomized positions: the random set (see the Appendix B in de
Carvalho  _et al._ (2018)); clearly these random sets are subjected to the
null-test to see if they are really random and no clustering signal is inside.
We refer this data set to as BAOtr.

  * •

SH0ES. A gaussian prior on the Hubble constant as measured by the SH0ES
collaboration Riess  _et al._ (2021), _i.e._ , $H_{0}=(73.04\pm 1.04)$
km/s/Mpc. We refer to this data set as $H_{0}$.

We use CLASS+MontePython code Lesgourgues (2011); Audren  _et al._ (2013);
Brinckmann and Lesgourgues (2019) with Metropolis-Hastings mode to derive
constraints on cosmological parameters for the IDE model baseline from several
combinations of the data sets defined above, ensuring a Gelman-Rubin
convergence criterion of $R-1<10^{-2}$ in all the runs. In what follows, we
discuss the main results of our analyses.

For the purpose of model comparison, we compute the Bayes factor $\ln B_{ij}$
to estimate the Bayesian Evidence of the IDE model with respect to the
$\Lambda$CDM scenario, through the publicly available package
MCEvidence,333github.com/yabebalFantaye/MCEvidence Heavens  _et al._ (2017a,
b)., and report the results in the last row of Table 1. We use the convention
of a negative value if the IDE model is preferred against the $\Lambda$CDM
scenario, or vice versa, and we refer to the revised Jeffrey’s scale by Trotta
Kass and Raftery (1995); Trotta (2008), to interpret the results. We will say
that the evidence is inconclusive if $0\leq|\ln B_{ij}|<1$, weak if $1\leq|\ln
B_{ij}|<2.5$, moderate if $2.5\leq|\ln B_{ij}|<5$, strong if $5\leq|\ln
B_{ij}|<10$, and very strong if $|\ln B_{ij}|\geq 10$.

##  III Results

Table 1: Constraints at 68% (95%) [99%] CL on selected parameters of the IDE
model obtained from various data set combinations as indicated in the upper
section of the Table. Bayes factors $\mathcal{B}_{ij}$ given by
$\ln\mathcal{B}_{ij}=\ln\mathcal{Z}_{\rm LCDM}-\ln\mathcal{Z}_{\rm IDE}$, are
also displayed for the different analyses, so that a negative value indicates
a preference for the IDE model against the $\Lambda$CDM scenario.

Parameter | Planck |  | \+ lensing
---
Planck + BAO |  | \+ lensing
---
Planck + BAOtr |  | \+ lensing
---
Planck + BAOtr + $H_{0}$ |  | \+ lensing
---
$\xi$ | $-0.43^{+0.28}_{-0.21}$ | $-0.40^{+0.23}_{-0.20}$ | $>-0.207$ | $>-0.210$ | $-0.683^{+0.088}_{-0.11}$ | $-0.683^{+0.087}_{-0.12}$ | $-0.58\pm 0.11$ | $-0.53\pm 0.11$
| $(>-0.775)$ | $(-0.40^{+0.40}_{-0.32})$ | $(>-0.389)$ | $(>-0.411)$ | $(-0.68^{+0.21}_{-0.19})$ | $(-0.68^{+0.23}_{-0.20})$ | $(-0.58^{+0.22}_{-0.21})$ | $(-0.53^{+0.19}_{-0.20})$
| $[>-0.819]$ | $[>-0.743]$ | $[>-0.486]$ | $[>-0.527]$ | $[-0.68^{+0.29}_{-0.23}]$ | $[-0.68^{+0.37}_{-0.27}]$ | $[-0.58^{+0.31}_{-0.29}]$ | $[-0.53^{+0.39}_{-0.25}]$
$H_{0}$ [Km/s/Mpc] | $71.7^{+2.3}_{-2.7}$ | $71.6\pm 2.1$ | $68.93^{+0.79}_{-1.2}$ | $69.08^{+0.74}_{-1.3}$ | $75.2^{+1.2}_{-0.75}$ | $75.3^{+1.3}_{-0.75}$ | $73.99\pm 0.88$ | $73.45^{+0.71}_{-0.59}$
$S_{8}$ | $1.109^{+0.063}_{-0.28}$ | $1.053^{+0.079}_{-0.21}$ | $0.891^{+0.025}_{-0.062}$ | $0.893^{+0.021}_{-0.065}$ | $1.49^{+0.24}_{-0.29}$ | $1.49\pm 0.26$ | $1.23^{+0.11}_{-0.22}$ | $1.15^{+0.10}_{-0.14}$
$r_{s}$ [Mpc] | $147.08\pm 0.30$ | $147.12\pm 0.27$ | $147.03\pm 0.25$ | $147.05\pm 0.25$ | $147.32\pm 0.27$ | $147.35\pm 0.29$ | $147.31^{+0.25}_{-0.29}$ | $147.32^{+0.26}_{-0.29}$
$\ln B_{ij}$ | $0.85$ | $-0.17$ | $1.60$ | $0.60$ | $-9.22$ | $-11.68$ | $-14.04$ | $-15.21$

Table 2: Constraints at 68% CL on selected parameters of the $\Lambda$CDM
model obtained from various data set combinations.

Parameter | Planck |  | \+ lensing
---
Planck + BAO |  | \+ lensing
---
Planck + BAOtr |  | \+ lensing
---
Planck + BAOtr + $H_{0}$ |  | \+ lensing
---
$H_{0}$ [Km/s/Mpc] | $67.32\pm 0.62$ | $67.32\pm 0.53$ | $67.65\pm 0.44$ | $67.60\pm 0.43$ | $69.01\pm 0.51$ | $68.85\pm 0.55$ | $69.88\pm 0.48$ | $69.65\pm 0.44$
$S_{8}$ | $0.832\pm 0.016$ | $0.834\pm 0.013$ | $0.825\pm 0.012$ | $0.827\pm 0.011$ | $0.794\pm 0.013$ | $0.802\pm 0.012$ | $0.774\pm 0.013$ | $0.7871^{+0.0095}_{-0.011}$
$r_{s}$ [Mpc] | $147.06\pm 0.30$ | $147.04\pm 0.27$ | $147.21^{+0.23}_{-0.26}$ | $147.13\pm 0.23$ | $147.75\pm 0.26$ | $147.64\pm 0.26$ | $148.06\pm 0.25$ | $147.91\pm 0.24$

Figure 1: 2D contours at 68%, 95%, and 99% CL and 1D posteriors for the cases
without lensing only. The grey vertical region refers to the value of $H_{0}$
measured by the SH0ES collaboration ($H_{0}=73.04\pm 1.04$ km/s/Mpc at 68%
CL).

Figure 2: Left panel: Best fit values for the $\Lambda$CDM and IDE models
obtained from Planck + BAOtr joint analysis, and $\Lambda$CDM from best fit
values from Planck only, compared against the BAOtr (i.e.,
$\theta_{\text{BAO}}$) sample. Right panel: Same as in the left panel, but for
a specific $z$ range in the sample.

In Table 1, we report the summary of the statistical analyses of the main
parameters of interest considering the IDE model from several data
combinations. In addition, we also show the constraints on the $\Lambda$CDM
model using the same data sets in Table 2 in order to compare the results.

We first consider Planck data only and its combination with the lensing data.
In these analyses, the dark coupling parameter $\xi$ is non-null at 1$\sigma$
confidence level (CL) from both the analyses. Also, it is well known that
$\xi$ is strongly anti-correlated with $H_{0}$. Thus, we can notice a high
value for $H_{0}$ when inferred from CMB analysis only, in such a way that by
this perspective, this class of IDE models can alleviate the $H_{0}$ tension
Di Valentino  _et al._ (2017); Kumar and Nunes (2017). For these analyses, the
Bayesian evidence is inconclusive, and the models, IDE and $\Lambda$CDM,
cannot be statistically distinguished from each other. When BAO data are
added, we notice that the values of $H_{0}$ are pulled to low values totally
compatible with Planck + BAO and Planck + BAO + lensing as predicted for the
$\Lambda$CDM model. The parameter $\xi$ attains lower bounds
$\xi>-0.389,>-0.411$ at 95% CL from Planck + BAO, Planck + BAO + lensing,
respectively. Again, the statistical evidence is inconclusive and weak. The
analysis with this BAO sample represents an update of the previous works.

Now, we consider the addition of transversal BAO sample. From both analyses,
i.e., Planck + BAOtr and Planck + BAOtr + lensing, notably we find $\xi<0$ at
more than 3$\sigma$. The constraints on the Hubble constant read as
$75.2^{+1.2}_{-0.75}$ km/s/Mpc at 68% CL and $75.3^{+1.3}_{-0.75}$ km/s/Mpc at
68% CL from Planck + BAOtr and Planck + BAOtr + lensing, respectively. These
constraints are compatible with direct local measurements by the SH0ES team
within 2$\sigma$. The Bayesian evidence for the IDE in this case is very
strong. Here we can notice a clear difference when considering different BAO
methodologies/samples to constrain the IDE framework. Since the $H_{0}$ values
are now compatible with SH$0$ES measurements, ultimately, we perform a joint
analysis considering the $H_{0}$ prior, that is equivalent to consider a
$M_{B}$ prior for this model Nunes and Di Valentino (2021). In this final
case, the constraint become slightly stronger than those reported without the
$H_{0}$ prior, but with the same conclusion, i.e, a very strong evidence in
favor of the IDE model. Figure 1 displays the parametric space on the plane
$\xi$-$H_{0}$ at 68%, 95%, and 99% CL for the analyses considered in this
work.

To understand what happens in these constraints, in Figure 2 we show the
theoretical prediction of $\theta_{\text{BAO}}(z)$ from the best-fit values of
Planck + BAOtr for the IDE and $\Lambda$CDM models. The comoving sound horizon
at the baryon drag epoch, $r_{s}$, is the main parameter which controls the
amplitude of the angular scale $\theta_{\text{BAO}}(z)$, but as shown in
Tables 1 and 2, for both models this scale is practically the same. This is
well predicted and expected, as the IDE model does not modify the physics at
early times. On the other hand, the possible theoretical presence for $\xi<0$,
will make the universe to expand faster at late times, and consequently,
decrease the angular diameter distance in $\theta_{\text{BAO}}(z)$ prediction,
making the amplitude of $\theta_{\text{BAO}}(z)$ to increase, without the need
to change $r_{s}$ scale. Figure 2 on the right panel shows the measurements in
the range $0.35<z<0.65$, where an excess in amplitude is noted in these
measurements in relation to the Planck-$\Lambda$CDM best-fit. Therefore, the
IDE framework has the ability to fit better to the $\theta_{\text{BAO}}(z)$
estimates than $\Lambda$CDM model, as the $\theta_{\text{BAO}}(z)$
measurements have an excess in amplitude in relation to the
Planck-$\Lambda$CDM best-fit prediction, where the $\Lambda$CDM model is not
able to fully fit. On the other hand, even in $\Lambda$CDM framework, the
Planck + BAOtr joint analysis constrains the Hubble constant to be
$H_{0}=69.01\pm 0.51$ km/s/Mpc at 1$\sigma$, which is higher than predicted by
Planck and Planck + BAO analyses, and at the same time prefers a lower value
for $S_{8}$, also helping with this tension. This difference occurs precisely
to try to fit the excess in amplitude in the range $0.35<z<0.65$. However,
this is not enough to fit all data points, as CMB data in combination with
BAOtr prefer to keep low $H_{0}$ values in $\Lambda$CDM model, while for IDE
the opposite is true. As already mentioned earlier, reducing the information
from 3D to 2D increases the error bars on the measurements. These perspectives
were investigated for $\Lambda$CDM and curvature in Nunes and Bernui (2020)
independently of CMB data and local distance ladder method. The 2D BAO sample
allows higher values of $H_{0}$, despite that the 2D BAO data set alone does
not have robustness to constrain the full parameter space of cosmological
models. In addition to all details mentioned above, we know that in the IDE
context, CMB data prefer higher $H_{0}$ values, and the 2D BAO data set also
does the same. Thus, the combination Planck + BAOtr, also leads to higher
$H_{0}$ values for this class of IDE, as a natural and expected behavior of
this model. We reinforce that new 2D BAO measurements with more accuracy are
necessary to investigate the $H_{0}$ tension, and to better constrain the
cosmological models. Hence we can conclude that the difference between the
addition of 3D or 2D BAO data is much less pronounced in $\Lambda$CDM than in
extended models, questioning the reliability and robustness of BAO
measurements in these cases, and the importance of their assumptions on the
fiducial model in the data reconstruction process.

Further, we can notice that $S_{8}$ values are higher in the IDE model than in
the $\Lambda$CDM (see Table 1 and Table 2). Also, the $S_{8}$ values in the
IDE model are higher than the ones estimated by weak lensing and galaxy
clustering surveys data assuming the $\Lambda$CDM framework Asgari  _et al._
(2021) (though they agree within the error bars Di Valentino  _et al._
(2020a)). However, this comparison is not relevant because $S_{8}$ value is
model-dependent. In fact, the $S_{8}$ value estimated by weak lensing and
galaxy clustering surveys data should be compared with the one estimated by
the other data (such as Planck) assuming the same underlying model.
Furthermore, it is important to mention that the accurate modeling of IDE
framework on weak lensing and galaxy clustering data, especially with regard
to the dynamics on non-linear scales and its application on the related
observables, has not yet been addressed in the literature. Therefore, more
conclusive findings cannot be made regarding the $S_{8}$ tension in this class
of IDE models. In conclusion, the IDE model considered in the present work,
solves the $H_{0}$ tension, but leaves the $S_{8}$ tension open for further
investigation.

##  IV Final Remarks

The main aim of this work is to answer the question: What is the impact of the
BAO measurements besides the usual traditional approach of the 3D BAO sample
on models beyond the $\Lambda$CDM model, when applied on models that can have
significantly different dynamical behavior with respect to the $\Lambda$CDM
prediction? In this regard, we have used transversal BAO measurements, which
are weakly dependent on a cosmological model to obtain new observational
constraints on IDE cosmologies. For a specific class of models, we find that
3D BAO and 2D BAO measurements can generate very different observational
constraints on the coupling parameter quantifying an interaction between DE
and DM. This difference is much more significant that the one obtained by
comparing the constraints of the $\Lambda$CDM model using 3D BAO and 2D BAO
data. We actually find a very strong evidence for IDE model when 2D BAO are
taken into account. Moreover, unlike Planck + BAO joint analysis, where it is
not possible to solve the $H_{0}$ tension, we note that the joint analysis
Planck + BAOtr is in perfect agreement with the direct local distance ladder
measurements by the SH0ES team. A different transversal BAO sample of the
adopted here is presented in Menote and Marra (2022). We check that, in
principle, this sample does not change our main conclusions. Therefore, we
conclude that the minimal baseline Planck + BAOtr in the context of IDE
framework solves the Hubble constant tension.

###### Acknowledgements.

The authors thank the referees for useful comments. AB thanks a CNPq
fellowship. EDV is supported by a Royal Society Dorothy Hodgkin Research
Fellowship. SK gratefully acknowledges support from the Science and
Engineering Research Board (SERB), Govt. of India (File No. CRG/2021/004658).
This article is based upon work from COST Action CA21136 Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse) supported by COST (European Cooperation in Science and
Technology). RCN thanks the CNPq for partial financial support under the
project No. 304306/2022-3.

## References

  * Abdalla  _et al._ (2022) E. Abdalla _et al._ , JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO] .
  * Aghanim  _et al._ (2020a) N. Aghanim _et al._ (Planck), Astron.Astrophys. 641, A6 (2020a), arXiv:1807.06209 [astro-ph.CO] .
  * Riess  _et al._ (2021) A. G. Riess _et al._ , (2021), arXiv:2112.04510 [astro-ph.CO] .
  * Riess  _et al._ (2022) A. G. Riess, L. Breuval, W. Yuan, S. Casertano, L. M. Macri, J. B. Bowers, D. Scolnic, T. Cantat-Gaudin, R. I. Anderson, and M. C. Reyes, Astrophys. J. 938, 36 (2022), arXiv:2208.01045 [astro-ph.CO] .
  * Cuceu  _et al._ (2019) A. Cuceu, J. Farr, P. Lemos, and A. Font-Ribera, JCAP 10, 044 (2019), arXiv:1906.11628 [astro-ph.CO] .
  * Schöneberg  _et al._ (2019) N. Schöneberg, J. Lesgourgues, and D. C. Hooper, JCAP 10, 029 (2019), arXiv:1907.11594 [astro-ph.CO] .
  * Schöneberg  _et al._ (2022) N. Schöneberg, L. Verde, H. Gil-Marín, and S. Brieden, JCAP 11, 039 (2022), arXiv:2209.14330 [astro-ph.CO] .
  * Aiola  _et al._ (2020) S. Aiola _et al._ (ACT), JCAP 12, 047 (2020), arXiv:2007.07288 [astro-ph.CO] .
  * Dutcher  _et al._ (2021) D. Dutcher _et al._ (SPT-3G), Phys. Rev. D 104, 022003 (2021), arXiv:2101.01684 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021a) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021a), arXiv:2103.01183 [astro-ph.CO] .
  * Perivolaropoulos and Skara (2022) L. Perivolaropoulos and F. Skara, New Astron. Rev. 95, 101659 (2022), arXiv:2105.05208 [astro-ph.CO] .
  * Schöneberg  _et al._ (2021) N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, and J. Lesgourgues, (2021), arXiv:2107.10291 [astro-ph.CO] .
  * Wang  _et al._ (2016) B. Wang, E. Abdalla, F. Atrio-Barandela, and D. Pavon, Rept. Prog. Phys. 79, 096901 (2016), arXiv:1603.08299 [astro-ph.CO] .
  * Yang  _et al._ (2021a) W. Yang, S. Pan, E. Di Valentino, O. Mena, and A. Melchiorri, JCAP 10, 008 (2021a), arXiv:2101.03129 [astro-ph.CO] .
  * Gao  _et al._ (2021) L.-Y. Gao, Z.-W. Zhao, S.-S. Xue, and X. Zhang, JCAP 07, 005 (2021), arXiv:2101.10714 [astro-ph.CO] .
  * Yao and Meng (2021) Y. Yao and X.-H. Meng, Phys. Dark Univ. 33, 100852 (2021), arXiv:2011.09160 [astro-ph.CO] .
  * Martinelli  _et al._ (2019) M. Martinelli, N. B. Hogg, S. Peirone, M. Bruni, and D. Wands, Mon. Not. Roy. Astron. Soc. 488, 3423 (2019), arXiv:1902.10694 [astro-ph.CO] .
  * Lucca and Hooper (2020) M. Lucca and D. C. Hooper, Phys. Rev. D 102, 123502 (2020), arXiv:2002.06127 [astro-ph.CO] .
  * Yang  _et al._ (2020a) W. Yang, S. Pan, R. C. Nunes, and D. F. Mota, JCAP 04, 008 (2020a), arXiv:1910.08821 [astro-ph.CO] .
  * Di Valentino  _et al._ (2020a) E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Dark Univ. 30, 100666 (2020a), arXiv:1908.04281 [astro-ph.CO] .
  * Pan  _et al._ (2019a) S. Pan, W. Yang, C. Singha, and E. N. Saridakis, Phys. Rev. D 100, 083539 (2019a), arXiv:1903.10969 [astro-ph.CO] .
  * Kumar  _et al._ (2019) S. Kumar, R. C. Nunes, and S. K. Yadav, Eur. Phys. J. C 79, 576 (2019), arXiv:1903.04865 [astro-ph.CO] .
  * Yang  _et al._ (2018) W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi, and D. F. Mota, JCAP 09, 019 (2018), arXiv:1805.08252 [astro-ph.CO] .
  * Kumar and Nunes (2016) S. Kumar and R. C. Nunes, Phys. Rev. D 94, 123511 (2016), arXiv:1608.02454 [astro-ph.CO] .
  * Murgia  _et al._ (2016) R. Murgia, S. Gariazzo, and N. Fornengo, JCAP 04, 014 (2016), arXiv:1602.01765 [astro-ph.CO] .
  * Kumar and Nunes (2017) S. Kumar and R. C. Nunes, Phys. Rev. D 96, 103511 (2017), arXiv:1702.02143 [astro-ph.CO] .
  * Di Valentino  _et al._ (2017) E. Di Valentino, A. Melchiorri, and O. Mena, Phys. Rev. D 96, 043503 (2017), arXiv:1704.08342 [astro-ph.CO] .
  * Yang  _et al._ (2019a) W. Yang, S. Pan, L. Xu, and D. F. Mota, Mon. Not. Roy. Astron. Soc. 482, 1858 (2019a), arXiv:1804.08455 [astro-ph.CO] .
  * Yang  _et al._ (2019b) W. Yang, O. Mena, S. Pan, and E. Di Valentino, Phys. Rev. D 100, 083509 (2019b), arXiv:1906.11697 [astro-ph.CO] .
  * Pan  _et al._ (2019b) S. Pan, W. Yang, E. Di Valentino, E. N. Saridakis, and S. Chakraborty, Phys. Rev. D 100, 103520 (2019b), arXiv:1907.07540 [astro-ph.CO] .
  * Di Valentino  _et al._ (2020b) E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Rev. D 101, 063502 (2020b), arXiv:1910.09853 [astro-ph.CO] .
  * Di Valentino and Mena (2020) E. Di Valentino and O. Mena, Mon. Not. Roy. Astron. Soc. 500, L22 (2020), arXiv:2009.12620 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021b) E. Di Valentino, A. Melchiorri, O. Mena, S. Pan, and W. Yang, Mon. Not. Roy. Astron. Soc. 502, L23 (2021b), arXiv:2011.00283 [astro-ph.CO] .
  * Rezazadeh  _et al._ (2022) K. Rezazadeh, A. Ashoorioon, and D. Grin, “Cascading dark energy,”  (2022).
  * Gómez-Valent  _et al._ (2020) A. Gómez-Valent, V. Pettorino, and L. Amendola, Phys. Rev. D 101, 123513 (2020), arXiv:2004.00610 [astro-ph.CO] .
  * Yang  _et al._ (2020b) W. Yang, E. Di Valentino, O. Mena, S. Pan, and R. C. Nunes, Phys. Rev. D 101, 083509 (2020b), arXiv:2001.10852 [astro-ph.CO] .
  * Yao and Meng (2020) Y.-H. Yao and X.-H. Meng, Phys. Dark Univ. 30, 100729 (2020).
  * Pan  _et al._ (2020) S. Pan, W. Yang, and A. Paliathanasis, Mon. Not. Roy. Astron. Soc. 493, 3114 (2020), arXiv:2002.03408 [astro-ph.CO] .
  * Di Valentino (2020) E. Di Valentino, Mon. Not. Roy. Astron. Soc. 502, 2065 (2020), arXiv:2011.00246 [astro-ph.CO] .
  * Kumar (2021) S. Kumar, Phys. Dark Univ. 33, 100862 (2021), arXiv:2102.12902 [astro-ph.CO] .
  * Yang  _et al._ (2021b) W. Yang, S. Pan, L. Aresté Saló, and J. de Haro, Phys. Rev. D 103, 083520 (2021b), arXiv:2104.04505 [astro-ph.CO] .
  * Lucca (2021a) M. Lucca, Phys. Rev. D 104, 083510 (2021a), arXiv:2106.15196 [astro-ph.CO] .
  * Halder and Pandey (2021) A. Halder and M. Pandey, (2021), arXiv:2101.05228 [astro-ph.CO] .
  * Gariazzo  _et al._ (2022) S. Gariazzo, E. Di Valentino, O. Mena, and R. C. Nunes, Phys. Rev. D 106, 023530 (2022), arXiv:2111.03152 [astro-ph.CO] .
  * Nunes and Di Valentino (2021) R. C. Nunes and E. Di Valentino, Phys. Rev. D 104, 063529 (2021), arXiv:2107.09151 [astro-ph.CO] .
  * Akarsu  _et al._ (2021) O. Akarsu, S. Kumar, E. Özülker, and J. A. Vazquez, Phys. Rev. D 104, 123512 (2021), arXiv:2108.09239 [astro-ph.CO] .
  * Akarsu  _et al._ (2022) O. Akarsu, S. Kumar, E. Özülker, J. A. Vazquez, and A. Yadav, (2022), arXiv:2211.05742 [astro-ph.CO] .
  * Yang  _et al._ (2022) W. Yang, S. Pan, O. Mena, and E. Di Valentino, (2022), arXiv:2209.14816 [astro-ph.CO] .
  * Pourtsidou and Tram (2016) A. Pourtsidou and T. Tram, Phys. Rev. D 94, 043518 (2016), arXiv:1604.04222 [astro-ph.CO] .
  * Lucca (2021b) M. Lucca, (2021b), arXiv:2105.09249 [astro-ph.CO] .
  * Poulin  _et al._ (2022) V. Poulin, J. L. Bernal, E. Kovetz, and M. Kamionkowski, “The sigma-8 tension is a drag,”  (2022).
  * Naidoo  _et al._ (2022) K. Naidoo, M. Jaber, W. A. Hellwing, and M. Bilicki, (2022), arXiv:2209.08102 [astro-ph.CO] .
  * Nunes  _et al._ (2022) R. C. Nunes, S. Vagnozzi, S. Kumar, E. Di Valentino, and O. Mena, Phys. Rev. D 105, 123506 (2022), arXiv:2203.08093 [astro-ph.CO] .
  * Camarena and Marra (2021) D. Camarena and V. Marra, Mon. Not. Roy. Astron. Soc. 504, 5164 (2021), arXiv:2101.08641 [astro-ph.CO] .
  * Efstathiou (2021) G. Efstathiou, Mon. Not. Roy. Astron. Soc. 505, 3866 (2021), arXiv:2103.08723 [astro-ph.CO] .
  * Lemos  _et al._ (2019) P. Lemos, E. Lee, G. Efstathiou, and S. Gratton, Mon. Not. Roy. Astron. Soc. 483, 4803 (2019), arXiv:1806.06781 [astro-ph.CO] .
  * Cai  _et al._ (2022) R.-G. Cai, Z.-K. Guo, S.-J. Wang, W.-W. Yu, and Y. Zhou, Phys. Rev. D 105, L021301 (2022), arXiv:2107.13286 [astro-ph.CO] .
  * Sánchez  _et al._ (2011) E. Sánchez, A. Carnero, J. García-Bellido, E. Gaztañaga, F. de Simoni, M. Crocce, A. Cabré, P. Fosalba, and D. Alonso, Mon. Not. Roy. Astron. Soc. 411, 277 (2011), arXiv:1006.3226 [astro-ph.CO] .
  * Gavela  _et al._ (2010) M. B. Gavela, L. Lopez Honorez, O. Mena, and S. Rigolin, JCAP 11, 044 (2010), arXiv:1005.0295 [astro-ph.CO] .
  * Aghanim  _et al._ (2020b) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A5 (2020b), arXiv:1907.12875 [astro-ph.CO] .
  * Aghanim  _et al._ (2020c) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A8 (2020c), arXiv:1807.06210 [astro-ph.CO] .
  * Alam  _et al._ (2021) S. Alam, M. Aubert, S. Avila, C. Balland, J. E. Bautista, M. A. Bershady, D. Bizyaev, M. R. Blanton, A. S. Bolton, J. Bovy, and et al., Physical Review D 103 (2021), 10.1103/physrevd.103.083533.
  * Nunes  _et al._ (2020) R. C. Nunes, S. K. Yadav, J. F. Jesus, and A. Bernui, Monthly Notices of the Royal Astronomical Society 497, 2133 (2020).
  * de Carvalho  _et al._ (2021) E. de Carvalho, A. Bernui, F. Avila, C. P. Novaes, and J. P. Nogueira-Cavalcante, Astron. Astrophys. 649, A20 (2021), arXiv:2103.14121 [astro-ph.CO] .
  * de Carvalho  _et al._ (2018) E. de Carvalho, A. Bernui, G. C. Carvalho, C. P. Novaes, and H. S. Xavier, JCAP 04, 064 (2018), arXiv:1709.00113 [astro-ph.CO] .
  * Lesgourgues (2011) J. Lesgourgues, (2011), arXiv:1104.2932 [astro-ph.IM] .
  * Audren  _et al._ (2013) B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, JCAP 02, 001 (2013), arXiv:1210.7183 [astro-ph.CO] .
  * Brinckmann and Lesgourgues (2019) T. Brinckmann and J. Lesgourgues, Phys. Dark Univ. 24, 100260 (2019), arXiv:1804.07261 [astro-ph.CO] .
  * Heavens  _et al._ (2017a) A. Heavens, Y. Fantaye, E. Sellentin, H. Eggers, Z. Hosenie, S. Kroon, and A. Mootoovaloo, Phys. Rev. Lett. 119, 101301 (2017a), arXiv:1704.03467 [astro-ph.CO] .
  * Heavens  _et al._ (2017b) A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie, S. Kroon, and E. Sellentin, (2017b), arXiv:1704.03472 [stat.CO] .
  * Kass and Raftery (1995) R. E. Kass and A. E. Raftery, J. Am. Statist. Assoc. 90, 773 (1995).
  * Trotta (2008) R. Trotta, Contemp. Phys. 49, 71 (2008), arXiv:0803.4089 [astro-ph] .
  * Nunes and Bernui (2020) R. C. Nunes and A. Bernui, The European Physical Journal C 80 (2020), 10.1140/epjc/s10052-020-08601-8.
  * Asgari  _et al._ (2021) M. Asgari _et al._ (KiDS), Astron. Astrophys. 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Menote and Marra (2022) R. Menote and V. Marra, Mon. Not. Roy. Astron. Soc. 513, 1600 (2022), arXiv:2112.10000 [astro-ph.CO] .


# Higher genus nonorientable maximal surfaces in the Lorentz-Minkowski 3-space

Shoichi Fujimori The first author was supported in part by Grant-in-Aid for
Scientific Research (C) No. 21K03226 from Japan Society for the Promotion of
Science.
2020 Mathematics Subject Classification. Primary 53A10; Secondary 53C42,
53C50.
Key words and phrases. maximal surface, nonorientable surface, higher genus
surface.  Shin Kaneda

(September 9, 2021)

###### Abstract

We study nonorientable maximal surfaces in Lorentz-Minkowski 3-space. We prove
some existence results for surfaces of this kind with high genus and one end.

## Introduction

The first example of a complete nonorientable minimal surface in Euclidean
3-space $\mathbb{R}^{3}$ was found by W. H. Meeks ${\rm{III}}$ in 1981. He
constructed a minimal Möbius strip [M](See Figure 1:Left). Then, in 1993, F.
J. López constructed a minimal once-punctured Klein bottle [L](See Figure
1:Right).

---
minimal Möbius strip minimal once-punctured Klein bottle
Figure 1: Nonorientable complete minimal surfaces (top) and their intersection
with the $(x_{1},x_{2})$-plane (bottom).

López’s example was generalized to complete nonorientable minimal surfaces
with arbitrary genus [LM](See Figure 2). Moreover, many nonorientable minimal
surfaces were found in [Ku], [MW], [O] and so on.

---
Figure 2: Higher genus nonorientable minimal surfaces (top) and their
intersection with the $(x_{1},x_{2})$-plane (bottom).

A maximal surface in Lorentz-Minkowski 3-space $\mathbb{L}^{3}$ is a spacelike
surface with zero mean curvature. Like in the case of minimal surfaces, there
is a Weierstrass-type representation formula [Ko] by using holomorphic data.
So, maximal surfaces share many local properties with minimal surfaces.
However, their global behavior is quite different. In fact, the only complete
maximal surface in $\mathbb{L}^{3}$ (without singularities) is the plane [C,
CY]. Maximal surfaces with singularities are investigated in [ER], [FeL],
[FLS1], [FLS2], [FSUY], [UY] and so on. M. Kokubu and M. Umehara investigated
co-orientability of maximal surfaces in [KU].

The first author and F. J. López studied some basic aspects of the global
theory of nonorientable maximal surfaces with singularities in [FuL]. Among
other things, they exhibited some fundamental examples (See Figure 3) and
obtained some classification results; the construction of higher genus
examples was postponed.

maximal Möbius strips

maximal once-punctured Klein bottles

Figure 3: Nonorientable maximal surfaces (top) and their intersection with the
$(x_{1},x_{2})$-plane (bottom)

In this paper we construct nonorientable maximal surfaces of high genus based
on [LM]. The main difficulty is to solve the period problem associated to the
Weierstrass data, that now is posed on a Riemann surface with higher homology
group.

This paper is organized as follows: In Section 1, we recall the definition of
nonorientable maximal surface with singularities and introduce some basic
properties of these surfaces. Section 2 is devoted to the construction of
higher genus nonorientable maximal surfaces. In Section 3, we discuss the
symmetry groups of the surfaces.

##  1 Preliminaries

Let $\mathbb{L}^{3}$ be the three dimensional Lorentz-Minkowski space with the
metric $\langle{~{}},{~{}}\rangle:=dx_{1}^{2}+dx_{2}^{2}-dx_{3}^{2}$. An
immersion $f:M\to\mathbb{L}^{3}$ of a 2-manifold $M$ into $\mathbb{L}^{3}$ is
called spacelike if the induced metric $ds^{2}=f^{*}\langle{~{}},{~{}}\rangle$
is positive definite. A spacelike immersion $f:M\to\mathbb{L}^{3}$ is said to
be maximal if the mean curvature vanishes identically.

######  Definition 1.1 (Maxfaces [UY], see also [FKKRSUYY]).

Let $M$ be a Riemann surface and $f:M\to\mathbb{L}^{3}$ a $C^{\infty}$-map.
$f$ is called a maxface if there exists an open dense subset $W$ of $M$ so
that the restriction $f|_{W}:W\to\mathbb{L}^{3}$ is a conformal maximal
immersion and the rank of $df$ at $p$ is positive for all $p\in M$. A point
$p\in M$ is said to be singular if rank$(df)<2$ at $p$.

######  Theorem 1.2 (Weierstrass-type representation [Ko, UY]).

Let $(g,\eta)$ be a pair consisting of a meromorphic function $g$ and a
holomorphic differential $\eta$ on a Riemann surface $M$ such that

(1.1) $(1+|g|^{2})^{2}\eta\bar{\eta}$

gives a Riemannian metric on $M$. Set

(1.2) $\Phi=\begin{pmatrix}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{pmatrix}=\begin{pmatrix}(1+g^{2})\eta\\\ i(1-g^{2})\eta\\\
2g\eta\end{pmatrix}.$

Suppose that

(1.3) $\operatorname{Re}\oint_{\gamma}\Phi={\bf{0}}$

hols for any $\gamma\in H_{1}(M,\mathbb{Z}).$ Then

(1.4)
$f=\operatorname{Re}\int_{z_{0}}^{z}\Phi:M\to\mathbb{L}^{3},\qquad(z_{0}\in
M)$

is a maxface. The singular set of f corresponds to $\\{p\in
M\>|\>|g(p)|=1\\}$, and the first and second fundamental forms of $f$ are
given by

$(1-|g|^{2})^{2}\eta\bar{\eta}\quad and\quad\eta dg+\bar{\eta}d\bar{g}$

respectively. We call $(M,g,\eta)$ $($or $(g,\eta)$$)$ the Weierstrass data of
$f:M\to\mathbb{L}^{3}$ $($We define the Weierstrass data of a nonorientable
maxface later$)$.

######  Remark 1.3.

The condition (1.3) is called the period problem, which guaranties the well-
defindness of $f$. It is equivalent to

(1.5) $\oint_{\gamma}g^{2}\eta+\overline{\oint_{\gamma}\eta}=0$

and

(1.6) $\operatorname{Re}\oint_{\gamma}g\eta=0$

for any $\gamma\in H_{1}(M,\mathbb{Z})$.

######  Definition 1.4 ([UY, Definition 1.3]).

A maxface $f:M\to\mathbb{L}^{3}$ is said to be complete if there exists a
compact set $C$ and a symmetric $(0,2)$-tensor $T$ on $M$ such that $T$
vanishes on $M\backslash C$ and $ds^{2}+T$ is a complete Riemannian metric.

######  Proposition 1.5 ([UY, Proposition 4.5]).

Let $f:M\to\mathbb{L}^{3}$ be a complete maxface with the Weierstrass data
$(M,g,\eta)$. Then the Riemann surface $M$ is biholomorphic to a compact
Riemann surface $\overline{M}$ minus a finite number of points
$\\{p_{1},\dots,p_{n}\\}$. Moreover, $g$ and $\eta$ extend meromorphically to
$\overline{M}$.

We say that a complete maxface
$f:\overline{M}\backslash\\{p_{1},\dots,p_{n}\\}\to\mathbb{L}^{3}$ is of genus
$k$ if $\overline{M}$ is a compact Riemann surface of genus $k$.

Let $M^{\prime}$ be a nonorientable surface with conformal coordinates. We
denote $\pi:M\to M^{\prime}$ the orientable conformal double cover of
$M^{\prime}$.

######  Definition 1.6 ([FuL, Definition 2.1]).

A conformal map $f^{\prime}:M^{\prime}\to\mathbb{L}^{3}$ is said to be a
nonorientable maxface if the composition

$f=f^{\prime}\circ\pi:M\to\mathbb{L}^{3}$

is a maxface. In addition, $f^{\prime}$ is said to be complete if $f$ is
complete.

We say that a complete nonorientable maxface
$f^{\prime}:M^{\prime}\to\mathbb{L}^{3}$ is of genus $k$ $(k\geq 1)$ if the
double cover $M$ of $M^{\prime}$ is biholomorphic to
$\overline{M}\backslash\\{p_{1},\dots,p_{n}\\}$, where $\overline{M}$ is a
compact Riemann surface of genus $k-1$.

Let $f^{\prime}:M^{\prime}\to\mathbb{L}^{3}$ be a nonorientable maxface, and
let $I:M\to M$ denote the antiholomorphic order two deck transformation
associated with the orientable double cover $\pi:M\to M^{\prime}$. Since
$f\circ I=f$, we have

(1.7) $g\circ I=\frac{1}{\bar{g}}\quad{\it{and}}\quad
I^{*}\eta=\overline{g^{2}\eta}.$

Conversely, if $(g,\eta)$ is the Weierstrass data of an orientable maxface
$f:M\to\mathbb{L}^{3}$ and $I$ is an antiholomorphic involution without fixed
points in $M$ satisfying (1.7), then the unique map
$f^{\prime}:M^{\prime}=M/\langle I\rangle\to\mathbb{L}^{3}$ satisfying that
$f=f^{\prime}\circ\pi$ is a nonorientable maxface. We call $(M,I,g,\eta)$ the
Weierstrass data of $f^{\prime}:M^{\prime}\to\mathbb{L}^{3}$.

##  2 Existence

Let $\overline{M_{k}}$ be the compact Riemann surface of genus $k$ defined by

$\overline{M_{k}}=\left\\{(z,w)\in\overline{\mathbb{C}}^{2}\;\left|\;w^{k+1}=\dfrac{z(z-r)}{rz+1}\right.\right\\},\qquad
r\in\mathbb{R}\backslash\\{0,1\\},$

where $\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$, and set
$M_{k}=\overline{M_{k}}\backslash\\{(0,0),(\infty,\infty)\\}$. Define

(2.1) $\displaystyle\begin{aligned}
I:\overline{M_{k}}\to\overline{M_{k}},\qquad
I(z,w)=\left(-\dfrac{1}{\bar{z}},\dfrac{1}{\bar{w}}\right),\\\
g=\frac{w^{k}(z+1)}{z(z-1)},\qquad\eta=i\dfrac{(z-1)^{2}}{zw^{k}}dz.\end{aligned}$

Note that $I$ has no fixed points, and $g$ and $\eta$ satisfy (1.7) and (1.1)
gives a Riemannian metric on $M_{k}$. We see that

$\deg(g)=2(k+1).$

See Table 1.

$(z,w)$ | $(-1,*)$ | $(-1/r,\infty)$ | $(0,0)$ | $(1,*)$ | $(r,0)$ | $(\infty,\infty)$
---|---|---|---|---|---|---
$g$ | $0^{1}\times(k+1)$ | $\infty^{k}$ | $\infty^{1}$ | $\infty^{1}\times(k+1)$ | $0^{k}$ | $0^{1}$
$\eta$ |  | $0^{2k}$ | $\infty^{k+1}$ | $0^{2}\times(k+1)$ |  | $\infty^{k+3}$
Table 1: Orders of zeros and poles of $g$ and $\eta$

######  Remark 2.1.

In the case $k=1$, changing the coordinate $\zeta=1/z$ and $\omega=iw/z$ in
$\overline{M_{k}}$ and rotating $-\pi/2$ about $x_{3}$-axis, the Riemann
surface $\overline{M_{1}}$ and the antiholomorphic involution $I$ and $g$ and
$\eta$ become

$\displaystyle\overline{M_{1}}=\left\\{(\zeta,\omega)\in\overline{\mathbb{C}}^{2}\;\left|\;\omega^{2}=\dfrac{\zeta(r\zeta-1)}{\zeta+r}\right.\right\\},\qquad
I(\zeta,\omega)=\left(-\frac{1}{\bar{\zeta}},-\frac{1}{\bar{\omega}}\right),$
$\displaystyle
g=\omega\frac{\zeta+1}{\zeta-1},\qquad\eta=i\frac{(\zeta-1)^{2}}{\zeta^{2}\omega}d\zeta.$

They coincide with the Weierstrass data of the maximal Klein bottle with one
end constructed in [FuL].

######  Theorem 2.2 (Existence).

For each $k\geq 1$, there are exactly two real values
$r_{1},r_{2}\in\mathbb{R}\backslash\\{0\\}$ for which the maxface

$f=\operatorname{Re}\int_{z_{0}}^{z}\Phi:M_{k}\to\mathbb{L}^{3}$

is well-defined and induces a one-ended complete nonorientable maxface
$f^{\prime}:M_{k}/\langle I\rangle\to\mathbb{L}^{3}$ of genus $k+1$, where
$\Phi$ is the $\mathbb{C}^{3}$-valued holomorphic differential given by
$(\ref{eq:wr})$. See Figure 4.

---
$k=2$, Left: $r\approx 0.478169$, Right: $r\approx 0.807158$. $k=3$, Left:
$r\approx 0.615965$, Right: $r\approx 0.859345$.
Figure 4: Nonorientable maximal surfaces of genus $k+1$ with one end (top) and
their intersection with the $(x_{1},x_{2})$-plane (bottom).

###### Proof.

Once we solve the period problem for some $r$, the arising immersion is
clearly a complete maxface; just observe that $|g|\neq 1$ at the two ends of
$M_{k}$. In the case $k=1$, the proof is given in [FuL]. Here we consider the
case $k\geq 2$. $f$ is well-defined if and only if $\phi_{j}$, $j=1,2,3$, have
no real periods on loops in $M_{k}$. We first observe that

(2.2) $\phi_{3}=2g\eta=d\left(\frac{2i(z^{2}+1)}{z}\right)$

is an exact 1-form on $\overline{M_{k}}$ and (1.6) is satisfied. It is easy to
check that $\phi_{1},\phi_{2}$ have no residues at the topological ends of
$M_{k}$ (see (3.1)). So, it suffices to prove (1.5) for any homology basis of
$\overline{M_{k}}$.

On the other hand, for any loop $\gamma$ in $\overline{M_{k}}$,

$\oint_{\gamma}\phi_{j}=\oint_{I_{*}(\gamma)}I^{*}(\phi_{j})=\oint_{I_{*}(\gamma)}\overline{\phi_{j}}$

and so

$2\operatorname{Re}\oint_{\gamma}\phi_{j}=\oint_{\gamma+I_{*}(\gamma)}\phi_{j}.$

Therefore, $f$ is well-defined on $M_{k}$ if and only if

(2.3) $\oint_{\gamma+I_{*}(\gamma)}\phi_{j}=0,\quad\gamma\in
H_{1}(\overline{M_{k}},\mathbb{Z}),\quad j=1,2.$

Define conformal maps $\kappa_{j}:\overline{M_{k}}\to\overline{M_{k}}$
$(j=1,2)$ as follows:

(2.4) $\kappa_{1}(z,w)=(z,e^{\frac{2\pi
i}{k+1}}w),\qquad\kappa_{2}(z,w)=(\bar{z},\bar{w}).$

Then we have the following:

(2.5)
$\kappa_{1}^{*}\Phi=K_{1}\Phi,\qquad\kappa_{2}^{*}\Phi=K_{2}\overline{\Phi},$

where

(2.6)
$K_{1}=\begin{pmatrix}\cos\frac{2\pi}{k+1}&\sin\frac{2\pi}{k+1}&0\\\\[4.0pt]
-\sin\frac{2\pi}{k+1}&\cos\frac{2\pi}{k+1}&0\\\\[4.0pt]
0&0&1\end{pmatrix},\qquad K_{2}=\begin{pmatrix}-1&0&0\\\\[4.0pt]
0&1&0\\\\[4.0pt] 0&0&-1\end{pmatrix}.$

Let $\gamma_{1}$ and $\gamma_{2}$ be two loops in $\overline{M_{k}}$ whose
projections to the $z$-plane are illustrated in Figure 5 as in [LM]. The
winding number of $\gamma_{1}$ around $0,-1/r$ is 1. The winding number of
$\gamma_{2}$ around 0 is 1, around $r$ is $-1$. The set

$\\{(\kappa_{j})^{m}_{*}(\gamma_{l})\;|\;j,l\in\\{1,2\\},m\in\\{1,\dots,k+1\\}\\}$

contains a homology basis of $\overline{M_{k}}$.

$-1/r$$0$$r$$\gamma_{1}$$\gamma_{2}$

Figure 5: Projection to the $z$-plane of $\gamma_{1},\gamma_{2}\in
H_{1}(\overline{M_{k}},\mathbb{Z})$.

Moreover, a straightforward computation gives that

$I_{*}(\gamma_{1})=\gamma_{1},\qquad
I_{*}(\gamma_{2})=\gamma_{1}-\gamma_{2}+(\kappa_{1})^{k}_{*}(\gamma_{1}).$

Therefore, (2.3) is equivalent to

$\oint_{\gamma_{1}}\phi_{j}=0\qquad(j=1,2).$

In other words, $f$ is well-defined on $M_{k}$ if and only if

$\oint_{\gamma_{1}}(1+g^{2})\eta=\oint_{\gamma_{1}}(1-g^{2})\eta=0$

holds, that is to say,

$\oint_{\gamma_{1}}\eta=\oint_{\gamma_{1}}g^{2}\eta=0$

holds. However,

$\oint_{\gamma_{1}}\eta=\oint_{I_{*}(\gamma_{1})}I^{*}\eta=\oint_{\gamma_{1}}\overline{g^{2}\eta},$

and hence $f$ is well-defined on $M_{k}$ if and only if

$\oint_{\gamma_{1}}g^{2}\eta=\oint_{\gamma_{1}}\frac{w^{k}(z+1)^{2}}{z^{3}}dz=0.$

We take the exact 1-form $dF$, where

$\displaystyle F$
$\displaystyle=\frac{(k+1)(z-r)(2rz^{2}-((k+1)r^{2}-2(k+2)r+k)z+r)}{(k+2)rwz}.$

Then we have

$\dfrac{w^{k}(z+1)^{2}}{z^{3}}dz+dF=\dfrac{a_{0}+a_{1}rz}{(k+2)rw}dz,$

where

$a_{0}=-(k+1)(k+2)r^{2}+2k(k+2)r-k(k-1),\qquad a_{1}=2(2k+1).$

Since the right-hand side is a holomorphic differential on $M_{k}$, the loop
$\gamma_{1}$ can be collapsed over the interval $[-1/r,0]$ by the Stokes
theorem. Thus $f$ is well-defined on $M_{k}$ if and only if

(2.7) $p(r):=\int_{-1/r}^{0}\frac{a_{0}+a_{1}rz}{r|w|}dz=0.$

Changing the coordinate $z=-t/r$, the equation (2.7) becomes

(2.8) $p(r)=|r|^{\frac{-2k}{k+1}}\int_{0}^{1}\frac{a_{0}-a_{1}t}{W(t)}dt,$

where

(2.9) $W(t)=\sqrt[k+1]{\frac{t(t+r^{2})}{1-t}}.$

Straightforward computations give that

$\displaystyle p_{+}(0):=\lim_{r>0,r\to 0}p(r)=-\infty,$ $\displaystyle
p(+\infty):=\lim_{r\to\infty}p(r)=-(k+1)(k+2)\int_{0}^{1}\sqrt[k+1]{\dfrac{1-t}{t}}dt<0,$
$\displaystyle p_{-}(0):=\lim_{r<0,r\to 0}p(r)=-\infty,$ $\displaystyle
p(-\infty):=\lim_{r\to-\infty}p(r)=-(k+1)(k+2)\int_{0}^{1}\sqrt[k+1]{\dfrac{1-t}{t}}dt<0.$

We want to prove that $p(k/(k+1))>0$ for any $k\geq 2$. It is clear that

$p(k/(k+1))=\frac{(k+1)(2k+1)}{k}\int_{0}^{1}\frac{1-2(1+1/k)t}{\sqrt[k+1]{t(1+(1+1/k)^{2}t)/(1-t)}}dt.$

Define the function $q(k)$ by

$q(k):=\int_{0}^{1}\frac{1-2(1+1/k)t}{V(t)}dt,$

where

$V(t)=\sqrt[k+1]{\frac{t(1+(1+1/k)^{2}t)}{1-t}}.$

A straightforward computation gives that

$\frac{1-2(1+1/k)t}{V(t)}dt-d\left(\frac{(k+1)t(1-t)}{kV(t)}\right)=\frac{(k+1)^{2}(1-t)t}{k\left((k+1)^{2}t+k^{2}\right)V(t)}dt.$

Hence we see that

$q(k)=\int_{0}^{1}\frac{1-2(1+1/k)t}{V(t)}dt=\int_{0}^{1}\frac{(k+1)^{2}(1-t)t}{k\left((k+1)^{2}t+k^{2}\right)V(t)}dt>0.$

Therefore, we get $p(k/(k+1))>0$ for any $k\geq 2$. It is clear that

$p(1)=-2\int_{0}^{1}\frac{(2k+1)t-k+1}{U(t)}dt,$

where

$U(t)=\sqrt[k+1]{\frac{t(1+t)}{1-t}}.$

Since

$\frac{(2k+1)t-k+1}{U(t)}dt+d\left(\frac{(k^{2}-1)t(1-t)}{kU(t)}\right)=\frac{t((2k+1)t+3)}{k(1+t)U(t)}dt,$

it is easy to check

$p(1)=-2\int_{0}^{1}\frac{(2k+1)t-k+1}{U(t)}dt=-2\int_{0}^{1}\frac{t((2k+1)t+3)}{k(1+t)U(t)}dt<0.$

As a consequence, $p$ has at least two roots in $(0,1)$.

Let us show that $p$ has exactly two real roots on
$\mathbb{R}\backslash\\{0,1\\}$.

Define $A_{0},A_{1}:\mathbb{R}\to\mathbb{R}_{>0}$ by

$A_{0}(r)=\int_{0}^{1}\frac{dt}{W(t)},\qquad
A_{1}(r)=\int_{0}^{1}\frac{t}{W(t)}dt,$

where $W(t)$ is given in (2.9). It is clear that (2.8) becomes

$p(r)=|r|^{\frac{-2k}{k+1}}(a_{0}A_{0}-a_{1}A_{1}).$

Suppose that $r_{0}\in\mathbb{R}$ satisfies $p(r_{0})=0$, then

$A_{1}(r_{0})=\frac{a_{0}(r_{0})}{a_{1}(r_{0})}A_{0}(r_{0})=-\frac{(k+1)(k+2)r_{0}^{2}-2k(k+2)r_{0}+k(k-1)}{2(2k+1)}A_{0}(r_{0}),$

hence we see that

(2.10)
$0\leq\frac{k(k+2)-\sqrt{k(k+2)(2k+1)}}{(k+1)(k+2)}<r_{0}<\frac{k(k+2)+\sqrt{k(k+2)(2k+1)}}{(k+1)(k+2)}.$

Therefore, we can conclude $r_{0}\in(0,\infty)$. On the other hand,

$A_{0}^{\prime}(r)=-\dfrac{2r}{k+1}\int_{0}^{1}\dfrac{dt}{(t+r^{2})W(t)},\qquad
A_{1}^{\prime}(r)=-\dfrac{2r}{k+1}\int_{0}^{1}\dfrac{t}{(t+r^{2})W(t)}dt.$

Since

$\displaystyle-\dfrac{2r}{(k+1)(t+r^{2})W(t)}dt+d\left(\dfrac{2t(1-t)}{r(r^{2}+1)W(t)}\right)$
$\displaystyle\phantom{-}=\frac{2\left(-1+k-r^{2}\right)}{(k+1)\left(r^{3}+r\right)W(t)}dt-\frac{2(2k+1)t}{(k+1)\left(r^{3}+r\right)W(t)}dt$

and

$\displaystyle-\dfrac{2rt}{(k+1)(t+r^{2})W(t)}dt-d\left(\dfrac{2rt(1-t)}{(r^{2}+1)W(t)}\right)$
$\displaystyle\phantom{-}=-\frac{2kr}{(k+1)(r^{2}+1)W(t)}dt+\frac{2r(2k+1)t}{(k+1)(r^{2}+1)W(t)}dt,$

it is straightforward to see that

$\displaystyle p^{\prime}(r)=b_{0}A_{0}+b_{1}A_{1}\qquad(r>0),$ $\displaystyle
p^{\prime\prime}(r)=c_{0}A_{0}+c_{1}A_{1}\qquad(r>0),$

where

$\displaystyle b_{0}$
$\displaystyle=\frac{2kr^{\frac{-3k-1}{k+1}}\left(-(k+1)(k+2)r^{3}+(k-1)r^{2}+(k+2)(k-1)r+k-1\right)}{(k+1)\left(r^{2}+1\right)},$
$\displaystyle b_{1}$
$\displaystyle=\frac{2k(2k+1)r^{\frac{-3k-1}{k+1}}\left((k+1)r^{2}-2(k+2)r+k+1\right)}{(k+1)\left(r^{2}+1\right)},$
$\displaystyle c_{0}$
$\displaystyle=\frac{2kr^{\frac{-4k-2}{k+1}}\left(2(k+1)^{2}(k+2)r^{3}-(k+1)(4k^{2}+5k-3)r^{2}\right)}{(k+1)^{2}\left(r^{2}+1\right)}$
$\displaystyle\phantom{=}\quad+\frac{2kr^{\frac{-4k-2}{k+1}}\left(-2(k^{2}+k-2)r-(k+3)(k-1)\right)}{(k+1)^{2}\left(r^{2}+1\right)},$
$\displaystyle c_{1}$
$\displaystyle=\frac{2k(2k+1)r^{\frac{-4k-2}{k+1}}\left((k+1)^{2}r^{2}+2(k+1)(k+2)r-(3k^{2}+6k-1)\right)}{(k+1)^{2}\left(r^{2}+1\right)}.$

If $p(r_{0})=0$, then $A_{1}(r_{0})=(a_{0}(r_{0})/a_{1}(r_{0}))A_{0}(r_{0})$,
and so

(2.11) $\displaystyle p^{\prime}(r_{0})$
$\displaystyle=\frac{k(k+2)r_{0}^{-\frac{3k+1}{k+1}}Q_{1}(r_{0})}{(1+k)(r_{0}^{2}+1)}A_{0}(r_{0}),$
(2.12) $\displaystyle p^{\prime\prime}(r_{0})$
$\displaystyle=\frac{k(k+2)r_{0}^{\frac{-4k-2}{k+1}}Q_{2}(r_{0})}{(k+1)\left(r_{0}^{2}+1\right)}A_{0}(r_{0}),$

where

$\displaystyle Q_{1}(r)$ $\displaystyle=-(k+1)^{2}r^{4}+2(k+1)(2k+1)r^{3}$
$\displaystyle\phantom{=}\quad-2(k+1)(3k+1)r^{2}+2(2k-1)(k+1)r-(k-1)^{2},$
$\displaystyle Q_{2}(r)$
$\displaystyle=-(k+1)^{2}r^{4}+2(k+1)(3k+1)r^{2}-4(k+1)(2k-1)r+3(k-1)^{2}.$

Since $Q_{1}^{\prime\prime}(r)<0$ for all $r\in\mathbb{R}$, $Q_{1}$ has at
most two real roots. Straightforward computations give that

$\displaystyle Q_{1}(0)=-(k-1)^{2}<0,$ $\displaystyle
Q_{1}(r_{0}^{-})=\frac{4(k^{2}+k+1)(-1-2k+\sqrt{k(k+2)(2k+1)})}{(k+1)(k+2)^{2}}>0,$
$\displaystyle Q_{1}\left(\frac{k}{k+1}\right)=-\frac{1+2k}{(k+1)^{2}}<0,$

where

(2.13) $r_{0}^{-}:=\frac{k(k+2)-\sqrt{k(k+2)(2k+1)}}{(k+1)(k+2)}.$

Hence the intermediate value theorem yields the existence of the exactly two
real numbers $s_{0}\in(0,r_{0}^{-})$ and $s_{1}\in(r_{0}^{-},k/(k+1))$ such
that $Q_{1}(s_{0})=Q_{1}(s_{1})=0$. Moreover, $Q_{1}|_{(0,s_{0})}<0$,
$Q_{1}|_{(s_{0},s_{1})}>0$, and $Q_{1}|_{(s_{1},\infty)}<0$ hold. As
$A_{0}(r_{0})>0$, the sign of $p^{\prime}(r_{0})$ coincides with that of
$Q_{1}(r_{0})$. Recall that $0\leq r_{0}^{-}<r_{0}$ by (2.10) and (2.13).
Since $p^{\prime}(r_{0})>0$ while $r_{0}\in(r_{0}^{-},s_{1})$, $p$ has at most
one root lying in $(r_{0}^{-},s_{1})$. Similarly, since $p^{\prime}(r_{0})<0$
while $r_{0}\in(s_{1},\infty)$, $p$ has at most one root lying in
$(s_{1},\infty)$.

It only remains to prove $r_{0}\neq s_{1}$ or $Q_{2}(s_{1})<0$ holds. In the
case of $k=2$,

$\displaystyle Q_{1}(r)$ $\displaystyle=-9r^{4}+30r^{3}-42r^{2}+18r-1,$
$\displaystyle Q_{2}(r)$ $\displaystyle=-9r^{4}+42r^{2}-36r+3.$

A direct computation shows that

$\displaystyle s_{1}$
$\displaystyle=\frac{5}{6}-\frac{1}{6}\sqrt{A}+\frac{1}{6}\sqrt{-A+\frac{62}{\sqrt{A}}-9}\approx
0.599176,\quad Q_{2}(s_{1})\approx-4.65184<0,$

where $A=2\sqrt[3]{3\sqrt{197}+46}+2\sqrt[3]{46-3\sqrt{197}}-3$.

In the case of $k\geq 3$, the discriminant $D(Q_{2})$ of $Q_{2}$ satisfies

$D(Q_{2})=-256(k+1)^{6}\left(432k^{4}-1448k^{3}+1011k^{2}-162k+19\right).\\\ $

A straightforward computation gives that

$\displaystyle 432k^{4}-1448k^{3}+1011k^{2}-162k+19$
$\displaystyle\quad=432(k-3)^{4}+3736(k-3)^{3}+11307(k-3)^{2}+13464(k-3)+4528.$

Therefore $D(Q_{2})<0$ and hence $Q_{2}$ has two real roots and two complex
conjugate roots. On the other hand, since

$\displaystyle Q_{2}(0)$ $\displaystyle=3(k-1)^{2}>0,$ $\displaystyle
Q_{2}\left(\frac{k}{k+1}\right)$ $\displaystyle=4-4k-\frac{1}{(k+1)^{2}}<0,$
$\displaystyle Q_{2}\left(r_{0}^{-}\right)$
$\displaystyle=\dfrac{-4(k^{5}+2k^{4}-3k^{3}-7k^{2}-8k-3)}{(k+1)^{2}(k+2)^{2}}$
$\displaystyle\phantom{=}+\dfrac{4(k^{3}-2k^{2}-3k-2)\sqrt{k(k+2)(2k+1)}}{(k+1)^{2}(k+2)^{2}}<0,$

$Q_{2}$ cannot have a real root in ($r_{0}^{-},k/(k+1)$), and hence
$Q_{2}(s_{1})<0$.

This proves that $p$ has exactly two real roots $r_{1}$ and $r_{2}$ on
$\mathbb{R}\backslash\\{0,1\\}$. ∎

By the above proof, we see that both $r_{1}$ and $r_{2}$ satisfy
$r_{1},r_{2}\in(0,1)$. The values $r_{1}$ and $r_{2}$ can be estimated using
the Mathematica software. See Figure 6 for $k=2,3,4$.

---
$k=2$ $k=3$ $k=4$
Figure 6: The period function $p(r)$.

######  Remark 2.3.

For $k=2$ and $r=r_{1}\approx 0.478169$, we see that the singular set of
$f^{\prime}$ consists of 12 swallowtails, 15 cuspidal cross caps, and cuspidal
edges by applying the criteria in [UY] and [FSUY] numerically. See Figure 7.

$\bigcirc$$\bigcirc$$\bigcirc$$\bigcirc$$\bigcirc$$\bigtriangleup$$\bigtriangleup$$\bigtriangleup$$\bigtriangleup$

Figure 7: The singular set of $f^{\prime}$ ($k=2,r=r_{1}$) in the unit disk of
the projection to the $z$-plane of $M_{k}$. The thick curves indicate the
singular points. $\bigcirc$ indicates a cuspidal cross cap and
$\bigtriangleup$ indicates a swallowtail. The other singularities are cuspidal
edges.

##  3 Symmetry

We consider the symmetry of the surfaces constructed in Theorem 2.2. Let
$f^{\prime}:M^{\prime}\to\mathbb{L}^{3}$ be the nonorientable complete maxface
constructed in Theorem 2.2 with the maxface
$f=f^{\prime}\circ\pi:M_{k}\to\mathbb{L}^{3}$. An isometry $S:M_{k}\to M_{k}$
is said to be a symmetry of $f$ if there exists a Lorentzian isometry
$\tilde{S}:\mathbb{L}^{3}\to\mathbb{L}^{3}$ such that $f\circ S=\tilde{S}\circ
f$. We denote by Sym($M_{k}$) the symmetry group of $M_{k}$.

######  Proposition 3.1.

The surface $f:M_{k}\to\mathbb{L}^{3}$ intersects the $(x_{1},x_{2})$-plane in
$(k+1)$ straight lines which meet at equal angles at the origin, and the
symmetry group Sym$(M_{k})$ of the surface $f:M_{k}\to\mathbb{L}^{3}$ is equal
to the dihedral group $D(k+1)$ generated by $\kappa_{1}$ and $\kappa_{2}$ in
$(\ref{eq:k1k2})$.

###### Proof.

We will determine Sym$(M_{k})$. The elements of the dihedral group $D(k+1)$
generated by $\kappa_{1},\kappa_{2}$ in (2.4) can be described as follows:

  * •

$K_{1}^{j}$ is rotation by $-2\pi j/(k+1)$ about the $x_{3}$-axis,
$j=1,\dots,k+1$,

  * •

$K_{1}^{j}K_{2}$ is the reflectional symmetry with respect to the straight
line $x_{2}=-\tan((k-1)j\pi/(k+1))x_{1},x_{3}=0,j=1,\dots,k+1$,

where $K_{1},K_{2}$ are given in (2.6). Thus $D(k+1)\subset{\rm{Sym}}(M_{k})$.

Next, we observe the asymptotic behavior of the ends. At $(z,w)=(0,0)$, $w$ is
a local coordinate for the Riemann surface $\overline{M_{k}}$, and then

$z=z(w)=w^{k+1}\left(-\frac{1}{r}+\mathcal{O}(w^{k+1})\right).$

We have

(3.1) $\displaystyle g$
$\displaystyle=\frac{\alpha_{0}}{w}+\mathcal{O}(w^{k}),$ $\displaystyle\eta$
$\displaystyle=\left(\frac{i\alpha_{1}}{w^{k+1}}+\mathcal{O}(1)\right)dw,$
$\displaystyle g\eta$
$\displaystyle=\left(\frac{i\alpha_{2}}{w^{k+2}}+\mathcal{O}(w^{k})\right)dw,$
$\displaystyle g^{2}\eta$
$\displaystyle=\left(\frac{i\alpha_{3}}{w^{k+3}}+\frac{\alpha_{4}}{w^{2}}+\mathcal{O}(w^{k-1})\right)dw,$

where $\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}$ and
$\alpha_{4}\in\mathbb{C}$ are constants. Note that $g\eta$ is exact (see
(2.2)). This implies that the end of $f$ is asymptotic to the end of the
surface determined by the Weierstrass data

$(g_{0},\eta_{0})=\left(\frac{1}{z},\frac{i}{z^{k+1}}dz\right)\text{\quad on
\quad$\mathbb{D}_{\varepsilon}^{*}=\\{z\in\mathbb{C}\>|\>0<|z|<\varepsilon\\}$}.$

Moreover, let $f_{0}:\mathbb{D}_{\varepsilon}^{*}\to\mathbb{L}^{3}$ be the end
of maxface with the Weierstrass data $(g_{0},\eta_{0})$. Then

$\displaystyle f_{0}$
$\displaystyle=\left(\frac{-\sin(k\theta)}{\rho^{k}k}-\frac{\sin(k+2)\theta}{\rho^{k+2}(k+2)},\frac{\cos(k\theta)}{\rho^{k}k}-\frac{\cos(k+2)\theta}{\rho^{k+2}(k+2)},\frac{-2\sin(k+1)\theta}{\rho^{k+1}(k+1)}\right)$
$\displaystyle=(x_{1},x_{2},x_{3}),$

where $z=\rho e^{i\theta}$. Since

$x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=\frac{1}{(k+2)^{2}\rho^{2(k+2)}}+\frac{1}{\rho^{2k+2}}\left(\frac{-2\cos
2\theta}{k(k+2)}+\frac{4\sin^{2}(k+1)\theta}{(k+1)^{2}}+\mathcal{O}(\rho^{2})\right),$

we have

$\mathcal{D}=f_{0}^{-1}(\\{(x_{1},x_{2},x_{3})\in\mathbb{L}^{3}\>|\>x_{1}^{2}+x_{2}^{2}-x_{3}^{2}\geq
R\\})\cup\\{0\\}\subset\overline{\mathbb{D}_{\varepsilon}}$

is a conformal disk if $R$ is large enough. Since Sym($M_{k}$) leaves
$\mathcal{D}$ invariant and fixes $0$, the group
$\\{S|_{\mathcal{D}}\>|\>S\in{\rm{Sym}}(M_{k})\\}$ is included in $D(k+1)$.
Therefore ${\rm{Sym}}(M_{k})=D(k+1)$. ∎

## Acknowledgements

The authors would like to thank Professor Francisco J. López for valuable
comments and suggestions.

## References

  * [C] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math. 15 (1970), 223–230.
  * [CY] S.-Y. Cheng and S.-T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. 104 (1976), 407–419.
  * [ER] F. J. M. Estudillo and A. Romeo, Generalized maximal surfaces in Lorentz-Minkowski space $\mathbb{L}^{3}$, Math. Proc. Camb. Phil. Soc. 111 (1992), 515–524.
  * [FeL] I. Fernández and F. J. López, Periodic maximal surfaces in the Lorentz-Minkowski space $\mathbb{L}^{3}$, Math. Z. 256 (2007), 573–601.
  * [FLS1] I. Fernández, F. J. López and R. Souam, The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space, Math. Ann. 332 (2005), 605–643.
  * [FLS2] I. Fernández, F. J. López and R. Souam, The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space $\mathbb{L}^{3}$, Manuscripta Math. 122 (2007), 439–463.
  * [FKKRSUYY] S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada, and S.-D. Yang, Zero mean curvature surfaces in Lorentz-Minkowski $3$-space and $2$-dimensional fluid mechanics, Math. J. Okayama Univ. 57 (2015), 173–200.
  * [FuL] S. Fujimori and F. J. López, Nonorientable maximal surfaces in the Lorentz-Minkowski 3-space, Tohoku Math. J. 62 (2010), 311–328.
  * [FSUY] S. Fujimori, K. Saji, M. Umehara, and K. Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), 827–848.
  * [Ko] O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space $\mathbb{L}^{3}$, Tokyo J. Math. 6 (1983), 297–309.
  * [KU] M. Kokubu and M. Umehara, Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces, Math. Nachr. 284 (2011), 1903–1918.
  * [Ku] R. Kusner, Conformal Geometry and complete minimal surfaces, Bull. Amer. Math. Soc. (N. S.) 17 (1987), 291–295.
  * [L] F. J. López, A complete minimal Klein bottle in $\mathbb{R}^{3}$, Duke Math. J. 71 (1993), 23–30.
  * [LM] F. J. López and F. Martín, Complete nonorientable minimal surfaces and symmetries, Duke Math. J. 79 (1995), 667–686.
  * [M] W. H. Meeks ${\rm{III}}$, The classification of complete minimal surfaces in $\mathbb{R}^{3}$ with total curvature greater than $-8\pi$, Duke Math. J. 48 (1981), 523–535.
  * [MW] W. H. Meeks ${\rm{III}}$ and M. Weber, Bending the helicoid, Math. Ann. 339 (2007), 783–798.
  * [O] M. Elisa G. G. de Oliveira, Some new examples of nonorientable minimal surfaces, Proc. Amer. Math. Soc. 98 (1986), 629–636.
  * [UY] M. Umehara and K. Yamada, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006), 13–40.

Shoichi Fujimori
Department of Mathematics
Hiroshima University
Higashihiroshima, Hiroshima 739-8526, Japan
E-mail address<EMAIL_ADDRESS>

Shin Kaneda
Department of Mathematics
Hiroshima University
Higashihiroshima, Hiroshima 739-8526, Japan
E-mail address<EMAIL_ADDRESS>