11institutetext: Departament de Física, Universitat Politècnica de Catalunya,
c/Esteve Terrades 5, 08860 Castelldefels, Spain 22institutetext: Institute for
Space Studies of Catalonia, c/Gran Capità 2–4, Edif. Nexus 104, 08034
Barcelona, Spain

# White dwarf Random Forest classification through _Gaia_ spectral
coefficients

Enrique Miguel García-Zamora 11 Santiago Torres, Email;
<EMAIL_ADDRESS>Alberto Rebassa-Mansergas 1122

###### Abstract

Context. The third data release of Gaia has provided approximately 220 million
low resolution spectra. Among these, about 100 000 correspond to white dwarfs.
The magnitude of this quantity of data precludes the possibility of performing
spectral analysis and type determination by human inspection. In order to
tackle this issue, we explore the possibility of utilising a machine learning
approach, based on a Random Forest algorithm.

Aims. To analyze the viability of the Random Forest algorithm for the spectral
classification of the white dwarf population within 100 pc from the Sun, based
on the Hermite coefficients of Gaia spectra.

Methods. We utilized the assigned spectral type from the Montreal White Dwarf
Database for training and testing our Random Forest algorithm. Once validated,
our algorithm model is applied to the rest of unclassified white dwarfs within
100 pc. First, we started by classifying the two major spectral type groups of
white dwarfs: hydrogen-rich (DA) and hydrogen-deficient (non-DA). Next, we
explored the possibility of classifying the various spectral subtypes,
including in some cases the secondary spectral types.

Results. Our Random Forest classification presented a very high recall (¿80%)
for DA and DB white dwarfs, and a very high precision (¿90%) for DB, DQ and DZ
white dwarfs. As a result we have assigned a spectral type to 9 446 previously
unclassified white dwarfs: 4 739 DAs, 76 DBs (60 of them DBAs), 4 437 DCs, 132
DZs and 62 DQs (9 of them DQpec).

Conclusions. Despite the low resolution of Gaia spectra, the Random Forest
algorithm applied to the Gaia spectral coefficients proves to be a highly
valuable tool for spectral classification.

###### Key Words.:

(Stars:) white dwarfs, atmospheres, catalogs

††offprints: S. Torres

##  1 Introduction

White dwarfs are the remnants of stars with initial masses $\la$8-10 M⊙ (e.g.
Althaus et al., 2010). They are basically composed of a degenerate core of
typically half a solar mass that is surrounded by a thin partially degenerate
atmospheric layer. Since nuclear reactions have practically ceased, the energy
source in the deep interior of white dwarfs is primarily derived from
gravothermal energy released by the ions and eventually provided by core
crystallization, phase separation, and other processes such as sedimentation
of minor species (see Isern et al. 2022 for a recent review). The heat
generated in the core of the white dwarf is radiated through the atmospheric
envelope. Thus, this thin layer plays a capital role in the cooling of the
white dwarf. In the canonical model, the outermost layer of a white dwarf is
primarily composed of helium with a mass around $10^{-2}$ M⊙, accounting for
less than 2% of the total white dwarf mass. However, in the majority of cases
(approximately 80%), there is an additional thinner layer of hydrogen with a
mass between $10^{-15}$ to $10^{-4}$ M⊙, which overlays the helium layer.

From an observational point of view, spectroscopic analysis of white dwarf
atmospheres enables the identification of atomic and molecular lines and
bands. This fact has allowed a spectral classification of white dwarfs
attending the presence of certain lines (Sion et al., 1983). Basically, white
dwarfs are divided into those that present Balmer lines (i.e. hydrogen-rich
white dwarfs, or DAs), and those that do not (generically called non-DAs).
Among this last group, we may also find white dwarf spectra that exhibit
absorption helium lines, He I or He II, called DB and DO, respectively; carbon
features, either atomic or molecular, named DQ; metallic lines such as Ca II
or Fe II, named DZ; or very weak lines or no features at all, thus showing a
continuous spectrum, named DC. This general spectral classification relates to
what is referred to as the primary spectral type (see Table 2 from Sion et
al., 1983). However, it is common to identify lines from different elements in
white dwarf spectra. For instance, we may find a DA with weaker helium lines
or metallic lines additionally present, in which case these objects will be
labelled as DAB or DAZ, respectively. The presence of a magnetic field or
variability in the white dwarf spectrum, will add a secondary H or V,
respectively to the primary spectral class.

Spectral classification of white dwarfs is of paramount importance for the
determination of their stellar parameters such as temperature, surface
gravity, mass or luminosity. Moreover, our understanding of the physical
evolution of the white dwarf population depends on the proper identification
of their atmospheres. For instance, processes like convective mixing or
convective dilution in spectral evolution (e.g Blouin et al., 2019; Cunningham
et al., 2020), the presence of carbon in hydrogen-deficient atmospheres as a
possible explanation of the Gaia color-magnitude bifurcation (Camisassa et
al., 2023; Blouin et al., 2023), the high ratio of DQ white dwarfs in the so-
named Q branch (Tremblay et al., 2019) or the origin of accreted material in
white dwarfs (e.g Zuckerman et al., 2007; Farihi et al., 2010), are a few
examples where a detailed identification of white dwarf spectra is required
for a proper understanding of these issues. However, spectroscopic follow-up
of white dwarfs is a time cost demanding task. A volume complete spectroscopic
sample is achieved up to 40 pc from the Sun (Tremblay et al., 2020; McCleery
et al., 2020; O’Brien et al., 2023), but this is not the case up to 100 pc,
where the percentage of spectral labelled white dwarfs is roughly 20% (e.g.
Kilic et al., 2020).

Nevertheless, the third Gaia mission Data Release (Gaia Collaboration et al.,
2023) has provided astrometric data for nearly two billion objects and mean
low resolution BP and RP spectra of approximately 220 million sources (De
Angeli et al., 2022). Of these, almost 100 000 correspond to candidates for
white dwarf objects (Gentile Fusillo et al., 2021).

This enormous quantity of data prevents spectral classification by human
inspection. With the recent increasing growth of large astronomical databases,
other approaches based on machine learning artificial intelligence algorithms
are absolutely necessary. These techniques are widely used nowadays in
astrophysics, and particularly in the field of white dwarfs. Since the
pioneering work of Torres et al. (1998) on the use of self-organizing maps for
the identification of halo stars, up to the most recent ones using the Random
Forest algorithm in Galactic component identification (Torres et al., 2019),
or their spectral identification (Echeverry et al., 2022; Montegriffo et al.,
2023), or through deep learning techniques (Kong et al., 2018; Vincent et al.,
2023), all of these approaches have been proven to be reliable methods in the
automatic analysis of large white dwarf databases. Additional statistical
classification methods have been performed, in particular in the spectral
classification of white dwarfs. For instance, in Jiménez-Esteban et al. (2023)
and Torres et al. (2023), the Virtual Observatory Spectral energy distribution
Analyzer tool (Bayo et al., 2008) was used to conduct an automated spectral
energy distribution (SED) fitting of the 100 pc and 500 pc Gaia white dwarf
samples, respectively, to different atmospheric models. These works allowed
the authors to classify the samples into DA and non-DA white dwarfs with an
accuracy of over 90 per cent.

In this work we apply a Random Forest algorithm specifically developed to
classify, for the first time, the whole 100-pc white dwarf Gaia sample into
their spectral types. Focusing the analysis to objects identified within this
distance limit is capital, since it represents a nearly-complete volume-
limited sample, which potentially allows to derive accurate percentages of the
different spectral type classes among white dwarfs. This approach based on
artificial intelligence techniques represents a clear advantage respect other
approaches we performed in Jiménez-Esteban et al. (2023) and Torres et al.
(2023) since it does not require the use of theoretical atmospheric models.
The models are subject to substantial uncertainties for temperatures below 5
500 K, which implies that unreliable classifications result for such cool
white dwarfs. Instead, the Random Forest algorithm presented here relies on
previously spectral type labelled white dwarfs covering all possible values of
effective temperatures. Thus, we aim to obtain as much spectroscopic
information as possible from the Gaia spectral coefficients. This includes not
only a classification of the white dwarfs into their primary spectral types,
but also attempting to classify them into different subcategories.

In Section 2 we explain the methodology applied. In Section 3, the validation
tests performed on a subset of white dwarfs with spectral data assigned and
their results are detailed. In Section 4, we apply the algorithm to the
classification of white dwarfs in a 100-pc radius around the Sun. In Section 5
we identify the most relevant spectral coefficients used in the classification
process. The performance of our Random Forest algorithm is compared in Section
7 to other recent classification methods. Finally, in Section 8, we present
our conclusions.

##  2 The method: Random Forest classification of Gaia spectral coefficients

The Random Forest (Breiman, 2001) is a widely used machine learning algorithm.
From a set of labelled data, which is used to train the algorithm, an ensemble
of decision trees, which is called a Random Forest, is created. Once this
ensemble has been obtained, it can be used to classify new data in the given
categories.

This algorithm has been widely used for the classification of stellar objects
(see, for instance, Li et al. 2019, Plewa 2018 or Dubath et al. 2011) and, in
particular, to the study of the white dwarf population: some examples already
show the feasibility of using Random Forest for the identification of
different Galactic white dwarf populations using Gaia data as input parameters
(Torres et al., 2019) or distinguishing between spectra of isolated white
dwarfs, main sequence objects and white dwarf-main sequence binaries
(Echeverry et al., 2022). Moreover, a Random Forest algorithm has also been
used for the selection of white dwarfs in the Gaia sample (Gaia Collaboration
et al., 2021). Besides, a first attempt to classify white dwarfs into DAs and
non-DAs using the spectral coefficients was performed by Montegriffo et al.
(2023).

Here, following the line of the previous works, we aimed to apply the Random
Forest algorithm for the spectral classification of the Gaia white dwarf
population in a 100-pc radius around the Sun. In particular, our effort is
employed in classifying the different sub-populations of the non-DA sample
identified by Jiménez-Esteban et al. 2023 as well as to extend the
classification to cool white dwarfs ($\la 5\,500$ K) that the previous work
did not consider due to the lack of accurate atmospheric models.

The Gaia spectra have low-resolution ($\lambda/\Delta\lambda\approx 100$) and
cover the 3 300-10 500 Å wavelength range (3 300-6 800 Å by the Blue
Photometer (BP) and 6 400-10 500 Å by the Red Photometer (RP); Carrasco et al.
2021). One particularity of the Gaia spectra is that they are not provided as
a typical series of flux values for certain wavelengths but rather as a set of
55 coefficients for each of the BP and RP spectrographs (i.e. 110 coefficients
in total). These coefficients refer to the Hermite functions that act as the
basis for the spectral representation (Carrasco et al., 2021). The spectra are
internally calibrated in a pseudo-pixel scale, and they can also be
transformed to an external calibration (i.e. flux versus wavelength
representation) by using the specifically designed Python package
GaiaXPy111https://gaia-dpci.github.io/GaiaXPy-website/.

As input data for the Random Forest algorithm, we use the 110 Hermite
coefficients. It was demonstrated in Montegriffo et al. (2023) that the use of
the coefficients provides better performance of the classification algorithm
than when other input passbands were used. Moreover, the coefficient procedure
can be considered totally appropriate, as the different white dwarf spectral
types are defined by their specific spectral features, and all this
information is contained in the coefficients (see, for instance, Weiler et al.
2023 for a mathematical description applied to hydrogen lines). As a
consequence, no external calibration was applied, since this process may
introduce what is known as ‘wiggles’, or oscillatory behaviour. In our data,
this effect would be more prominent at both ends of the externally calibrated
spectra. These ‘wiggles’ are produced by the mathematical process used to
obtain the spectra (De Angeli et al., 2022).

##  3 Training and validating the algorithm: the Montreal White Dwarf database

The Montreal White Dwarf Database is a
virtual222https://www.montrealwhitedwarfdatabase.org/ database containing
astrometric, photometric and spectroscopic data including a spectral type
classification for tens of thousands of white dwarfs (Dufour et al., 2017). A
total of 41 570 white dwarfs are classified into their different spectral
types, 2 905 of them within 100 pc from the Sun. In this work, the MWDD
spectroscopic white dwarf classification within 100 pc is used as the input
labelled sample for the training and validation tests of our Random Forest
algorithm.

For the cross-validation, we adopt the stratified $k$-fold method. It consists
in dividing the whole set into $k$ folds, where $k$ is a variable number (in
this work we chose $k=10$). Each fold has approximately the same number of
objects, and the category ratio (the proportion of objects assigned to
different spectral types) is kept as close as possible to the original set
category ratio. For each fold, a Random Forest is trained with all nine
remainder folds, and tested on it. The advantages over the random training-
test split consists in avoiding the randomness of the subset divisions, the
constancy of the subset proportions and the fact that the whole set is used
for both training and testing.

As it is well established, the Random Forest performance tends to be optimal
for balanced data sets (e.g. Breiman, 2001). Consequently, the validation
strategy followed consisted of keeping the classification samples as close as
possible to a balanced sample. Thus, our first validation test consisted in
classifying white dwarfs into DA and non-DA spectral types. The second one,
focused on those labelled as non-DA, and classified them into DB, DC, DQ and
DZ types. Finally, the third one consisted of classifying the white dwarfs of
an specific type into its different subtypes. For instance, DA white dwarfs
are divided into DA, DAB, DAH, DAP and DAZ, and similarly for the other
spectral types.

In order to create the random forests and obtain the confusion matrices and
classification metrics, the Python package scikit-learn (Pedregosa et al.,
2011) was used.

Figure 1: Confusion matrices for our validation tests: DA vs non-DA (top
panel), non-DA types (middle panel) and DB subtypes (bottom panel). As true
label (rows) we adopted the MWDD classification, while the predicted label
(columns) is the one resulting from our Random Forest algorithm.

###  3.1 First validation test: classifying the Gaia population with a MWDD
type into DA and non-DA types

In the first place, we classify the whole sample of white dwarfs with labels
in the MWDD within 100 pc into DA and non-DA types (2,905 objects; 1993 as DAs
and 912 as non-DAs). Although the ratio of DA and non-DA, 68.6% and 31.4%,
respectively, is not strictly balanced, the proportion of the two groups is
large enough to ensure an optimal performance of the Random Forest algorithm
and avoid extreme imbalance effects.

The resulting confusion matrix of our Random Forest model is shown in the top
panel of Figure 1. True labels (rows) correspond to the MWDD classification,
while the label predicted by our algorithm is shown as columns. The total
number of objects are indicated in each element of the matrix, and in the line
below, we indicate the percentage of objects relative to a group. An ideal
classification case would correspond to a diagonal matrix. The hyperparameters
used for this validation test are shown in the center column of Table 1 and
the resulting metrics are collected in Table 2. For a description of the
metrics, we refer the reader to Appendix A in Echeverry et al. (2022).

The analysis of the results indicates that the performance of the Random
Forest algorithm presents and excellent recall333The recall of sub-class $i$
is defined as $r_{i}={\frac{a_{ii}}{\sum_{i=1}^{n}a_{ij}}}$, where $a_{ij}$
indicate the number of objects of true class $i$ classified as class $j$. for
DA white dwarfs (95%), and a very good recall for non-DAs (79%). A global
accuracy of 0.90 is achieved. Similar values are obtained in Jiménez-Esteban
et al. (2023) and Torres et al. (2023).

Table 1: Hyperparameters and optimal values adopted in the first two validation tests. | Optimal value
---|---
Hyperparameter | Test 1 | Test 2
Number of estimators | 10000 | 10000
Maximum depth | 50 | 70
Minimum samples split | 20 | 2
Minimum samples leaf | 1 | 1
Maximum features | sqrt | sqrt

###  3.2 Second validation test: classifying the Gaia non DA white dwarf
population with a MWDD type into their subtypes

In our second validation test, we classify the non-DA white dwarfs (912
objects in total) into their spectral types (DB, DC, DQ and DZ). The resulting
confusion matrix is shown in the middle panel of Figure 1. The hyperparameters
used and the metrics obtained can be found in Tables 1 and 2, respectively.
The results obtained reveal a very good performance with an accuracy of 0.81.
In particular, the algorithm presents a very good recall for DB white dwarfs
(82%), an excellent recall for DCs (98%), and low recalls for both DQs and DZs
(¡50%). Two main reasons can be identified to account for these facts. First,
the low recall may be caused by the low resolution spectra inherent to Gaia.
That is, not very prominent spectral lines might be unnoticed in the Gaia low
resolution spectra, which would result in the algorithm treating them as
featureless, continuous spectra characteristic of DC white dwarfs. Second, DQ
and DZ classes represent 12.8% and 13.7%, respectively, of the non-DA
population used for training. Thus, imbalanced effects, worsening the
performance of the algorithm, are likely to start to manifest.

However, it must be noted that, in spite of the low recall for DQ and DZ white
dwarfs, their precision444Analogously to the recall, the precision of sub-
class $i$ is defined as $p_{i}={\frac{a_{ii}}{\sum_{j=1}^{n}a_{ij}}}$ (as well
as for DB white dwarfs) is excellent, i.e. 92% for the three types. False
positives are almost absent. This implies that, while the algorithm does not
find all DZ and DQ white dwarfs, the probability of a white dwarf belonging to
the type it has been classified into is very high. This makes our algorithm
highly useful for efficiently identifying white dwarfs of these spectral types
within an unclassified population.

Finally, as a verification exercise, we have attempted to classify the entire
sample into their primary subtypes; DA, DB, DC, DQ and DZ. An excellent recall
is achieved for DAs (97 %), DBs show a very good recall (80%); and DQ and DZ
recalls are certainly improvable (25 and 34%, respectively). Once more, DBs,
DQs and DZs show an excellent precision (89% for DBs and 91% for DQs and DZs).
However, the scoring values are lower than the values obtained in the first
two validation tests. The conclusion we extract from this result is that
better results are obtained when the workflow includes a first DA/non-DA
classification and a second, specific non-DA classification.

Table 2: Classification metrics for the first validation test in which we classify white dwarfs of the MWDD into DA and non-DA classes, and the second validation test in which non-DAs are classified into DB, DC, DQ and DZ. Metric | Validation test | DA | non-DA | DB | DC | DQ | DZ
---|---|---|---|---|---|---|---
Recall | Test 1 | 0.95 | 0.79 | - | - | - | -
Recall | Test 2 | - | - | 0.82 | 0.98 | 0.31 | 0.46
Precision | Test 1 | 0.91 | 0.87 | - | - | - | -
Precision | Test 2 | - | - | 0.92 | 0.78 | 0.92 | 0.92
F1 score | Test 1 | 0.93 | 0.83 | - | - | - | -
F1 score | Test 2 | - | - | 0.87 | 0.87 | 0.46 | 0.61

###  3.3 Third validation test: classifiying the Gaia white dwarf population
with a MWDD type into their secondary types

So far, we have demonstrated that the Random Forest algorithm based on the
coefficients of Gaia spectra is a feasible tool for classifying white dwarfs
into their primary spectral types. Now, we explore the possibility to classify
the secondary types. Several factors prevent us from being optimistic about
such performance. First, Gaia low-resolution spectra appear to have limited
capability to discern detailed features, such as distorted Balmer lines in DAH
white dwarfs or weak lines in other atmospheres. Second, subtype classes
represent in many cases clearly imbalanced samples with respect to the
predominant subtype, thus worsening the performance of the algorithm.

Nevertheless, even if we expect a very low recall in the classification of the
majority of spectral subtypes, if a high precision is achieved, it would imply
the identification of valuable objects. Consequently, the five considered
spectral types (DA, DB, DC, DZ, and DQ) were divided into different subtypes
and analyzed separately.

####  3.3.1 DA subtype classification

The DA type was divided into pure DA, DAB, DAH, DAP and DAZ. The
classification test reveals the disappointing, but not unexpected, result that
only one DAH white dwarf is correctly classified as such (1% recall). This
shows that, in almost all cases, the fine magnetic splitting of the spectral
lines produced by the intense magnetic field is too small to be discerned in
the low resolution Gaia spectra. Additionally, one DA is missclassified as a
DAH, resulting in a precision of 50% for DAHs.

The rest of the subtypes are not recognized in the classification and are all
missclassified as DAs. Low prominent lines and the spectral resolution, as
well as their low number compared to the initial sample, would explain these
results.

####  3.3.2 DB subtype classification

In this test, the DB type was divided into pure DB, DBA, DBAH, DBP, DBQA, DBZ
and DBZA. Except for DBAs, which comprise 47% of the DB sample, the other
subtypes are only a residual part. Imbalanced effects are therefore expected.

The confusion matrix (bottom panel of Figure 1) shows that, except for the DBA
type, all subtypes are misclassified. DBAH, DBP, DBZ and DBQA subtypes are
incorrectly classified as DBA; and DBZA are missclassified as 33% DB and 67%
DBA. In the DB/DBA classification, 50% of the DB and 74% of DBAs are correctly
classified.

####  3.3.3 DC subtype classification

Considered DC subtypes include pure DC, DCP and DCQ. However, the algorithm
is, as expected, unable to distinguish between these different subtypes: all
objects in this subset are classified as DC. Two possible explanations are
proposed for this result. First, the Gaia spectral resolution is likely too
low for the subtype-defining spectral classes to be noticeable. Second, the
extremely low number of DCP and DCQ objects (two for each subtype against 569
objects labelled as DC) is not enough for the algorithm to be able to properly
differentiate between them.

####  3.3.4 DQ subtype classification

The results of the Random Forest applied to the DQ subtype reveals that the
algorithm is only able to distinguish DQpec white dwarfs, and even then only
two out of seventeen (11% recall). The other subtypes (DQA, DQZ and DQZA),
which comprise at most two objects for each, are indistinguishable from DQs.

It is also worth noting, however, that precision is also perfect for the DQpec
subtype (100%), with no false positives from other subtypes. This implies that
the few DQpec stars the algorithm may find have a very high probability of
belonging to this subtype.

####  3.3.5 DZ subtype classification

Regarding DZs, the Random Forest algorithm is not able to distinguish DZH and
DZP subtypes from DZ white dwarfs. On the other hand, a single DZA (8% recall)
is properly classified. However, three DZs are mislabelled as DZA, negatively
impacting the precision for this subtype (25%).

####  3.3.6 Spectral subtype classification summary

From the analysis of the results of the Random Forest algorithm applied to the
different spectral subtypes, we can conclude that the algorithm is mostly
unable to classify secondary spectral subtypes, whether due to the numerical
imbalances or the inherent low resolution of the Gaia spectra that prevents
their spectral lines from being recognised by the algorithm.

A possible exception is the subtype DQpec, which, although shows a low recall,
also presents a perfect precision. Its situation among DQ subtypes is similar
to the situation of DQs among non-DA white dwarfs: low recall, but very high
precision that might allow us to find candidates with a very high probability
of actually belonging to the group is has been classified into.

Furthermore, although the DB/DBA classification may seem possible due to the
good recovery of DBA white dwarfs, we must take this result cautiously. While
the recall is reasonably good for DBAs (74%), it is just 50% for DBs.
Additionally, the precision is only slightly superior to 50% for both
subtypes. Consequently, we cannot assume that a white dwarf identified as a DB
or DBA has a high probability of really being one.

##  4 Classifying the Gaia  non-DA 100-pc white dwarf population

Once our Random Forest algorithm has been tested and validated, it can be
applied to the unclassified Gaia 100-pc white dwarf population. A subgroup of
these white dwarfs, namely those with ${\rm BR-RP}<0.86$ (equivalent to white
dwarfs hotter than $\ga$5 500 K), has already been classified into DAs and
non-DAs by Jiménez-Esteban et al. (2023). To that end, synthetic photometry of
all white dwarfs was generated using their spectra and the J-PAS (Benitez et
al., 2014) filter system (Marín-Franch et al., 2012). These spectra were
fitted using a collection of DA and DB atmospheric models, and a probability
for each them belonging to the DA type was computed from the $\chi^{2}$
arising from the best fits. In this exercise, Jiménez-Esteban et al. (2023)
adopted two approaches: model fits using all Gaia spectral coefficients and
model fits using the truncated coefficients. The former case, defined as the
VOSA-GJP estimator, provided better results, with an overall accuracy of 91%.
This value is slightly higher than the accuracy we have obtained here using
our Random Forest algorithm (90%). Although both classification performances
can be considered practically equivalent, we hereafter adopt the DA and non-DA
VOSA-GJP classification of Jiménez-Esteban et al. (2023) for all white dwarfs
with ${\rm BR-RP}<0.86$.

Thus, in this section we first apply our Random Forest model to those white
dwarfs classified into DA and non-DA by Jiménez-Esteban et al. (2023) with the
aim to obtain their spectral subtypes. Then, we expand the classification to
those unclassified objects with colour ${\rm BR-RP}>0.86$, i.e. the cooler
white dwarfs that we failed to identify in Jiménez-Esteban et al. (2023) due
to the lack of accurate atmospheric models.

###  4.1 White dwarfs identified by VOSA-GJP

In this section, we analyse the objects classified in Jiménez-Esteban et al.
(2023) with the aim of obtaining their spectral syb-types. In that
classification, as mentioned before, white dwarfs were assigned a probability,
$P_{\rm DA}$, of being DAs. Those with $P_{\rm DA}>0.5$ were classified as
DAs, while those with $P_{\rm DA}<0.5$ were classified as non-DAs. A total of
5 823 white dwarfs with ${\rm BR-RP}>0.86$ are considered in this section; 4
157 of them classified as DAs and 1 666 classified as non-DAs.

####  4.1.1 DA white dwarfs identified by VOSA-GJP

Figure 2: HR diagram of the classified Gaia DA 100-pc white dwarf population
in Jiménez-Esteban et al. (2023) Two DAH (blue dots) are identified by our
algorithm.

Despite the poor performance of the algorithm in classifying secondary
spectral types found in subsection 3.3.1, we attempted to find possible DAH or
DAZ candidates among the group of 4 157 white dwarfs classified as DA in
Jiménez-Esteban et al. (2023).

The result reveals that only two DAH candidates are found. Nonetheless,
despite the very low number of DAHs, we consider it as a success for our
algorithm, specially since the effect of magnetic fields in spectral lines is
fine magnetic splitting, which is not easily noticeable in low-resolution
spectra. In Figure 2 we show the location of these two DAH candidates in the
Gaia Hertzsprung-Russell (HR) diagram.

####  4.1.2 Non-DA white dwarfs identified by VOSA-GJP

We analyze now the white dwarfs that have been classified as non-DAs in
Jiménez-Esteban et al. (2023) via adopting the VOSA-GJP estimator. In order to
train the set, we once more resort to the MWDD. From the whole set, we derive
a subset that mimics the conditions of the objects that will be classified
(i.e. non-DA white dwarfs with ${\rm BR-RP}<0.86$). This left us with only 509
objects in the training set for the Random Forest algorithm, contrasting with
912 non-DA white dwarfs classified in the whole training set.

The classifying algorithm is then applied to the rest 1 666 objects in the
test subset. The classification yields the following results: 76 objects are
identified as DBs, 1 429 as DCs, 40 as DQs and 121 as DZs. The corresponding
HR diagram of these classified objects is shown in Figure 3.

The HR diagram not only serves to illustrate the composition of the classified
population, but it also allows us to check for consistency with expected white
dwarf characteristics. For instance, no DB white dwarfs should be found below
a certain temperature ($\approx 10\,000\,$K). In Figure 3, DBs appear
restricted to the top left, hotter region of the white dwarf sequence, while
some DQ white dwarfs appear in the DQ branch. All these factors reinforce the
idea that our classification is essentially correct, and no spectral types
appear outside of their expected locations.

Figure 3: HR diagram of the classified Gaia non-DA white dwarf population
within 100 pc in Jiménez-Esteban et al. (2023), divided into their different
subtypes found in this work. Figure 4: HR diagram of the classified pure DB
and DBA white dwarfs found in this work.

Furthermore, as seen in our third validation test (see Section 3.3), the
Random Forest algorithm is able to identify (although with low recall but with
high precision) secondary subtypes of DBs and DQs. As we do not expect to find
more DBs in the cooler region that remains to be analyzed, we apply our
classification algorithm to the set of 76 DB white dwarfs identified so far.
The Random Forest identified 16 pure DB and 60 DBA objects. No other secondary
subtypes (DBAH, DBAZ, DBP, DBQA, DBZ, DBZA and DBe) were identified. In Figure
4 we depict the HR diagram location for the identified pure DBs and DBAs. We
can check that no pure DBs are found with colors redder than ${\rm BP-
RP}\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}-0.1$ (i.e. effective
temperatures cooler than $\approx$12 000 K), this fact being clearly
indicative of the presence of hydrogen in pure-helium atmospheres due to
convective mixing (e.g. see Bergeron et al., 2019, and references therein).

The sample of identified DQs is analyzed into its secondary types in Section
4.3.

###  4.2 White dwarfs not identified by VOSA-GJP

Once those white dwarfs classified in Jiménez-Esteban et al. (2023) as DA and
non-DA have been further classified in their different subtypes, it was
decided to explore the cold region of the HR-diagram, i.e. ${\rm BR-RP}>0.86$,
which had not been analyzed in the aforementioned work.

As described in Section 3, our strategy consists in first classifying the cold
white dwarf sample into DAs and non-DAs; then, the non-DAs are classified into
DBs, DCs, DQs and DZs. Finally, we look for possible secondary spectral type
candidates (although this last step will probably be impracticable, as the
spectra in this region have a very low signal-to-noise ratio).

####  4.2.1 DA vs non-DA classification.

The number of white dwarfs present in the 100 pc sample from Jiménez-Esteban
et al. (2023) and with colors ${\rm BR-RP}>0.86$ (that is, with no VOSA-GJP
classification) is 3 623 objects. To that sample we apply our Random Forest
algorithm, once trained with those objects labelled in the MWDD (2 905
objects). It is worth saying that the MWDD sample contains 192 DA and 293 non-
DA white dwarfs with colors ${\rm BR-RP}>0.86$ and within 100 pc. This set of
white dwarfs guarantees the reliability of our method, as there are enough
labeled objects to train the algorithm in the HR region of interest.

The results of applying our Random Forest to the cold sample of Jiménez-
Esteban et al. (2023) can be seen in the HR diagram presented in Figure 5. The
vast majority of white dwarfs (3 041; 84%) are classified as non-DAs (blue
dots), while DAs (magenta dots) comprise only a small fraction (582; 16%) and
none of them has colour ${\rm BR-RP}>1.25$.

This classification is consistent with the expected behaviour at temperatures
lower than $\simeq$5 000 K, since at this range the hydrogen in the white
dwarf atmosphere remains mostly in the ground state. Thus, Balmer spectral
lines would become too weak (or they simply disappear) to be detected in Gaia
low resolution spectra and, consequently, the object will be classified as a
featureless DC.

Figure 5: HR diagram of the classified Gaia DA and non-DA 100-pc white dwarf
population with color BP-RP$>0.86$.

####  4.2.2 DA secondary type classification

As we have seen in Section 4.1.1, our Random Forest algorithm was able to find
two DAHs. Thus, we applied the algorithm to the classified cold DA white
dwarfs. From the 582 objects, none was classified as a DAH or DAZ; all of them
were classified as DAs. This result is not entirely unexpected as it was
already known that the Gaia resolution was, in almost all cases, insufficient
for this purpose.

Figure 6: As Fig. 5, but showing the classification of non-DAs into their
different spectral subtypes. Figure 7: HR diagram of the classified DQ and
DQpec white dwarfs found in this work.

####  4.2.3 Non-DAs subtype classification.

Once the identification between DAs and non-DAs has been completed, the 3 041
found non-DAs were classified into DC, DQ and DZ categories. DBs were
discarded, as none can be found at these low temperatures. As in subsection
4.2.1, the whole MWDD classified set was used as the training data. In
particular, we used 248 DCs, 19 DQ and 26 DZ with colours ${\rm BR-RP}>0.86$.

The results shown in Figure 6 reveal, as expected, that the most prominent
group are DCs: 3 008 objects representing 98.9% of the sample. Despite the low
Gaia resolution, and possibly low S/R, which impair the algorithm’s ability to
correctly identify any spectral feature at this low temperature regime, the
Random Forest algorithm was able to identify 22 DQs and 11 DZs. Taking into
account that only 19 DQs and 26 DZs white dwarfs with colours ${\rm BR-
RP}>0.86$ form the training sample, these newly found objects represent a
115.6% and 42.3% increment, respectively.

###  4.3 DQ secondary type classification

In our analysis, we have found 62 DQ so far. Of them, 40 with colors bluer
than BP-RP$=0.86$ (see Section 4.1.2) and 22 redder than that value (previous
section). As demonstrated in our third validation test (see Section 3.3), the
Random Forest is capable to identified certain secondary DQ spectral types,
although at the expense of low recall.

We apply our Random Forest algorithm to the 62 DQ-identified white dwarfs,
with the aim of classifying them into the secondary spectral types (i.e. DQ,
DQA, DQP, DQZ, DQZA, DQp, DQpec and DQpecP). The result reveals that objects
are thus classified only into two groups: 53 DQ and 9 DQpec white dwarfs.
Figure 7 shows the corresponding HR diagram. Those objects classified as DQpec
are tipically cold, with BP-
RP$\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$}0.6$,
indicative that Swan bands are more easily distorted at low temperatures (e.g
Blouin & Dufour, 2019).

Figure 8: Feature importance as a function of the Gaia spectral coefficients
for DA vs non-DA classification (top panel) and the non-DA classification into
the different spectral subtypes (bottom panel). An importance threshold of
0.8% is represented by a black horizontal line.

##  5 Feature importance

As an ensemble learning method, the Random Forest algorithm constructs
multiple decision trees combining their predictions to achieve the more
accurate and stable result. In this construction, some features (variables or
parameters of the sample) play a more remarkable role than others. Even more,
one can remove some features without significantly altering the result. In our
case, the features are the 110 Gaia spectral coefficients. We aim to analyse
which of them have the highest importance for each classification. The method
used to compute the feature importance was the mean decrease in impurity
(MDI), which is based on the decrease of node impurity averaged over the whole
Random Forest. This can be understood as follows. When a decision tree is
generated, decision nodes are created. Node impurity is a measurement of the
amount of classes in a certain decision node. They are said to be pure if they
only comprise one class. Therefore, the most important features in our
analysis are the ones that reduce the node impurity the most across the
forest. These will, of course, be dependent on the set that is being
classified. For instance, the coefficients that rule the Balmer lines are
capital in a DA non-DA classification, but of no importance in a non-DA
classification.

In Figure 8 we show the feature importance obtained by the MDI method as a
function of the Gaia spectral coefficients for the DA versus non-DA (left
panel) and the non-DA subtype (right panel) classification.

Figure 9: Gaia HR diagrams showing DA white dwarfs. Left panels: DA white
dwarfs classified in this work. Right panels: entire population of DA white
dwarfs (i.e. those classified in this work and those labelled in MWDD).

Regarding the DA versus non-DA classification, the most important coefficients
are approximately the 15 first red coefficients and the 20 first blue
coefficients. Moreover, if we consider a 0.8% threshold (marked as black line
in Fig 8), we can eliminate most of the low-significant spectral coefficients,
representing the remaining ones 73.6% of the information.

This result implies that a greater importance is placed in the BP information;
indeed, all Balmer lines except H${\alpha}$, He I lines, Swan bands and most
metallic lines fall in the BP wavelength range. The most important feature,
however, corresponds to the RP. We identify it with the Balmer H${\alpha}$
line. Since DAs show H features, it is predictable that the algorithm
considers this spectral line as the most important to distinguish between DAs
and non-DAs.

With respect to the non-DA classification into its spectral subtypes, the
feature importance distribution (right panel) reveals that blue coefficients
are the most relevant. Applying the same 0.8% threshold that in the previous
case, approximately the first 30 coefficients contain the 52% of the
information. As most of the type-characteristic spectral lines (for instance,
most He I lines, the Swan bands, or Ca II lines) appear in the wavelength
range covered by the BP, rather than in the RP range, this result is both
expected and consistent with our previous knowledge.

##  6 The Gaia 100-pc sample classification summary

In this section, we present a summary of our white dwarf spectral
classification. From the 9 446 classified white dwarfs, 4 737 have been
classified as DA, 2 as DAH, 76 as DB, 4 437 as DC, 62 as DQ and 132 as DZ. The
original, labelled MWDD sample used as training comprises 2 905 objects; 1 845
DA, 90 DAH, 97 DB, 573 DC, 117 DQ and 125 DZ.

Consequently, the number of classified objects within 100 pc from the Sun has
been increased by 257% for DAs, 2.2% for DAHs, 78.4% for DBs, 774% for DCs,
53% for DQs and 105.6% for DZs.

Figure 9 show the Gaia HR diagrams for the white dwarfs classified in this
work as DA and its secondary types (left panels), while in Figure 10 are
represented the corresponding for objects classified as DB, DC, DQ and DZ by
our algorithm (left panels). For completeness, we also show the HR diagrams
including those white dwarfs previously classified in the MWDD (right panels).

Additionally, all the objects studied here are collected in a list, where we
provide their corresponding spectral classification among other Gaia
parameters. A representative excerpt of this catalogue is presented in Table
3. The whole catalogue can be found in the electronic version of the paper.
Moreover, for illustrative purposes, in Annex 9 we show some examples of Gaia
spectra corresponding to white dwarfs of different spectral types classified
by our algorithm. These spectra are compared to the Gaia spectra of white
dwarfs labelled in the MWDD.

Figure 10: As Fig. 9 but for the different spectral subtypes of non-DAs. Table 3: Gaia 100 pc white dwarf sample catalogue classified by our Random Forest algorithm into spectral types. Gaia Source ID | RA | Dec | $M_{G}$ | BP-RP | $d$ | $S_{\rm type}$
---|---|---|---|---|---|---
| (deg) | (deg) | (mag) | (mag) | (pc) |
6617996741403360128 | 328.75 | -27.84 | 14.42 | 0.91 | 23.5 | DA
3022956969731332096 | 88.79 | -4.18 | 15.19 | 1.13 | 6.44 | DA
5314177402013456256 | 141.20 | -49.26 | 14.59 | 0.89 | 22.57 | DA
863131372427958912 | 176.43 | 63.10 | 14.29 | 0.92 | 24.01 | DA
4789516154317811456 | 65.99 | -45.85 | 14.29 | 0.89 | 29.94 | DA
3501922067493606272 | 109.10 | -23.30 | 13.24 | 0.88 | 41.89 | DA
5142336825646176256 | 25.67 | -17.24 | 14.36 | 0.90 | 40.05 | DA
6906197808698867712 | 306.88 | -7.60 | 14.63 | 1.00 | 51.38 | DA
5058635403471767680 | 49.30 | -29.19 | 14.74 | 0.96 | 31.50 | DA
5432789383518999168 | 144.40 | -38.87 | 14.27 | 0.86 | 34.50 | DA
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
5945252202546434432 | 266.95 | -52.12 | 12.72 | -0.11 | 38.92 | DAH
3630648387747801088 | 204.92 | -7.22 | 12.57 | -0.01 | 56.33 | DAH
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
6600365522695593472 | 336.84 | -34.19 | 10.81 | -0.22 | 53.20 | DB
6570892323240774144 | 329.51 | -43.47 | 11.10 | -0.19 | 50.59 | DB
2446993162322393088 | 354.82 | -4.41 | 11.41 | -0.15 | 89.28 | DB
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
2628943473222829440 | 339.03 | -1.68 | 12.74 | 0.15 | 39.02 | DBA
3375135698070213632 | 93.44 | 20.84 | 11.98 | 0.26 | 57.98 | DBA
5657351404992422784 | 148.25 | -26.96 | 10.97 | -0.19 | 83.64 | DBA
5719723160586005888 | 124.65 | -18.55 | 11.53 | -0.17 | 87.66 | DBA
1608497864040134016 | 216.90 | 53.81 | 11.43 | -0.14 | 51.54 | DBA
184735992329821312 | 75.43 | 33.40 | 11.72 | -0.10 | 74.98 | DBA
5099182265566440320 | 52.15 | -8.26 | 11.49 | -0.21 | 87.54 | DBA
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
4647914239368623616 | 87.62 | -76.81 | 14.63 | 1.09 | 85.91 | DC
4983839647522981504 | 21.02 | -42.68 | 13.36 | 0.72 | 18.45 | DC
4108828945319007744 | 256.67 | -26.73 | 14.73 | 0.72 | 13.05 | DC
1196295211098415616 | 237.68 | 16.06 | 14.41 | 0.91 | 91.01 | DC
4923825240566074624 | 3.01 | -54.02 | 15.37 | 1.36 | 73.78 | DC
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
5717278911884258176 | 115.09 | -17.42 | 13.16 | 0.34 | 9.15 | DQ
5332606522595645952 | 176.46 | -64.84 | 13.09 | 0.29 | 4.64 | DQ
946030529073021440 | 107.56 | 37.67 | 13.58 | 0.49 | 24.37 | DQ
6797171060323993728 | 305.11 | -30.45 | 12.39 | 0.10 | 17.46 | DQ
1932612039116771456 | 341.64 | 40.41 | 11.53 | -0.21 | 48.79 | DQ
51090628850761088 | 57.83 | 19.63 | 14.29 | 1.02 | 79.04 | DQ
6812238900812142720 | 329.21 | -24.88 | 14.41 | 0.87 | 67.19 | DQ
2201461976641376512 | 335.06 | 60.36 | 14.25 | 1.01 | 94.86 | DQ
3030820432081929088 | 115.74 | -12.80 | 14.45 | 1.47 | 76.83 | DQ
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
5271072526109138176 | 124.31 | 124.31 | 15.21 | 1.36 | 38.88 | DQpec
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
6054148143441683072 | 184.36 | -63.50 | 13.17 | 0.30 | 37.52 | DZ
3327488430402704000 | 96.91 | 10.04 | 12.88 | 0.22 | 52.91 | DZ
6598883720324218240 | 335.91 | -34.64 | 12.98 | 0.15 | 57.02 | DZ
2645295955612242688 | 350.31 | 1.04 | 15.07 | 1.21 | 61.06 | DZ
6429838617917056512 | 306.32 | -63.40 | 14.25 | 1.01 | 61.70 | DZ
2067245446933399936 | 305.98 | 39.44 | 14.92 | 1.01 | 67.25 | DZ
6505113009316709760 | 339.00 | -55.82 | 14.73 | 1.03 | 31.91 | DZ
4823007896975577088 | 81.84 | -34.46 | 14.91 | 1.17 | 65.17 | DZ
4217729529025009024 | 306.73 | -6.43 | 12.56 | 0.13 | 51.72 | DZ
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$

##  7 Comparison to other automatic classification methods.

To assess the quality of the performance of our Random Forest algorithm, in
this section, we compare it with other similar automated classification
methods described in the literature. In particular, we analyze the results
obtained by Vincent et al. (2023) in their white dwarf spectral classification
using neural networks.

Figure 11: Confusion matrix of objects that appear both in our classification
and in Vincent et al. (2023).

Although the methodology is not the same, and neither the classification
sample, the training sample, nor the input data are identical, we can
establish a certain comparative analysis of the results. For instance, our
work is focused on the Gaia 100 pc white dwarf sample for mainly primary
spectral types (DA, DB, DC, DQ and DZ) classification, while the work by
Vincent et al. (2023) consists in a more general approach for white dwarf
candidate selection and spectroscopic classification. This includes primary
spectral types, and also other subtypes, such as DO, hot DQ, DAH, PG 1159
objects and various types of subdwarfs, as well as white dwarfs plus main
sequence binaries. Moreover, the input data used in Vincent et al. (2023)
comes mainly from both the Gaia parameter database and Sloan Digital Sky
Survey spectra, while in our study, we only focused on the Hermite
coefficients from Gaia spectra.

Nevertheless, for comparative purposes, we have constructed a confusion matrix
with the spectral classification of the objects that appear in both the
present work and Vincent et al. (2023). Both catalogues were cross-matched and
$1\,103$ objects were found in both tables. Of them, six were classified in
Vincent et al. (2023) as cataclismic variables and two as hot subdwarf stars;
these objects were disregarded in the construction of the confusion matrix. As
such, the resulting confusion matrix, shown in Figure 11, contains 1 095
objects.

The obtained confusion matrix is nearly diagonal, which indicates a general
good agreement (86% accuracy considering the Vincent et al. (2023)
classification as true labels). The only remarkable exception are magnetic DAH
white dwarfs. For the 22 white dwarfs classified as DAH in Vincent et al.
(2023), 19 of them are classified by our algorithm as DAs and 3 as DCs. In
Section 3.3.1 we showed that our algorithm is practically unable to
distinguish DAs from DAHs. The reason is due to the lack of the necessary Gaia
spectra resolution to resolve the fine magnetic splitting. Therefore, this
result for DAHs is entirely understandable.

Similarly, but to a lesser extent, the 104 DZ candidates in Vincent et al.
(2023) are broadly in agreement with our classification (61 objects, 59%), but
our algorithm classifies the discrepancies as DC, DA, or DQs. Once again, the
low-resolution of Gaia spectra prevent to more accurate classification.

We can conclude that, despite the quality of the input data (which, obviously,
the better the quality data, the better the performance), our Random Forest
algorithm is a feasible tool with very low-cost computer time consuming and
model-independent tuning parameters, allowing a reliable and robust
classification of white dwarf spectra

##  8 Conclusions

By using Artificial Intelligence techniques based on a Random Forest
algorithm, we have analyzed the information contained in the coefficients of
Gaia spectra. Even though these spectra are of low resolution, we have
verified their usefulness in classifying the population of white dwarfs into
their different spectral types. In particular, we have classified the full 100
pc Gaia white dwarf population into their primary spectral types (i.e. DA, DB,
DC, DZ and DQ) and also finding some secondary types (DAH, DBA and DQpec).

A summary of the main findings is as follows:

  1. 1.

The Random Forest algorithm is able to classify DA and DC white dwarfs with
excellent recall (¿97%), DBs with very good recall (¿80%), and DQs and DZs
with improvable recall (¡50%).

  2. 2.

In spite of the low recall, DQ and DZ white dwarfs are classified with an
excellent precision (¿90%).

  3. 3.

While the algorithm performance is certainly improvable at correct
identification of DQ and DZ white dwarfs, its high precision for these
spectral types, as well as DB, allows us to use the classifying algorithm as a
white dwarf finder for these specific types.

  4. 4.

With the possible exception of the DBA and DQpec subtypes, spectral subtypes
do not seem to be recognised by the algorithm. Low resolution inherent to Gaia
mean spectra seems to be the limiting factor for classification, as non-
prominent spectral lines are not expected to be detected in them.

  5. 5.

Our algorithm has identified 76 DB (most of them, 60, DBA), 60 DQ (9 of them
DQpec), 132 DZ and 2 DAH candidates in a 100-pc radius around the Sun. For
comparison, the MWDD classified sample used in validation tests and as
training material contained 117 DQ and 125 DZ white dwarfs.

In conclusion, this initial classification of the entire white dwarf
population within 100 pc opens the door to more precise studies of mass
distribution and luminosity function, among others, based on the spectral
classification of these objects. In parallel, we have initiated a
spectroscopic follow-up of a large sample of candidate objects to confirm
their classification.

###### Acknowledgements.

We acknowledge support from MINECO under the PID2020-117252GB-I00 grant and by
the AGAUR/Generalitat de Catalunya grant SGR-386/2021. EMGZ also acknowledges
financial support from Banco de Santander, under a Becas Santander
Investigación/Ajuts de Formació de Professorat Universitari (2022_FPU‐UPC_16)
grant.

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##  9 Annex

In this Annex, we show examples of Gaia white dwarf spectra classified by our
Random Forest algorithm. In all of them, the expected location of Balmer
spectral lines are shown, as well as some characteristic spectral lines for
every spectral type. These include He I spectral lines for DB white dwarfs,
Swan bands for DQs and a selection of metallic lines for DZs, detailed in
subsection 9.6. For comparative purposes, we accompanied each classified
spectrum obtained by our algorithm with one spectrum corresponding to a white
dwarf classified with the same type from the MWDD.

All shown spectra have been obtained from Gaia internally calibrated spectra,
using the GaiaXPy Python package to transform them into wavelength-flux
externally calibrated spectra. Oscillatory behaviour in the spectra, as
explained in Section 2 at the blue and red extremes, are characteristic of
Gaia externally calibrated spectra, and are caused by the behaviour of the
Hermite polynomials used to convey them near the extremes

###  9.1 DA spectra

In this section, we show a DA spectrum from a registered white dwarf in the
MWDD and a spectrum classified as such in this work in Figure 12. Balmer lines
(H$\alpha$, H$\beta$, H$\gamma$, H$\delta$ and H$\epsilon$) are marked in both
spectra. These lines are very prominent, which explains the excellent recall
for DAs (see confusion matrix in top panel of Figure 1.)

Figure 12: Examples of Gaia spectra. Left panel: of a white dwarf classified
as DA by MWDD. Right panel: of a white dwarf classified as DA by our
algorithm.

###  9.2 DAH spectra

Gaia spectra for those objects classified as DAH are shown in this section, in
Figure 13. As hydrogen-dominated DAs, Balmer lines are prominent; Zeeman
effect magnetic splitting is not noticeable in the objects labeled as such in
MWDD nor in those identified by our Random Forest algorithm. This serves to
illustrate the role of Gaia spectral low resolution in explaining the low
recall for DAH white dwarfs by our algorithm.

Figure 13: As Fig. 12 but for a DAH classified in MWDD (left panel) and a DAH
classified by our algorithm (right panel).

###  9.3 DB spectra

The Gaia spectra of a white dwarf classified as DB by the MWDD and by our
algorithm are presented in Figure 14. Neutral helium lines are present and
noticeable in the shown spectra, and the He I lines at 4 471, 5 015, 5 875 and
6 678 $\AA$ have been marked.

Figure 14: As Fig. 12 but for a DB classified in MWDD (left panel) and a DB
classified by our algorithm (right panel).

###  9.4 DC spectra

In Figure 15 we show the Gaia DC spectra of a MWDD and here classified white
dwarf. It can clearly be visualized a characteristic, featureless spectra
(except for oscillations caused by Hermite polynomials behaviour). No spectral
lines could be found; wavelengths corresponding to H I Balmer lines have been
marked nonetheless to stress the featurelessness of these spectra.

Figure 15: As Fig. 12 but for a DC classified in MWDD (left panel) and a DC
classified by our algorithm (right panel).

###  9.5 DQ spectra

The defining characteristic of DQ white dwarfs is the presence of carbon
spectral lines. Most atomic carbon lines are outside of the Gaia BP and RP
spectral coverage. However, Swan bands, vibrational bands characteristic of
diatomic carbon (C2) are in the visible spectra. These are marked in the shown
spectra (Figure 16). Even though Swan bands comprise a high number of
vibrational transitions, for clarity they have been marked at 4 600 and 5 050
$\AA$.

Figure 16: As Fig. 12 but for a DQ classified in MWDD (left panel) and a DQ
classified by our algorithm (right panel).

Figure 17: As Fig. 12 but for a DZ classified in MWDD (left panel) and a DZ
classified by our algorithm (right panel).

###  9.6 DZ spectra

Metallic lines are present in DZ spectra. In these example (Figure 17) we have
marked spectral lines of elements that frequently form planetary matter:
calcium (Ca II at 3 933, 3 968, 8 542 and 8 662 $\AA$), magnesium (Mg II at 4
481 $\AA$), oxygen (O I at 7 772, 7 774 and 7 775 $\AA$), iron (Fe I at 4 578,
5 167, 5 227 and 5 269 $\AA$) and silicon (Si II at 6 347 and 6 371 $\AA$).


# A novel model of non-singular oscillating cosmology on flat Randall-Sundrum
II braneworld

Rikpratik Senguptaa111∗Corresponding author.
E-mail addresses: rikpratik.sengupta@gmail.com(RS) aDepartment of Physics,
Aliah University, Kolkata 700 160, West Bengal, India

###### Abstract

We obtain a novel model of oscillating non-singular cosmology on the spatially
flat Randall-Sundrum (RS) II brane. At early times, the universe is dominated
by a scalar field with an inflationary emergent potential
$V(\phi)=A(e^{B\phi}-1)^{2}$, $A$ and $B$ being constants. Interestingly, we
find that such a scalar field can source a non-singular bounce, replacing the
big bang on the brane. The turnaround again happens naturally on the brane
dominated by a phantom dark energy (favoured by observations[1, 2, 3] at late
times), thus avoiding the big rip singularity and leading upto the following
non-singular bounce via a contraction phase. There is a smooth non-singular
transition of the brane universe through both the bounce and turnaround,
leading to alternate expanding and contracting phases. This is the first model
where a single braneworld of positive tension can be made to recycle as
discussed in details in the concluding section.

###### keywords:

oscillating universe, braneworld, bounce, turnaround, phantom.

##  1 Introduction

The standard big bang model of cosmology is plagued by an initial singularity
(at time $t=0$), where the Friedmann equations describing the time evolution
of the universe fails to render a plausible physical description of the space-
time dynamics due to the diverging Hubble parameter $H$, which is a
consequence of the infinitely large energy density prevailing in the very
early universe. The scalar curvature $R$ also diverges, implying that the
initial singularity is a curvature singularity of the Ricci type, which are
characterized by diverging energy densities. As shown in their famous
singularity theorems by Hawking and Penrose, the initial big bang singularity
cannot be avoided in a general relativistic (GR) setup provided the energy
conditions are obeyed by matter filling up the universe[4, 5]. One of the key
problems of the standard inflationary cosmology, besidies the ambiguity
regarding the nature of the inflaton field is that, within the GR framework,
inflation can not be past eternal[6]. So, if the inflation is preceeded by a
radiation dominated phase, then the origin of the universe is singular.
However, today many cosmologists are unhappy about the initial singularity and
consider it to be a limitation of GR to describe space-times involving very
large energy densities.

If there is no big bang, then the possibilities are that the universe either
experiences a quantum creation where there is a quantum mechanical tunelling
into an inflationary phase, or it may be that the universe existed for an
eternally long time in a quasi static state followed by an emergent
inflationary phase, or there is a non-singular bounce replacing the singular
big bang prior to which the universe contracts and following which the
universe expands. In the context of GR, the emergent and bouncing scenarios
can be realised effectively only for a spatially closed universe ($k=1$). A
fully consistent treatment of the first possibility considering quantum
creation will most probably require a quantum gravity (QG) treatment. However,
there is no fully understood and developed QG theory at the moment and the two
most accepted theories being worked on in this context are M-theory[7]
involving extra dimensions and Loop Quantum Gravity (LQG)[8]. M- theory
requires eleven space-time dimension for its quantum consistency while LQG
quantizes space-time itself in the usual four dimensions. Effective theories
from both the scenarios have become popular in recent times in the form of the
extra dimensional braneworld models[9, 10] and the effective loop quantum
cosmology (LQC)[11, 12] models.

It is interesting to note that although the premises of the background QG
theories are completely different from each other, there are some similarities
and identical features that may be obtained from a class of the effective
braneworld and LQC models, that may hint at some hidden correspondance between
the two contrasting approaches. In this letter, we shall only talk about
models that introduce corrections to standard GR at the ultraviolet (UV) scale
as we are concerned with resolution of the initial singularity. Braneworld
models have the feature that our universe is represented by a
$(3+1)$-dimensional hypersurface known as the ’brane’ (which are objects
appearing in M-theory) embedded in a higher dimensional bulk space-time. The
Randall Sundrum single brane model (RS-II) is one such model with a spacelike
extra dimension, indicating that the signature of the bulk space is
Lorentzian. If the signature of the bulk deviates from the former, then we can
have a bulk signature $(-,-,+,+,+)$ such that the braneworld has a timelike
extra dimension. The standard model particles and fields are confined to the
brane, while gravity is free to propagate in the bulk.

##  2 The cosmological model

The Einstein Field Equations (EFE) on the brane have the general form

$G_{\mu\nu}+\Lambda h_{\mu\nu}=8\pi
G_{eff}T_{\mu\nu}+\frac{\epsilon}{1+\beta}\bigg{[}\frac{S_{\mu\nu}}{M^{6}}-W_{\mu\nu}\bigg{]}$
(1)

The parameter $\epsilon=\pm 1$, where the + sign implies a spacelike and -
sign a timelike extra dimension. $G_{\mu\nu}$ and $T_{\mu\nu}$ denote the
Einstein and stress-energy tensors on the brane, respectively while
$h_{\mu\nu}$ is the induced metric on the brane. We have $8\pi
G_{eff}=\frac{\beta}{\beta+1}\frac{1}{m^{2}}$ and
$\Lambda=\frac{\Lambda_{br}}{\beta+1}$, where the parameter
$\beta=\frac{2\epsilon\sigma m^{2}}{3M^{6}}$ and
$\Lambda_{br}=\frac{\Lambda_{bulk}}{2}+\frac{\epsilon\sigma^{2}}{3M^{6}}$.
Here $G_{eff}$ is the effective Gravitational constant on the brane,
$\Lambda_{br}$ and $\Lambda_{bulk}$ denotes the effective cosmological
constant on the brane and the bulk cosmological constant, respectively. $m$
and $M$ denote the Planck masses in $4$ and $5$ dimensions, respectively and
$\sigma$ represents the brane tension. In the RS limit, we have $m\rightarrow
0$ implying $\beta\rightarrow 0$, such that the effective gravitational
constant is given by $G_{eff}=\frac{\epsilon\sigma}{12\pi M^{6}}$. The sign of
$\epsilon$ can also be interpreted as the sign of $M$ or alternatively as the
sign of the brane tension as either $M$ or $\sigma$ has to be negative for
$\epsilon=-1$ to make the gravitational constant positive. Thus, a braneworld
with a timelike extra dimension is characterized by a negative brane tension
in order to localize the bulk gravity in the vicinty of the brane. Also,
$\Lambda_{bulk}>0$ for such a brane, as contrasted to the RS II brane which is
characterized by an anti-deSitter ($AdS_{5}$) bulk. To sum up, in the RS
limit, the modified EFE is written with $\Lambda$ replaced by $\Lambda_{br}$,
$8\pi G_{eff}$ replaced by $\frac{2\epsilon\sigma}{3M^{6}}$ and $\beta=0$.
Finally, coming to the tensors $S_{\mu\nu}$ and $W_{\mu\nu}$, they represent
the local and non-local corrections on the brane, where the former represents
quadratic corrections in stress-energy components given as
$S_{\mu\nu}=\frac{TT_{\mu\nu}}{12}-\frac{T_{\mu\alpha}T^{\alpha}_{\nu}}{4}+\frac{g_{\mu\nu}}{24}\left[3T_{\alpha\beta}T_{\alpha\beta}-(T^{\alpha}_{\alpha})^{2}\right]$
and the later represents the projected bulk Weyl tensor on the brane given by
$W_{\mu\nu}=C_{ABCD}^{bulk}n^{C}n^{D}h_{\mu}^{A}h_{\nu}^{B}$, where
$C_{ABCD}^{bulk}$ is the bulk Weyl tensor and $n^{C}$ is unit vector normal to
the surface.

For the FRW line element describing an isotropic, homogenous universe, the
modified EFE on the brane has the form

$H^{2}+\frac{k}{a^{2}}=\frac{\Lambda_{bulk}}{6}+\frac{\epsilon\sigma^{2}}{9M^{6}}+\frac{2\epsilon\sigma\rho}{9M^{6}}+\frac{\epsilon^{2}\rho^{2}}{9M^{6}}+\frac{C}{a^{4}}.$
(2)

Observations suggest a spatially flat universe[13, 14, Sv2, 16] enabling us to
put $k=0$ and also we can consider $C=0$ due to the high symmetry of the
space-time to obtain a simpler solution, without much loss of generality. RS
also considered the effective cosmological constant on the brane
($\Lambda_{br}$) to be $zero$ as the bulk cosmological constant and the
parameter $\epsilon$ have opposite signs and so the first two terms on the
right hand side of Equation (3) ($=\frac{\Lambda_{br}}{3}$) can be fine tuned
to vanish. After some algebraic simplification, the modified EFE on the brane
takes the form

$H^{2}=\frac{8\pi
G_{eff}}{3}\rho\bigg{(}1+\frac{\epsilon\rho}{2|\sigma|}\bigg{)}$ (3)

Coincidentally, Equation (3) for a timelike extra dimension ($\epsilon=-1$)
gives the same modified EFE as obtained from the LQC perspective, although the
frameworks are inspired from completely different background theories, as
discussed[10, 11]. The factor involving the brane tension $|\sigma|$ in the
denomenator of the second term on the RHS acts as a critical density
$\rho_{cr}$ in LQC. A very interesting feature of such models is that they can
remove the initial singularity which can be understood very simply from
Equation (3) as follows: the energy density cannot rise beyond the critical
density as for $\rho=\rho_{cr}=2|\sigma|$, the Hubble parameter vanishes. So,
the big bang singularity is replaced by a non-singular big bounce even in a
spatially flat universe, which is not possible in standard GR. This can be
attributed to some repulsive force that is generated as the energy density
increases, solely due to the extra dimension being timelike or due to the
quantization of spacetime. The identical effect and result may indicate a deep
underlying connection between the different mechanisms.

However, in this letter we shall try to construct a ${novel}$ non-singular
cosmological model of oscillating nature on the spatially flat RS II brane
with a spacelike extra dimension ($\epsilon=1$). For a spacelike extra
dimension $\epsilon$ being positive, the Hubble parameter does not vanish
naturally if the energy density $\rho$ rises upto 2$\sigma$. So, RS II
cosmology does not contain any inherent feature through which a non-singular
bounce can be naturally realized. A Weyl curvature singularity due to
infinitely large tidal forces at the wormhole throat can be resolved on the RS
II brane with ordinary matter[17]. We shall try to induce the bounce using an
ingredient that is quite familiar in early universe cosmology and finds
application in achieving the inflationary mechanism- a scalar field. The
scalar field is minimally coupled meaning that there is no coupling between
the scalar field and gravity. In such a model there may exist an infinite
number of cycles containing expansion and contraction phases. However, another
additional mechanism is required besides a non-singular bounce to generate
such an oscillating universe. The universe must begin to contract at late
times following an expanding phase before the next cycle begins. This
mechanism is known as the turnaround. We shall use two different mechanisms to
generate the bounce and turnaround on the flat RS II brane.

The conservation equation on the brane is the same as in GR given by
$\dot{\rho}+3H(p+\rho)=0$, where for the scalar field $\phi$, the energy
density is expressed as $\rho=\frac{\dot{\phi}^{2}}{2}+V(\phi)$ and the
pressure has the form $p=\frac{\dot{\phi}^{2}}{2}-V(\phi)$. Here $V(\phi)$
represents the potential of the scalar field. So, the conservation for the
scalar field on the brane has a form identical to standard GR and is given as

$\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0$ (4)

We shall consider a scalar field of potential having a flat wing from which an
inflationary emergent scenario can be realized in GR. Such a well known
potential having the form $V(\phi)=A(e^{B\phi}-1)^{2}$ was used by Ellis et
al.[18] to construct an emergent universe scenario in the relativistic setting
for a spatially closed universe. The potential as a function of the field has
been plotted in Figure 1, where $A$ and $B$ denote constant parameters. It can
be considered that the scalar field is weakly decelerated
($|\ddot{\phi}|<|\frac{dV}{d\phi}|$) as it is the left branch of the potential
that can possibly lead to a non-singular cosmology. Under this consideration,
the $\ddot{\phi}$ in the equation of motion for the scalar field is of
negligible significance.

Making use of the modified Friedmann equation and the conservation equation
for a weakly decelerating scalar field of potential $V(\phi)$ having the form
as mentioned above, a solution for the field $\phi(t)$ is obtained to be given
by

$\phi(t)=\frac{1}{3}\frac{2B^{2}\sqrt{6\sigma}t+3BC_{1}\kappa+3LambertW\left(\frac{2B^{2}\sqrt{6\sigma}t-3BC_{1}\kappa}{3\kappa}-1\right)\kappa}{B\kappa},$
(5)

where $\kappa=\sqrt{8\pi G_{eff}}$.

Figure 1: Variation of the scalar field potential $V(\phi)$ with the field
($\phi$)and the field $\phi(t)$ with time $t$.

The time evolution of the scalar field has also been plotted in Figure 1. As
evident from the figure (estimating the time of the bounce from Figure 2),
negative values of the field play an important role in the bounce. So, from
the evolution of the potential, it may be said that the bounce happens for the
flatter branch of the potential that leads to an emergent cosmology in a
closed relativistic Friedmann universe.

We know that the Hubble parameter is defined as $H=\frac{\dot{a}}{a}$ and the
scale factor $a$ can be expressed in tetrm of the scalar field $\phi(t)$ as

$a(\phi)=C_{2}e^{-\frac{1}{4}\frac{A\kappa^{2}}{B\sigma}\left[\frac{e^{2B\phi}}{2B}-\phi-3\frac{e^{B\phi}}{B}+\frac{e^{-B\phi}}{B}\right]},$
(6)

where $C_{2}$ is a constant of integration.

Using Equations (5) and (6), we have

$a(t)=C_{2}e^{-\frac{1}{4}\frac{A\kappa^{2}}{B\sigma}}\left[\frac{e^{\frac{4B^{2}\sqrt{6\sigma}t+3BC_{1}\kappa+3LambertW\left(\frac{2B^{2}\sqrt{6\sigma}t-3BC_{1}\kappa}{3\kappa}-1\right)\kappa}{3\kappa}}}{2B}\right.\\\
\left.-\frac{1}{3}\frac{2B^{2}\sqrt{6\sigma}t+3BC_{1}\kappa+3LambertW\left(\frac{2B^{2}\sqrt{6\sigma}t-3BC_{1}\kappa}{3\kappa}-1\right)\kappa}{B\kappa}\right.\\\
\left.-3\frac{e^{\frac{2B^{2}\sqrt{6\sigma}t+3BC_{1}\kappa+3LambertW\left(\frac{2B^{2}\sqrt{6\sigma}t-3BC_{1}\kappa}{3\kappa}-1\right)\kappa}{3\kappa}}}{B}+\frac{e^{-\frac{2B^{2}\sqrt{6\sigma}t+3BC_{1}\kappa+3LambertW\left(\frac{2B^{2}\sqrt{6\sigma}t-3BC_{1}\kappa}{3\kappa}-1\right)\kappa}{3\kappa}}}{B}\right]$
(7)

Figure 2: Variation of the scale factor $a(t)$ and the energy denisty
$\rho(t)$ with time $t$ showing the non-singular bounce.

As we can see from the variation of the scale factor with time in Figure 2, a
scalar field with the potential of the form $V(\phi)=A(e^{B\phi}-1)^{2}$ can
cause the brane universe to bounce without reaching a singular state. The
contracting universe bounces into an expanding phase before a singular state
of diverging energy density and scalar curvature is reached. The varaiation of
the energy density of the scalar field dominated universe with time has also
been shown in Figure 2 around the time of the bounce. At the bounce, the
energy density reaches a maximum as expected, making the $\rho^{2}$ term in
the UV corrected EFE significant, before falling off again following it. The
effectiveness of the scalar field in averting the initial singularity and
generating the bounce can be realized through this quadratic energy density
term reflecting the higher dimensional effect due to the braneworld scenario.
We shall not go into the details of whether there is a subsequent inflationary
phase to seed cosmological perturbations, although it is known that inflation
can be obtained from both the branches (flat and steep) of the potential that
we have used. We shall focus only on the beginning of the cycle through the
bounce and its completion through a turnaround leading to the beginning of the
next cycle. The detailed evolution of the cycle including the generation of
large scale structures from seed perturbations is beyond the scope of this
letter and shall be taken up in a following work.

We now move on to the late time where the universe is in an accelerating
phase. Such a phase has been inferred from astronomical observations[19, 20].
With the discovery of the accelerating universe, there was a resurrection of
the $\Lambda$-term in cosmology. However, there are certain inconsistencies
with the $\Lambda$ dark energy (DE) that has lead to a wide range of models
including quintessence involving scalar fields[21, 22], Chaplygin gas
involving fluids with non-linear equation of states[23, 24] (EoS), the phantom
with an exotic EoS[25, 26] and geometrical models which effectively modify the
matter sector through geometrical contributions at the infrared (IR) scale[27,
28, 29] but not the real source matter. Here, we have already used a framework
in which the geometrical contributions modify the matter sector effectively at
the UV scale through the $\rho^{2}$ term. If such a term has to be significant
in the universe at late times to source a turnaround through its possible non-
conventional behaviour ($H\propto\rho$), then there has to be a mechanism to
make the energy density of the universe grow sufficiently large enough. There
is a possibility to achieve this using one of the possible DE candidates that
is observationally well favoured-the phantom. The phantom is an exotic fluid
with a supernegative EoS ($\omega<-1$) violating the Null Energy condition
(NEC), proposed by Caldwell[25] to fit the observational data. The fact tha
the phantom fits quite well with the observational data ($-1.61<\omega<-0.78$)
has latter been verified by a number of groups[1, 2, 3].

We have already discussed that the conservation equation on the brane is the
same as in GR. For a perfect fluid described the the EoS $p=\omega\rho$, the
conservation equation can be used to obtain the evolution of the energy
density in terms of the scale factor as $\rho\simeq a^{-3(1+\omega)}$. As for
the phantom $\omega<-1$, so the factor in the power of $a$ is always a
positive number, resulting in $\rho(t)$ growing with $a$ for a phantom
dominated universe. This makes the UV corrected $\rho^{2}$ term significant
also at late times. In a relativistic DE scenario with a phantom fluid, there
appears a few bizzare features. If we consider $t^{\prime}$ to be the time
when the density of ordinary matter dominating the contacting phase of the
universe is equal to that of the phantom DE, before DE domination
(accelerating expansion) takes over, then the scale factor during the later
phase evolves as
$a(t)=a(t^{\prime})\bigg{[}(1+\omega)\frac{t}{t^{\prime}}-\omega\bigg{]}^{\frac{2}{3(1+\omega)}}$,
which blows up to infinity within finite time $t=\frac{\omega
t^{\prime}}{1+\omega}$, making the universe singular. This singularity is
termed as the big rip. At the same finite time, the energy density of the
phantom dominated universe
$\rho(t)\simeq\bigg{[}(1+\omega)\frac{t}{t^{\prime}}-\omega\bigg{]}^{-2}$ also
blows up to infinity making the Hubble parameter diverge. The curvature scalar
also diverges resulting in a curvature singularity like the big bang. However,
as we shall see, the quadratic correction term in the modified EFE can avoid
this singularity too, by making the universe turnaround and enter a
contracting phase before the energy density diverges, resulting in the next
non-singular bounce dominated by a scalar field with the potential discussed.

Again making use of the UV corrected Friedmann equation on the brane, we
obtain

$\int\frac{da}{\bigg{[}a^{\frac{-1-3\omega}{2}}+\frac{\alpha}{2\sigma}a^{\frac{-4-6\omega}{2}}\bigg{]}}=\sqrt{\frac{\kappa^{2}\alpha}{3}}\int
dt,$ (8)

where $\alpha$ is a constant.

We obtain analytical solutions for three different values of the EoS
parameter, considering a phantom dominated universe.

$\displaystyle(\omega=-\frac{4}{3}):a(t)=\frac{1}{\bigg{[}LambertW(\frac{1}{2}+2t^{2})+1\bigg{]}^{2}}$
$\displaystyle(\omega=-\frac{3}{2}):a(t)=\frac{1}{\bigg{[}LambertW(\frac{1}{2}+\frac{9}{8}t^{2})+1\bigg{]}^{\frac{4}{3}}}$
$\displaystyle(\omega=-3):a(t)=\frac{1}{\bigg{[}LambertW(\frac{1}{2}+6t^{2})+1\bigg{]}^{\frac{1}{3}}}$
(9)

Figure 3: Variation of the scale factor $a(t)$ and the energy denisty
$\rho(t)$ with time $t$ showing the turnaround for different values EoS
parameter $\omega$ of the dominant cosmic fluid.

As we find from Figure 3 on plotiing the scale factor with time, the
accelerating phantom dominated universe enters into a contracting phase before
the scale factor diverges. Once the universe begins to contract, it is
dominated by radiation or matter until the scalar field begins to dominate the
universe that causes it to bounce before the scale factor reaches a zero
value. So, the scale factor maintains a non zero finite value all along the
evolution of the universe, never reaching a singular state. The energy density
of the universe has also been plotted in Figure 3 as it evolves with time
close to the turnaround. It is found to reach a peak just before the
turnaround as it keeps on increasing in the phantom dominated eoch, before
falling off again. The energy density is large enough at both early and late
times to make the quadratic correction term in the modified EFE significant,
but never diverges. It starts dropping following both the bounce and
turnaround.

##  3 Discussion

The two most important components of our oscillating cosmological model on the
brane are the bounce and turnaround mechanisms. The bounce mechanism is
sourced by a scalar field with the inflationary emergent potential
$V(\phi)=A(e^{B\phi-1})^{2}$ that may be followed by an inflationary phase (or
some other mechanism leading to an increase in amplitude of each successive
cycle) to seed cosmological perturbations resulting in the formation of large
scale structures in the universe. This shall be a subject of future concern.
However, if there is an inflationary phase, it can be sourced from either of
the two branches (flat and steep) of the potential mentioned above for
suitable values of the parameters $A$ and $B$. However, the left branch leads
to an emergent cosmology and also to a bouncing cosmolgy as obtained by us.
The bounce is possible due to the effective violation of NEC by the scalar
field on the brane. So, no additional source may be required to seed
perturbations in the early universe. The turnaround mechanism is sourced by
the phantom energy described a supernegative EoS ($\omega<-1$). Such a
component can not only explain the accelerating phase of the universe but also
source the turnaround due to the unique property that the energy density grows
with time making the UV correction term significant also at late times, that
ultimately leads to the turnaround on the brane. So, no additional component
is required for this purpose also. Moreover, we obtain the cyclic scenario for
an observationally favoured spatiall flat universe[14, 15, 16].

When the universe is contracting, the energy density grows and ultimately
diverges making the scalar curvature and the Hubble parameter to diverge as
well. This can be understood by the fact that the scale factor vanishes in the
Friedmann equation. Bounce is the mechanism that simply prevents the initial
singularity from forming by making the scale factor begin to increase before
it can reach zero or making the energy density drop off before it can diverge.
The condition for bounce is achieved through $\ddot{a}>0$, such that the
contracting universe starts expanding. Alternatively, in turnaround, the
expanding universe must begin to start contracting so that both the scale
factor and energy density donot diverge in the finite future and this can be
achieved through the condition $\ddot{a}<0$. Both at the bounce and
turnaround, the Hubble parameter vanishes rather than diverging. The the scale
factor neither reaches zero or infinite value as the effective energy density
on the brane remains finite. Thus, the universe transits smoothly through both
the bounce and turnaround. For cosmology on the brane, both the mechanisms can
be achieved by a minimum of requisite components- a scalar field with an
inflationary potential and a DE component that violates the NEC.

The problem with the phantom is that such an exotic fluid has a number of
theoretical inconsitencies and pathologies at the quantum level that makes
their existence questionable. The problem with the future singularities can be
cured from the correction term on the brane as we have found, but such a fluid
may also lead to the vacuum being unstable. Attempts to construct dynamical
scalar field models of the phantom have led to a negative kinetic term[25]
that in turn results in quantum instabilities[30]. However, there exists a
cosmological model of DE with vanishing $\Lambda$ where the vacuum energy
obtained from the quantization of a free scalar field having low mass is
described by a supernegative EoS and the model is free of pathologies at the
quantum level[31]. Two problems in general feature in most oscillating
cosmology models. We shall discuss them very briefly without going into much
details before concluding the letter. The first problem is posed by the
continued existence of singular objects like black holes from the area
theorems of Hawking. However, before the turnaround leading up to the next
bounce (via a contraction phase) in a phantom dominated universe, such
structures may well be dissolved due to the extremely large gravitationally
repulsive effects[32], thus being prevented from disrupting the evolution of
the universe during the contracting phase following the turnaround. Infact, it
has been shown that[33] the Hawking area theorems may not hold true if the NEC
($\rho+p\geq 0$) is violated, as is the case for a phantom dominated universe.
Any surviving remnant microscopic black holes may act as possible dark matter
candidates. Moreover, the black hole singularity may also be resolved in the
UV corrected picture just like the initial big bang and the big rip
singularities and also, there may exist non-singular black hole mimickers like
gravastar on the brane[34], leading to a complete resolution of the problem at
once. It is worth mentioning in this context that the RSII braneworld has also
been used in explaining a recent GW event GW170817[35] and the recent
observation of the dark shadow of M87∗[36]. The second problem is associated
with the entropy of the universe, which we think remains the same in a
periodic manner after the bounce over each cycle, such that the possible
increase in entropy during the expanding phase being compensated by a possible
decreasing during the radiation/matter dominated expanding phase. This
prevents the enrtopy to increase to infinitely large values limiting the
number of cycles. We are however concerned mainly with the bounce and
turnaround in a single cycle in this letter.

We have constructed a novel model of non-singular oscillating cosmology on the
RSII brane where the universe in a contracting phase bounces into expansion
sourced by a scalar field before the energy density diverges, thus resolving
the big bang singularity and is taken over by a phantom fluid at the late
times (causing the universe to accelerate as observed and with an increasing
energy density) making the $\rho^{2}$ term in the EFE on the brane contribute,
thus avoiding a big rip singularity from forming in the finite future (via the
universe entering into a contracting phase which continues but again re-enters
into an expanding phase, due to the non-singular bounce occuring in an
oscillating manner). In between the scalar field and phantom fluid is the
standard radiation and matter dominated epochs. Thus, there are alternate
expanding and contracting epochs.

This is the first model that can avert the initial singularity using a single
brane approach with a positive brane tension. Braneworlds that have a space-
like extra dimension like the one we have considered here are characterized by
a positive brane tension (as the effective gravitational constant on the brane
needs to be positive to explain the attractive nature of gravity) but such a
setup could not resolve the big bang singularity. The non-singular models of
brane cosmology till date have either resorted to a single brane with time-
like extra dimension where no scalar field has to be invoked to generate the
bounce which happens naturally from the cosmological dynamics[10], but the
brane tension must be negative for the same reason of obtaining a positive
effective gravitational constant, or alternatively, introduce a second
braneworld with a negative tension parallel to the positive tension brane with
finite separation between the branes. The advantage of introducing the
parallel negative tension braneworld is two-fold: firstly, negative tension
branes have the unique feature of reduced inertia on matter with positive
energy density being dumped onto it helping the dynamical realization, and
secondly the two brane setup comes with the benefit of a scalar field known as
the radion which modulates the inter-brane separation and can both source the
bounce at early times as well as behave like phantom dark energy at late times
dure to the non-canonical kinetic term evolving to have a negative value[37].
However, there are some tachyonic instabilities associated with negative
tension braneworlds which can be possibly resolved in M-theory but has not
been explored well enough and requires further formal developments in M-theory
(although the properties are really appealing). On the contrary, the
ingredients of our model are well explored and the physics is more well
understood in terms of a single positive tension brane. The phantom dark
energy also does not lead up to the big rip as the quadratic correction to
stress energy becomes significant before the singularity can is reached.

Also, the scalar field we have used is a physically well motivated one, since
it can accommodate the inflationary scenario naturally and its potential need
not be reconstructed to explain the generation of seed cosmological
perturbations. Most models of cosmology with a non-singular bounce have either
to resort to alternative mechanisms to generate seed perturbations that are
not very well understood physically, or have to reconstruct the potential on
an ad hoc basis to generate the perturbations, but in our model an
inflationary epoch following the bounce driven by the scalar field with an
inflationary emergent potential already has all the ingredients responsible
for generating these perturbations, and is well understood. We may think of it
as a toy model, not because the scenario is physically ill-motivated, but
since we have not tested the model against observations. We plan to analyze
the primordial observables like the amplitude of scalar perturbations, tensor
to scalar ratio and the spectral index and test them against latest
observations in a follow up work in the recent future.

## Acknowledgments

RS is thankful to the Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India for providing work facilities during a visit where this
work was done.

Data Availability Statement This is a theoretical work and does not make use
of any data for arriving at our results. However, the observation supporting
our analysis have been cited in each case in the manuscript.

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