# Using Physiological Information to Classify Task Difficulty in Human-Swarm
Interaction

Joseph P. Distefano1, Hemanth Manjunatha1, Souma Chowdhury1,
Karthik Dantu2, David Doermann2, and Ehsan T. Esfahani1∗ This work was
supported by National Science Foundation Award IIS-1927462 and the DARPA
Cooperative Agreement HR00111920030.1 JD, HM, SC, and EE are with the
Department of Mechanical and Aerospace Engineering, University at Buffalo,
Buffalo, NY, 14260 USA.2 KD and DD are with the Department of Computer Science
and Engineering, University at Buffalo, Buffalo, NY, 14260 USA.∗Corresponding
Author<EMAIL_ADDRESS>
###### Abstract

Human-swarm interaction has recently gained attention due to its plethora of
new applications in disaster relief, surveillance, rescue, and exploration.
However, if the task difficulty increases, the performance of the human
operator decreases, thereby decreasing the overall efficacy of the human-swarm
team. Thus, it is critical to identify the task difficulty and adaptively
allocate the task to the human operator to maintain optimal performance. In
this direction, we study the classification of task difficulty in a human-
swarm interaction experiment performing a target search mission. The human may
control platoons of unmanned aerial vehicles (UAVs) and unmanned ground
vehicles (UGVs) to search a partially observable environment during the target
search mission. The mission complexity is increased by introducing adversarial
teams that humans may only see when the environment is explored. While the
human is completing the mission, their brain activity is recorded using an
electroencephalogram (EEG), which is used to classify the task difficulty. We
have used two different approaches for classification: A feature-based
approach using coherence values as input and a deep learning-based approach
using raw EEG as input. Both approaches can classify the task difficulty well
above the chance. The results showed the importance of the occipital lobe (O1
and O2) coherence feature with the other brain regions. Moreover, we also
study individual differences (expert vs. novice) in the classification
results. The analysis revealed that the temporal lobe in experts (T4 and T3)
is predominant for task difficulty classification compared with novices.

##  I Introduction

Human-swarm interaction (HSI) has gained a lot of attention in recent years
due to its many applications, including exploration, search, rescue, and
surveillance [1]. The combination of mobile swarm robotics and human-in-the-
loop systems generates an opportunity to leverage the best of both worlds [2].
However, increased task difficulty decreases the efficiency of the human
operator, which in turn decreases the overall efficiency of the human-swarm
team. Thus it is required to maintain an optimal level of task difficulty and
intelligently adapt the task allocation between human operators and robot
swarms. Hence identifying the operator’s perceived task difficulty [3]
occupies a central role in efficient HSI.

However, the task difficulty cannot be predicted in real-time as relevant task
information is not known prior to the interaction. Frequently during human-
swarm interaction, the state of the swarms or swarm mission does not provide
adequate information to completely quantify the nature of the task. In this
direction, physiological information such as brain activity and eye movement
can augment the swarm states and provide secondary information about the task
and state of the operator. Thus providing a means to infer task difficulty.
Specifically, by understanding the various features in the brain activity that
contribute to predicting task difficulty, we allow an opportunity to predict
when the operator may need assistance or is not getting enough feedback. This
analysis could also potentially be used to provide insight on game balance or
the training of operators [4]. Moreover, due to individual differences in
human brain cognition, the perceived task difficulty depends on individual
differences [5]. These individual differences can be identified in a skilled
marksman vs. a novice marksman. For instance, the human’s visuomotor
performance and efficiency have been shown to affect cortico-cortical
communications in rifle shooting [6]. Thus, identifying different features
that could be utilized for expert vs. novice operators could provide a better
classification of task difficulty. In this regard, we explore the use of brain
activity (measured through electroencephalography) to classify the task
difficulty of a human operator interacting with a robot swarm.

To study task difficulty prediction using EEG in HSI, we have built a
simulation platform that allows for human supervisory control of swarm
robotics while completing missions with increasing difficulty levels. In this
study, task difficulty is represented through the game difficulty levels. The
simulation platform has a graphical user interface that allows the users to
interact with the swarms. While the human interacts with the swarm, an
electroencephalogram (EEG) is used to record the human’s brain activity. We
consider two approaches to classify the task difficulty level using EEG: 1)
Features-based approach: Here, a set of well-known features (e.g., coherence
values) are extracted and used to classify the task difficulty using a support
vector machine (SVM). 2) Deep learning approach: Here, raw EEG epochs are used
as input to a convolution neural network (CNN), where the output is the task
difficulty level. When human participants are involved, the individual
differences in skill level affect the perceived task difficulty. Hence, we
consider individual differences by categorizing the participants as experts or
novices and separately classifying task difficulty in each category. The
results show that both feature-based SVM (71.60%) and CNN (83.80%) approaches
can classify the task difficulty well above the chance, with CNN performing
better than the SVM classifier. In terms of experts and novices, the
classification accuracy is higher among experts when compared to novices.

##  II Method and Materials

###  II-A Human Subject Study Framework

The framework for the human-swarm interaction simulation platform consisted of
three separate modules (Fig. 1). The three modules are the gaming framework,
the physiological data recording, and the lab streaming layer. These modules
exchange information and offer different functionalities to create synchronous
data recording during a human supervisory control swarm mission. These three
modules run parallel using an open-source Ray framework [7].

The gaming framework includes two separate components that work in parallel
throughout the game. The first component is the simulation environment which
was developed using the open-source library pybullet [8]. The simulation
environment is run through a real-time 3D physics engine. The second component
is the graphical user interface that displays the information from the
simulation environment. The interactive GUI presents all information needed
from the simulation for the user to complete the mission.

Comprised in the simulation environment is the rendered map, which
incorporates roads and buildings. In the simulation environment, the robotic
swarms can execute three primitives with a high level of autonomy: formation
control, path planning, and task allocation. The user is only required to
input the UxV target location. In the environment, the human is supervising
three UGV and three UAV platoons. More information on the design of the
simulation environment can be found in [9]. This simulation environment was
also used for learning swarm tactics over complex environments in [10]. The
lab streaming layer module is the terminal location for all data being
recorded. This data includes all the environment, EEG, game, and user data.
Every instance of information is inputted into the lab streaming layer,
labeled with a time stamp, and stored for post-processing.

Figure 1: The framework consists of three components: Gaming environment,
physiological measurement, and lab streaming layer for time synchronization.
Figure 2: The GUI presents all information to the user needed to complete the
mission. It allows for the control of UxV platoons through the main window.
Each platoon has a panel window. The user feedback panel provides information
about enemy encounters and the remaining time in the game.

###  II-B Graphical User Interface

The simulation environment was portrayed as a 2D interactive graphical user
interface using the open-source pygame library. The design of the GUI was
based on popular real-time strategy games such as starcraft. The GUI consists
of a partially observable environment map, platoon panels to display an up-
close overview of the platoons, a user feedback panel that notifies the user
of game information, and a user controls panel (Fig. 2). The GUI allows for
user control allowing complete freedom to allocate the platoons anywhere on
the map. More information on GUI components can be found in [9].

###  II-C Physiological Measurement

Participant’s brain activity is recorded through a non-invasive B-Alert X24
EEG headset from Advanced Brain Monitoring© at 20 channels and 2 reference
electrodes with a sampling rate of 256 Hz. The sensor locations are O1, O2,
P4, POz, P3, Pz, Cz, C3, C4, Fz, F3, F4, T6, T4, F8, Fp1, Fp2, F7, T5, and T3
(Fig. 3).

Figure 3: (a) Experimental setup for monitoring the human physiological states
while supervising a group of UAVs and UGVs in target search task. (b)
International EEG 10-20 electrode placements.

###  II-D Experiments

Students who had experience playing real-time strategy games were recruited
from the University at Buffalo. The international review board approved the
human subject study (IRB # STUDY00003659) before experimentation. All
participants had normal or correct vision. The participants were asked to play
the game before to show proficiency in controls. After gaining adequate
practice time, the participants underwent individual difference assessments.

After taking the individual difference and EEG baselines, the participants
played three randomized games of different difficulties. The game’s
difficulties vary depending on the different complexities in the environment.
The easiest task included a fully observable environment with no adversarial
teams. The second level included a static adversarial team that blocked the
paths and target buildings. Finally, the hardest level included a dynamic
adversarial team whose position changed over time. The adversarial teams were
initially hidden and made visible when the user’s platoon was within the
predetermined range in both static and dynamic levels. If the user’s platoons
and adversarial teams collided, a battle would result in the loss of platoons
in a stochastic manner. If a platoon is defeated (zero members), it will
disappear from the map, and the user may no longer use it.

Figure 4: This figure shows the game being played both with complexities and
without complexities. In a) you can see the human has no adversarial units
where as in b) adversarial units (red) are blocking the paths.

###  II-E Individual difference

When working with human participants, individual differences are inevitable.
Furthermore, individual differences affect the cognitive ability of humans and
their perception of task difficulty. Thus it is desirable to capture this
information and use it in analysis. We have used visual search (VS) and multi-
object traction (MOT) to capture the individual differences [11]. These
differences allow us to categorize the participants as experts and novices.

VS and MOT are EEG baseline procedures that assess individual skill levels and
categorize the participant as an expert or a novice. In MOT, a participant
should track a set of pre-marked circles. These circles are then moved around
a space, and the user must select the marked circle at the end of the trial
(Fig. 5(a)). VS is tested by flashing a screen with multiple shifted ‘T’s and
asking the user to identify the direction of the correct T within a 2-second
window (Fig. 5(b)). The user’s scores depend on their ability to identify the
circles and find the direction of the T. If the participant scored well on
both tests, they are considered an expert. Out of 9 participants, 5 were
labeled as experts, and the remaining were labeled as novices.

Figure 5: (a) Multi-object tracking experiment. Target objects are highlighted
briefly and the participants must track them during random movement and
identify them when they stop. (b) Visual search experiment. Participants are
required to identify the direction of the target shape (‘T’) among the
detractors by pressing the arrow key corresponding to the direction. (c)
Identifying experts and novices using MOT and VS scores.

##  III Data Analysis

The data recorded during the human subject study underwent post-processing and
further analysis. The details of the data analysis techniques are elaborated
in the section below.

###  III-A Classification

Classifying task difficulties is posed as a three-class classification
problem. The baseline mission data with no adversarial team is the least
difficult level. The difficulty level is increased by introducing static and
dynamic adversarial teams. Before classification, the class labels were
balanced. We explored two classification methods, support vector machine (SVM)
classifier with EEG features and Convolutional neural network (CNN) approach
with raw EEG.

We have considered three types of data splitting schemes. Due to the
individual differences in EEG research, it is important to explore different
training data sets to obtain accurate feature selection results and compare
the results obtained from different training sets. 1) Subject independent:
pool all the participant’s data, balance, and randomly split 66% for training
and 33% for testing. 2) Subject dependent: Out of 10 participants, training is
done in 9 participants, and testing is done on the remaining participants
(leave-one-out). The process is repeated until every participant is considered
for testing. 3) Individual difference-based: The participants are split into
experts and novices depending on the individual difference score (see Section
II-E). Then the classification is carried separately for experts and novices.
The discussed data splitting scheme is shown in Fig. 6. Note that the data is
split after the feature selection step, and the scheme is the same for SVM and
CNN.

Figure 6: Data splitting scheme for classification.

###  III-B Feature Extraction

The EEG signals were bandpass filtered (0.1-70 Hz) to remove the electrical
noise of the headset and the slow drift caused by the electrodes moving on the
scalp. EEG signals were epoched into 2-second windows, and epochs with large
artifacts were dropped. Increasing the window length would increase the noise,
and decreasing the length would reduce the data points available for learning.
Hence, we heuristically choose a window length of 2 seconds. The eye blink and
drastic body movement artifacts were also removed using independent component
analysis with the Picard algorithm in MNE-Python [12].

Once a cleaned epoched EEG signal was obtained, the features were extracted.
The first set of features extracted were from the b-alert live software
(Advanced Brain Monitoring©. Carlsberg, CA, USA). These features include
mental workload, engagement, and distraction. Additional details on feature
extraction and baselines used can be found in [13].

The second set of features were coherence values from the brain activity
frequency bands. Coherence measures the similarity of two signals in a
predefined frequency band for a given period. In this paper, we have used 6
frequency bands; delta [4-8 Hz], low alpha [8-10 Hz], high alpha [11-13 Hz],
low beta [14-22 Hz], high beta [23-35 Hz] and gamma [36-44 Hz]. The coherence
values (as given by equation 1) were calculated for each electrode pair and
frequency band.

$coh=\frac{\lvert E[Sxy]\rvert}{\sqrt{E[Sxx]*E[Syy]}}$ (1)

where $S_{xx}$, $S_{yy}$ are power spectral densities of the channels, and
$S_{xy}$ is the cross power spectral density of the channel being used.

###  III-C Feature Selection

Since the number of features is larger, the classification algorithm can over-
fit by learning noise. To avoid over-fitting, we perform feature selection.
The present study recorded EEG from 20 channels using a 10-20 international
system and considered six frequency bands for feature extraction. This results
in 1200 coherence features. To reduce the number of features even further, we
explored the existing literature on task difficulty. We found that the
frontal, parietal, motor, and occipital regions, especially during visual-
motor tasks, are predominantly correlated with task difficulty [14]. Thus we
selected T4, T3, O1, P3, Pz, F3, Fz, F4, C4, P4, C3, Cz, and O2 electrodes to
use.

A recursive feature elimination (RFE) was performed to obtain the best set of
features. The RFE takes the initial set of features used as the input for the
machine learning algorithm and ranks the features. The features deemed
significant stay in the data set, and the features that have small effects on
the classification are removed. This is repeated until a user-defined number
of features are identified. RFE is stochastic in nature. As a result,
different features can be selected in a different trial. To reduce the effect
of randomness, we ran RFE 10 times and selected the most common features. For
RFE, we have used linear SVM with RBF kernel.

###  III-D Convolutional neural network

Figure 7: Architecture of the deep learning model. K: Kernel size, S: Stride,
and N: Number of filters.

The architecture of the convolution neural network (CNN) used for
classification, as shown in Fig. 7 is inspired by [15] and [16]. The first two
linear layer operation is given by

$x^{\prime}=(x-W_{sh}\bar{x})\odot(W_{sc}\tilde{x}_{sh})$ (2)

Where $x$ is the input epoch, $\bar{x}$ is the mean value calculated across
each electrode and $\tilde{x}_{sh}$ is the standard deviation of
$(x-W_{sh}\bar{x})$. Note $W_{sh}$ and $W_{sc}$ are the learnt weights of the
linear layers and the $\odot$ represents element-wise multiplication (Hadamard
product). Note that if$W_{sh}$ is identity and $W_{sc}$ is the inverse
covariance matrix (precision matrix), then the equation 2 reduces to a
standard z-score. The linear layers are succeeded by two convolution
operations; kernel size $1\times 30$ and $20\times 40$ (stride of 1). The
number of filters is kept as 10 for both convolution layers. After two
convolution layers, the output is element-wise squared, followed by average
pooling. The kernel for average is $1\times 150$, and stride is $1\times 30$.
After pooling, the $log$ function is applied to the output, followed by a
dropout. The dropout percentage is kept at 20% to avoid over-fitting. Finally,
a convolution operation with kernel size $1\times 10$ (stride=1) is applied to
reduce the feature size to 3. Finally, a softmax function is used to convert
the output to class probability.

##  IV Results and Discussion

In this section, we provide the classification results and discuss the
importance of selected features. Note the discussion of selected features is
in the context of the SVM classifier. Table I shows the classification results
using two different methods and different data splitting schemes (see Section
III-A). For subject-independent classification, deep learning performed better
than the feature-based SVM classifier. However, for subject-dependent
classification, SVM performance was better than the deep learning approach.
Nonetheless, both classifier’s performance is poor for subject-dependent
classification. The main reason for poor performance can be attributed to
individual differences among participants. Interestingly, when the
classification is performed separately between expert and novice, the
classification rate is higher for experts than novices. This trend holds both
for the SVM and CNN approaches. A probable explanation might be that experts
make more efficient use of neural resources leading to different brain
dynamics when compared with novices [5].

TABLE I: Classification results Classification
Method |  Subject
Independent |  Subject
Dependent |  Expert |  Novice
---|---|---|---|---
CNN | 83.80$\pm$1.42 | 42.19$\pm$2.88 | 90.11$\pm$4.87 | 86.13$\pm$5.63
SVM | 71.60$\pm$2.49 | 50.1$\pm$1.23 | 77.98$\pm$1.65 | 72.14$\pm$1.05

The significant features extracted from the feature selection are listed in
Table II. Among the selected features, the Pz-O2 and O1-P4 coherence values
show the inner connectivity in the parietal and occipital lobes of the brain.
The F3-C3 and F3-O2 coherence values portray the connectivity between the
prefrontal cortex with the brain’s motor cortex and occipital lobe. These
features signify which sections of the brain interact differently during
different task difficulties.

TABLE II: Significant Features Coherence Electrodes | Frequency Band
---|---
Pz-O2 | 8-10 Hz (Low Alpha)
F3-C3 | 14-22 and 23-35 Hz (Low and high beta)
F3-O2 | 14-22 and 23-35 Hz (Low and high beta)
O1-P4 | 14-22 and 35-44 Hz (Low beta and gamma)
TABLE III: Expert vs Novice Significant Features Expert Features | Novice Features
---|---
T4-O2 (High beta 23-35 Hz) | O1-C4 (Low Beta 14-22 Hz)
T3-O1 (High beta 23-35 Hz) | Cz-O2 (Low Beta 14-22 Hz)
Fz-F4 (Gamma 36-44 Hz) | Pz-C4 (Low Beta 14-22 Hz)

The expert features show a strong correlation between the temporal lobe and
the occipital lobe. These are unique features that did not show up in the
previous feature selection. The novice features all contained electrodes in
the motor cortex. Fig. 8 shows the visual representation of the significant
features on the EEG 10-20 electrode locations. As you can see, all
participant’s features connect the prefrontal cortex and the occipital lobe.
However, the expert’s connectivity features are unique because they connect
the occipital lobe to the temporal side lobes.

Figure 8: The electrodes that have significant coherence are displayed on the
EEG 10-20 system.

In terms of most selected features, the Pz-O2 coherence is representative of
the occipital and parietal lobe connectivity. In previous research, the low
alpha band has reports of synchronization and variance when performing
different tasks. [17], [18]. It was also found that the Pz electrode spectral
density drastically changes due to task load in human-computer interaction
with an event-related potential [19]. The beta frequency band’s F3-C3 and
F3-O2 coherence values represent the prefrontal cortex connectivity with the
motor cortex and the occipital lobe. The occipital lobe activation indicates
the visual demand of humans, and the frontal lobe signifies engagement and
data processing [20]. In this human swarm interaction mission, both the visual
demand and the engagement would increase due to the added adversarial
complexities. On similar lines, the O1-P4 electrode can also be accredited to
the increase in visual demand.

The selected features between experts and novices are significantly different.
The expert features are unique due to the involvement of the temporal lobe.
Previous research shows that electrodes Fz, T4, and T3 power spectral
densities strongly correlate with psychomotor efficiency. This has been shown
both in rifle shooting by experts and novices and through a physical human-
robot interaction task difficulty analysis [21, 6]. Therefore, these three
electrodes are included in the expert features but are lacked in the novice
features. Although the Fz, T4, and T3 coherence features are not selected, the
connection with the occipital lobe could be from the visual intensity of the
human swarm interaction task.

##  V Conclusion

This paper presents an analysis of brain activity during a human swarm
interaction study to quantify task-difficulty in a target search mission. The
participants played three separate games to find the target location with
increasing task difficulty due to adversarial complexities in a partially
observable map. While the participants provided supervisory commands, their
brain activity was measured, which is used to quantify the task difficulty
level.

Task difficulty quantification was posed as a three class classification
problem using EEG data as the input. The three-class problem corresponds to
the three different levels of difficulty in the mission: no adversarial team,
static adversarial team, and dynamic adversarial team. We explored two
classification methods: support vector machine (SVM) classifier with EEG
features and deep learning approach with raw EEG. The EEG features consisted
of coherence values extracted from 2-second epochs from T4, T3, O1, P3, Pz,
F3, Fz, F4, C4, P4, C3, Cz, and O2 sites and in the alpha, beta, and gamma
frequency bands. For the deep learning approach, the raw epoch was directly
used as an input. Irrespective of the classification method, the EEG signals
were filtered and cleaned with ICA. Both feature-based and deep learning
approaches were able to classify task difficulty well above the chance (33%).
Most of the prominent features in the classification showed a connection
between the prefrontal cortex with the motor cortex and occipital lobes.

Due to the individual differences in the skill level, the participants were
separated into experts and novices based on the multi-object tracking and
visual search tasks scores. Among the experts, coherence values at the
temporal lobe (T3 and T4) were most significant, indicating that the experts
have more psychomotor efficiency than novices. However, most of the selected
coherence features involved the occipital lobe regardless of expert or novice.
In future work, we wish to explore the relationship between task difficulty
classification and the cognitive workload of the participants. We also hope to
explore more complex features including the human brain connectivity graphs.

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# Period-Age-Metallicity and Period-Age-Color-Metallicity relations for
Classical Cepheids: an application to the Gaia EDR3 sample

Giulia De Somma,12 Marcella Marconi,1 Santi Cassisi,34 Vincenzo Ripepi,1
Adriano Pietrinferni,3 Roberto Molinaro,1 Silvio Leccia1 and Ilaria Musella1
1 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131
Napoli, Italy
2 Istituto Nazionale di Fisica Nucleare (INFN)-Sez. di Napoli, Compl. Univ.di
Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy
3 INAF-Osservatorio Astronomico d’Abruzzo, Via Maggini sn, 64100 Teramo, Italy
4 Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Pisa, Universitá
di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy E-mail<EMAIL_ADDRESS>
(Accepted 2021 September 10. Received 2021 September 10; in original form 2021
July 01)

###### Abstract

Based on updated pulsation models for Classical Cepheids, computed for various
assumptions about the metallicity and helium abundance, roughly representative
of pulsators in the Small Magellanic Cloud ($Z$=$0.004$ and $Y$=$0.25$), Large
Magellanic Cloud ($Z$=$0.008$ and $Y$=$0.25$), and M31 ($Z$=$0.03$ and
$Y$=$0.28$), and self-consistent updated evolutionary predictions, we derived
Period-Age and multi-band Period-Age-Color relations that also take into
account variations in the Mass-Luminosity relation. These results, combined
with those previously derived for Galactic Cepheids, were used to investigate
the metallicity effect when using these variables as age indicators. In
particular, we found that a variation in the metal abundance affects both the
slope and the zero point of the above-mentioned relations. The new relations
were applied to a sample of Gaia Early Data Release 3 Classical Cepheids. The
retrieved distribution of the individual ages confirms that a brighter Mass-
Luminosity relation produces older ages and that First Overtone pulsators are
found to be concentrated towards older ages with respect to the Fundamental
ones at a fixed Mass-Luminosity relation. Moreover, the inclusion of a
metallicity term in the Period-Age and Period-Age-Color relations slightly
modifies the predicted ages. In particular, the age distribution of the
selected sample of Galactic Cepheids is found to be shifted towards slightly
older values, when the F-mode canonical relations are considered, with respect
to the case at a fixed solar chemical composition. A marginally opposite
dependence can be found in the noncanonical F-mode and canonical FO-mode
cases.

###### keywords:

stars: evolution — stars: variables: Cepheids — stars: oscillations — stars:
distances

††pubyear: 2021††pagerange: Period-Age-Metallicity and Period-Age-Color-
Metallicity relations for Classical Cepheids: an application to the Gaia EDR3
sample–Period-Age-Metallicity and Period-Age-Color-Metallicity relations for
Classical Cepheids: an application to the Gaia EDR3 sample

##  1 Introduction

Classical Cepheid (CC) Period-Luminosity (PL) and Period-Luminosity-Color
(PLC) relations are the basis for the absolute calibration of the
extragalactic distance scale, which leads to a local evaluation of the Hubble
constant ($H_{0}$) (see e.g. Freedman et al., 2001; Riess et al., 2018; Riess
et al., 2021; Saha et al., 2006). This role of CCs has known a renewed
interest in the last few years due to the debate on the so-called $H_{0}$
tension, the discrepancy between early Universe measurements of $H_{0}$ based
on the Cosmic Microwave background analysis, and late Universe estimates of
the constant, based on the extragalactic distance scale (see e.g. Verde et
al., 2019, and references therein).

In addition, CCs are also known to obey Period-Age (PA) and Period-Age-Color
(PAC) relations (see e.g. Anderson et al., 2016; Bono et al., 2005; Efremov,
2003, and references therein). Indeed, the combination of the period-mean
density relation and the Stefan-Boltzmann Law leads to a period-luminosity-
mass-effective temperature relation that, once it assumes a mass-luminosity
relation and a transformation into the observational plane, leads to a period-
luminosity-color relation. As the luminosity is related to the mass and the
evolving mass is anticorrelated with the stellar age, a period-age-color
relation arises. The period-age relation is then found by averaging the period
over the color extension of the instability strip for any given age.
Therefore, this class of pulsating stars represents a powerful tool for
measuring individual stellar ages, and hence, inferring tight constraints on
the star formation history of the stellar populations to which they belong.

A PA relation for CCs can be obtained via a semi-empirical approach combining
the measurement of the pulsational properties of a sample of CCs in a star
cluster, with the corresponding age estimate obtained via the isochrone
fitting technique applied to the cluster Color-Magnitude Diagram (CMD) (see,
for a recent analysis, Medina et al., 2021; Zhou & Chen, 2021, and references
therein).

A completely different approach (Bono et al., 2005) relies on a theoretical
scenario in which the Mass-Luminosity (ML) relation and evolutionary lifetimes
during the core He-burning stage, as predicted by stellar models (see e.g.
Bono et al., 2000; Chiosi et al., 1993, and references therein), are connected
with the pulsation relation (linking the period with the stellar mass,
luminosity and effective temperature) obtained via suitable pulsational models
so as to derive the PA relation. The use of the PAC relation with respect the
PA one represents an improvement, applied in order to overcome the problem
related to the finite color width of the instability strip (IS) (Caputo et al.
(2000); Marconi et al. (2005); Marconi et al. (2010) and De Somma et al.
(2020a, hereafter DS20a)).

In a recent work (De Somma et al., 2020b, hereafter DS20b), we computed
updated PA and PAC relations - in the Gaia DR2 photometric passbands (Gaia
Collaboration et al., 2018, 2016) \- for solar metallicity CCs, by combining
an extended and homogeneous set of nonlinear convective pulsation models
obtained for several assumptions on the ML relation, and the efficiency of
superadiabatic convection (we refer to DS20a for more details), with the
updated BaSTI library of stellar evolutionary predictions presented in Hidalgo
et al. (2018).

The obtained PA and PAC relations were applied to a subset of Gaia Data
Release 2 Galactic Cepheids (see ; , for details). The retrieved age
distributions confirm that the effect of a variation in the efficiency of
superadiabatic convection in the pulsational computations is negligible,
whereas older ages are obtained when a brighter mass-luminosity relation, due
to a combination of mild overshooting and/or rotation, and/or mass-loss, is
adopted.

The aim of the present investigation is to extend the theoretical calibration
of the PA and PAC relations to other chemical compositions, representative of
the typical chemical composition of the Small and Large Magellanic Clouds
($Z$=$0.004$ and $Z$=$0.008$ with $Y$=$0.25$), and the Andromeda (M31) galaxy
($Z$=$0.03$ with $Y$=$0.28$).

A complete analysis of the pulsational models corresponding to these chemical
abundances, including the investigation of the effects related to changes in
the adopted ML relation and efficiency of super-adiabatic convection, will be
presented in a companion paper (De Somma et al. 2021, in prep.). Here, we
focus our analysis on the derivation of PA and PAC relations in several
photometric systems, and of the first Period-Age-Metallicity (PAZ) and Period-
Age-Color-Metallicity (PACZ) relationships.

The structure of the paper is as follows: in Section 2, we present the new
metal-dependent theoretical scenario for CCs, based on an extended set of
evolutionary tracks and nonlinear convective pulsation models; in Section 3,
we derive the new PA and multi-filter PAC relations by varying the chemical
compositions which are then applied to a subset of Gaia Early Data Release 3
(EDR3) CCs in Section 4; the conclusions and possible future developments of
the presented investigation are discussed in Section 5.

##  2 A metal dependent theoretical scenario

In order to retrieve the theoretical PA and PAC calibrations, it is mandatory
to combine the stellar evolutionary framework with the pulsation model
predictions:

  * •

Evolutionary models provide the relevant information about the morphology of
the evolutionary tracks during the core He-burning phase (the so-called blue
loop), and the evolutionary lifetimes in this phase;

  * •

The pulsation scenario gives crucial information about the IS boundaries, and
the relation between the pulsation period and stellar evolutionary properties
such as current mass, luminosity and effective temperature.

In the following subsections, we provide a concise description of the main
characteristics of the adopted evolutionary and pulsation frameworks.

###  2.1 The evolutionary framework

The sets of stellar models adopted in the present investigation are fully
consistent with those adopted for deriving the PA and PAC relations for a
solar chemical composition by DS20b. These stellar model sets correspond to
the updated version of the BaSTI stellar evolutionary library for the solar
scaled heavy element distribution, presented by Hidalgo et al. (2018) and
available at the URL:http://basti-iac.oa-abruzzo.inaf.it.

A complete description of the physical inputs and assumptions adopted for
computing these models is provided by Hidalgo et al. (2018), while a brief
summary of the features more relevant for the investigation of the CC
properties can be found in De Somma et al. (2020b). Here, it is enough to
mention that for this analysis we selected stellar models in the $4M_{\odot}$
and $11M_{\odot}$ mass range for the following chemical compositions: Z=0.004,
Y=0.252; Z=0.008, Y=0.257; and Z=0.03, Y=0.284111The chemical compositions of
the stellar model sets are quite consistent with those adopted for the
pulsational computations. The extremely small differences in the initial He
abundances have no impact at all on present investigation.

The BaSTI library provides evolutionary predictions for both stellar models,
accounting for a moderate core convective overshooting during the core
H-burning stage (noncanonical models), and completely neglecting the
occurrence of any physical process able to increase the size of the convective
core above the canonical value predicted by the Schwarzschild criterion
(canonical scenario) (we refer to Hidalgo et al., 2018, for a detailed
discussion on this topic).

A subsample of the stellar evolutionary tracks adopted in this work is shown
in Fig. 1.

In order to allow the derivation of the PAC relation, the selected sets of
stellar models have been converted from the theoretical Hertzsprung-Russell
(HR) diagram to various photometric systems, such as the Johnson-Cousin, the
Gaia EDR3 (Gaia Collaboration et al., 2021) and the James Webb Space Telescope
(JWST) NIRCam ones. As such, the procedure described in detail in Hidalgo et
al. (2018) has been adopted.

Figure 1:  The location, in the HR diagram, of a sub-sample of massive and
intermediate-mass stellar models from the BaSTI-IAC library, for various
assumptions of the initial chemical compositions (see labels). The predicted
boundaries of the IS, fundamental blue edge (FBE), fundamental red edge (FRE),
first overtone blue edge (FOBE) and first overtone red edge (FORE), for both F
(solid line) and FO-mode pulsators (dashed line) are also shown. Left panels
refer to the canonical scenario, while the right panels correspond to the
noncanonical scenario (see text for more details). Figure 2: Upper panels:The
F (left panel) and, when available, FO (right panel) instability strip
boundaries for $Z$ = $0.004$ (dashed lines), $Z$ = $0.008$ (solid lines), $Z$
= $0.02$ (dotted lines) and $Z$ = $0.03$ (dash-dotted lines), obtained by
adopting the canonical ML relation and the standard efficiency for the
superadiabatic convection. Bottom panels: The same as in the upper panels but
for noncanonical models.

###  2.2 The pulsational framework

To investigate the metallicity effect on the pulsational properties of CCs, as
well as to pave the way for future theoretical calibrations of the Cepheid-
based extragalactic distance scale, a theoretical analysis of the CC pulsation
scenario was extended to chemical compositions different from the solar one
already analyzed in DS20a. In particular, we considered the following chemical
compositions: $Z$=$0.004$, $Y$= $0.25$; $Z$=$0.008$, $Y$= $0.25$; and
$Z$=$0.03$, $Y$= $0.28$, to be representative of CCs observed in the Small
Magellanic Cloud, the Large Magellanic Cloud, and M31, respectively.
For each chemical composition, a wide range of masses (3<$M$/$M_{\odot}\,$<11)
and effective temperatures ($3600<T_{eff}$[K]$<7200$), as well as two
assumptions of the ML relation were considered. Models with a canonical
luminosity level, i.e. models computed by neglecting core convective
overshooting, rotation, and mass loss, were labeled ‘case A‘, while those with
a noncanonical luminosity level, obtained by increasing the canonical
luminosity level by $\Delta\log(L$/$L_{\odot})$=$~{}0.2\;dex$, were labeled
‘case B‘. An analysis of the impact of a change in the super-adiabatic
convection efficiency was also performed for consistency with the analysis
performed at solar chemical abundance (DS20a).
This notwithstanding, we note that for the present analysis, a standard
convective efficiency value (i.e. $\alpha_{ml}$ = 1.5) was accounted for
because both the PA and PAC relations were found to be barely affected by the
choice of the mixing length parameter (see DS20b, for more details).

The adopted input parameters are listed in Table LABEL:param_puls_models. For
each selected model and pulsation mode, the system of nonlinear dynamical and
convective equations was integrated until a stable limit cycle of pulsation
was achieved.

The relation of the effective temperature of each IS boundary as a function of
the luminosity, obtained by performing a linear fit, for both the F and FO
pulsators, is provided in Tables 2, to 4, for $Z$=$0.004$, $Z$=$0.008$ and
$Z$=$0.03$, respectively.

The linear relations for the F and FO-mode (when it was stable) boundaries
obtained for the canonical ML relation and the standard convective efficiency
are plotted in Fig. 2. For the sake of comparison, the same figure also shows
the IS boundaries for the solar chemical composition ($Z$=$0.02$, $Y$=$0.28$).
These plots show that, in agreement with previous studies (Bono et al., 1999,
1997), a change in the chemical composition does significantly affect the
topology of the instability strip. As the metallicity increases from
$Z$=$0.004$ to $Z$=$0.03$, at a fixed mixing length parameter and ML relation,
the IS gets redder. This occurrence is due to both the decreased hydrogen
abundance that makes pulsation less efficient at the blue edge of the IS,
delaying the onset of pulsation to lower effective temperatures, and the
increased contribution of iron bump opacity to pulsation at the IS red
boundary, thereby delaying the quenching of pulsation due to convection.

Following the same approach adopted for the solar chemical composition case, a
linear regression of the period as a function of the luminosity, the mass, and
the effective temperature (PMLT relation) was carried out. The coefficients of
the relations for F and FO pulsators, for each chemical composition, are
reported in Table 5. A quick inspection of their values clearly shows that the
metallicity has a larger effect on the zero-point and on the term related to
the effective temperature than on the other coefficients. In agreement with
previous investigations (see e.g. Bono et al., 2000), longer periods are
expected for higher metal abundance models, when all the other parameters are
fixed.

Table 1: The intrinsic stellar parameters for the cases $Z$=$0.004$, $Z$=$0.008$, $Z$=$0.02$ and $Z$=$0.03$ of the F and FO-mode pulsation models adopted for computing the PA and PAC relations. The various columns list the metallicity, helium content, mass, luminosity level, the ML label, and the pulsation mode. Z | Y |  $M$/$M_{\odot}\,$ |  $\log$L/$L_{\odot}\,$ | ML | mode
---|---|---|---|---|---
0.004 | 0.25 | 4.0 | 2.91 | A | F/FO
0.004 | 0.25 | 4.0 | 3.11 | B | F/FO
0.004 | 0.25 | 5.0 | 3.24 | A | F/FO
0.004 | 0.25 | 5.0 | 3.44 | B | F/FO
0.004 | 0.25 | 6.0 | 3.50 | A | F/FO
0.004 | 0.25 | 6.0 | 3.70 | B | F
0.004 | 0.25 | 7.0 | 3.73 | A | F/FO
0.004 | 0.25 | 7.0 | 3.93 | B | F
0.004 | 0.25 | 8.0 | 3.92 | A | F
0.004 | 0.25 | 8.0 | 4.12 | B | F
0.004 | 0.25 | 9.0 | 4.09 | A | F
0.004 | 0.25 | 9.0 | 4.29 | B | F
0.004 | 0.25 | 10.0 | 4.25 | A | F
0.004 | 0.25 | 10.0 | 4.45 | B | F
0.004 | 0.25 | 11.0 | 4.39 | A | F
0.004 | 0.25 | 11.0 | 4.59 | B | F
0.008 | 0.25 | 4.0 | 2.81 | A | F/FO
0.008 | 0.25 | 4.0 | 3.01 | B | F/FO
0.008 | 0.25 | 5.0 | 3.14 | A | F/FO
0.008 | 0.25 | 5.0 | 3.34 | B | F/FO
0.008 | 0.25 | 6.0 | 3.40 | A | F/FO
0.008 | 0.25 | 6.0 | 3.60 | B | F
0.008 | 0.25 | 7.0 | 3.63 | A | F/FO
0.008 | 0.25 | 7.0 | 3.83 | B | F
0.008 | 0.25 | 8.0 | 3.82 | A | F
0.008 | 0.25 | 8.0 | 4.02 | B | F
0.008 | 0.25 | 9.0 | 3.99 | A | F
0.008 | 0.25 | 9.0 | 4.19 | B | F
0.008 | 0.25 | 10.0 | 4.14 | A | F
0.008 | 0.25 | 10.0 | 4.34 | B | F
0.008 | 0.25 | 11.0 | 4.28 | A | F
0.008 | 0.25 | 11.0 | 4.48 | B | F
0.02 | 0.28 | 4.0 | 2.74 | A | F/FO
0.02 | 0.28 | 4.0 | 2.94 | B | F/FO
0.02 | 0.28 | 5.0 | 3.07 | A | F/FO
0.02 | 0.28 | 5.0 | 3.27 | B | F
0.02 | 0.28 | 6.0 | 3.33 | A | F/FO
0.02 | 0.28 | 6.0 | 3.53 | B | F
0.02 | 0.28 | 7.0 | 3.56 | A | F
0.02 | 0.28 | 7.0 | 3.76 | B | F
0.02 | 0.28 | 8.0 | 3.75 | A | F
0.02 | 0.28 | 8.0 | 3.95 | B | F
0.02 | 0.28 | 9.0 | 3.92 | A | F
0.02 | 0.28 | 9.0 | 4.12 | B | F
0.02 | 0.28 | 10.0 | 4.08 | A | F
0.02 | 0.28 | 10.0 | 4.28 | B | F
0.02 | 0.28 | 11.0 | 4.21 | A | F
0.02 | 0.28 | 11.0 | 4.41 | B | F
0.03 | 0.28 | 4.0 | 2.68 | A | F/FO
0.03 | 0.28 | 4.0 | 2.88 | B | F
0.03 | 0.28 | 5.0 | 3.01 | A | F
0.03 | 0.28 | 5.0 | 3.21 | B | F
0.03 | 0.28 | 6.0 | 3.27 | A | F
0.03 | 0.28 | 6.0 | 3.47 | B | F
0.03 | 0.28 | 7.0 | 3.50 | A | F
0.03 | 0.28 | 7.0 | 3.70 | B | F
0.03 | 0.28 | 8.0 | 3.69 | A | F
0.03 | 0.28 | 8.0 | 3.89 | B | F
0.03 | 0.28 | 9.0 | 3.86 | A | F
0.03 | 0.28 | 9.0 | 4.06 | B | F
0.03 | 0.28 | 10.0 | 4.02 | A | F
0.03 | 0.28 | 10.0 | 4.22 | B | F
0.03 | 0.28 | 11.0 | 4.15 | A | F
0.03 | 0.28 | 11.0 | 4.35 | B | F
Table 2:  The coefficients of the relation $\log T_{eff}$ = a +b $\log$L/$L_{\odot}\,$, for both the F and FO-mode IS boundaries, for $Z$ = $0.004$ and $Y$ = $0.25$, $\alpha_{ml}$ = 1.5 and various assumptions of the ML relation. The last two columns represent the root-mean-square deviation ($\sigma$) and the R-squared ($R^{2}$) coefficients.  ML | a | b | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---
FBE |  |  |  |  |  |
A | 3.825 | -0.016 | 0.012 | 0.003 | 0.005 | 0.7696
B | 3.849 | -0.022 | 0.011 | 0.003 | 0.005 | 0.8879
FRE |  |  |  |  |  |
A | 3.958 | -0.074 | 0.014 | 0.004 | 0.006 | 0.9814
B | 3.937 | -0.069 | 0.033 | 0.009 | 0.014 | 0.9031
FOBE |  |  |  |  |  |
A | 3.941 | -0.044 | 0.012 | 0.004 | 0.003 | 0.9796
B | 3.945 | -0.045 | 0.005 | 0.002 | 0.001 | 0.9988
FORE |  |  |  |  |  |
A | 3.844 | -0.026 | 0.043 | 0.013 | 0.010 | 0.5588
B | 3.888 | -0.042 | 0.099 | 0.032 | 0.010 | 0.6320
Table 3:  The same as in Table 2 but for $Z$ = $0.008$ and $Y$ = $0.25$ ML | a | b | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---
FBE |  |  |  |  |  |
A | 3.833 | -0.020 | 0.012 | 0.003 | 0.005 | 0.8362
B | 3.865 | -0.029 | 0.017 | 0.005 | 0.007 | 0.8444
FRE |  |  |  |  |  |
A | 3.975 | -0.081 | 0.014 | 0.004 | 0.006 | 0.9839
B | 3.987 | -0.087 | 0.022 | 0.006 | 0.009 | 0.9692
FOBE |  |  |  |  |  |
A | 3.916 | -0.037 | 0.021 | 0.007 | 0.005 | 0.9073
B | 3.969 | -0.054 | 0.013 | 0.004 | 0.001 | 0.9932
FORE |  |  |  |  |  |
A | 3.864 | -0.034 | 0.029 | 0.009 | 0.007 | 0.8122
B | 3.879 | -0.039 | 0.078 | 0.026 | 0.008 | 0.6879
Table 4:  The same as in Table 2 but for $Z$ = $0.03$ and $Y$ = $0.28$. ML | a | b | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---
FBE |  |  |  |  |  |
A | 3.979 | -0.070 | 0.007 | 0.002 | 0.002 | 0.9953
B | 3.976 | -0.070 | 0.009 | 0.002 | 0.003 | 0.9937
FRE |  |  |  |  |  |
A | 4.009 | -0.096 | 0.015 | 0.004 | 0.005 | 0.9892
B | 4.048 | -0.111 | 0.013 | 0.004 | 0.004 | 0.9940
Table 5:  The coefficients of the PMLT relations $\log P$ = a +b$\log T_{eff}$ \+ c $\log$ ($M$/$M_{\odot}\,$) + d $\log$L/$L_{\odot}\,$for both F and FO pulsators as a function of the assumed $\alpha_{ml}$ parameter for $Z$=$0.004$ and $Y$ = $0.25$, $Z$ = $0.008$ and $Y$ = $0.25$, and $Z$ = $0.03$ and $Y$ = $0.28$. For comparison, the relations for $Z$ = $0.02$ and $Y$ = $0.28$ taken by DS20b are also reported. a | b | c | d | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma_{c}$ | $\sigma_{d}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---|---|---|---
|  |  |  | Z=0.004 | Y= 0.25 |  |  |  |
F |  |  |  |  |  |  |  |  |
10.711 | -3.315 | -0.776 | 0.918 | 0.109 | 0.028 | 0.017 | 0.005 | 0.013 | 0.9990
FO |  |  |  |  |  |  |  |  |
12.042 | -3.636 | -0.574 | 0.799 | 1.271 | 0.325 | 0.112 | 0.037 | 0.032 | 0.9867
|  |  |  | Z=0.008 | Y= 0.25 |  |  |  |
F |  |  |  |  |  |  |  |  |
10.482 | -3.254 | -0.773 | 0.920 | 0.103 | 0.026 | 0.017 | 0.005 | 0.013 | 0.9991
FO |  |  |  |  |  |  |  |  |
10.880 | -3.337 | -0.622 | 0.816 | 0.122 | 0.031 | 0.009 | 0.003 | 0.003 | 0.9999
|  |  |  | Z=0.02 | Y= 0.28 |  |  |  |
F |  |  |  |  |  |  |  |  |
10.268 | -3.192 | -0.758 | 0.919 | 0.001 | 0.025 | 0.015 | 0.005 | 0.011 | 0.9995
FO |  |  |  |  |  |  |  |  |
10.595 | -3.253 | -0.621 | 0.804 | 0.002 | 0.067 | 0.014 | 0.005 | 0.003 | 0.9996
|  |  |  | Z=0.03 | Y= 0.28 |  |  |  |
F |  |  |  |  |  |  |  |  |
10.414 | -3.227 | -0.765 | 0.918 | 0.119 | 0.029 | 0.023 | 0.008 | 0.007 | 0.9998

##  3 The dependence of the period-age and period-age-color relations on the
metallicity

By combining our pulsation model results for the aforementioned chemical
compositions with the corresponding stellar evolution model predictions, and
adopting the derived PMLT relation, the periods for each combination of mass,
luminosity, and effective temperature, along the portions of the evolutionary
tracks within the predicted instability strip boundaries (see Fig. 1), were
derived.
A linear regression of the periods estimated from the PMLT relation, the age
and mean colors - in the various selected photometric systems - of the
corresponding points along the evolutionary tracks, allowed us to derive
accurate PA and multi-filter PAC relations, including the first theoretical
PAC in the Gaia bands for the new chemical compositions (see Table
LABEL:param_puls_models). The coefficients of the PA relation are reported in
Table 6.

Fig. 3 shows the canonical (left panel) and noncanonical (right panel) PA
relations obtained for the standard convective efficiency for $Z$ = $0.004$
(dashed blue line), $Z$ = $0.008$ (solid green line), $Z$ = $0.02$ (dotted
cyan line) and $Z$ = $0.03$ (dash-dotted magenta line). The colored areas
represent the $1\sigma$ errors on these relationships.

For any given initial mass, noncanonical models during the blue loop phase are
older and brighter with respect to the canonical ones as a consequence of the
larger convective core during the central H-burning stage, and of the larger
He core during the central He-burning phase. As a result, PA relations -
regardless of the metallicity - based on the noncanonical scenario, predict
older ages than the corresponding canonical PA relation as also shown at solar
metallicity by DS20b, for any given period.

Regardless of the selected evolutionary scenario, i.e. either the canonical or
the noncanonical one, as the metal abundance increases, the PA relation gets
flatter. This occurrence becomes more evident when moving from $Z$=$0.008$ to
$Z$=$0.02$ and then to $Z$=$0.03$.

Fig. 4 shows the same comparison as Fig. 3 but for FO-mode models. In this
case, due to the reduced number of available models, only the canonical cases
for $Z$=$0.004$, $Z$=$0.008$ and $Z$=$0.02$ are represented. As FO-mode
pulsators have masses and periods smaller than F pulsators, their ages are
systematically older than the F ones for each chemical composition. Again, we
note a non-negligible metallicity effect on both the slope and the zero point
that becomes important, particularly when moving from $Z$=$0.004$ to
$Z$=$0.02$. This difference is related to the decreasing number of FO-mode
pulsators as the metallicity increases.

Similar to the PL relation, the PA is affected by the intrinsic dispersion,
due to the finite width of the instability strip. This implies that the use of
PA relations provides individual CC ages that are affected by systematic
errors related to the actual position of the pulsator inside the IS. In order
to take into account this occurrence, as done in DS20b, we derived the PAC
relation for various photometric systems, such as the Gaia EDR3, Johnson-
Cousin and JWST NIRCam photometric systems.

The coefficients of the F and FO-mode PAC relation for the Gaia
$(G_{BP}-G_{RP})$ color, as well as $(V-I)$, $(V-K)$, $(J-K)$ and
$(F115W-F150W)$, $(F150W-F200W)$ and $(F115W-F200W)$ colors are listed in
Tables 7 and 8, respectively.222To validate the inclusion of the color term to
the Period-Age relation, an F-test analysis was performed (Press et al., 2003,
and references therein). Assuming a standard significance level of
$\alpha$=$0.05$, and comparing it with the p-values obtained from the F-test,
lead us to assert that for the 77 Period-Age-Color relations listed in Tables
7 and 8 the color term is favoured and therefore, the Period-Age-Color
relation is preferred to the Period-Age relation.

The availability of a theoretical scenario spanning a wide metallicity range
allowed us to obtain the Period-Age-Metallicity (PAZ) and Period-Age-Color-
Metallicity (PACZ) relations. The coefficients of the PAZ relations are
reported in Table 9 while Table 10 shows the coefficients of the PACZ relation
for both F and FO-mode pulsators. These PAZ and PACZ relations are the first
metallicity-dependent theoretical calibration of the CC PA and PAC
relations.333Following a referee’s suggestion, we verified that present
results are barely affected, if any, by the adopted density grid for both
pulsation and evolutionary models. This is because the density of the
pulsation model grid has been optimized to properly define the location of the
instability strip boundaries. From the point of view of the evolutionary
models, the density of the grid - both in terms of the number of evolutionary
tracks and time steps for each stellar model - is very fine to properly trace
any evolutionary features during the core He-burning stage. This
notwithstanding, we decided to perform a test by deriving the PA, PAC, PAZ and
PACZ relations after halving the number of the adopted evolutionary models. We
found that this assumption increases the errors on the slope and intercept as
well as the standard deviation of the relations by about 35%; but the average
values of the slope and intercept in each relation remain consistent within
the errors.

Table 6: The coefficients of the F and FO-mode PA relations in the form $\log
t$ = $a$ \+ $b\log P$, for $Z$=$0.004$, $Y$= $0.25$; $Z$=$0.008$, $Y$= $0.25$
and $Z$=$0.03$, $Y$= $0.28$ derived by assuming linear IS boundaries and
adopting both case A and B ML relations. The last two columns represent the
root-mean-square deviation ($\sigma$) and the R-squared ($R^{2}$) coefficients

. For comparison, the relations for $Z$=$0.02$ and $Y$=$0.28$ taken by DS20b
are also reported.  ML a b $\sigma_{a}$ $\sigma_{b}$ $\sigma$ $R^{2}$ Z=0.004
Y= 0.25 Fundamental mode A 8.455 -0.800 0.013 0.013 0.090 0.849 B 8.570 -0.692
0.019 0.014 0.089 0.767 First overtone mode A 8.342 -0.891 0.024 0.063 0.054
0.585 Z=0.008 Y= 0.25 Fundamental mode A 8.398 -0.776 0.016 0.013 0.097 0.681
B 8.503 -0.688 0.010 0.007 0.085 0.839 First overtone mode A 8.280 -0.777
0.023 0.039 0.065 0.770 Z=0.02 Y= 0.28 Fundamental mode A 8.393 -0.704 0.008
0.009 0.084 0.916 B 8.480 -0.626 0.010 0.009 0.080 0.866 First overtone mode A
8.120 -0.396 0.020 0.057 0.052 0.506 Z=0.03 Y= 0.28 Fundamental mode A 8.336
-0.673 0.009 0.010 0.076 0.887 B 8.356 -0.555 0.012 0.009 0.082 0.822

Table 7: The coefficients of the F-mode canonical and noncanonical PAC relations $\log t$ = $a$ \+ $b\log P$ \+ $c$ CI), for $Z$=$0.004$, $Y$= $0.25$; $Z$=$0.008$, $Y$= $0.25$; $Z$=$0.02$, $Y$=$0.28$; and $Z$=$0.03$, $Y$= $0.28$. The last two columns represent the root-mean-square deviation ($\sigma$) and the R-squared ($R^{2}$) coefficients. Z | Y | color | ML | a | b | c | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma_{c}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---|---|---|---|---|---
Fundamental mode |  |  |  |  |  |  |  |  |  |  |
0.004 | 0.25 | $G_{BP}-G_{RP}$ | A | 8.794 | -0.571 | -0.543 | 0.036 | 0.026 | 0.055 | 0.082 | 0.868
0.004 | 0.25 | V-I | A | 8.808 | -0.578 | -0.618 | 0.038 | 0.026 | 0.063 | 0.082 | 0.868
0.004 | 0.25 | V-K | A | 8.767 | -0.576 | -0.254 | 0.033 | 0.025 | 0.025 | 0.081 | 0.869
0.004 | 0.25 | J-K | A | 8.715 | -0.579 | -0.864 | 0.027 | 0.024 | 0.082 | 0.081 | 0.870
0.004 | 0.25 | F115W-F150W | A | 8.769 | -0.583 | -1.087 | 0.032 | 0.024 | 0.102 | 0.080 | 0.871
0.004 | 0.25 | F150W-F200W | A | 8.616 | -0.574 | -1.954 | 0.020 | 0.025 | 0.190 | 0.081 | 0.869
0.004 | 0.25 | F115W-F200W | A | 8.716 | -0.579 | -0.702 | 0.027 | 0.024 | 0.066 | 0.081 | 0.870
0.004 | 0.25 | $G_{BP}-G_{RP}$ | B | 8.517 | -0.758 | 0.135 | 0.024 | 0.024 | 0.035 | 0.068 | 0.772
0.004 | 0.25 | V-I | B | 8.080 | -0.394 | 0.047 | 0.015 | 0.014 | 0.027 | 0.071 | 0.421
0.004 | 0.25 | V-K | B | 8.096 | -0.379 | 0.006 | 0.013 | 0.013 | 0.011 | 0.071 | 0.421
0.004 | 0.25 | J-K | B | 8.103 | -0.370 | -0.006 | 0.011 | 0.012 | 0.034 | 0.071 | 0.421
0.004 | 0.25 | F115W-F150W | B | 8.159 | -0.319 | -0.227 | 0.014 | 0.012 | 0.045 | 0.071 | 0.423
0.004 | 0.25 | F150W-F200W | B | 8.097 | -0.416 | 0.253 | 0.008 | 0.013 | 0.072 | 0.071 | 0.422
0.004 | 0.25 | F115W-F200W | B | 8.114 | -0.353 | -0.048 | 0.011 | 0.012 | 0.028 | 0.071 | 0.421
0.008 | 0.25 | $G_{BP}-G_{RP}$ | A | 8.772 | -0.313 | -0.769 | 0.026 | 0.030 | 0.045 | 0.085 | 0.733
0.008 | 0.25 | V-I | A | 8.728 | -0.345 | -0.794 | 0.026 | 0.030 | 0.051 | 0.087 | 0.724
0.008 | 0.25 | V-K | A | 8.651 | -0.332 | -0.321 | 0.022 | 0.030 | 0.020 | 0.086 | 0.728
0.008 | 0.25 | J-K | A | 8.574 | -0.333 | -1.086 | 0.018 | 0.029 | 0.064 | 0.086 | 0.731
0.008 | 0.25 | F115W-F150W | A | 8.691 | -0.327 | -1.497 | 0.022 | 0.027 | 0.082 | 0.084 | 0.738
0.008 | 0.25 | F150W-F200W | A | 8.407 | -0.330 | -2.168 | 0.015 | 0.031 | 0.136 | 0.087 | 0.726
0.008 | 0.25 | F115W-F200W | A | 8.578 | -0.323 | -0.896 | 0.018 | 0.029 | 0.051 | 0.085 | 0.734
0.008 | 0.25 | $G_{BP}-G_{RP}$ | B | 8.783 | -0.401 | -0.533 | 0.030 | 0.030 | 0.054 | 0.082 | 0.846
0.008 | 0.25 | V-I | B | 8.600 | -0.581 | -0.216 | 0.030 | 0.031 | 0.062 | 0.085 | 0.840
0.008 | 0.25 | V-K | B | 8.620 | -0.526 | -0.131 | 0.024 | 0.030 | 0.024 | 0.084 | 0.841
0.008 | 0.25 | J-K | B | 8.608 | -0.499 | -0.527 | 0.018 | 0.029 | 0.078 | 0.084 | 0.842
0.008 | 0.25 | F115W-F150W | B | 8.772 | -0.373 | -1.207 | 0.024 | 0.026 | 0.098 | 0.081 | 0.850
0.008 | 0.25 | F150W-F200W | B | 8.512 | -0.627 | -0.337 | 0.011 | 0.031 | 0.165 | 0.085 | 0.839
0.008 | 0.25 | F115W-F200W | B | 8.633 | -0.455 | -0.525 | 0.018 | 0.029 | 0.063 | 0.083 | 0.844
0.02 | 0.28 | $G_{BP}-G_{RP}$ | A | 8.416 | -0.711 | -0.019 | 0.037 | 0.022 | 0.049 | 0.080 | 0.917
0.02 | 0.28 | V-I | A | 8.338 | -0.740 | 0.090 | 0.040 | 0.023 | 0.062 | 0.080 | 0.921
0.02 | 0.28 | V-K | A | 8.351 | -0.738 | 0.033 | 0.033 | 0.023 | 0.024 | 0.080 | 0.921
0.02 | 0.28 | J-K | A | 8.360 | -0.735 | 0.109 | 0.027 | 0.022 | 0.084 | 0.080 | 0.921
0.02 | 0.28 | F115W-F150W | A | 8.347 | -0.736 | 0.156 | 0.035 | 0.022 | 0.115 | 0.080 | 0.921
0.02 | 0.28 | F150W-F200W | A | 8.377 | -0.735 | 0.201 | 0.016 | 0.023 | 0.162 | 0.080 | 0.921
0.02 | 0.28 | F115W-F200W | A | 8.360 | -0.736 | 0.089 | 0.027 | 0.022 | 0.068 | 0.080 | 0.921
0.02 | 0.28 | $G_{BP}-G_{RP}$ | B | 8.439 | -0.702 | 0.102 | 0.026 | 0.017 | 0.034 | 0.079 | 0.858
0.02 | 0.28 | V-I | B | 8.462 | -0.692 | 0.082 | 0.023 | 0.017 | 0.036 | 0.079 | 0.857
0.02 | 0.28 | V-K | B | 8.473 | -0.690 | 0.031 | 0.019 | 0.017 | 0.014 | 0.079 | 0.857
0.02 | 0.28 | J-K | B | 8.482 | -0.686 | 0.099 | 0.017 | 0.016 | 0.049 | 0.079 | 0.857
0.02 | 0.28 | F115W-F150W | B | 8.457 | -0.693 | 0.182 | 0.022 | 0.016 | 0.068 | 0.079 | 0.858
0.02 | 0.28 | F150W-F200W | B | 8.499 | -0.685 | 0.170 | 0.012 | 0.017 | 0.093 | 0.079 | 0.857
0.02 | 0.28 | F115W-F200W | B | 8.478 | -0.690 | 0.091 | 0.017 | 0.016 | 0.039 | 0.079 | 0.857
0.03 | 0.28 | $G_{BP}-G_{RP}$ | A | 8.211 | -0.752 | 0.181 | 0.036 | 0.024 | 0.050 | 0.074 | 0.890
0.03 | 0.28 | V-I | A | 8.227 | -0.746 | 0.186 | 0.033 | 0.023 | 0.054 | 0.074 | 0.890
0.03 | 0.28 | V-K | A | 8.251 | -0.743 | 0.071 | 0.027 | 0.023 | 0.021 | 0.074 | 0.890
0.03 | 0.28 | J-K | A | 8.268 | -0.739 | 0.247 | 0.022 | 0.022 | 0.075 | 0.074 | 0.889
0.03 | 0.28 | F115W-F150W | A | 8.238 | -0.742 | 0.359 | 0.029 | 0.022 | 0.102 | 0.074 | 0.890
0.03 | 0.28 | F150W-F200W | A | 8.306 | -0.738 | 0.443 | 0.013 | 0.023 | 0.142 | 0.074 | 0.889
0.03 | 0.28 | F115W-F200W | A | 8.268 | -0.741 | 0.201 | 0.022 | 0.022 | 0.060 | 0.074 | 0.890
0.03 | 0.28 | $G_{BP}-G_{RP}$ | B | 8.369 | -0.539 | -0.027 | 0.021 | 0.023 | 0.035 | 0.080 | 0.822
0.03 | 0.28 | V-I | B | 8.374 | -0.526 | -0.050 | 0.017 | 0.022 | 0.034 | 0.080 | 0.822
0.03 | 0.28 | V-K | B | 8.368 | -0.530 | -0.018 | 0.015 | 0.022 | 0.014 | 0.080 | 0.822
0.03 | 0.28 | J-K | B | 8.365 | -0.529 | -0.068 | 0.014 | 0.021 | 0.050 | 0.080 | 0.822
0.03 | 0.28 | F115W-F150W | B | 8.368 | -0.538 | -0.067 | 0.018 | 0.021 | 0.073 | 0.080 | 0.822
0.03 | 0.28 | F150W-F200W | B | 8.354 | -0.530 | -0.118 | 0.012 | 0.022 | 0.095 | 0.080 | 0.822
0.03 | 0.28 | F115W-F200W | B | 8.363 | -0.535 | -0.044 | 0.014 | 0.021 | 0.041 | 0.080 | 0.822
Table 8:  The same as in Table 7 but for FO-mode canonical PAC relations. Z | Y | color | ML | a | b | c | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma_{c}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---|---|---|---|---|---
First overtone mode |  |  |  |  |  |  |  |  |  |  |
0.004 | 0.25 | $G_{BP}-G_{RP}$ | A | 8.002 | -1.097 | 0.668 | 0.107 | 0.088 | 0.206 | 0.047 | 0.614
0.004 | 0.25 | V-I | A | 7.995 | -1.088 | 0.741 | 0.110 | 0.087 | 0.230 | 0.047 | 0.613
0.004 | 0.25 | V-K | A | 8.005 | -1.094 | 0.333 | 0.107 | 0.088 | 0.103 | 0.047 | 0.614
0.004 | 0.25 | J-K | A | 8.061 | -1.089 | 1.173 | 0.090 | 0.087 | 0.363 | 0.047 | 0.613
0.004 | 0.25 | F115W-F150W | A | 8.054 | -1.071 | 1.266 | 0.094 | 0.084 | 0.399 | 0.047 | 0.612
0.004 | 0.25 | F150W-F200W | A | 8.034 | -1.125 | 4.024 | 0.096 | 0.093 | 1.211 | 0.046 | 0.615
0.004 | 0.25 | F115W-F200W | A | 8.049 | -1.084 | 0.964 | 0.094 | 0.086 | 0.300 | 0.047 | 0.613
0.008 | 0.25 | $G_{BP}-G_{RP}$ | A | 8.044 | -0.601 | 0.180 | 0.100 | 0.076 | 0.179 | 0.064 | 0.421
0.008 | 0.25 | V-I | A | 8.040 | -0.599 | 0.208 | 0.104 | 0.075 | 0.207 | 0.064 | 0.421
0.008 | 0.25 | V-K | A | 8.042 | -0.601 | 0.095 | 0.100 | 0.075 | 0.092 | 0.064 | 0.422
0.008 | 0.25 | J-K | A | 8.058 | -0.599 | 0.336 | 0.085 | 0.074 | 0.326 | 0.064 | 0.422
0.008 | 0.25 | F115W-F150W | A | 8.055 | -0.594 | 0.367 | 0.090 | 0.072 | 0.364 | 0.064 | 0.421
0.008 | 0.25 | F150W-F200W | A | 8.050 | -0.611 | 1.161 | 0.087 | 0.079 | 1.044 | 0.064 | 0.422
0.008 | 0.25 | F115W-F200W | A | 8.054 | -0.598 | 0.279 | 0.089 | 0.074 | 0.270 | 0.064 | 0.422
0.02 | 0.28 | $G_{BP}-G_{RP}$ | A | 8.082 | -0.452 | 0.069 | 0.142 | 0.097 | 0.223 | 0.050 | 0.332
0.02 | 0.28 | V-I | A | 8.082 | -0.451 | 0.078 | 0.144 | 0.095 | 0.258 | 0.050 | 0.331
0.02 | 0.28 | V-K | A | 8.083 | -0.452 | 0.036 | 0.135 | 0.095 | 0.115 | 0.050 | 0.332
0.02 | 0.28 | J-K | A | 8.090 | -0.451 | 0.131 | 0.115 | 0.094 | 0.416 | 0.050 | 0.332
0.02 | 0.28 | F115W-F150W | A | 8.090 | -0.448 | 0.139 | 0.121 | 0.090 | 0.471 | 0.050 | 0.331
0.02 | 0.28 | F150W-F200W | A | 8.094 | -0.452 | 0.364 | 0.108 | 0.100 | 1.224 | 0.050 | 0.331
0.02 | 0.28 | F115W-F200W | A | 8.091 | -0.449 | 0.101 | 0.118 | 0.093 | 0.340 | 0.050 | 0.331
Table 9: The coefficients of the F and FO-mode PAZ relations in the form $\log t$ = $a$ \+ $b\log P$ \+ c [Fe/H] obtained by using all the computed models for $Z$ = $0.004$, $Z$ = $0.008$, $Z$= $0.02$ and $Z$ = $0.03$. The root-mean-square deviation ($\sigma$) and the R-squared ($R^{2}$) coefficients are reported in the last two columns. ML | a | b | c | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma_{c}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---|---|---
Fundamental mode |  |  |  |  |  |  |  |
A | 8.419 | -0.775 | -0.015 | 0.006 | 0.006 | 0.007 | 0.083 | 0.845
B | 8.423 | -0.642 | -0.067 | 0.006 | 0.004 | 0.006 | 0.081 | 0.842
First overtone mode |  |  |  |  |  |  |  |
A | 8.236 | -0.762 | -0.079 | 0.014 | 0.028 | 0.015 | 0.041 | 0.686
Table 10: The coefficients of the F and FO PACZ relations $\log t$ = $a$ \+ $b\log P$ \+ $c$ CI + d [Fe/H] ), derived by adopting both cases A and B ML relations. color | ML | a | b | c | d | $\sigma_{a}$ | $\sigma_{b}$ | $\sigma_{c}$ | $\sigma_{d}$ | $\sigma$ | $R^{2}$
---|---|---|---|---|---|---|---|---|---|---|---
Fundamental mode |  |  |  |  |  |  |  |  |  |  |
$G_{BP}-G_{RP}$ | A | 8.711 | -0.559 | -0.446 | -0.048 | 0.018 | 0.014 | 0.026 | 0.026 | 0.042 | 0.858
V-I | A | 8.676 | -0.577 | -0.460 | -0.038 | 0.017 | 0.014 | 0.029 | 0.007 | 0.078 | 0.856
V-K | A | 8.628 | -0.572 | -0.186 | -0.023 | 0.014 | 0.013 | 0.011 | 0.007 | 0.078 | 0.857
J-K | A | 8.581 | -0.570 | -0.650 | -0.001 | 0.011 | 0.013 | 0.036 | 0.007 | 0.077 | 0.859
F115W-F150W | A | 8.651 | -0.558 | -0.921 | -0.012 | 0.013 | 0.012 | 0.046 | 0.007 | 0.076 | 0.862
F150W-F200W | A | 8.490 | -0.589 | -1.177 | 0.030 | 0.007 | 0.013 | 0.075 | 0.007 | 0.079 | 0.856
F115W-F200W | A | 8.584 | -0.566 | -0.530 | 0.006 | 0.011 | 0.013 | 0.029 | 0.007 | 0.077 | 0.859
$G_{BP}-G_{RP}$ | B | 8.560 | -0.519 | -0.231 | -0.014 | 0.013 | 0.011 | 0.020 | 0.020 | 0.043 | 0.847
V-I | B | 8.360 | -0.494 | 0.077 | -0.022 | 0.011 | 0.010 | 0.020 | 0.005 | 0.080 | 0.670
V-K | B | 8.374 | -0.467 | 0.054 | -0.021 | 0.009 | 0.010 | 0.008 | 0.004 | 0.080 | 0.671
J-K | B | 8.366 | -0.457 | 0.215 | -0.029 | 0.007 | 0.010 | 0.026 | 0.004 | 0.080 | 0.671
F115W-F150W | B | 8.318 | -0.419 | -0.246 | -0.004 | 0.010 | 0.009 | 0.036 | 0.004 | 0.076 | 0.617
F150W-F200W | B | 8.265 | -0.450 | -0.147 | 0.008 | 0.006 | 0.010 | 0.054 | 0.004 | 0.076 | 0.616
F115W-F200W | B | 8.290 | -0.430 | -0.113 | 0.003 | 0.008 | 0.010 | 0.022 | 0.004 | 0.076 | 0.617
First overtone mode |  |  |  |  |  |  |  |  |  |  |
$G_{BP}-G_{RP}$ | A | 7.958 | -0.832 | 0.410 | -0.155 | 0.073 | 0.033 | 0.106 | 0.106 | 0.040 | 0.698
V-I | A | 7.965 | -0.832 | 0.463 | -0.134 | 0.073 | 0.033 | 0.125 | 0.020 | 0.040 | 0.698
V-K | A | 7.979 | -0.837 | 0.208 | -0.123 | 0.070 | 0.034 | 0.056 | 0.018 | 0.040 | 0.698
J-K | A | 8.023 | -0.836 | 0.734 | -0.110 | 0.059 | 0.034 | 0.202 | 0.016 | 0.040 | 0.697
F115W-F150W | A | 8.022 | -0.827 | 0.798 | -0.105 | 0.061 | 0.033 | 0.226 | 0.016 | 0.040 | 0.697
F150W-F200W | A | 8.003 | -0.853 | 2.462 | -0.121 | 0.061 | 0.036 | 0.639 | 0.018 | 0.040 | 0.699
F115W-F200W | A | 8.016 | -0.833 | 0.605 | -0.109 | 0.061 | 0.034 | 0.167 | 0.016 | 0.040 | 0.697
Figure 3: Fundamental PA relations for $Z$ = $0.004$ (dashed lines), $Z$ =
$0.008$ (solid lines), $Z$ = $0.02$ (dotted lines) and $Z$ = $0.03$ (dash-
dotted lines), obtained by adopting the canonical ML relation (left panel) and
noncanonical ML relation (right panel) and the standard efficiency for the
superadiabatic convection. The colored areas represent the $1\sigma$ errors on
these relations. Figure 4: First overtone PA relations for $Z$ = $0.004$
(dashed lines), $Z$ = $0.008$ (solid lines), $Z$ = $0.02$ (dotted lines) and
$Z$ = $0.03$, obtained by adopting the canonical ML relation and the standard
efficiency for the superadiabatic convection.

###  3.1 Comparison with literature results

In this Section we compare the derived PA and PAC relations with those
obtained by other authors (see Table 4 in DS20b for more details). For the
chemical composition $Z$ = $0.004$ and $Y$ = $0.25$, Fig. 5 shows the
comparison between our canonical PA relation derived for F-mode pulsators
(magenta solid line) with the ones derived by Bono et al. (2005) (blue dashed
line). Fig. 6 shows the same comparison but for FO-mode pulsators.

These figures show that, for F pulsators, within the error, a good agreement
does exist, whereas in the case of the FO-mode, our relation is steeper than
that provided by Bono et al. (2005); although the two slopes are still
consistent within the errors. This difference can be related to, the small
number of FO pulsators with respect to the F-mode models, a slight difference
in the assumed ML relation, and the selected mass range because we only
selected masses equal to or greater than four solar masses while in Bono et
al. (2005) masses less than four solar masses were also included.

In Fig. 7, the left panel displays the comparison between our canonical PA
relations obtained for $Z$=$0.008$ and $Y$=$0.25$ (magenta solid line) and the
one derived by Bono et al. (2005) (blue dashed line), while the right panel
shows our noncanonical PA relation for the same chemical composition compared
with the Efremov (2003) results (green dash-dotted line). Our relation for
$Z$=$0.008$ is in excellent accord with the result by Bono et al. (2005), and
slightly steeper than the Efremov (2003) relation, but still consistent within
the corresponding errors.

The comparison between our canonical PA relation for FO-mode pulsators with
$Z$= $0.008$ and the PA relation by Bono et al. (2005) is shown in Fig. 8.
Here, the difference in the slope is still more significant than at
$Z$=$0.004$ because of the smaller number of FO-mode pulsators as the
metallicity increases. In Fig. 9, we compare our noncanonical PA relation for
$Z$=$0.004$ and $Z$=$0.008$ with the relation derived by Anderson et al.
(2016) for $Z$=$0.006$. The Anderson et al. (2016) PA relation provides ages
that are systematically older than the ones predicted by our relations. This
discrepancy is due to the brighter noncanonical ML relation adopted by
Anderson et al. (2016) which includes both overshooting and rotation in their
models.

In Fig. 10, our noncanonical PA relation, derived for $Z$=$0.03$ and $Y$ =
$0.28$ (solid magenta line) is compared with the one derived by Magnier et al.
(1997) (dotted red line) and the ones derived by Senchyna et al. (2015) for
M31 CCs (orange triangle marker) and M31 clusters (grey star marker). While
the agreement with Magnier et al. (1997) is very good, our relations show
fainter/brighter zero points than the Senchyna et al. (2015) results based on
cluster Cepheids. In any case, their associated errors are so large that our
predictions are still consistent with the results by these authors. The upper
panels of Figs. 5-10 show the relative age difference between our PA relations
and the literature ones (see labels).

Figure 5: Bottom panel: comparison between F-mode PA relations derived in the
present work for $Z$=$0.004$ (solid magenta line); obtained by varying the
adopted ML relation (see labels) with the canonical PA relations derived by
Bono et al. (2005). The colored areas represent the $1\sigma$ errors on these
relationships. Upper panel: the relative difference, with the expected
uncertainty, between the age predictions obtained from the PA relationships
compared in the bottom panel. Figure 6: The same as in Fig. 5 but for FO-mode
pulsators.

Figure 7:  Bottom panels: comparison between present F-mode PA relations
obtained for $Z$=$0.008$ (solid magenta line) by varying the adopted ML
relation (see labels), with similar predictions from the literature: the
dashed blue line shows the canonical PA relations by Bono et al. (2005) (left
panel) and the dashed green line the noncanonical PA relation by Efremov
(2003) (right panel). The $1\sigma$ errors on these relationships as provided
by the authors are represented with the colored areas. Upper panels: the
relative age difference between the age predictions obtained by current PA
relations for the canonical case (left panel) and the noncanonical one (right
panel) and those obtained with the PA relations taken from the literature.
Figure 8: Bottom panel: comparison between present FO-mode PA relation (solid
magenta line) obtained for $Z$=$0.008$ and ML case A, and the theoretical FO-
mode GCC PA relation obtained by Bono et al. (2005) (dashed blue line). Upper
panel: the relative difference between the age estimates provided by these PA
relations. Figure 9: Bottom panel: comparison between F-mode noncanonical PA
relations for $Z$=$0.004$ (solid brown line) and $Z$ = $0.008$ (dashed line)
derived in the present work and the noncanonical PA relation provided by
Anderson et al. (2016) for $Z$ = $0.006$. Upper panel: the relative difference
between the age estimates provided by these PA relations. Figure 10: Bottom
panel: comparison between present noncanonical F-mode PA relation (solid
magenta line) obtained for $Z$=$0.03$ and ML case B, and the PA relation
obtained by Magnier et al. (1997) (dotted red line). The relations marked with
triangles and stars represent the PA relations for M31 CCs and M31 clusters,
respectively; both derived by Senchyna et al. (2015). Upper panel: the
relative age difference provided by these PA relations.

##  4 Individual ages of Gaia Early Data Release 3 Classical Cepheids

In this Section, we apply the theoretical relations discussed above to a
subsample of Gaia Early Data Release 3 (EDR3) Galactic Cepheids with bayesian
distances (see Bailer-Jones et al., 2021, for more details) and complementary
metal abundance data (Gaia Collaboration et al., 2017; Groenewegen, 2018;
Ripepi et al., 2020), in order to apply both the PAZ, PACZ relations and the
PA, PAC, at a fixed solar metallicity, to the Cepheid sample. The resulting
sample consists of 416 F-mode and 56 FO-mode pulsators.

###  4.1 Individual age estimates from PAZ and PACZ relations

By applying the PAZ and PACZ relations derived above, we estimated the
individual ages of all the Cepheids in the selected sample. Figs. 11 and 12
show the distribution of the individual ages computed by using PAZ and PACZ
relations, respectively. Inspection of these plots suggests that, in both
cases, the age distribution depends on the assumed ML relation, with the
brighter ML relation (case B) providing a shift of the age distribution
towards older ages. This trend confirms previous results by Anderson et al.
(2016), Bono et al. (2005) and DS20b. Furthermore, a comparison between Fig.
11 and 12 confirms that similar results for the main peak of the age
distribution are inferred when using the PAZ or the PACZ relations.

For the sake of quantifying the impact of the metallicity term on the previous
relations, we also plotted, in Fig. 13, the individual age distributions
obtained when applying the F and FO-mode PAC relations derived for the solar
metallicity case. Regardless of the selected ML relation, the peaks of the
retrieved F-mode CC age distributions appear to be shifted towards slightly
older ages with respect to the case in which the appropriate CC metallicity is
accounted for. For the FO-mode CCs, the age distribution appears quite more
peaked than the one obtained from the PACZ relation.

Assuming the canonical ML relation, the age distribution for fundamental
pulsators (see blue bars in the left panels of Figs. 11 and 12), made using
the PAZ and the PACZ relations, peaks around 7.86 and 7.89 Myrs, respectively.
The minimum and maximum age values derived from the PAZ relation are 6.99 Myr
and 8.20 Myr, respectively. However, the application of the PACZ provides a
minimum age of 7.13 Myr and a maximum age of 8.23 Myr. In the noncanonical
case (brown bars), the PAZ relation provides a peak age around 7.92 Myr while
the PACZ’s is around 7.96 Myr. The minimum and maximum age values from the PAZ
and PACZ relations are, 7.24 Myr and 8.28 Myr, and 7.32 Myr and 8.24 Myr,
respectively. As expected, the entire FO-mode GCC age distribution (see
magenta bars in the right panels of Figs. 11 and 12) is shifted towards older
ages with a peak around 7.86 Myr and 7.93 Myr from the PAZ and PACZ relations,
respectively; and minimum and maximum age values of 7.61 Myr and 8.05 Myr from
the PAZ application and 7.63 Myr and 8.03 Myr from the PACZ one. Since the age
errors are smaller in the case of the PACZ, we decided to provide only the
ages derived using the PACZ relation. The computed ages of the fundamental
pulsators for canonical and noncanonical ML assumptions are partly reported in
Tables 11 and 12, respectively. The retrieved ages for the canonical first
overtone pulsators are partly listed in Table 13. Complete tables are
available in the supplementary material.

For comparison, Fig. 13 shows the age distributions derived by applying the
PAC relation for the solar metallicity to the same Cepheid sample; that is,
neglecting any difference in the individual metal abundances. In particular,
the left panel, showing the age distribution for the F-mode pulsators,
highlights that the age distributions for canonical (red bars) and
noncanonical (green bars) pulsators each peaks around 7.85 Myr and 8.0 Myr,
respectively. This is slightly different from the metal-dependent value (7.89
and 7.96 Myr, respectively). The minimum and maximum age values are 7.07 Myr
and 8.19 Myr (against 7.13 and 8.23 for the metal-dependent case) for the
canonical case, and 7.26 Myr and 8.32 Myr (against 7.32 and 8.24 for the
metal-dependent case) for the noncanonical case. The FO-mode pulsator age
distribution (right panel) peaks around 7.90 Myr with minimum and maximum age
values of 7.77 Myr and 8.01 Myr (against 7.63 and 8.03 for the metal-dependent
case).

Figure 11:  The predicted age distribution, derived by applying the PAZ
relation to the selected sample of F (left panel) and FO-mode (right panel)
CCs, for the assumed cases A and B of the ML relation (see labels).

Figure 12:  The predicted age distribution, derived by applying the PACZ
relation to the selected sample of F (left panel) and FO-mode (right panel)
CCs, for the assumed cases A and B of the ML relation (see labels).

Figure 13:  The predicted age distribution, derived by applying the PAC
relation for $Z$=$0.02$ to the selected sample of F (left panel) and FO-mode
(right panel) CCs, for the assumed cases A and B of the ML relation (see
labels).

###  4.2 The age distribution maps

Fig. 14 shows the age distribution map (color-code scale), inferred from the
application of the canonical fundamental (left panel), canonical first
overtone (middle panel) PACZ relation and the noncanonical fundamental PACZ
relation (right panel) to the selected sample of Galactic Cepheids. To build
the figure we adopted the photogeometric distances by Bailer-Jones et al.
(2021) based on Gaia EDR3 astrometry and transformed the galactic coordinates
into galactocentric cartesian coordinates, assuming that the investigated
Cepheids are located on the Galactic disk and that the distance of the Sun
from the Galactic centre is 8.3 kpc. The plotted maps show that the selected
sample is concentrated at a relatively low Galactocentric distance so that the
dependence of the inferred age on this parameter is not evident as in our
previous investigation (DS20b). This is due to the limited number of Cepheids
in the Gaia EDR3 catalogue with individual metal abundances. To investigate
the effect of the metallicity term in the adopted PAC relation, Fig. 15 shows
the direct comparison between the individual ages inferred from the PACZ
relation and the ones obtained from the PAC relation at solar chemical
composition, for canonical (upper panel) and noncanonical (middle panel)
F-mode and canonical FO-mode (bottom panel) assumptions.

We notice that in the case of the canonical F-mode relations, the inclusion of
the metallicity term produces, on average, a slight shift towards older ages,
especially moving towards longer periods. On the other hand, for the F-mode
noncanonical case, the individual ages predicted by the PACZ relation appear
to be, on average, smaller than in the fixed $Z$=$0.02$ case. Finally, in the
case of the FO-mode, the difference is significant at shorter periods, with
the ages predicted by the PACZ being systematically smaller than the ones
obtained from the PAC relation. The comparisons shown in Fig. 15 suggest that
the inclusion of the metallicity term is relevant and it can have a
significant effect on the application of PAC relations to constrain the star
formation history of Cepheid host galaxies.

Figure 14: Distribution of the selected Gaia EDR3 Cepheids on the Galactic
plane plotted in polar coordinates for canonical F-mode pulsators (left
panel), canonical FO-mode pulsators (middle panel) and noncanonical F-mode
pulsators (right panel). The Galactic center is in the middle. The black ‘X’
marks the Sun’s position. In each panel, the colored circles show the
predicted individual ages obtained using the PACZ relations, according to the
logarithmic color-bar axis Figure 15: Comparison between the individual ages
obtained by using the PACZ relation and the PAC relation for $Z$=$0.02$ to the
selected sample of canonical F-mode (upper panel), noncanonical F-mode (middle
panel) and canonical FO-mode GCCs (bottom panel). The black solid line is the
1:1 line.

##  5 Conclusions and Future Developments

We have presented an updated evolutionary and pulsational scenario for CCs by
varying the pulsation mode, the ML relation and the chemical composition. The
predicted evolutionary paths and instability strip boundaries were combined to
infer updated PA and multi-filter PAC relations, including the first
theoretical PAC in the Gaia bands for chemical compositions different from the
solar one (see Table LABEL:param_puls_models).

We found that the predicted PA relations, based on the noncanonical scenario,
predict older ages than the corresponding canonical PA relations, for any
given period and chemical composition; confirming previous results for solar
metallicity (DS20b). Moreover, in both the case of the canonical and
noncanonical assumptions, as the metal abundance increases, the F-mode PA
relation gets flatter. Whereas, no clear trend can be noticed for the FO-mode
PA relation apart from a non-negligible variation of its coefficient when
varying the chemical composition. A similar dependence on metallicity is found
for the PAC coefficients. Therefore, since we rely on a theoretical scenario
spanning a wide metallicity range, we derived metal dependent PA and PAC
relations that we named PAZ and PACZ relations, respectively.

These PAZ and PACZ relations were applied to a subset of Galactic Cepheids
from the Gaia EDR3 catalogue with information on the metal abundance. The
inferred individual ages show a distribution that confirms that the predicted
age is higher for FO-mode pulsators and increases as the ML relation gets
brighter. Moreover, the derived age distributions for F and FO-mode pulsators
are compared with similar results based on $Z$=$0.02$ models only. As a result
we can conclude that, even if the selected sample is concentrated at a
relatively low Galactocentric distance the inclusion of the metallicity term
produces slightly different individual ages both in the F-mode and FO-mode
cases. In particular, in the F-mode canonical case the adoption of the PACZ
relation produces, on average, a slight shift towards older ages, especially
at longer periods. Whereas, an opposite trend is found in the F-mode
noncanonical case and in the FO-mode canonical case.

Table 11:  Individual ages for the F-mode GCC in our sample obtained by using
the canonical PAZ and PACZ relations. The full table is available as
supplementary material.

Gaia EDR3 Source Id | RA[deg] | DEC[deg] | P[d] | [Fe/H] |  $\sigma$ [Fe/H] | $G[mag]$ |  $G_{BP}$-$G_{RP}$[mag] | E($G_{BP}$-$G_{RP}$)[mag] |  $\sigma$E($G_{BP}$-$G_{RP}$)[mag] |  $t_{PAZ}$[Myr] |  $\sigma$ $t_{PAZ}$[Myr] |  $t_{PACZ}$[Myr] |  $\sigma$ $t_{PACZ}$[Myr]
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14)
4757601523650165120 | 83.40631 | -62.48977 | 9.84229 | -0.06 | 0.12 | 3.59 | 0.95 | 0.1 | 0.02 | 44.69 | 8.54 | 54.50 | 5.27
4240272953377646592 | 298.11824 | 1.00562 | 7.17680 | 0.14 | 0.12 | 3.75 | 0.95 | 0.19 | 0.02 | 56.70 | 10.84 | 63.24 | 6.12
3366754155291545344 | 106.02718 | 20.57029 | 10.14860 | -0.11 | 0.12 | 3.54 | 0.93 | 0.06 | 0.03 | 43.72 | 8.36 | 54.87 | 5.31
2200153454733285248 | 337.29288 | 58.41521 | 5.36625 | 0.09 | 0.12 | 3.85 | 0.87 | 0.1 | 0.03 | 71.15 | 13.60 | 81.31 | 7.86
4057701830728920064 | 266.89007 | -27.83083 | 7.01302 | -0.21 | 0.12 | 4.33 | 0.84 | 0.26 | 0.03 | 58.42 | 11.17 | 74.89 | 7.24
4050309195613114624 | 271.25513 | -29.58011 | 7.59498 | 0.08 | 0.12 | 4.59 | 0.96 | 0.15 | 0.02 | 54.37 | 10.39 | 61.03 | 5.90
4514145288240593408 | 284.56145 | 17.36087 | 4.47092 | 0.1 | 0.12 | 5.17 | 0.79 | 0.26 | 0.03 | 81.93 | 15.66 | 97.45 | 9.42
3435571660360952704 | 97.14203 | 30.49298 | 3.72833 | 0.1 | 0.12 | 5.34 | 0.75 | 0.08 | 0.03 | 94.31 | 18.03 | 112.74 | 10.90
1820309639468685824 | 299.00526 | 16.63476 | 8.38234 | 0.14 | 0.12 | 5.46 | 0.93 | 0.16 | 0.02 | 50.27 | 9.61 | 59.18 | 5.72
4096107909387492992 | 275.34576 | -18.86003 | 5.77338 | 0.11 | 0.12 | 5.48 | 0.89 | 0.27 | 0.03 | 67.18 | 12.84 | 76.06 | 7.36
… | … | … | … | … | … | … | … | … | … | … | … | … | …

Table 12:  The same as in Table 11 but for noncanonical F-mode models. The
full table is available as supplementary material.

Gaia EDR3 Source Id | RA[deg] | DEC[deg] | P[d] | [Fe/H] |  $\sigma$ [Fe/H] | $G[mag]$ |  $G_{BP}$-$G_{RP}$[mag] | E($G_{BP}$-$G_{RP}$)[mag] |  $\sigma$E($G_{BP}$-$G_{RP}$)[mag] |  $t_{PAZ}$[Myr] |  $\sigma$ $t_{PAZ}$[Myr] |  $t_{PACZ}$[Myr] |  $\sigma$ $t_{PACZ}$[Myr]
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14)
4757601523650165120 | 83.40631 | -62.48977 | 9.84229 | -0.06 | 0.12 | 3.59 | 0.95 | 0.1 | 0.02 | 61.58 | 11.54 | 67.04 | 6.64
4240272953377646592 | 298.11824 | 1.00562 | 7.17680 | 0.14 | 0.12 | 3.75 | 0.95 | 0.19 | 0.02 | 73.13 | 13.71 | 78.43 | 7.77
3366754155291545344 | 106.02718 | 20.57029 | 10.14860 | -0.11 | 0.12 | 3.54 | 0.93 | 0.06 | 0.03 | 60.85 | 11.40 | 66.68 | 6.60
2200153454733285248 | 337.29288 | 58.41521 | 5.36625 | 0.09 | 0.12 | 3.85 | 0.87 | 0.1 | 0.03 | 88.82 | 16.65 | 95.33 | 9.44
4057701830728920064 | 266.89007 | -27.83083 | 7.01302 | -0.21 | 0.12 | 4.33 | 0.84 | 0.26 | 0.03 | 78.34 | 14.68 | 84.96 | 8.41
4050309195613114624 | 271.25513 | -29.58011 | 7.59498 | 0.08 | 0.12 | 4.59 | 0.96 | 0.15 | 0.02 | 71.18 | 13.34 | 75.84 | 7.51
4514145288240593408 | 284.56145 | 17.36087 | 4.47092 | 0.1 | 0.12 | 5.17 | 0.79 | 0.26 | 0.03 | 99.71 | 18.69 | 109.22 | 10.81
3435571660360952704 | 97.14203 | 30.49298 | 3.72833 | 0.1 | 0.12 | 5.34 | 0.75 | 0.08 | 0.03 | 112.04 | 21.00 | 122.79 | 12.16
1820309639468685824 | 299.00526 | 16.63476 | 8.38234 | 0.14 | 0.12 | 5.46 | 0.93 | 0.16 | 0.02 | 66.19 | 12.41 | 73.13 | 7.24
4096107909387492992 | 275.34576 | -18.86003 | 5.77338 | 0.11 | 0.12 | 5.48 | 0.89 | 0.27 | 0.03 | 84.49 | 15.84 | 90.62 | 8.97
… | … | … | … | … | … | … | … | … | … | … | … | … | …

Table 13:  The same as in Table 11 but for canonical FO-mode models. The full
table is available as supplementary material.

Gaia EDR3 Source Id | RA[deg] | DEC[deg] | P[d] | [Fe/H] |  $\sigma$ [Fe/H] | $G[mag]$ |  $G_{BP}$-$G_{RP}$[mag] | E($G_{BP}$-$G_{RP}$)[mag] |  $\sigma$E($G_{BP}$-$G_{RP}$)[mag] |  $t_{PAZ}$[Myr] |  $\sigma$ $t_{PAZ}$[Myr] |  $t_{PACZ}$[Myr] |  $\sigma$ $t_{PACZ}$[Myr]
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14)
6058439910929477120 | 187.91793 | -59.42394 | 3.34265 | -0.11 | 0.12 | 5.27 | 0.70 | 0.14 | 0.03 | 70.04 | 6.79 | 67.07 | 6.18
5506374096132016512 | 114.57584 | -48.60141 | 5.69453 | -0.11 | 0.12 | 5.47 | 0.76 | 0.14 | 0.02 | 46.67 | 4.52 | 45.60 | 4.20
5519380081746387328 | 122.99983 | -46.64432 | 4.22727 | 0.09 | 0.12 | 5.59 | 0.74 | 0.05 | 0.03 | 56.47 | 5.47 | 53.50 | 4.93
1853025642297186688 | 316.62604 | 31.18463 | 2.49921 | 0.16 | 0.12 | 5.65 | 0.67 | 0.09 | 0.03 | 83.22 | 8.06 | 75.29 | 6.93
1964855904803120640 | 319.84243 | 38.23746 | 3.33282 | 0.09 | 0.12 | 5.72 | 0.70 | 0.03 | 0.0 | 67.69 | 6.56 | 62.47 | 5.75
541716257882226560 | 42.99483 | 68.88847 | 1.95151 | 0.12 | 0.12 | 5.73 | 0.63 | 0.37 | 0.04 | 101.22 | 9.81 | 90.72 | 8.36
3409635486731094400 | 69.31155 | 18.54301 | 3.14804 | 0.15 | 0.12 | 6.27 | 0.76 | 0.41 | 0.04 | 69.93 | 6.78 | 67.70 | 6.24
5960623272099513856 | 264.40651 | -40.81355 | 3.38033 | 0.18 | 0.12 | 7.07 | 0.72 | 0.34 | 0.04 | 65.87 | 6.38 | 61.00 | 5.62
5848500161483878400 | 222.62620 | -67.49763 | 3.06522 | 0.14 | 0.12 | 7.14 | 0.71 | 0.5 | 0.04 | 71.49 | 6.93 | 66.67 | 6.14
5877460679352962048 | 221.67487 | -61.46196 | 2.39810 | -0.01 | 0.12 | 7.33 | 0.70 | 0.35 | 0.06 | 88.58 | 8.58 | 84.94 | 7.82
… | … | … | … | … | … | … | … | … | … | … | … | … | …

## Acknowledgements

The authors thank the anonymous Referee for the pertinent and useful comments
that significantly improved the content and readability of the manuscript. We
acknowledge the financial support from ASI-Gaia (“Missione Gaia Partecipazione
italiana al DPAC – Operazioni e Attività di Analisi dati”), and the project
Mainstream INAF "Stellar evolution and asteroseismology in the context of
PLATO space mission" (PI: S.Cassisi). GDS thanks the support from Istituto
Nazionale di Fisica Nucleare (INFN), Naples section-Specific Initiative
Moonlight2.

Data availability statement:The data underlying this article are available in
the article and in its online supplementary material.

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