# The RR Lyrae Delay-Time Distribution: A Novel Perspective on Models of Old
Stellar Populations

Sumit K. Sarbadhicary Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  Mairead Heiger  Pittsburgh Particle
Physics, Astrophysics and Cosmology Center (PITT PACC), University of
Pittsburgh, 3941 O’Hara St, Pittsburgh, PA 15260, USA  Carles Badenes
Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC),
University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA 15260, USA  Cecilia
Mateu Departamento de Astronomía, Facultad de Ciencias, Universidad de la
República, Iguá 4225, 14000, Montevideo, Uruguay  Jeffrey Newman  Pittsburgh
Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of
Pittsburgh, 3941 O’Hara St, Pittsburgh, PA 15260, USA  Robin Ciardullo
Department of Astronomy & Astrophysics, The Pennsylvania State University,
University Park, PA 16802, USA  Na’ama Hallakoun Department of Particle
Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 7610001,
Israel  Dan Maoz  School of Physics and Astronomy, Tel-Aviv University, Tel-
Aviv 6997801, Israel  Laura Chomiuk Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA

###### Abstract

The delay-time distribution (DTD) is the occurrence rate of a class of objects
as a function of time after a hypothetical burst of star formation. DTDs are
mainly used as a statistical test of stellar evolution scenarios for supernova
progenitors, but they can be applied to many other classes of astronomical
objects. We calculate the first DTD for RR Lyrae variables using 29,810 RR
Lyrae from the OGLE-IV survey and a map of the stellar-age distribution (SAD)
in the Large Magellanic Cloud (LMC). We find that $\sim 46\%$ of the OGLE-IV
RR Lyrae are associated with delay-times older than 8 Gyr (main-sequence
progenitor masses less than 1 M⊙), and consistent with existing constraints on
their ages, but surprisingly about $51\%$ of RR Lyrae appear have delay times
$1.2-8$ Gyr (main-sequence masses between $1-2$ M⊙ at LMC metallicity). This
intermediate-age signal also persists outside the Bar-region where crowding is
less of a concern, and we verified that without this signal, the spatial
distribution of the OGLE-IV RR Lyrae is inconsistent with the SAD map of the
LMC. Since an intermediate-age RR Lyrae channel is in tension with the lack of
RR Lyrae in intermediate-age clusters (noting issues with small-number
statistics), and the age-metallicity constraints of LMC stars, our DTD result
possibly indicates that systematic uncertainties may still exist in SAD
measurements of old-stellar populations, perhaps stemming from the
construction methodology or the stellar evolution models used. We described
tests to further investigate this issue.

RR Lyrae variable stars (1410); Large Magellanic Cloud (903); Stellar
populations (1622); Stellar evolution (1599); Stellar evolutionary models
(2046); Horizontal branch (2048); Stellar ages (1581); Stellar pulsations
(1625); Hertzsprung Russell diagram (725);

##  1 Introduction

A detailed understanding of stellar evolution remains one of the most sought-
after goals in astrophysics. Popular stellar evolution codes such as Geneva
(Schaller et al., 1992; Schaerer et al., 1993), Y2 (Kim et al., 2002; Yi et
al., 2003; Demarque et al., 2004), BaSTI (Pietrinferni et al., 2004, 2006;
Hidalgo et al., 2018), Darthmouth (Dotter et al., 2008), PARSEC (Bressan et
al., 2012; Chen et al., 2014) and MESA (Paxton et al., 2011, 2013, 2015, 2018)
are powerful tools for interpreting observations of stellar populations.
However, many essential topics in stellar evolution are still not well
understood, and/or not properly taken into account in even the most state-of-
the-art models. Examples of such topics include convection, mass loss and mass
transfer, common envelope evolution, and binary interaction. Often, these
three-dimensional phenomena are approximated by simplified parametric models
tuned to specific observables and integrated into one-dimensional stellar
evolution codes. These uncertainties limit our understanding of many important
phases of stellar evolution, such as the horizontal branch, the asymptotic
giant branch (AGB) and post-AGB phase, planetary nebulae, and supernovae (see
discussions in Gallart et al., 2005; Conroy et al., 2009; Conroy, 2013).

The delay-time distribution (DTD) is a promising method for testing stellar
evolution models in complex stellar populations (Maoz & Mannucci, 2012; Maoz
et al., 2014). The DTD is defined as the occurrence rate of a class of
astronomical object as a function of time since a hypothetical brief burst of
star formation; it is equivalent to the impulse response, or Green’s function.
The DTD constrains the evolutionary timescale and formation efficiency of the
object’s progenitors, and theoretical DTDs are common predictions of stellar
population synthesis models (Mennekens et al., 2010; Nelemans et al., 2013;
Toonen et al., 2013; Zapartas et al., 2017). Observationally, DTDs can be
derived from surveys of objects, provided that the stellar age distributions
(SADs) of their host galaxies are measured (Gal-Yam & Maoz, 2004; Totani et
al., 2008; Maoz & Sharon, 2010; Maoz et al., 2011, 2012; Graur et al., 2014;
Maoz & Graur, 2017; Friedmann & Maoz, 2018).

More recently it was shown that DTDs can be a useful stellar evolution
diagnostic in Local Group galaxies with high-quality observations of their
resolved stellar populations (Badenes et al., 2010; Maoz & Badenes, 2010;
Badenes et al., 2015). Using this approach, Badenes et al. (2015) measured the
first DTD for planetary nebula, showing evidence of two distinct populations
of planetary nebula progenitors: one with ages of 5–8 Gyrs, and another with
ages of 35–800 Myrs. The key advantage of a DTD is that it constrains the
evolutionary timescales for the progenitors of the entire population of
objects in a galaxy, taking into account the variety of star-formation
histories that these objects have evolved from. It can be a powerful tool for
identifying or ruling out the presence of specific formation channels,
measuring their efficiency, and identifying physical mechanisms that are not
part of canonical progenitor models.

Figure 1: The spatial distribution of LMC RR Lyrae (red dots) in the OGLE-IV
sample overlaid on the spatial cells (blue) from the Harris & Zaritsky (2009)
SAD map of the galaxy. Figure 2: $I$-band luminosity function of the OGLE-IV
RR Lyrae. The cumulative histograms show the fraction of RR Lyrae dimmer than
a certain $I$-band brightness. The luminosity function of the full RR Lyrae
sample from Figure 1 is shown in black, and that of RR Lyrae in the ‘LMC503’
field of the OGLE-IV survey, a region with dense stellar crowding (Udalski et
al., 2015), is shown in blue. The inset shows the location of the LMC503
region on an $r$-band continuum map (as reference) of the LMC Bar from the
Magellanic Cloud Emission-Line Survey (MCELS, Winkler et al., 2005; Pellegrini
et al., 2012). The $I$-band completeness limit of the LMC503 region is shown
with the dashed line.

In this paper, we will use a SAD map of the Large Magellanic Cloud (LMC) to
derive the DTD of RR Lyrae stars— pulsating horizontal branch stars with
periods between 0.2 and 1 day (Smith, 2004). We chose to test the DTD method
on RR Lyrae for several reasons. Firstly, the sample size of RR Lyrae in the
LMC is quite large (see Section 2.1), allowing us to measure a DTD with high
significance. Secondly, there is strong evidence that RR Lyrae are mostly
ancient stars, older than 10 Gyrs, given their pulsational properties (Smith,
2004; Marconi et al., 2015) and abundance in old globular clusters (Clement et
al., 2001; Soszyński et al., 2014, 2016). Measuring an RR Lyrae DTD therefore
provides a rigorous test of the DTD method for the recovery of progenitor age-
distributions, as well as star-formation history measurements of old resolved
stellar populations. Lastly, a DTD analysis provides an opportunity to test
stellar evolution models of RR Lyrae in a completely new way. While there is
consensus on the interpretation of RR Lyrae as ancient stars, the lower-limit
on their ages has been somewhat unconstrained. For example, the absence of RR
Lyrae in the SMC cluster Lindsey 1 ($t\approx 9$ Gyr), compared to their
presence in NGC 121 ($t\approx$10–11 Gyr), is generally cited as evidence of a
lower limit of 10 Gyr for progenitor age of RR Lyrae (Olszewski et al., 1996;
Glatt et al., 2008). However, growing evidence of thin disk RR Lyrae in the
Milky Way raises questions about whether this limit might be lower, and if an
intermediate-age progenitor channel can exist (Layden, 1995; Zinn et al.,
2019; Prudil et al., 2020). Additionally uncertainties may exist in the
evolutionary models of RR Lyrae stars themselves: since RR Lyrae are
horizontal-branch stars, their positions on the color-magnitude diagram depend
on an unknown combination of factors (commonly known as the second-parameter
phenomenon) like metallicity, age, mass-loss on the red-giant branch, stellar
rotation, core structure, and chemical abundance (see Fusi Pecci & Bellazzini,
1997; Catelan, 2009; Dotter, 2013, for reviews).

The paper is organized as follows: Section 2 briefly describes the two
ingredients for calculating the RR Lyrae DTD – the OGLE-IV survey of RR Lyrae,
and the Harris & Zaritsky (2009) SAD map of LMC, Section 3 describes the
measurement of the RR Lyrae DTD from the OGLE-IV survey, Section 4 checks
whether the measured DTD is consistent with the observed spatial distribution
of RR Lyrae in the OGLE-IV survey, incompleteness in the SAD map and RR Lyrae
in star clusters, Section 5 discusses the two possible interpretations of the
DTD result – that RR Lyrae may have a previously-unidentified younger
progenitor channel, or unknown systematics still exist in SAD measurements of
resolved stellar populations older than a Gyr.

##  2 Ingredients for Calculating The DTD

###  2.1 OGLE-IV Sample of LMC RR Lyrae

We use the Soszyński et al. (2016) catalog of 39,082 RR Lyrae stars from the
Optical Gravitational Microlensing Experiment (OGLE-
IV)111http://ogle.astrouw.edu.pl/ survey of the LMC (Udalski et al., 2015).
These RR Lyrae were selected by the OGLE-IV pipeline from the full database of
OGLE $I$-band light curves with periods between 0.2 and 1 day, and then
further classified as fundamental (RRab), first-overtone (RRc), and mixed mode
pulsators based on their periods, Fourier amplitudes, and light curve shapes.
We excluded catalog entries that were flagged as Galactic RR Lyrae or
eclipsing variables, objects with uncertain classification, and sources that
fall outside the SAD maps. This produced a final sample of 29,810 RR Lyrae
(Figure 1). The photometric completeness of the sample is quite high, as
evidenced by the I-band luminosity function of RR Lyrae in the most crowded
OGLE-IV field (LMC503, see Figure 2). The field has a completeness limit of
$I\approx 20.5$ (Udalski et al., 2015), whereas the RR Lyrae sample in the
field has a median magnitude $\bar{I}=18.82$ and standard deviation
$\sigma_{I}=0.4$. The median magnitude of the RR Lyrae is nearly
4.2$\sigma_{I}$ above the completeness limit (almost 99.9$\%$ of the RR Lyrae
have I-band magnitude above the completeness limit). Thus even in the most
crowded region, the RR Lyrae sample can be considered photometrically
complete. Also as seen in Figure 2, the I-band luminosity function of the full
OGLE-IV sample inside the Harris & Zaritsky (2009) region (see Section 2.2) is
also well above the completeness limit of the most crowded region. Such high
completeness is important for measuring unbiased rates and DTDs (Maoz &
Badenes, 2010; Badenes et al., 2010).

###  2.2 Stellar Age Distribution Map of LMC

We use the SAD map of the LMC constructed by Harris & Zaritsky (2009)
(hereafter, HZ09). This map provides the best-fit stellar mass formed as a
function of lookback time in spatial cells resolving the central
$8.5^{\circ}\times 7.5^{\circ}$ of the galaxy (Zaritsky et al., 2004). HZ09
also provides the associated 1$\sigma$ upper and lower limits to the SAD in
each cell, which we will incorporate into the DTD uncertainties in Section
3.2. The SADs were calculated using data from the Magellanic Cloud Photometric
Survey (MCPS) of nearly 4 million stars collected with the 1m Swope telescope,
down to a completeness of V=20–21 mag (Zaritsky et al., 1997, 2004). The MCPS
region was divided into 1376 cells, each measuring $24^{\prime}\times
24^{\prime}$, or $12^{\prime}\times 12^{\prime}$ if the field contained more
than 25,000 stars. The contours of these cells are shown in Figure 1. SADs
were derived using the StarFISH algorithm by fitting the color magnitude
diagram in each cell with a linear combination of isochrones (Harris &
Zaritsky, 2001). After accounting for extinction and photometric errors, each
cell’s SAD was fit using 16 logarithmically-spaced bins spanning the ages
between 4 Myr and 20 Gyr, and four metallicity bins ($Z=0.008$, 0.004, 0.0025,
and 0.001). For ages younger than 100 Myr, a single metallicity of Z=0.008 was
used because the different metallicity isochrones are almost
indistinguishable.

Note that while the lower limit on the age of RR Lyrae stars is generally
quoted as 10 Gyr, the SAD map has a single indivisible age-bin of 8–12 Gyr,
and so we refer to this lower limit as 8 Gyr in the rest of the paper.

##  3 The RR Lyrae Delay-Time Distribution

###  3.1 Method

The RR Lyrae catalog and SAD maps of the LMC are used to estimate the RR Lyrae
DTD using the non-parametric method described in Badenes et al. (2015)
(hereafter, B15), although we improve on some aspects of it. The RR Lyrae DTD
is the number of RR Lyrae formed per unit stellar mass as a function of the
time-delay between star-formation and the RR Lyrae phase. The convolution of
the DTD with the SAD in each spatial cell predicts the number of RR Lyrae that
will be produced by the stellar population in that cell. For each SAD cell
$i$, the number of RR Lyrae predicted ($\lambda_{i}$) is:

$\lambda_{i}=\sum_{j=1}^{N}M_{ij}\left(\Psi T_{vis}\right)_{j}$ (1)

where $M_{ij}$ is the stellar mass formed in age-bin $j$ and cell $i$, $\Psi$
is the RR Lyrae formation rate (RR Lyrae per year per unit M⊙), and Tvis is
the duration of the RR Lyrae phase (both $\Psi$ and Tvis are function of age-
bin). The widths of the age-bins, $j$, are selected using the methodology
described in Appendix A and provide the best compromise between detection
significance and temporal resolution.

In this paper, we retain the notation of B15 and refer to the quantity
$\left(\Psi T_{vis}\right)$ as the DTD (with units of RR Lyrae per unit M⊙).
The DTD ($\Psi T_{vis}$) is determined by minimizing the difference between
the predicted and observed number of RR Lyrae across spatial cells. This is
carried out in a similar manner to B15 using the Markov Chain Monte Carlo
(MCMC) solver emcee (Foreman-Mackey et al., 2013). We denote
$\mathbf{N}=[N_{i}]$ as the vector representing the number of RR Lyrae in each
spatial cell, and $\mathbf{\Psi T_{vis}}=[\left(\Psi T_{vis}\right)_{j}]$ as
the vector of predicted number of RR Lyrae per stellar mass for each age-bin.
The posterior is calculated as

$p\left(\mathbf{\Psi
T_{vis}}|\mathbf{N}\right)\propto\mathcal{L}\left(\mathbf{N}|\mathbf{\Psi
T_{vis}}\right)\pi\left(\mathbf{\Psi T_{vis}}\right)$ (2)

where $\mathcal{L}$ is the likelihood and $\pi$ is the prior. We assume
$\mathcal{L}$ is either a product of Poisson or Gaussian probabilities
depending on Ni,

$\mathcal{L}\left(\mathbf{N}|\mathbf{\Psi
T_{vis}}\right)=\begin{cases}\displaystyle\prod_{i=1}^{K}\frac{\mathrm{e}^{-\lambda_{i}}\lambda_{i}^{N_{i}}}{N_{i}!}&\quad
N_{i}\leq 25\\\
\displaystyle\prod_{i=1}^{K}\frac{1}{2\pi\lambda_{i}}\left(\frac{(\lambda_{i}-N_{i})^{2}}{2\lambda_{i}}\right)&\quad
N_{i}>25\end{cases}$ (3)

where $K$ is the number of SAD cells. Although Poisson distributions converge
to Gaussian for large $N_{i}$, we specify the likelihoods separately to avoid
computational issues. The prior is defined such that the $(\Psi T_{vis})_{j}$
values have uniform probabilities in log-space:

$\pi\left(\mathbf{\Psi T_{vis}}\right)=\begin{cases}\mathbf{\left(\Psi
T_{vis}\right)^{-1}}&\quad(\Psi T_{vis})_{j}>0\\\ 0&\quad(\Psi
T_{vis})_{j}\leq 0\end{cases}$ (4)

###  3.2 Estimating Uncertainties

We propagate the errors on the $M_{ij}$ values in the SAD map into
uncertainties on $\mathbf{\Psi T_{vis}}$. B15 calculated DTDs for the best-fit
SAD, the upper-limit on the SAD, and the lower-limit on the SAD. The
differences between the best-fit and upper/lower limits were treated as the
DTD’s 1$\sigma$ uncertainties.

In this paper, we use an improved method of propagating SAD uncertainties into
the DTD. We randomly generate 100 mock SADs, assuming $M_{ij}$ has the
normally-distributed uncertainties given by HZ09, and calculate a DTD for each
mock SAD. We combine the MCMC posterior chains from these 100 DTDs into a
single chain, and estimate the $95\%$ credible interval on this chain using a
highest posterior density criterion. We define the mode of the distribution
minus the upper and lower-limits of this interval as our $2\sigma_{+}$ and
$2\sigma_{-}$ confidence intervals, respectively. We define a “signal”
detection in each bin $j$ of the DTD as a value of $\Psi T_{vis}$ in that bin
that is $\geq$ $2\sigma_{-}$ above 0. Non-detections are presented as
$2\sigma$ limits on the DTD signal in a particular age bin.

###  3.3 Sample Contribution per Age-bin

We estimate the contribution of each age-bin to the total sample by
multiplying the value of $\Psi T_{vis}$ with the total stellar mass formed in
each age-bin. The contribution percentage will then have uncertainties due to
both the DTD and the total stellar masses formed. We estimate these
uncertainties with a Monte Carlo method. We draw DTD values from the recovered
posterior probability distributions (called $\left(\Psi T_{vis}\right)_{j}$),
multiply by the total stellar mass per age-bin in each randomized SAD map
(called Mj), and get the number of RR Lyrae contributed by age-bin,
$\lambda_{j}$. The contribution percentages (=
$\lambda_{j}/\Sigma\lambda_{j}$) and their 1$\sigma$ uncertainties are listed
in Table 1.

Figure 3: The delay-time distribution, in units of number of RR Lyrae per M⊙,
as a function of delay-time in Myr. The grey histogram represents the DTD for
the full OGLE-IV sample, while the colored histograms are DTDs for RR Lyrae
subtypes RRab and RRc Filled histograms represent bins with a statistically
significant signal from the MCMC analysis. The error bars are 1$\sigma$
uncertainties that include uncertainties from the SAD maps. The arrows
represent 2$\sigma$ upper limits in bins that do not have a statistically
significant signal. At top, the progenitor mass-scales for two different
metallicities, calculated using PARSEC, are shown for reference.

###  3.4 Results

The DTD for the full OGLE-IV RR Lyrae sample and the two main RR Lyrae
subtypes is shown in Figure 3, and the values are tabulated in Table 1. We
detect signal in the DTD at a high significance ($>5\sigma$) for all age-bins
older than 1.3 Gyrs. The detections with the highest significance are found in
the 2–3, 5–8 and $>12$ Gyr bins. Most of this signal is contributed by the
RRab stars, which are the most common subtype in the OGLE-IV sample. The RRc
subclass contributes to the DTD mostly above 2 Gyr. Although RRc’s have been
susceptible to confusion with other variable sources in time domain-surveys
(Kinman & Brown, 2010; Mateu et al., 2012; Drake et al., 2014), Figure 3 shows
that the total DTD is dominated by RRab objects in all age-bins, making it
unlikely that the DTD is biased by sample contamination. The detected signal
of the full DTD is relatively flat above 1.2 Gyr, with about $1-3$ RR Lyrae
produced per 105 M⊙ of stellar mass formed. While there is signal in the DTD
below 0.8 Gyrs, it falls below our 2$\sigma_{-}$ threshold, and we only show
the 2$\sigma_{+}$ upper limit.

Our detection of a DTD signal below 8 Gyrs is a surprising result. About
$46\%$ of the LMC’s RR Lyrae stars are produced from populations older than 10
Gyr, the age range generally inferred for RR Lyrae in star clusters. But about
$51\%$ of the DTD signal comes from progenitors with ages between 1.3 and 8
Gyr, and this result has high ($>5\sigma$) significance. If we re-calculate
the DTD without assuming normal errors on $M_{ij}$ (i.e., using the same
method as B15, see Appendix B), we still detect a strong signal below 8 Gyr,
but with a total contribution of 41$\%$. A comparison of these timescales with
those of the PARSEC222https://people.sissa.it/~sbressan/parsec.html. We use
PARSEC because it is one of the latest stellar evolution codes with both
publicly available main-sequence and horizontal branch tracks. We get similar
results with the MIST evolutionary tracks of Choi et al. (2016). models for
the onset of helium burning leads to the conclusion that RR Lyrae can arise
from main-sequence progenitors as massive as $\sim 2$ M⊙ at LMC metallicity.
Incidentally, this upper limit is similar to the mass at which stars
transition between igniting He under degenerate conditions (the “He-flash”) to
burning helium under stable, non-degenerate conditions (Bildsten et al., 2012;
Mosser et al., 2014). We also tested the dependence of the DTD as a function
of RR Lyrae star period and brightness by sub-dividing the OGLE-IV data by
pulsation time (3 bins with $<0.45$ d, $0.45-0.58$ d, and $>0.58$ d) and
$I$-band magnitude (3 bins with I$<19.2$, $19.2-19.4$ and $>19.4$). By sub-
dividing the sample in this way, we ensured that each period and magnitude bin
contained enough RR Lyrae for a robust measurement of their DTDs. We found
that all the sub-samples have significant ($>2\sigma$) DTD signals in the
range 1 to 8 Gyr, with no discernible trend in the shape of the DTD with
magnitude or period.

Although, in principle, it is possible to derive a metallicity-dependent DTD
with the HZ09 SAD maps, we defer that study to a future work. However, it is
well-known that LMC RR Lyrae are generally metal-poor, with [Fe/H]$<-0.5$ and
a peak in the metallicity distribution function at [Fe/H]$\sim-1.5$ (Skowron
et al., 2016). We can therefore check to see if there is DTD signal below 8
Gyrs under the assumption that OGLE-IV RR Lyrae are only produced by the HZ09
SAD in the two metal-poor bins, i.e., for $Z<0.0025$ or [Fe/H]$\lesssim-1.02$
(Bertelli et al., 1994). Star-formation in this metallicity range dominated
the LMC SAD until $\sim 2$ Gyrs, as seen in Figure 4, and the resulting DTD is
shown in Figure 4. Even if we restrict our analysis to the metal-poor SAD, the
signal below 8 Gyrs persists. This implies that, even if we assume that all
LMC RR Lyrae are produced by metal-poor stars, progenitors younger than 8 Gyrs
are still needed to explain their distribution.

Figure 4: (a) The global (summed over all SAD cells) stellar mass formed
versus age ($M_{ij}$ in Eq (1)) with $Z<0.0025$ (red) and $Z>0.0025$ (blue) in
the LMC as given by the HZ09 SAD. The black histogram shows the sum of the red
and blue histograms. Overlapping regions appear in a darker shade. (b) The DTD
measured from the OGLE-IV survey using only the SAD in the $Z<0.0025$ bin
(grey; see Section 3.4 for details). For comparison, the DTD from Figure 3 is
shown in red.

Before delving into the implications for RR Lyrae studies, we carry out a more
detailed analysis of the recovered DTD to assess the robustness of our
results.

Table 1: The RR Lyrae DTD calculations, with lifetimes, significance of detection and contribution to the OGLE-IV RR Lyrae sample considered in this study. Delay-Times | DTD ($\Psi T_{\rm{vis}}$) | Significance | Contribution
---|---|---|---
(Gyr) | (N/105 M⊙) | ($\sigma$) | ( $\%$)
$<$ 0.8 | $<0.75$ | $-$ | $-$
0.8–1.3 | $2.12^{+0.61}_{-0.78}$ | $3.8$ | $2.9\pm 0.8$
1.3–2.0 | $1.36^{+0.2}_{-0.21}$ | $5.6$ | $8.4\pm 1.2$
2.0–3.2 | $2.05^{+0.11}_{-0.22}$ | $20.5$ | $20.7\pm 1.2$
3.2–5.0 | $1.10^{+0.18}_{-0.23}$ | $5.6$ | $6.0\pm 1.0$
5.0–7.9 | $2.17^{+0.19}_{-0.36}$ | $11.6$ | $15.6\pm 1.3$
7.9–12.6 | $1.08^{+0.17}_{-0.29}$ | $6.8$ | $7.5\pm 1.1$
12.6–20.0 | $1.18^{+0.07}_{-0.1}$ | $18.5$ | $39.0\pm 1.8$

##  4 On the robustness of the recovered RR Lyrae DTD

###  4.1 Mock DTD Test

The HZ09 SAD map of the LMC allows us to test whether the traditional DTD for
RR Lyrae (i.e., one in which the progenitors are always older than 8 Gyr) is
consistent with the spatial distribution of RR Lyrae observed by OGLE. This is
equivalent to inverting the DTD recovery process. To do this, we generate mock
RR Lyrae maps by convolving the SAD map with a DTD that is non-zero only in
the 2 oldest age-bins, 8–12 and 12–20 Gyr. The total stellar mass that formed
in these age-bins was $1.23\times 10^{9}$ M⊙.

Assuming all 29,810 RR Lyrae were produced by progenitors in these age-bins
results in a DTD of the form,

$\Psi T_{vis}=\begin{cases}2.42\times 10^{-5}\mathrm{\ RRL\
M_{\odot}^{-1}}&\quad t\geq 8\ \mathrm{Gyrs}\\\ 0&\quad t<8\
\mathrm{Gyrs}\end{cases}$ (5)

We generate 100 mock RR Lyrae maps using this DTD, where the number of RR
Lyrae per cell, $N_{i}$ is drawn from the Poisson distribution in Equation (3)
with $\lambda_{i}$ given by Equation (1).

Figure 5 shows the DTD recovered from this analysis. Our MCMC solver correctly
measures the input mock DTD with strong detections only in the two oldest age-
bins. The younger bins have 2$\sigma$ upper limits that are almost an order of
magnitude lower than the DTD recovered from the OGLE-IV sample. Moreover, the
difference between populations of RR Lyrae with the OGLE-IV and mock DTDs can
be seen visually in Figure 6. The measured OGLE-IV DTD predicts a distribution
of RR Lyrae stars elongated along the LMC Bar and declining smoothly with
radius (center bottom panel of Figure 6); this is very similar to what is seen
in the OGLE-IV map (left panel). In contrast, our mock distribution from a
uniformly old RR Lyrae progenitor population (Equation (5)) has more structure
and leaves larger residuals in the difference map (top center and right panels
of Figure 6). To identify significant regions of discrepancy in the OGLE-IV
DTD and mock old DTD maps, we generate 104 maps of RR Lyrae, where in each map
the number of RR Lyrae in each cell is drawn from a Poisson distribution with
mean number of RR Lyrae in that cell given by the DTD according to Eq 1. We
then estimate the mean and standard deviation of the number of RR Lyrae per
cell, and identify cells with black squares in Figure 6 where the observed RR
Lyrae is greater than 5 times the standard deviation from the mean. We find
the mock old DTD map has a larger number of cells where the RR Lyrae number
count is discrepant, and these cells are mostly located in the Bar region,
with a few located outside (note that this is only for the purpose of
visualizing the discrepancy; the actual DTD is constrained by the joint
likelihood measured from _all_ the cells as in Eq 3.)

Figure 5: Result of our mock DTD test in Section 4.1. The red histogram shows
the DTD measured with the original OGLE-IV survey in Figure 3. The black
histogram shows the mock input DTD defined in Eq 5. The blue histogram with
arrows shows the DTD recovered from the mock input DTD and the SAD maps of
HZ09.

We conclude that if the OGLE-IV RR Lyrae had indeed been produced exclusively
from the old ($t>8$ Gyr) stars of the HZ09 SAD map, our method would have
recovered the correct DTD, without spurious signals at younger ages. The DTD
signal we recover at ages between 0.8 and 8 Gyr must either be real, or an
artifact produced by systematics in the HZ09 SADs. Next, we investigate the
impact of these SAD systematics on the DTD more thoroughly.

Figure 6: Comparison between the spatial distribution of RR Lyrae in OGLE-IV
(left, large panel) and the distributions predicted by convolving the HZ09 SAD
with the ‘mock’ old DTD (upper middle panel) and our recovered RR Lyrae DTD
(lower middle panel). The right panels show “residuals”, or the difference
between the observed distribution and DTD-predicted distribution of RR Lyrae
for the ‘mock’ old DTD (top right) and measured DTD (bottom right). The black
squares show the cells where the difference between predicted and observed
number of RR Lyrae differ by 5$\sigma$ (see Section 4.1 for details)

.

###  4.2 Effect of incompleteness and crowding in the SAD map

We explored if any incompleteness in the MCPS photometry (from which the SAD
was measured) could be driving the intermediate-age signal. The photometric
completeness limit is around V=21 mag, with the Bar region being almost 1 mag
shallower as a result of the stellar crowding. Because of this, StarFISH could
reliably solve for the number of stars per age-bin only for ages younger than
4 Gyr (although we note that our intermediate-age DTD signal is detected
younger than 4 Gyrs). Populations older than 4 Gyr were traced by their giant
branch stars, and assigned by StarFISH to a single age-bin. To anchor the
‘shape’ of the SAD beyond 4 Gyrs, stellar ages were measured in a few isolated
_HST_ -fields in the LMC Bar (Holtzman et al., 1999; Olsen, 1999; Smecker-Hane
et al., 2002). As shown in Figure 7 of HZ09, the SADs of each _HST_ Bar field
are characterized by star-formation at look-back times of 10 Gyrs and 5 Gyrs,
with a period of quiescence in between. This consistency allowed HZ09 to adopt
a common shape for the SAD in the rest of the Bar region.

We first checked if an underestimation of the total mass formed at old stellar
ages due to any photometric incompleteness could be driving the intermediate-
age signal. Obtaining a complete census of old stellar mass is generally
difficult without including infra-red data (Conroy, 2013), so we studied the
changes to our DTD by manually changing the old SADs. We recalculated the RR
Lyrae DTD with the same SAD map, except with each cell’s 8–12 Gyr and 12–20
Gyr stellar mass multiplied by factors of 2, 4 and 10. We find that regardless
of how much mass is added, the DTDs still find signals for ages between 0.8
and 8 Gyrs with $>5\sigma$ confidence. Since the incompleteness in the SAD map
is dominant in the Bar region, we also tried re-calculating the DTD without
the LMC Bar (i.e., by removing the SAD cells and their RR Lyrae from our
calculations). We show the RR Lyrae DTDs without the “Inner Bar” in Figure 7,
and without both the “Inner” and “Outer Bar” in Figure 8 (the cells of these
regions are defined according to Figure 6 in HZ09). The recovered DTDs outside
the excluded regions are similar to the DTD obtained from the full OGLE-IV
sample. For the case with both the Inner and Outer Bar removed, the number of
RR Lyrae and the number of SAD cells are smaller, leading to a higher upper
limit on the DTD older than $0.8$ Gyr, and a non-detection in the 3–5 Gyr bin.
However, there is still significant signal at ages younger than 8 Gyr, and
particularly below 4 Gyr where the ages are determined by the main-sequence
turnoff in the MCPS photometry. It therefore appears unlikely that missing
stellar masses at ages $>8$ Gyrs due to incompleteness, or crowding in the Bar
region are the only factors driving the intermediate-age signal in the DTD.

Figure 7: RR Lyrae DTD of the OGLE-IV survey, but excluding the SAD cells and
RR Lyrae of the Inner Bar. The excluded cells of HZ09 are shown in the inset
plot, overlaid on the $r$-band continuum map from MCELS. Figure 8: Same as
Figure 7, but now excluding both the Inner and Outer Bar. Note that the
excluded cells for both Inner and Outer Bar are based on Figure 6 of HZ09.

###  4.3 Comparison with RR Lyrae in LMC star clusters

Figure 9: (_Left_): Comparison of ages of the 62 clusters in Baumgardt et al.
(2013) older than 1 Gyr, and their $\Delta n=(n_{o}-n_{b})-n_{p}$, the
difference between the background-subtracted number of RR Lyrae observed per
cluster, and the number predicted by the DTD given the cluster mass and age
(see Eq (6), (7) and Section 4.3 for details). Colors indicate the cluster
mass. The inset zooms-in on clusters between 1–4 Gyr. (_Right_): $\Delta n$ vs
metallicity for a subset of 29 clusters with metallicity information available
in the literature. Colorbar indicates the cluster age. The inset zooms-in on
the 1–4 Gyr clusters, which also happen to cluster in the metallicity range
$-1\lesssim$ [Fe/H] $\lesssim-0.25$.

Our measured DTD suggests that progenitors of RR Lyrae can be as young as
$\sim$1 Gyr, which is much smaller than the lower limit on RR Lyrae age of 10
Gyrs inferred from star clusters (Olszewski et al., 1996). We therefore
examine the number of OGLE-IV RR Lyrae in the LMC star clusters as a function
of age, realizing that cluster membership can only be confirmed with
spectroscopic and proper motion measurements that are beyond the scope of this
work.

To perform this test, we use the catalog of LMC clusters compiled by Baumgardt
et al. (2013) and only include the 296 systems inside the HZ09 area. Baumgardt
et al. (2013) compiled ages and masses for these clusters measured using
either isochrone fitting or broadband spectral energy distribution fitting of
data obtained by previous surveys (Pietrzynski & Udalski, 2000; Hunter et al.,
2003; Mackey & Gilmore, 2003; de Grijs & Anders, 2006; Milone et al., 2009;
Glatt et al., 2010; Popescu et al., 2012). We determine the cluster radii,
$r_{c}$ from their major ($a$) and minor axes ($b$) reported in Bica et al.
(2008) as $r_{c}=(a+b)/4$. Bica et al. (2008) notes that these are ‘apparent’
sizes (measured as far as the background limit in the images), but they most
likely enclose the majority of the cluster mass. We also obtained chemical
abundances for 29 clusters ([Fe/H], shown in Table 2 with references), and as
we show later, this is sufficient for understanding the correlation of cluster
RR Lyrae statistics with metallicity. Although there have been more recent
studies of the LMC star cluster population (e.g. Nayak et al., 2016; Bitsakis
et al., 2017; Piatti, 2017, 2018), the effects of field star contamination and
asterisms on cluster identification in these studies are unclear. Since
Baumgardt et al. (2013) compiles previously identified clusters above 5000 M⊙,
we rely on this catalog for our analysis. Additionally, this catalog (to the
best of the authors’ knowledge) is the only catalog of LMC clusters in the
literature with both age _and_ mass estimates of the clusters—both being
critical to our analysis.

Out of the 296 clusters, we investigate the RR Lyrae statistics of 62 clusters
that have ages above 1 Gyr in order to compare with the DTD. We define $n_{o}$
as the number of OGLE-IV RR Lyrae observed within a circle of radius $r_{c}$
centered on the cluster, and $n_{b}$ as the expected number of RR Lyrae within
the cluster area but unassociated with the cluster (we refer to these as
‘background’ RR Lyrae). Assuming $N$ is the number of RR Lyrae within radii
$2r_{c}$ and $4r_{c}$ of the cluster (we allow a buffer region of $r_{c}$ to
account for uncertainty in the actual extent of the cluster), we define
$n_{b}$ as,

$n_{b}=\frac{N\pi r_{c}^{2}}{\pi(4r_{c})^{2}-\pi(2r_{c})^{2}}=\frac{N}{12}$
(6)

Values for $n_{o}$ and $n_{b}$ are listed in Table 2 and shown in Figure 9.

LMC globular clusters older than 10 Gyr clearly host populations of multiple
RR Lyrae that exceed the background. The richest population is in NGC 1835
with over a 100 RR Lyrae, while NGC 2005, NGC 1928 and NGC 1939 have less than
10 RR Lyrae. In contrast the intermediate-age clusters (1–10 Gyr old) are
generally lacking in RR Lyrae. Out of the 53 intermediate-age clusters, 42
have 0 RR Lyrae and only three clusters (NGC 1978, SL180 and HS190) have more
than 1 RR Lyrae star after subtracting the background numbers (i.e.
$n_{o}-n_{b}$). This is also the case in the Milky Way, where studies of
variable stars in well known intermediate-age systems such as M67 (Pribulla et
al., 2008), NGC 188 (Zhang et al., 2004), NGC 6791 (de Marchi et al., 2007)
and NGC 6253 (Kaluzny et al., 2014) have found no RR Lyrae that are likely to
be cluster members. In contrast, the Milky Way’s globular cluster system
contains almost 2000 of such objects (Clement et al., 2001; Soszyński et al.,
2014).

While this deficit of RR Lyrae in intermediate-age clusters appears
contradictory to the 1–8 Gyr signal in the DTD, we note that the intermediate-
age clusters in the LMC are generally less massive than old globular clusters,
and are therefore less likely to host relatively short-lived objects such as
RR Lyrae stars. We quantify this with $n_{p}$, the expected number of RR Lyrae
per cluster given our measured DTD $\Psi T_{vis}$, the mass $M_{c}$, and age
$t_{c}$ of a cluster:

$n_{p}=M_{c}(\Psi T_{vis})|_{t=t_{c}}$ (7)

For each cluster, we measure $\Delta n=(n_{o}-n_{b})-n_{p}$, i.e. the
difference between the observed background-subtracted number of RR Lyrae
($n_{o}-n_{b}$) and the number predicted by the DTD. We estimate the
uncertainty in $\Delta n$ in the following way: we create 106 random samples
of the 62 clusters, each sample having the same $n_{o}$ per cluster as given
in Table 2 but with $n_{b}$ and $n_{p}$ generated from a Poisson distribution
with mean $n_{b}$ and $n_{p}$ values given in Table 2. For any cluster in
these samples with a negative value of $n_{o}-n_{b}$, we set its
$n_{o}-n_{b}=0$. The mean and standard deviation of this random sampling is
taken to be the value and error of $\Delta n$. These $\Delta n$ are shown as a
function of cluster age and metallicity in Figure 9. We see that $\Delta n$ is
consistent with 0 within 2$\sigma$ for most intermediate-age clusters in the
LMC. This is because the DTD predicts $\sim 1-3$ RR Lyrae per 105 M⊙ of stars,
whereas the intermediate-age clusters have an average mass of $4.3\times
10^{4}$ M⊙, so the majority will have less than 1 RR Lyrae star. This caveat
was also raised by Olszewski et al. (1996) in their study of the SMC: if the
RR-Lyrae-rich cluster NGC 121 ($t\approx 12$ Gyr) were scaled down to the
magnitude of the 9 Gyr system Lindsey 1 (which has no RR Lyrae), it would host
less than 1 RR Lyrae star. The intermediate-age clusters of the Milky Way,
including the ones listed above, have the same issue: all have masses less
than $10^{4}$ M⊙ (Kaluzny & Udalski, 1992; Mermilliod, 2000; Piskunov et al.,
2008; Grundahl et al., 2008; Elsanhoury et al., 2016; Kruijssen et al., 2019).
Thus, the traditional lower limit on the age of the RR Lyrae progenitor
population is subject to small number statistics.

We note that the globular clusters host a diversity of RR Lyrae populations.
NGC 1939, NGC 1916 and NGC 2005 are consistent with $\Delta n=0$ within their
uncertainties, and their production rates $(n_{o}-n_{b})/M_{c}=(1.8-2.5)\times
10^{-5}$ RR Lyrae per M⊙ are similar to the measured DTD. The other old
clusters, however, host RR Lyrae significantly in excess of their DTD-
predicted $n_{p}$ (i.e. $\Delta n\gg 0$), and have an average production rate
$\sim 10^{-4}$ RR Lyrae per M⊙, a factor $\sim 5$ times higher than the DTD.
The high production rate in the old clusters may be an effect of their low
metallicities, which is consistent with the differences in production rates of
RR Lyrae in the halo versus the bulge and disk of our Galaxy (Layden, 1995;
Dékány et al., 2018), although the large differences in $\Delta n$ for the
globular clusters could also be an effect of the second parameter phenomenon
in clusters (see Fusi Pecci & Bellazzini, 1997; Catelan, 2009; Dotter, 2013,
for reviews). A more detailed assessment of the field and cluster DTDs can be
done with a metallicity-dependent DTD that we reserve for future work.

Table 2: LMC Clusters older than 1 Gyr in the Baumgardt et al. (2013) catalog
and inside the Harris & Zaritsky (2009) region.

Name | Age (Gyr) | [Fe/H] |  aRef | Log Mass (M⊙) | $n_{o}$ | $n_{b}$ | $n_{p}$
---|---|---|---|---|---|---|---
SL569 | 1.2 $\pm$ 0.03 | -0.32 $\pm$ 0.05 | 6 | 4.29 | 0 | 0 | 0
KMK88-38 | 1.26${}^{+0.15}_{-0.14}$ | – | – | 3.71 | 0 | 0 | 0
BSDL880 | 1.26${}^{+0.15}_{-0.14}$ | – | – | 3.83 | 0 | 0 | 0
SL197 | 1.26${}^{+0.25}_{-0.21}$ | – | – | 3.93 | 0 | 0 | 0
HS102 | 1.26${}^{+0.15}_{-0.14}$ | – | – | 3.87 | 1 | 0 | 0
HS223A | 1.26${}^{+0.15}_{-0.14}$ | – | – | 4.02 | 0 | 0 | 0
NGC1795 | 1.29${}^{+0.16}_{-0.14}$ | -0.23 | 4 | 4.36 | 0 | 0 | 0
NGC1917 | 1.29${}^{+0.16}_{-0.14}$ | -0.21 | 4 | 4.42 | 0 | 1 | 1
KMHK898 | 1.32 $\pm$ 0.09 | – | – | 4.16 | 0 | 0 | 0
NGC1852 | 1.32${}^{+0.16}_{-0.14}$ | – | – | 4.51 | 1 | 0 | 0
SL282 | 1.35${}^{+0.35}_{-0.28}$ | – | – | 3.71 | 1 | 0 | 0
BSDL946 | 1.35${}^{+0.2}_{-0.17}$ | – | – | 4.17 | 0 | 0 | 0
NGC2154 | 1.41${}^{+0.17}_{-0.15}$ | -0.56 | 3 | 4.57 | 1 | 0 | 1
NGC1751 | 1.41${}^{+0.17}_{-0.15}$ | -0.44 $\pm$ 0.05 | 6 | 4.6 | 0 | 0 | 1
SL151 | 1.41${}^{+0.25}_{-0.21}$ | – | – | 4.1 | 0 | 0 | 0
HODGE7 | 1.48${}^{+0.18}_{-0.16}$ | – | – | 4.47 | 0 | 0 | 0
NGC1846 | 1.48${}^{+0.18}_{-0.16}$ | -0.4 | 1 | 5.1 | 1 | 1 | 2
NGC1783 | 1.51${}^{+0.18}_{-0.16}$ | -0.75 | 3 | 5.26 | 0 | 1 | 2
NGC1806 | 1.51${}^{+0.18}_{-0.16}$ | -0.71 $\pm$ 0.23 | 1 | 5.01 | 1 | 1 | 1
SL136 | 1.55${}^{+0.45}_{-0.35}$ | – | – | 4.39 | 0 | 0 | 0
NGC2213 | 1.58${}^{+0.46}_{-0.35}$ | -0.7 $\pm$ 0.1 | 1 | 4.56 | 0 | 0 | 0
SL180 | 1.58${}^{+0.41}_{-0.33}$ | – | – | 4.32 | 3 | 0 | 0
H1 | 1.58${}^{+0.41}_{-0.33}$ | -0.29 | 5 | 4.84 | 1 | 2 | 1
SL357 | 1.58${}^{+0.32}_{-0.27}$ | – | – | 4.75 | 0 | 1 | 1
SL390 | 1.58 $\pm$ 0.11 | -0.4 | 2 | 4.48 | 0 | 1 | 0
HS117 | 1.58${}^{+0.41}_{-0.33}$ | – | – | 4.15 | 2 | 1 | 0
BSDL1102 | 1.62${}^{+0.47}_{-0.36}$ | – | – | 3.77 | 0 | 0 | 0
SL66 | 1.7${}^{+0.7}_{-0.5}$ | – | – | 4.34 | 0 | 0 | 0
NGC1652 | 1.7${}^{+0.21}_{-0.18}$ | -0.46 | 3 | 4.29 | 0 | 0 | 0
BSDL2652 | 1.74${}^{+0.5}_{-0.39}$ | – | – | 3.73 | 0 | 0 | 0
HS87 | 1.78${}^{+0.22}_{-0.19}$ | – | – | 3.98 | 0 | 0 | 0
OGLE-LMC0114 | 1.78${}^{+0.46}_{-0.37}$ | – | – | 3.71 | 0 | 0 | 0
HS37 | 1.78${}^{+1.24}_{-0.73}$ | – | – | 4.02 | 1 | 0 | 0
HS177 | 1.78${}^{+0.22}_{-0.19}$ | – | – | 3.95 | 0 | 0 | 0
H2 | 1.78${}^{+0.46}_{-0.37}$ | -0.38 | 5 | 4.98 | 0 | 1 | 1
BSDL734 | 1.78${}^{+0.22}_{-0.19}$ | – | – | 3.8 | 1 | 0 | 0
KMHK355 | 1.86${}^{+0.96}_{-0.63}$ | – | – | 3.7 | 0 | 0 | 0
OGLE-LMC0531 | 2.0${}^{+0.24}_{-0.22}$ | – | – | 3.91 | 0 | 0 | 0
NGC1978 | 2.0${}^{+0.24}_{-0.22}$ | -0.38 $\pm$ 0.07 | 1 | 5.33 | 2 | 0 | 3
NGC1651 | 2.0${}^{+0.46}_{-0.37}$ | -0.53 $\pm$ 0.03 | 1 | 5.24 | 0 | 0 | 2
SL629 | 2.04${}^{+0.1}_{-0.09}$ | – | – | 4.0 | 0 | 0 | 0
KMHK1112 | 2.4${}^{+0.69}_{-0.54}$ | – | – | 3.78 | 1 | 0 | 0
HS88 | 2.45${}^{+1.09}_{-0.76}$ | – | – | 4.1 | 0 | 0 | 0
SL244 | 2.69 $\pm$ 0.06 | -0.7 $\pm$ 0.2 | 1 | 4.77 | 0 | 0 | 1
SL150 | 2.75 $\pm$ 0.06 | – | – | 4.34 | 0 | 0 | 0
HS190 | 2.75${}^{+3.41}_{-1.52}$ | – | – | 4.51 | 2 | 0 | 1
BSDL2300 | 2.95${}^{+0.21}_{-0.2}$ | – | – | 4.7 | 0 | 0 | 1
KMHK1188 | 3.02${}^{+0.14}_{-0.14}$ | – | – | 4.0 | 0 | 0 | 0
H88-93 | 3.02${}^{+4.06}_{-1.73}$ | – | – | 3.85 | 0 | 0 | 0
BSDL1334 | 3.02${}^{+4.74}_{-1.85}$ | -0.4 | 2 | 3.86 | 0 | 0 | 0
SL663 | 3.24${}^{+0.48}_{-0.42}$ | -0.7 $\pm$ 0.1 | 1 | 5.23 | 0 | 0 | 2
NGC2121 | 3.24${}^{+0.48}_{-0.42}$ | -0.4 $\pm$ 0.1 | 1 | 5.69 | 0 | 1 | 5
NGC2155 | 3.24${}^{+0.48}_{-0.42}$ | -0.7 $\pm$ 0.1 | 1 | 4.9 | 0 | 0 | 1
NGC1939 | 13.49${}^{+3.49}_{-2.77}$ | -2.1 $\pm$ 0.19 | 1 | 5.08 | 4 | 1 | 1
NGC1928 | 13.49${}^{+3.49}_{-2.77}$ | -1.27 $\pm$ 0.14 | 1 | 4.87 | 7 | 1 | 1
NGC1898 | 14.13${}^{+2.47}_{-2.1}$ | -1.37 $\pm$ 0.15 | 1 | 5.88 | 38 | 2 | 9
NGC1786 | 15.14${}^{+0.35}_{-0.34}$ | -2.1 $\pm$ 0.3 | 1 | 5.57 | 47 | 1 | 4
NGC1754 | 15.49${}^{+2.29}_{-2.0}$ | -1.42 $\pm$ 0.15 | 1 | 5.39 | 32 | 0 | 3
NGC1916 | 15.85${}^{+3.65}_{-2.97}$ | -2.08 | 4 | 5.79 | 14 | 3 | 7
NGC2005 | 16.6${}^{+6.31}_{-4.57}$ | -1.35 $\pm$ 0.15 | 1 | 5.49 | 9 | 2 | 4
NGC1835 | 16.6${}^{+2.9}_{-2.47}$ | -1.62 $\pm$ 0.15 | 1 | 5.83 | 105 | 2 | 8
NGC2019 | 17.78${}^{+3.6}_{-2.99}$ | -1.23 $\pm$ 0.15 | 1 | 5.68 | 49 | 1 | 6

Note. — _a_ – References for [Fe/H] measurement. (1) Harris & Zaritsky (2009)
(2) Palma et al. (2016), (3) Girardi & Marigo (2007) (4) Kontizas et al.
(1993) (5) Olszewski et al. (1991) (6) Grocholski et al. (2006)

* - These 2 clusters are listed as having ages $\sim 1.7$ Gyr in Baumgardt et al. (2013), but are more likely to be $\gtrsim 10$ Gyr based on HST WFPC2 observations (Mackey & Gilmore, 2004).

##  5 Implications of the RR Lyrae DTD

If we assume the DTD measured from the OGLE-IV survey is correct, there are
two possible interpretations of this result: 1) RR Lyrae can form from
progenitors younger than 10 Gyr _in addition_ to the conventional route via
older stars, and this result was undetected in previous studies due to various
observational limitations, or 2) all OGLE-IV RR Lyrae are older than 10 Gyr
stars, and our result is a product of significant (though not readily obvious)
systematics in the age derivation of older stellar populations. We discuss
both of these interpretations below.

###  5.1 Can LMC RR Lyrae have an intermediate-age channel?

Since the DTD is generally regarded as a reflection of the progenitor age
distribution (Maoz & Badenes, 2010; Badenes et al., 2015), it is tempting to
consider that RR Lyrae in the LMC are being produced by an as-of-yet
undiscovered intermediate-age progenitor channel between 1-8 Gyrs, in addition
to the usual channel older than 8 Gyrs.

However, looking at currently available evidence, the possibility of such an
undiscovered intermediate-age channel appears to be questionable. Although the
small-number statistics described in Section 4.3 is a factor, it is
nevertheless true that ancient globular clusters host abundant and rich
populations of RR Lyrae compared to the RR Lyrae-poor intermediate-age
clusters, a feature easily explained by an exclusively old channel for RR
Lyrae formation. Evidence that a small fraction of RR Lyrae may arise from
progenitors only a few Gyrs old was recently obtained from thin-disk metal-
rich RR Lyrae observed in the _Gaia_ data (Zinn et al., 2019; Prudil et al.,
2020; Iorio & Belokurov, 2020), and from measurements of companion masses
greater than 1 Msun in wide-orbit RR Lyrae binaries (Kervella et al., 2019a,
b). The thin disk, metal-rich RR Lyrae population was speculated by Iorio &
Belokurov (2020) to be manifestation of binary evolution pulsators (Karczmarek
et al., 2017), which will register as younger stars. However, the existence of
such an intermediate-age channel for LMC RR Lyrae (assuming single-stellar
evolution) would be in tension with the age-metallicity relation of the LMC
constrained by mutiple studies of field and cluster stars(Cole et al., 2005;
Carrera et al., 2008; Rubele et al., 2012; Meschin et al., 2014). According to
these studies, the LMC star-formation history between 2-8 Gyrs was associated
with [Fe/H] between -0.4 and -1. In contrast, the LMC RR Lyrae are
predominantly metal-poor, with [Fe/H]$<-1$ and peaking at [Fe/H]$\sim-1.5$, as
confirmed by spectroscopic studies of field RR Lyrae (Gratton et al., 2004;
Borissova et al., 2006), photometric light curves of RR Lyrae (Haschke et al.,
2012; Wagner-Kaiser & Sarajedini, 2013; Skowron et al., 2016), and RR Lyrae-
hosting globular clusters (Fig 9). Thus the LMC stellar population at ages
below 8 Gyrs, where we measure significant signal in the DTD, is more metal-
rich than the metallicity range measured for LMC RR Lyrae.

###  5.2 Systematic uncertainties in old SADs?

The DTD represents an empirical connection between the RR Lyrae sample and the
SAD map. Since the RR Lyrae sample is highly complete (Section 2.1) and has
strong independent evidence of originating from old stars (Section 5.1), and
since our DTD recovery method would have correctly recovered a purely old RR
Lyrae DTD signal from this SAD map (Section 4.1), it may be possible that the
intermediate-age signal in the DTD is indicative of some systematic
uncertainty in measuring older stellar ages in the LMC. The source and
magnitude of this uncertainty is not readily obvious. The global star-
formation history measured by HZ09 is broadly consistent with the interaction
history of the Magellanic Clouds derived from proper motion modeling (Lin et
al., 1995; Zaritsky & Harris, 2004; Besla et al., 2007) and with the star
formation and chemical enrichment history derived from star clusters
(Chilingarian & Asa’d, 2018). As shown in Section 4.2, we verified that any
incompleteness or statistical uncertainties in the MCPS photometry is unlikely
to be affecting our DTD because: 1) we propagate the reported uncertainties in
the SAD map into our DTDs; 2) the DTD retains signal below 8 Gyrs even when
measured outside the crowded LMC Bar; 3) wiping out the DTD signal of younger
progenitors would require an unreasonably large unseen stellar mass with age
$>$8 Gyr; and 4) we directly detect a signal in the DTD below 4 Gyrs, where
the main-sequence turnoff is detectable above the MCPS completeness limit.

On the other hand, the SAD solutions per region depend on the overall
methodology adopted by the study. An example of this can be found by comparing
the Harris & Zaritsky (2004) SAD map of the SMC, which was derived using
roughly the same methodology used for the LMC, with the Rubele et al. (2018)
SAD map of SMC derived from the deeper VISTA near-infrared survey of the
Magellanic System (VMC, Kerber et al., 2009; Cioni et al., 2011). The Harris &
Zaritsky (2004) 2–3 Gyr SADs has a distinct ring pattern in the SMC, which
they suspected was either the result of photometric incompleteness or
systematic uncertainties in the photometric zero-point of the central SMC
stars. In the VMC SAD, this ring pattern is absent in the 2–3 Gyr stellar
population. Figure 11 of Rubele et al. (2018) shows that the global stellar
mass formed at ages $>1$ Gyr in the SMC differ by more than a factor of 2
between Rubele et al. (2018) and Harris & Zaritsky (2004), and the bimodal
star-formation history at 2.5 and 10 Gyrs found by Harris & Zaritsky (2004) is
replaced with a single broad peak at 5 Gyr in Rubele et al. (2018). These
systematic differences could stem from differences in IMF and distance
assumed, the stellar isochrone models used, and/or age-metallicity binning.
For example, the age-metallicity relation measured by the two studies diverges
for stars older than 4 Gyrs; this is most likely because Rubele et al. (2018)
uses metallicity bins that extend to lower abundances than Harris & Zaritsky
(2004). Similar systematic differences in the LMC SADs may also exist. For
example, the star-formation history of the Northern Void region measured by
Meschin et al. (2014), using $V$\- and $I$-band photometry with the CTIO
Blanco 4 m telescope, differs from that of HZ09 for ages younger than 4 Gyr.
Partial estimates from upcoming SMASH data reveals a well-mixed SAD in the LMC
for ages

Another source of systematic uncertainty in the SADs may be due to the use of
single-stellar evolution isochrone models. According to Moe et al. (2019), the
close ($<10$ AU) binary fraction for Milky Way field stars of LMC metallicity
is $\sim 30\%$ (compared to $\sim 20\%$ for Solar metallicity), so the
influence of binary evolution physics in older populations may be non-
negligible. Stanway & Eldridge (2018) have shown that models of integrated
spectra and photometry of globular clusters and elliptical galaxies that
include binary evolution physics yield age estimates that are different by a
few Gyr compared to single-star models. A well-known observational
manifestation of binary interaction in old stellar populations is the
appearance of blue straggler stars, which are formed from the merger of $\sim
1M_{\odot}$ stars. Blue straggler stars can mimic younger stars in color-
magnitude diagrams as they appear brighter and bluer than the main-sequence
turnoff (Santana et al., 2016), and single stellar population models that
correct for the presence of blue straggler stars yield globular cluster ages
older by a few Gyrs (Fan & de Grijs, 2012). Correcting for blue stragglers
however is non-trivial as the frequency of blue straggler stars is likely a
function of stellar density (Weisz et al., 2014; Santana et al., 2013, 2016),
and the contribution of blue straggler stars at $\sim$Gyr ages have been
difficult to determine in composite stellar populations (Surot et al., 2019).

Leaving binary evolution aside, there are uncertainties even in the physics of
single-star evolution models that can affect age estimates. For example, Tayar
et al. (2017) showed that the mixing length parameter—commonly used to
approximate convection theory in 1D stellar models—appears to be correlated
with metallicity, and if left unaccounted for when estimating ages from the
giant branch (which is the case for ages $>4$ Gyr in the HZ09 maps), it can
lead to age uncertainties up to a factor of 2.

Any number of these reasons could distort the SAD solutions in a subset of the
cells, and therefore the final DTD, which is derived from these data.

###  5.3 Caveats and Future work

From our work, we have shown that DTD provides a new rigorous and quantitative
test of SADs of Local Group galaxies, in addition to its original purpose as a
stellar evolution diagnostic. Although it is possible that unknown sources in
systematic uncertainties in the SAD map may be driving our DTD result, we
would need further tests to verify the authenticity of this issue, which we
will perform in subsequent papers. Estimating the precise form of the HZ09 SAD
map that would be consistent with a purely old RR Lyrae DTD is non-trivial
because of the large number of SAD parameters involved in this exercise (the
stellar masses per age and metallicity bin per cell), and also because RR
Lyrae can only reliably constrain the oldest ages. In addition, the coarse age
and metallicity binning of the HZ09 map may be less than ideal for
constraining the production rate of RR Lyrae if they are coming from a
narrower range of ages and metallicities within each bin (as e.g. indicated in
Section 4.3. We can however check if our DTD result persists when calculated
with SADs derived from deeper photometric data with finer age and metallicity
resolution, such as the upcoming SMASH star-formation histories (Ruiz-Lara et
al., 2020), as well as in other Local Group galaxies like the SMC (Rubele et
al., 2018), M31 (Williams et al., 2017) and Local Group dwarfs (Weisz et al.,
2014). A more constraining test of the SADs at intermediate and younger ages
can also be obtained by calculating DTDs of younger variables stars with well-
constrained ages, such as $\delta$-Scutis (ages 1-3 Gyrs, Petersen &
Christensen-Dalsgaard, 1996) and Classical Cepheids (ages 70-200 Myrs, Bono et
al., 2005). We will pursue these in future papers.

##  6 Conclusions

We have calculated the first delay time-distribution (DTD) of RR Lyrae stars
using the large sample of LMC RR Lyrae from the OGLE-IV survey (Soszyński et
al., 2016) and the LMC’s SAD map from Harris & Zaritsky (2009). Our DTD, shown
in Figure 3 and Table 1, constrains the age-distribution of the full LMC RR
Lyrae population, given the measured SAD of the LMC. The OGLE-IV RR Lyrae
sample which overlaps the SAD map of HZ09 contains 29,810 objects, allowing us
to recover a DTD with an unprecedented balance of age resolution and detection
significance. We determined the DTD signal in each age-bin with an MCMC
solver, and used a randomization technique to propagate uncertainties in the
SAD map into the final DTD.

Our measured RR Lyrae DTD has statistically significant ($>5\sigma$) power in
all age-bins above 1.3 Gyrs, with about 51$\%$ of the RR Lyrae associated with
ages between 1.3 and 8 Gyr, and only 46$\%$ with ages above 8 Gyr (the
conventional lower limit to RR Lyrae age; note that while the lower limit
quoted in the literature is 10 Gyr, the SAD map has a single indivisible age-
bin of 8–12 Gyr, and so we refer to the lower limit as 8 Gyr in this paper).
This would imply that the progenitors of RR Lyrae have zero-age main-sequence
masses $\lesssim 2$ M⊙ at LMC metallicity, in contrast with existing
constraints.

We checked the DTD for possible sources of bias. Completeness of the RR Lyrae
sample is probably not an issue based on their $I$-band luminosity function
and the predominance of RRab (fundamental) pulsators, which are least
susceptible to confusion with other types of variables. We also tested our DTD
algorithm on fake RR Lyrae maps drawn from a DTD that assumes all RR Lyrae are
older than 8 Gyrs, and found that our MCMC algorithm recovers this old DTD
without any outlying detections at younger ages. The spatial distribution of
RR Lyrae from a purely old DTD is also inconsistent with the spatial
distribution of the OGLE-IV RR Lyrae. A possible caveat to our result is the
incomplete photometry in the MCPS data and heavy crowding in the LMC central
region, which limits the reliability of the SAD maps to ages younger than 4
Gyr; information about older populations are all based on _HST_ -derived SADs
in a few narrow fields. However it is not readily obvious how this is
producing an intermediate-age signal since: (1) we recover the DTD signal at
ages younger than 8 Gyr even after excluding the Bar region, (2) we measure a
DTD signal younger than 4 Gyr, and (3) we find that in order to affect the
solution, the estimates of the LMC’s old stellar mass must be more than an
order-of-magnitude greater than current measurements.

The direct interpretation of our result would be that RR Lyrae have an
intermediate-age progenitor channel of 1.3-8 Gyrs stars, in addition to its
conventional route via ancient stars, but this possibility is in tension with
existing constraints. Both in the Milky Way and Magellanic Clouds, RR Lyrae
are abundantly hosted in ancient globular clusters, in contrast with
intermediate-age clusters (although this conclusion is somewhat affected by
small-number statistics due to the lower-masses of intermediate-age clusters).
In addition, the 1.3-8 Gyr RR Lyrae population would likely have [Fe/H]$>$-1
based on existing constraints on the LMC age-metallicity relation, whereas the
metallicity distribution of LMC RR Lyrae measured from spectroscopy and
photometric light curves is in the range of [Fe/H]$<$-1 with a peak at
[Fe/H]$\sim-1.5$.

The other possibility of the intermediate-age DTD result is the presence of
unknown systematics in the LMC SAD map. This is not obvious because the global
star-formation history based on the SAD is consistent with the LMC–SMC–Milky
Way interaction histories and the chemical enrichment history of the LMC
derived from independent studies. However, comparison of the SMC SADs of
Harris & Zaritsky (2004) and Rubele et al. (2018) shows that spatial solutions
of the SAD maps can be influenced by the overall methodology adopted for their
construction (e.g., assumptions about the IMF, the spatial size of cells, the
age and metallicity bin size, and stellar isochrone models). Age estimation of
old stellar populations from color-magnitude diagrams can also be affected by
binary evolution processes, such as mass-transfer and mergers, as well as
approximations in single-star evolution models. Any combination of these
reasons could be skewing the SADs for old stellar populations, and this error
could be propagating into the DTD results. We laid out further tests to
investigate the physical nature of the systematics that are driving the
intermediate-age signal in the DTD, such as revisiting the RR Lyrae DTD once
LMC SADs from deeper photometric studies (e.g., SMASH) become available, and
also measuring DTDs in SMC and dwarf galaxies, which have less crowded star
fields and deep _HST_ -derived SADs. We will also continue to apply this
technique to other types of variable stars, such as Cepheids and Delta Scutis,
to probe other enigmatic phases of stellar evolution.

##  7 Acknowledgements

We are grateful to Horace Smith for reading the manuscript and providing many
helpful comments and insight on RR Lyrae observations and models. We also
acknowledge the detailed feedback from our anonymous referee on the
interpretation of RR Lyrae ages in the LMC and their relation with
metallicity, and Dennis Zaritsky, Knut Olsen, Thomas Matheson, Benjamin
Williams, J.J. Eldridge and Jay Strader for insightful discussions and
feedback on this work.

SKS, CB, and LC are grateful for the suppport of NSF grants AST-1412980 and
AST-1907790. CM acknowledges support from the DGAPA/UNAM PAPIIT program grant
IG100319 and from the ICC University of Barcelona visiting academic grants and
thanks the _Gaia_ -UB team for hosting her during part of this research. CM
also thanks the Polo de Desarrollo Universitario (PDU) en Ciencias Físicas at
CURE-UdelaR (Rocha), for their hospitality. DM acknowledges support by grants
from the Israel Science Foundation, the German Israeli Science Foundation and
the European Research Council (ERC) under the European Union’s FP7 Programme,
Grant No. 833031.

This work made use of the publicly available OGLE-IV variable star catalog
333http://ogledb.astrouw.edu.pl/ ogle/OCVS/. This research has made use of
NASA’s Astrophysics Data System, and the VizieR catalogue access tool, CDS,
Strasbourg, France. The original description of the VizieR service was
published in A&AS 143, 23. This work made use of the IPython package (Pérez &
Granger, 2007), SciPy (Jones et al., 2001–), NumPy (van der Walt et al.,
2011), matplotlib, a Python library for publication quality graphics (Hunter,
2007), and Astropy, a community-developed core Python package for Astronomy
(Astropy Collaboration et al., 2013). The Institute for Gravitation and the
Cosmos is supported by the Eberly College of Science and the Office of the
Senior Vice President for Research at The Pennsylvania State University.

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##  Appendix A Choice of Binning

Since we are calculating the DTD non-parametrically, the DTD depends on the
binning we choose for the ages. The SAD comes with a native resolution of 16
age-bins. We can decide on which combination of these bins provide the optimal
DTD through the Bayesian Information Criteria. These are defined as follows,

$BIC=k\ \mathrm{ln}(N)-2\ \mathrm{ln}(\mathcal{L}_{max})$ (A1)

where $k$ is the number of age-bins, $N$ is the number of SAD cells and ln
$\mathcal{L}_{max}$ is the maximum likelihood. The binning scheme that
minimizes BIC (i.e., the information loss) is favored. We calculate DTDs for
different age binnings using just the best-fit SAD (i.e., no randomized SADs)
and the original OGLE-IV sample inside the SAD area.

We show eight example binning schemes for our DTDs in Figure 10, which differ
in how the young and old ages are binned. The four binning schemes in the
bottom row have the smallest bin-sizes for ages $>0.8$ Gyr and varying bin-
sizes for young ages. The schemes in the top row have coarser resolution in
the oldest age-bins. More binning schemes similar to these are possible and
can be tested, but these eight provide a general sense of the impact of
binning. The DTDs measured in the detected bins vary at most by a factor of 2,
and are generally around $10^{-5}$ RR Lyrae per M⊙. Statistical errors in the
DTDs increase with the number of bins, because we are essentially increasing
the number of fitting parameters. Binning schemes that retain the highest
resolution in the oldest bins measure smaller values of BIC, irrespective of
the binning in the younger age-bins. This implies that the RR Lyrae DTD has
the strongest signal in the oldest bins, which is consistent with an older
stellar origin of RR Lyrae.

Apart from BIC, we also show the acceptance fraction of the MCMC solver,
$a_{f}$ in Figure 10. This is the fraction of new steps accepted by the emcee
walkers as the scan the multi-dimensional parameter space. While not a model
selection statistic like BIC, values of $a_{f}$ = 0.2–0.5 indicate that the
emcee algorithm is performing optimally (Foreman-Mackey et al., 2013). Values
of $a_{f}$ which are too low indicate multiple peaks in the posterior space
separated by valleys of “low probability”, while high values imply that the
walkers are simply random-walking, with no regard for the target probability
density. We note that while binning schemes in the second row have similar
values of BIC, they have different $a_{f}$, with the native resolution of the
SAD (last panel) having the smallest $a_{f}$.

Based on these tests, we choose the binning scheme 7 which has the smallest
value of BIC, and the largest value of $a_{f}$ = 0.28 in Figure 10.

Figure 10: RR Lyrae DTDs calculated with different age binning. The scheme
number is shown in the top right corner of each panel. The acceptance fraction
$a_{f}$ of the MCMC solver and the value of the BIC test (BIC) are shown in
each panel. Shaded grey regions mark age-bins with significant detections,
while arrows show 2$\sigma$ upper limits. We use the binning scheme plotted at
bottom right, as it has the smallest information loss, and also has the
highest acceptance fraction.

##  Appendix B Uncertainties using the B15 method

When generating the randomized SADs, we assumed that the uncertainties of the
stellar masses were normally distributed. However, this is only an
approximation, and the underlying shape of the probability distribution of
masses is unknown. We therefore also calculated the DTD uncertainties based on
the method in B15, which can be treated as a more conservative estimate of the
uncertainties. In B15, the 1 $\sigma$ uncertainty due to the SAD is equal to
the difference between the DTDs for the best fit SAD, and DTDs for the 68$\%$
upper and lower limits on the SADs. This difference is added in quadrature to
the statistical uncertainties in the best-fit DTD, and the total value is used
for assessing detectability. The uncertainties using this method are larger,
and as a result the DTD (Figure 11) is different from the DTD in Figure 3.
Nevertheless, it still shows significant signal below 10 Gyrs, particularly in
the age range 2-8 Gyrs. The signals in the other age-bins fall below the
2$\sigma$ limit. The total contribution below 10 Gyrs in this case is about
41.6$\%$. Our main science result,$-$ that the DTD has statistically
significant signal below 10 Gyrs $-$ is unchanged, even with our most
conservative estimates of the uncertainties.

Figure 11: Same as Figure 3, but with uncertainties calculated using the
method in Badenes et al. (2015), described in Section B.


# TURB-Rot. A large database of 3d and 2d snapshots from turbulent rotating
flows

L. Biferale
Dept. Physics and INFN
University of Rome Tor Vergata, Italy, and IIC-Paris, France
<EMAIL_ADDRESS>
&F. Bonaccorso
Center for Life Nano Science@La Sapienza
Istituto Italiano di Tecnologia and INFN
University of Rome Tor Vergata, Italy.
<EMAIL_ADDRESS>
&M. Buzzicotti
Dept. Physics and INFN
University of Rome Tor Vergata, Italy.
<EMAIL_ADDRESS>
&P. Clark Di Leoni
Department of Mechanical Engineering,
Johns Hopkins University, Baltimore, USA.
<EMAIL_ADDRESS>

###### Abstract

We present TURB-Rot, a new open database of 3d and 2d snapshots of turbulent
velocity fields, obtained by Direct Numerical Simulations (DNS) of the
original Navier-Stokes equations in the presence of rotation. The aim is to
provide the community interested in data-assimilation and/or computer vision
with a new testing-ground made of roughly 300K complex images and fields.
TURB-Rot data are characterized by multi-scales strongly non-Gaussian features
and rough, non-differentiable, fields over almost two decades of scales. In
addition, coming from fully resolved numerical simulations of the original
partial differential equations, they offer the possibility to apply a wide
range of approaches, from equation-free to physics-based models. TURB-Rot data
are reachable at http://smart-turb.roma2.infn.it.

##  1 Introduction

Turbulent flows under rotation are a physical system extremely relevant in
many areas of research dealing with both natural, e.g. astrophysical and
geophysical [1, 2], and industrial flows [3]. Rotating turbulence, is well
known for its complex dynamics resulting from the fact that the energy
injected from the external forcing produces large-scale cyclonic and/or anti-
cyclonic structures as well as small-scale intermittent homogeneous and quasi
isotropic highly non-Gaussian fluctuations, leading to the formation of
structures over many decades of scales (see [4] for a recent review and [5, 6]
for textbooks). In this short note, we present an open access data base of
rotating turbulence. We deploy a first dataset at medium numerical resolution,
planning to extend it to larger resolution and different datatype, including
Lagrangian information along particle trajectories in a later stage. The aim
is to have a huge set of 2d and 3d images/fields and trajectories that can be
used by people belonging to different communities and with different goals,
from data-assimilation and field reconstruction, interesting for numerical
weather prediction, to denoising and inpainting for computer vision tasks [7,
8, 9, 10, 11, 12]. A first application has been presented by us in [13].

In Sec. 2 we provide a brief description of the Direct Numerical Simulations
(DNS) performed to generated our TURB-Rot database while in Sec. 3 we discuss
how the 300K images have been selected and how it is possible to access to the
entire data-set.

##  2 Numerical Simulations

To generate the first dataset deployed on TURB-Rot database, we have performed
a DNS of the Navier-Stokes equations (NSE) for incompressible rotating fluid
in a triply periodic domain of size $L=2\pi$, using a fully dealiased parallel
3d pseudospectral code on a grid of $N=256^{3}$ collocation points. The time
integration has been implemented with a second-order Adams-Bashforth scheme
with viscous term integrated implicitly. The NSE for a fluid in the rotating
frame can be written as

$\begin{cases}\partial_{t}\bm{u}+\bm{u}\cdot\nabla\bm{u}+2{\bm{\Omega}}\times{\bm{u}}=-\nabla
p+\nu\Delta\bm{u}+\alpha\Delta^{-1}\bm{u}+\bm{f}\\\
\nabla\cdot\bm{u}=0,\end{cases}$ (1)

where $\nu$ is the kinematic viscosity, the term
$2{\bm{\Omega}}\times{\bm{u}}$ is the Coriolis force produced by rotation, and
$\bm{\Omega}=\Omega\hat{x}_{3}$ is the angular velocity with frequency
$\Omega$ around the rotation axis $\hat{x}_{3}$. The fluid density is constant
and absorbed into the definition of pressure $p$. The linear friction term
$\alpha\Delta^{-1}\boldsymbol{u}$, with hypo-viscosity coefficient
$\alpha=0.1$ is acting only on wave numbers with $|\boldsymbol{k}|\leq 2$ and
it is needed to prevent the formation of large-scale a condensate [4]. The
forcing mechanism, $\bm{f}$, is a Gaussian process delta-correlated in time,
with support in wavenumber space around $k_{f}=4$. In our DNS we fixed
$\Omega=8$, resulting in a Rossby number $Ro=E^{1/2}/k_{f}\Omega\sim 0.1$,
where $E$ is the flow kinetic energy. The dissipation in our simulation is
modeled by an hyperviscous term $\nu\nabla^{4}\bm{u}$, which replaces the
laplacian in (1), with $\nu=1.6\times 10^{-6}$.

The total energy evolution as a function of time evolved in the full
simulation is shown in Fig. 1(a), while in the inset of the same figure we
show a 3d snapshot of the turbulent velocity field taken at a fixed time in
the stationary regime, where we can see large columnar vortices as well as
other small-scale structures dispersed in the whole flow domain.

In Fig. 1(b) we show the energy spectrum
$E(k)=\frac{1}{2}\sum_{k\leq|\boldsymbol{k}|<k+1}|\boldsymbol{\hat{u}}(\boldsymbol{k})|^{2}$
averaged over time during the simulation, the shaded area identifies the
frequency where the forcing is acting. It is easy to see how both ends of the
spectrum get populated with active modes on almost two decades in wave-
numbers. This is due to the presence of an energy flux that cascades energy in
booth directions, namely towards smaller and higher scales. In the inset of In
Fig. 1(b) we plot the averaged energy flux across the scale $k$, defined as:
$\Pi(k)=-\sum_{|\boldsymbol{k}|\leq
k}ik_{j}\hat{u}^{*}_{i}(\boldsymbol{k})\sum_{\boldsymbol{p},\boldsymbol{q}}\hat{u}_{i}(\boldsymbol{p})\hat{u}_{j}(\boldsymbol{q})\delta(\boldsymbol{p}+\boldsymbol{q}-\boldsymbol{k})$,
where the $\delta$ function constrains the nonlinear interactions to closed
wave-vectors triads. The negative values of $\Pi(k)$ measured at scales
$k<k_{f}$ indicate that in this range of scales there is an inverse energy
cascade, while a positive, $\Pi(k)$ show the presence of a simultaneous
forward energy cascade for $k>k_{f}$ leading to the formation of small-scales
structures.

Figure 1: (Panel a) Energy evolution for the turbulent flow generated during
the simulation performed to generate the database of $600$ different
configurations. The velocity fields are extracted from time $t=276$ up to
$t=876$ every $t=1$. In the inset we show a 3d visualization of the velocity
amplitude $\boldsymbol{u}$ extracted from the simulation at a fixed time.
(Panel b) Log-log plot of the averaged energy spectrum. Inset: energy flux
$\Pi(k)$. The gray areas indicate the forced wavenumbers, while $k_{\eta}\sim
32$ is the Kolmogorov dissipative wavenumber, defined as the scale such that
for $k>k_{\eta}$ the velocity field becomes smooth [14].

##  3 DataBase Description

TURB-Rot database is extracted from the simulation described in the previous
section as follows:

  * •

During the simulation we have dumped a number of $600$ snapshots of the full
3d velocity field (see Fig. 1(a) for the total energy evolution). Snapshots
are chosen with large temporal separation in a way to decrease correlations in
time between two successive data-points. These configurations are the backbone
of our image dataset and they can be found under the data-class
$3d\\_256$cubed.

  * •

We provide also a set of 2d downsized images that can be easily used as input
to ML algorithms, under the name $2d\\_64$squared. In particular, the original
DNS resolution of $256^{3}$ grid points is downsized to $64^{3}$ grid, after
applying a Galerkin truncation in Fourier space where the maximum frequency
allowed is set to $|\boldsymbol{k}|\leq 32$.

  * •

For each configuration, $16$ horizontal planes $(x_{1},x_{2})$ are selected at
different $x_{3}$-levels (the rotation direction). Each of these $16$ planes
is shifted along both $x_{1}$ and $x_{2}$ using periodic boundary conditions
in $11$ different ways, by choosing randomly a new center of the plane, such
as to obtain a total of $16\times 11=176$ planes at each instant of time.

  * •

Finally, the dataset composed of $600\times 176=105600$ planes, is reordered
in time randomly to avoid any correlation between successive planes.

The database TURB-Rot is available for download using the SMART-Turb portal
http://smart-turb.roma2.infn.it. The portal is based on the concept of
"Dataset" to aggregate resources related to the same simulation: we have
released both the original full resolution of 3d DNS snapshot at $256^{3}$
grid points and the database of $105600$ ($x_{1},x_{2}$)-planes of size
$64\times 64$ for each velocity component leading to a total of roughly 300K
images. Details on how to access the data with a few examples can be found on
the portal. Other data-sets concerning rotating turbulence at higher
resolution, different rotation rate, $\Omega$, and including also Lagrangian
data for tracers and inertial particles will be made open and available soon.
Finally, In Fig. 2 we show a few examples of the images contained in TURB-Rot
made out of the velocity magnitude. In Figs 3-5 we show for the same snapshots
the images generated looking at the three different velocity components. In
Fig. 6 finally, we show the image obtained from the same field but looking at
the vorticity field, defined as $w_{3}=\partial_{1}u_{2}-\partial_{2}u_{1}$.

Figure 2: Example of $16$ different $64\times 64$ planes composing the TURB-
Rot database. The figures presented here are obtained measuring the amplitude
of the velocity fields $|\boldsymbol{u}|$. Figure 3: Same planes shown in Fig.
2 but for the ${v_{1}}$ component of the velocity field. Figure 4: Same planes
shown in Fig. 2 but for the ${v_{2}}$ component of the velocity field. Figure
5: Same planes shown in Fig. 2 but for the ${v_{3}}$ component of the velocity
field. Figure 6: Same planes shown in Fig. 2 but for the ${w_{3}}$ component
of the vorticity.

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