# Relative Ages of Nine Inner Milky Way Globular Clusters from Proper Motion
Cleaned Color-Magnitude Diagrams111Based on observations made with the
NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are
associated with program GO-15065.

Roger E. Cohen  Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Andrea Bellini  Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  Luca Casagrande  Research
School of Astronomy and Astrophysics, Australian National University,
Australia  Thomas M. Brown  Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA  Matteo Correnti  Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Jason S. Kalirai
Johns Hopkins University Applied Physics Laboratory, 11101 Johns Hopkins Road,
Laurel, MD 21723, USA

###### Abstract

Our picture of the age-metallicity relation for Milky Way globular clusters
(MWGCs) is still highly incomplete, and the majority of MWGCs lack self-
consistent age measurements. Here, we exploit deep, homogenous multi-epoch
Hubble Space Telescope (HST) imaging of nine MWGCs located towards the inner
Milky Way to measure their relative ages, in most cases for the first time.
Our relative age measurements are designed to be directly comparable to the
large set of MWGC ages presented by VandenBerg et al., 2013 (V13), using
identical filters, evolutionary models, and bolometric corrections, extended
to the higher extinction values relevant to our target clusters. Adopting the
V13 MWGC age scale, our relative age measurements imply that our target
clusters are consistently very old, with a mean age of 12.9$\pm$0.4 Gyr, with
the exception of the young metal-rich MWGC NGC 6342. We perform two tests to
validate the precision of our methodology, and discuss the implications of our
target cluster loci in the MWGC age-metallicity plane. In addition, we use our
fully self-consistent bolometric corrections to assess the systematic impact
of variations in the total-to-selective extinction ratio $R_{V}$ on relative
age measurements.

††facilities: HST (ACS/WFC,WFC3/UVIS)

##  1 Introduction

The ages and metallicities of MWGCs pose crucial constraints on Milky Way
formation and evolution. By using MWGCs as tracers, substantial progress
towards understanding the assembly history of the Milky Way has been made
possible by the combination of homogenous iron abundances (Carretta et al.,
2009a, hereafter C09, their Table A.1) and self-consistent age measurements
for large ensembles of MWGCs from high quality photometry (VandenBerg 2000,
Salaris & Weiss 2002, De Angeli et al. 2005, Marín-Franch et al. 2009, Dotter
et al. 2010, VandenBerg et al., 2013, hereafter V13, O’Malley et al. 2017).
The resulting age-metallicity relation (AMR) revealed a bifurcation (Marín-
Franch et al., 2009; Forbes & Bridges, 2010; Leaman et al., 2013) implying
that a substantial fraction of MWGCs were accreted early in the history of the
Milky Way, while many others likely formed in situ, and this scenario was
reinforced by then-available kinematics (e.g. Leaman et al., 2013).

A leap forward in linking MWGC properties to the global picture of Milky Way
formation has been made by combining age and metallicity information with
cluster kinematic properties. This recent progress is due in large part to the
quality of space-based MWGC absolute proper motions (e.g. Sohn et al., 2018),
in particular from the all-sky Gaia mission (Gaia Collaboration et al., 2016,
2020). These absolute proper motions, when combined with distances and a model
of the Milky Way potential, yield orbital parameters (Baumgardt et al., 2019;
Massari et al., 2019; Vasiliev & Baumgardt, 2021). Using the additional
information available from orbital criteria, individual MWGCs can be linked to
a specific progenitor accreted onto the Milky Way, or categorized as being
formed in situ (Myeong et al., 2018; Massari et al., 2019). Furthermore, such
information can be combined with state-of-the-art simulations to ascertain
which parameters are most useful to constrain the accretion history of the
Milky Way (Kruijssen et al., 2019a). By matching MWGCs to their progenitors,
information is gained not only about each cluster individually, but about the
formation history of the Milky Way via the nature (mass and time of accretion)
of the progenitors themselves (Kruijssen et al., 2019b, 2020).

While the power of a large ensemble of MWGCs to unlock key events in Milky Way
formation and evolution is clear, the full discriminatory power of MWGCs in
the inner Milky Way (where a substantial fraction of its MWGC population
actually resides) has not been realized. Most of these clusters have been
excluded from our current picture of the MWGC age-metallicity relation simply
because they lack age estimates and, in many cases, reliable metallicity
estimates. As an additional complication, distances from literature
compilations, used to calculate orbital properties, can be unreliable as well,
placing clusters on the wrong side of the Galactic bulge in some cases (e.g.
Alonso-García et al., 2015; Cadelano et al., 2020). In addition, we know that
not all MWGCs presently located in the inner Milky Way are confined to the
Galactic bulge, instead showing a kinematic diversity including interlopers
from the halo and thick disk (Ortolani et al., 2019; Cadelano et al., 2020;
Pérez-Villegas et al., 2020). Therefore, placing inner Milky Way globulars on
the MWGC age-metallicity relation is critical not only to determine
progenitors or discriminate interlopers on a case-by-case basis, but also to
constrain the global fraction of present-day MWGCs that were formed in situ
(e.g. Keller et al., 2020; Trujillo-Gomez et al., 2021).

Here, we exploit multi-epoch, homogenous space-based imaging of an ensemble of
nine inner Milky Way globular clusters to measure their ages. We have already
demonstrated the potential of this dataset from both a photometric and
astrometric perspective in Cohen et al. (2020), where we analyzed radial
density and proper motion dispersion profiles of our target clusters, finding
that they are preferentially highly dynamically evolved. Here, we use the
proper-motion-cleaned color-magnitude diagrams (CMDs) to gain additional
insight on the evolutionary history of these clusters to place them on the
MWGC age-metallicity relation in the context of their more well-studied
counterparts from Sarajedini et al. (2007). Importantly, because systematics
can play a significant role in MWGC age estimates (see e.g. the Appendix of
Massari et al. 2019), we conduct our analysis self-consistently with the oft-
cited MWGC age investigation of V13 from both observational and theoretical
standpoints, using an identical observing setup as well as identical models,
including self-consistent bolometric corrections, to obtain our MWGC age
estimates.

The remainder of this paper is organized as follows: In the next section, we
summarize the observations and data reduction used to produce differential-
reddening-corrected, proper-motion-cleaned photometric catalogs. In Sect. 3 we
describe the technique used to measure cluster ages, and our results are
discussed in Sect. 4 along with some outstanding issues. In the final section,
we present our conclusions.

##  2 Data

Our nine target clusters were selected as having extant archival HST imaging
of sufficient depth and spatial resolution to reach several magnitudes
faintward of the main sequence turnoff in the cluster cores. To measure
relative proper motions, we obtained a second epoch of deep imaging with
ACS/WFC onboard HST (GO-15065, PI:Cohen), using the F606W and F814W filters to
produce homogenous photometry with an identical observing setup as that used
for the ACS Galactic Globular Cluster Treasury Survey (Sarajedini et al.,
2007), on which several recent MWGC age compilations are based (Marín-Franch
et al. 2009, Dotter et al. 2010, V13). The details of both epochs of
observations for all target clusters, separated by time baselines of
$\gtrsim$9 yr, are given in Table 1 of Cohen et al. (2020)222The MAST data
used in this work are available at 10.17909/t9-na33-8504 (catalog
https://doi.org/10.17909/t9-na33-8504).

The tools and techniques we used to produce astrophotometric catalogs
including PSF photometry and relative proper motions for our target clusters
are identical to those used in other recent studies (Bellini et al., 2014,
2017a, 2018; Libralato et al., 2018), and further details, including
photometric quality cuts and extensive artificial star tests, are given in
Cohen et al. (2020). The resulting catalogs yield proper-motion-cleaned,
Vegamag-calibrated PSF photometry extending several magnitudes faintward of
the main sequence turnoff (MSTO) of all target clusters, and as far brightward
as the red giant branch (RGB), and in most cases the horizontal branch (HB),
limited only by the archival first epoch observations333While the brightest
cluster giants near the tip of the RGB are saturated in one or both epochs,
the saturation limit occurs significantly ($>$1 mag in all cases) brightward
of the magnitude range needed for our relative age analysis described in Sect.
3.3.. We have calculated differential reddening corrections using proper-
motion-selected members as described elsewhere (e.g. Milone et al., 2012;
Bellini et al., 2017b), and the photometry we analyze below consists of
proper-motion-selected cluster members, with differential reddening
corrections applied.

##  3 Relative Age Measurements

###  3.1 Models

To enable a direct comparison with the age measurements of V13, we use
identical stellar evolutionary tracks and the publicly available software from
VandenBerg et al. (2012) and VandenBerg et al. (2014) to compute theoretical
isochrones from the evolutionary tracks. We assume [$\alpha$/Fe]=+0.4 since
high-resolution spectroscopy reveals inner Milky Way globular clusters to be
enhanced in $\alpha$-elements (Muñoz et al. 2017; Villanova et al. 2017;
Barbuy et al. 2018; Johnson et al. 2018; also see table 7 of Johnson et al.
2016 and references therein), and assume a helium abundance of Y=0.25 for
[Fe/H]$\leq$1.0, slightly increasing towards higher [Fe/H] as
$\Delta$Y/$\Delta$Z$\sim$1.4 following V13. To convert these isochrones to the
observational plane, we have calculated bolometric corrections in the filters
relevant to our observations (ACS/WFC F606W and F814W) identically as
described in Casagrande & VandenBerg (2014), but extended to higher values of
$E(B-V)$$\leq$ 3 beyond the maximum value of $E(B-V)$=0.72 available through
their public software. These bolometric corrections are computed using MARCS
synthetic spectra (Gustafsson et al., 2008) and assume the extinction law of
Cardelli et al. (1989) and O’Donnell (1994) (the impact of variations in the
properties of interstellar extinction is discussed further in Sect. 4.2).

###  3.2 Fiducial Sequences

For each target cluster, we construct a fiducial sequence from the
differential-reddening-corrected, proper-motion-cleaned photometry in the
(F606W-F814W),F814W plane (since we fit color as a function of magnitude, the
subgiant branch is less perpendicular to the fitting axis than if we had used
the F606W filter on the vertical CMD axis).

The procedure we use to generate a fiducial sequence is quite similar to those
used in the recent literature (i.e. Marín-Franch et al., 2009; Wagner-Kaiser
et al., 2017): The median color is calculated in 0.1 mag bins (rotated to be
parallel to the fiducial sequence from the previous iteration if not the first
iteration) using a 2.5$\sigma$ clip to reject outliers (i.e. blue stragglers
and high-mass-ratio binaries), and the RGB is fit with a 3rd order
polynomial444The base of the RGB is defined as the maximum in the second
derivative of color with respect to magnitude.. This procedure is repeated 10
times, shifting the magnitude bins by one-tenth of the binsize (0.01 mag) each
time. The 10 resulting fiducial sequences are averaged, and this process is
repeated several times (typically three, with a maximum of five) until
convergence is indicated by a sum of absolute deviations from the previous
iteration of $<$0.001 mag. To quantify the uncertainties on this fiducial
sequence, we repeat the entire procedure for 10,000 monte carlo iterations,
where several sources of uncertainty are injected in each monte carlo
iteration: First, for each observed star, we apply color and magnitude offsets
randomly drawn from the nearest (in color, magnitude and distance from the
cluster center) artificial stars555Specifically, for each observed star,
artificial stars are drawn from those within 0.4 mag in F814W, 0.2 mag in
F606W-F814W (using their raw observed magnitudes before differential reddening
correction), and 20% in distance from the cluster center, giving thousands of
artificial stars per observed star.. By drawing directly from an ensemble of
artificial stars for each observed star, we require no a priori assumptions
about whether photometric error distributions are Gaussian, have null offsets
(bias), or whether color and magnitude errors are correlated. In addition,
differential reddening uncertainties are taken into account by offsetting each
star along the reddening vector by Gaussian deviates of the uncertainty on its
differential reddening correction, and lastly, an offset in each filter is
applied drawn from the (potentially correlated) distribution of photometric
zeropoint uncertainties. By sampling from error distributions of color,
magnitude, zeropoint, and differential reddening correction over each of the
monte carlo iterations, we obtain a fiducial sequence in each iteration, which
is interpolated to a fixed magnitude grid. To account for any potential
photometric bias, the final fiducial sequence color at each magnitude point is
the median color over the monte carlo iterations, and the uncertainty in the
fiducial sequence color is the 16th-84th percentile interval over the monte
carlo iterations. Fiducial sequences for our target clusters are shown in Fig.
1.

Figure 1: Proper motion cleaned, differential reddening corrected photometric
catalogs of our target clusters in the region near the MSTO, shown as Hess
diagrams (color-magnitude density plots, Log-scaled in this case). Fiducial
sequences are shown in red.

###  3.3 Relative Age Measurement Technique

We measure the relative ages of our target clusters using the classical MSTO-
to-RGB color difference as described by Vandenberg et al. (1990) and
implemented by V13 (also see Sarajedini & Demarque, 1990; Chaboyer et al.,
1996; Buonanno et al., 1998). This method is designed to assess the age
difference between a target cluster (with unknown age) and a reference cluster
(generally with a known age) through a CMD registration procedure, illustrated
for an example case in Fig. 2. To measure relative ages, the fiducial
sequences of both the target and reference cluster are shifted along the
horizontal axis so that their MSTO colors coincide, setting the zeropoint of
the horizontal axis in the registered CMD, shown as a red diamond in Fig. 2.
Next, the fiducial sequences are shifted vertically so that their magnitudes
coincide at the magnitude on the main sequence (i.e. faintward of the MSTO)
where the fiducial sequence color is 0.05 mag redder than the MSTO
color666Measuring the magnitude of a fiducial sequence 0.05 redward of the
MSTO (Buonanno et al., 1998) is much easier than attempting to measure the
MSTO magnitude, since the fiducial sequence is, by definition, vertical at the
MSTO. and this magnitude is set as the new zeropoint of the vertical axis in
the registered CMD, shown as a red circle in Fig. 2. The reason for
registering the fiducial sequences in this way, as discussed extensively by
e.g. Vandenberg et al. (1990) and V13, is that the age difference between the
target and reference clusters can now be directly read from the difference in
their registered RGB colors at a chosen registered magnitude, for example
$-$2.8 mag above the new (registered) magnitude zeropoint, shown as a dashed
horizontal red line in Fig. 2. In the case where the target and reference
clusters do not have identical values of [Fe/H], an isochrone-based correction
is applied to the color of the lower RGB. To illustrate the relative age-
dating procedure, the inset of Fig. 2 shows the difference between target and
reference cluster age that would be obtained as a function of the color of the
lower RGB, illustrating that bluer (redder) registered RGB colors correspond
to older (younger) target cluster ages relative to their reference cluster.

Normally, ages and their uncertainties from the above registration procedure
are straightforward to calculate because the registration procedure only
involves vertical and horizontal shifts of the cluster fiducial sequences in
the CMD. In our case, application of this technique comes with an additional
step, which is needed because our target clusters are typically at
substantially higher values of $E(B-V)$ than the well-studied reference
clusters to which we compare them. In particular, because the shape of an
isochrone changes as a function of total extinction, two clusters at
substantially different values of $E(B-V)$ cannot be compared by simply
shifting the fiducial sequence of one to match the other. Therefore, for each
reference cluster, we use the bolometric corrections we have calculated (see
Sect. 3.1) to correct the reference cluster fiducial sequence for the
difference in $E(B-V)$ between the target cluster and the reference cluster.
This correction can change the shape of the reference cluster fiducial
sequence quite significantly, as shown by the difference between the dotted
and dashed blue lines in Fig. 2. Only once the change in isochrone shape due
to the difference in $E(B-V)$ has been accounted for do we then apply an
isochrone-based correction to the color of the lower RGB to account for any
difference in [Fe/H] between target and reference clusters, shown as a solid
blue line segment along the RGB in Fig. 2. Importantly, this relative age
dating procedure, including the corrections for both the $E(B-V)$ difference
and [Fe/H] difference between target and reference clusters, does not assume
that the isochrones are correct in an absolute sense; rather they are used
only to apply relative corrections to empirically determined fiducial
sequences.

Figure 2: An example of the CMD registration procedure used to measure
relative ages. In this case the reference cluster, NGC 6205 (= M13), is being
registered to the target cluster NGC 6355. The target cluster fiducial
sequence is shown as a black line. The observed reference cluster fiducial
sequence (dotted blue line) has its shape corrected for the difference in
$E(B-V)$ between the target and reference clusters (dashed blue line). The two
fiducial sequences are then matched in color at their MSTO (red diamond) and
in magnitude at the point on the main sequence 0.05 mag redward of the MSTO
(red circle). Lastly, an isochrone-based correction to the RGB color (solid
blue line) is applied to account for any difference in [Fe/H] between the
target and reference clusters. The relative age is then directly measured from
the color difference between the RGB and MSTO (F606W-F814W)-(F606W-F814W)TO on
the lower RGB at a magnitude of F606W-F606WTOC+0.05=$-$2.8, shown using a
dashed red horizontal line. In the inset, we show a zoomed-in version of the
registered CMD near this RGB magnitude, where the dashed red lines indicate
the shift in the RGB color as a function of age relative to the reference
cluster predicted by the isochrones, labeled in red.

###  3.4 Target Cluster Parameters

Our relative age measurement technique requires assuming, for both the target
and reference clusters, per-cluster values and uncertainties for both the
color excess $E(B-V)$ (to correct the isochrone shape for the difference in
color excess between target and reference clusters) and the iron abundance
[Fe/H] (to correct the color of the RGB for any difference in [Fe/H] between
the target and reference clusters). Regarding assumed [Fe/H] values and
uncertainties, we have compiled in Table 1 recent literature estimates of
[Fe/H] for our target clusters from various sources. These include the
compilation by C09 (see their Table A.1), and several other studies that each
include multiple target clusters. These are the CaII study by Vásquez et al.,
2018 (denoted V18), low resolution optical spectroscopy by Dias et al., 2016
(D16), estimates from near-IR photometric indices by Valenti et al., 2010
(V10), and the values given in the catalog of Harris 1996 (2010 edition,
hereafter H10). In the last column, we give additional estimates from other
individual sources on a per-cluster basis, where values from high-resolution
optical spectroscopy of individual cluster giants, which we deem most
trustworthy, are given in boldface and adopted as our metallicity estimates
for those clusters (NGC 6342 and NGC 6558). However, for the majority of our
sample, metallicity estimates from high-resolution spectra are notably absent,
so we adopt an estimate and uncertainty based on the recent literature values
in Table 1. For target cluster $E(B-V)$ values, we draw from either H10 or
V10, averaging them when both are available, and conservatively assume a 15%
uncertainty on $E(B-V)$.

Table 1: Literature [Fe/H] Abundances for Target Clusters

Cluster | C09 |  V18aaAssuming their Saviane et al. (2012) calibration. | D16 | V10 | H10 | OtherbbValues from high resolution optical spectroscopy of multiple resolved member stars are given in boldface.
---|---|---|---|---|---|---
NGC 6256 | -0.62$\pm$0.09ccFor NGC 6256 and NGC 6380, the values given by C09 result from applying a fixed offset of 0.025 dex to [Fe/H] values from the then-most-recent (2003) edition of the Harris catalog, which are notably different from the more recent H10 values. | -1.61$\pm$0.14 | -1.05$\pm$0.13 | -1.63 | -1.02 |
NGC 6325 | -1.37$\pm$0.14 |  |  |  | -1.25 | -1.23$\pm$0.17 (1)
NGC 6342 | -0.49$\pm$0.14 |  |  | -0.71 | -0.55 | -0.60$\pm$0.01 (2); -0.53$\pm$0.11 (3)
NGC 6355 | -1.33$\pm$0.14 | -1.59$\pm$0.15 | -1.46$\pm$0.06 | -1.42 | -1.37 |
NGC 6380 | -0.40$\pm$0.09ccFor NGC 6256 and NGC 6380, the values given by C09 result from applying a fixed offset of 0.025 dex to [Fe/H] values from the then-most-recent (2003) edition of the Harris catalog, which are notably different from the more recent H10 values. |  |  | -0.87 | -0.75 | -0.72$\pm$0.17 (1); -0.73$\pm$0.06 (4)
NGC 6401 | -1.01$\pm$0.14 | -1.30$\pm$0.15 | -1.12$\pm$0.07 | -1.37 | -0.75 | -1.13$\pm$0.06 (5)
NGC 6453 | -1.48$\pm$0.14 | -1.67$\pm$0.13 | -1.54$\pm$0.09 | -1.38 | -1.50 |
NGC 6558 | -1.37$\pm$0.14 |  | -1.01$\pm$0.05 |  | -1.32 | -1.07$\pm$0.17 (1); -1.17$\pm$0.10 (6)
NGC 6642 | -1.19$\pm$0.14 |  |  | -1.20 | -1.26 |

Note. — References: (1)Mauro et al. (2014), using their cubic fit to C09c
values (their table 3). Error is their urms (unweighted rms) of this
calibration. (2)Origlia et al. (2005) (3)Johnson et al. (2016) (4)Fernández-
Trincado et al. (2021) (5)Tsapras et al. (2017) (6)Barbuy et al. (2018)

###  3.5 Reference Clusters

For our reference clusters, we choose MWGCs with ages from V13 and
metallicities similar to (and bracketing the range of) those of our target
clusters. The reference clusters have been intentionally selected to have
$\alpha$-enhancement consistent with that assumed for our target clusters
(Carretta et al., 2010; Johnson et al., 2017; Massari et al., 2017) and
generally low extinction, with the exception of NGC 6366 at the metal-rich
end, which high-resolution spectroscopy reveals to be essentially a chemical
twin of the target cluster NGC 6342 in terms of [Fe/H], but also O and
$\alpha$-elements (Johnson et al., 2016). Reference cluster fiducial sequences
and their uncertainties are determined identically as for our target clusters
(see Sect. 3.2), using photometric catalogs from the public database of
Sarajedini et al. (2007) and accompanying artificial star tests, restricted to
stars with valid proper motions and membership probabilities $p_{\rm mem}>$80%
from the catalogs of Nardiello et al. (2018). In addition, due to the
substantial impact of differential reddening on the CMD of NGC 6366,
differential reddening corrections were applied to this cluster identically as
for the target clusters. Photometry and fiducial sequences for our reference
clusters are shown in Fig. 3. The six reference clusters we adopt are listed
in Table 2, along with adopted values and uncertainties of [Fe/H] from C09
(also used by V13 to derive ages) and $E(B-V)$ from V13, for which we
conservatively assume an uncertainty of $\sigma$$E(B-V)$=0.01 mag or 15%,
whichever is larger.

Figure 3: Hess diagrams (color-magnitude density plots, Log-scaled as in Fig.
1) of our reference clusters spanning the metallicity baseline of our target
clusters, in order of decreasing [Fe/H]. Of stars in the catalogs of
Sarajedini et al. (2007), only those with membership probabilities $p_{\rm
mem}>$80% from Nardiello et al. (2018) are included. Fiducial sequences are
shown in red.  Table 2: Reference Cluster Parameters

Reference Cluster | [Fe/H] | $E(B-V)$ | Age
---|---|---|---
| dex | mag | Gyr
NGC6366 | -0.55$\pm$0.10 | 0.730$\pm$0.110 | 11.00$\pm$0.50
NGC0104 | -0.76$\pm$0.02 | 0.032$\pm$0.010 | 11.75$\pm$0.25
NGC6362 | -1.07$\pm$0.05 | 0.076$\pm$0.011 | 12.50$\pm$0.25
NGC0362 | -1.30$\pm$0.04 | 0.032$\pm$0.010 | 10.75$\pm$0.25
NGC6205 | -1.58$\pm$0.04 | 0.017$\pm$0.010 | 12.00$\pm$0.38
NGC5986 | -1.63$\pm$0.08 | 0.280$\pm$0.042 | 12.25$\pm$0.75

Note. — Reference cluster [Fe/H] is from C09, except NGC 6366, which is from
the high resolution spectroscopic study of Johnson et al. (2016). $E(B-V)$ is
from V13, and the uncertainties we assume are 15% or 0.01 mag, whichever is
larger. Relative ages and uncertainties are from V13.

###  3.6 Relative Age Uncertainties

To calculate the total uncertainties on the relative ages we measure, we take
into account several sources of uncertainty, both on the observations as well
as the assumed [Fe/H] and $E(B-V)$ for the target and reference clusters,
injecting these uncertainties and measuring a relative age for each of the
monte carlo iterations in Sect. 3.2:

  * •

$\sigma_{\rm obs}$ is the uncertainty in registered RGB color due to
(potentially correlated) errors in the photometry and photometric zeropoint,
described in Sect. 3.2.

  * •

$\sigma_{E(B-V)}$ is the contribution from the uncertainty in $E(B-V)$, which
affects the shape of the reference cluster fiducial sequence. Specifically, an
assumed value of $E(B-V)$ is needed to correct the reference cluster fiducial
sequence shape for the difference in $E(B-V)$ between the target and reference
clusters using our bolometric corrections.

  * •

$\sigma_{\rm[Fe/H]}$ is the contribution due to the uncertainty of the
metallicity correction to the RGB color. The uncertainties on target and
reference cluster [Fe/H] have additional, more subtle contributions to the
uncertainty budget because the assumed values of [Fe/H] affect both the
gradient of RGB color with relative age (see Sect. 4.2) and the bolometric
corrections used to account for a difference in $E(B-V)$ above.

The values we assume for $\sigma_{E(B-V)}$ and $\sigma_{\rm[Fe/H]}$ are given
in Table 2 for the reference clusters and Table 3 for our target clusters. All
of the above error sources are injected in each monte carlo iteration to
obtain an individual relative age estimate, with the standard deviation of
these estimates taken as the uncertainty on the target cluster age relative to
the reference cluster, listed in Table 3.

Table 3: Target Cluster Relative Ages  Target Cluster | [Fe/H] (assumed) |  $E(B-V)$ (assumed) | Reference Cluster |  $\Delta$(Age) (Target-Reference) | AgeaaThe ages given in the last column are not direct absolute age measurements, rather they are calculated based on our relative ages and the comparison cluster absolute ages given by V13, see text for details.
---|---|---|---|---|---
| dex | mag |  | Gyr | Gyr
NGC6256 |  $-$1.61$\pm$0.20 | 1.15$\pm$0.17 | NGC6205 | 0.8$\pm$0.7 | 12.9$\pm$1.0
NGC6325 |  $-$1.37$\pm$0.14 | 0.91$\pm$0.14 | NGC6205 | 0.5$\pm$0.6 | 12.5$\pm$0.9
NGC6342 |  $-$0.53$\pm$0.11 | 0.52$\pm$0.08 | NGC6366 | 0.5$\pm$1.0 | 11.5$\pm$1.3
NGC6355 |  $-$1.50$\pm$0.20 | 0.79$\pm$0.12 | NGC6205 | 1.1$\pm$0.6 | 13.2$\pm$0.9
NGC6380 |  $-$0.73$\pm$0.15 | 1.23$\pm$0.18 | NGC0104 | 1.2$\pm$0.8 | 12.9$\pm$1.1
NGC6401 |  $-$1.15$\pm$0.20 | 0.91$\pm$0.14 | NGC6362 | 0.6$\pm$0.9 | 13.2$\pm$1.2
NGC6453 |  $-$1.48$\pm$0.14 | 0.67$\pm$0.10 | NGC6205 | 1.2$\pm$0.5 | 13.3$\pm$0.8
NGC6558 |  $-$1.17$\pm$0.10 | 0.44$\pm$0.07 | NGC6362 |  $-$0.2$\pm$0.8 | 12.3$\pm$1.1
NGC6642 |  $-$1.20$\pm$0.15 | 0.50$\pm$0.08 | NGC0362 | 1.9$\pm$0.8 | 12.7$\pm$1.1

###  3.7 Testing the Method

We perform two tests as a check on the reliability of our age measurement
technique. First, for each target cluster we use an alternate reference
cluster, selecting the next nearest reference cluster in metallicity, which
has a metallicity difference opposite in sign to the original reference
cluster. Using this alternate set of reference clusters caused a change in our
target cluster ages of $<$0.1 Gyr in the mean, with a standard deviation of
$<$0.3 Gyr and an absolute value $<$0.5 Gyr in all cases. As a second test, we
apply our age measurement procedure within our set of reference clusters,
treating each one in turn as a target cluster and using the other reference
clusters with neighboring [Fe/H] to calculate ages. We find that we recover
the ages listed by V13 to within their uncertainties across all reference-
reference cluster pairs, and the difference between the ages we estimate
versus those listed by V13 has a mean of 0.1$\pm$0.2 Gyr. Given the results of
these two tests, the ages we measure for our target clusters should be
directly comparable with those given by V13 at the level of $\sim$0.3 Gyr or
better (modulo assumptions on chemical abundances, discussed below in Sect.
4.3), well within their individual uncertainties.

##  4 Results and Discussion

###  4.1 The Age-Metallicity Relation

To view our target cluster ages in the context of the MWGC age-metallicity
relation, in the left panel of Fig. 4 we show our targets in red, and in grey
we show the 55 clusters from V13, supplemented by 14 clusters from Forbes &
Bridges (2010) with [Fe/H]$\leq$$-$1.1 following Massari et al. (2019). The
reference cluster ages from V13 (see Table 2) have been used to place our
target cluster relative ages (Table 3) on their MWGC age scale, adding a
contribution of 0.3 Gyr (see Sect. 3.7) to the relative age uncertainties in
Table 3. This plot is reproduced in the right-hand panel of Fig. 4, but now
color-coding the literature clusters by their kinematically-identified
progenitor from Massari et al. (2019). We see that the metal-rich cluster NGC
6342 is fairly young, consistent with other metal-rich MWGCs formed in situ,
bolstering its classification from Massari et al. (2019). The remainder of our
sample is relatively old, to a remarkably consistent extent: Their mean age is
12.9$\pm$0.4 Gyr, at least as old as any other component of the MWGC age-
metallicity relation. This is in excellent agreement with a mean age of
12.86$\pm$0.36 Gyr for metal-intermediate
($-$1.5$\lesssim$[Fe/H]$\lesssim$$-$0.85) bulge globulars quoted by Oliveira
et al. (2020), who compiled literature ages to update a similar analysis by
Saracino et al. (2019). However, these age measurements are quite
heterogenous, derived using observations in different bandpasses, different
age measurement techniques, and in some cases different evolutionary models
and different bolometric corrections. We have attempted to mimimize
systematics by keeping all of these ingredients consistent with V13 to the
greatest extent possible to permit a comparison that is reliable at least in a
relative sense. Accordingly, we suggest that applications of heterogenous
techniques and observations to measure ages of highly extincted MWGCs ideally
should be validated on well-studied low-extinction MWGCs (e.g. Correnti et
al., 2016, 2018) before blindly applying them to MWGCs along heavily extincted
sightlines.

Fortuitously, much of our sample lies at [Fe/H] values of
$-$1.5$\lesssim$[Fe/H]$\lesssim$$-$1.1, where the discriminatory power is
maximized to separate clusters formed in situ, which are older in this
metallicity range, from the younger branch of clusters that were accreted from
various progenitors. The old ages of our target clusters in this metallicity
range are generally consistent with the kinematic classification of Massari et
al. (2019), who list seven of our nine target clusters as being formed in
situ, with NGC 6401 and NGC 6453 belonging to the low-energy structure dubbed
Kraken by Kruijssen et al. (2020). Meanwhile, the location of these two
clusters in Fig. 4 points towards an in situ origin as being perhaps more
likely, but especially in light of uncertainties on the chemical abundances of
these clusters, membership to Kraken cannot be strongly excluded. As
demonstrated by Massari et al. (2019) and Kruijssen et al. (2020), orbital
elements are key to assigning clusters to a progenitor on a case-by-case
basis, and these orbital elements rely in turn (in addition to accurate and
precise absolute proper motions) on heliocentric distances. These distances
may be particularly uncertain for inner Milky Way globulars due to the
variable extinction law towards these sightlines, and just as an example, an
assumption of $R_{V}$=2.5 instead of $R_{V}$=3.1 for clusters with
$E(B-V)$$~{}\sim$ 1 at the approximate distance of the Galactic bulge can
change their distances by $>$10% ($>$1 kpc), potentially placing them on the
opposite side of the bulge/bar.

Comparing the ages we measure with those in the literature, there are only two
clusters with extant age measurements: For NGC 6256, an isochrone-fitting
analysis of photometry from the first-epoch HST imaging was recently presented
by Cadelano et al. (2020), who found an age of 13.0$\pm$0.6 using presumably
identical Victoria-Regina isochrones but a different (maximum likelihood)
fitting technique, in excellent agreement with our age of 12.9$\pm$1.0 Gyr
(Table 3). Meanwhile, NGC 6342 is included in the relative age study by De
Angeli et al. (2005), who find a young relative age, consistent with our
results.

Figure 4: Left:  Age-metallicity relation from V13 supplemented with metal-
poor clusters from Forbes & Bridges (2010) following Massari et al. (2019)
(shown in grey), where horizontal error bars are [Fe/H] uncertainties from
C09. Our target clusters are overplotted in red and labeled with their NGC
number. Right: Same, but now literature clusters are color-coded according to
their Galactic component from Massari et al. (2019), and our target clusters
are overplotted using large grey diamonds.

Lastly, the location of NGC 6380 in the age-metallicity relation is, thus far,
unique. One possibility is that our assumed metallicity of [Fe/H]$\sim$$-$0.7
is erroneous, since literature estimates of the cluster metallicity vary: CaII
triplet calibrations based on near-IR photometry (Mauro et al., 2014) as well
as near-IR photometric indices (V10) give values of [Fe/H]$\sim$$-$0.7 to
$-$0.8, while earlier spectroscopic indices indicated a higher value of
[Fe/H]$\sim$$-$0.4 (C09). However, during the peer review process, elemental
abundances based on high-resolution near-IR spectroscopy of 12 RGB members
were published by Fernández-Trincado et al. (2021), who report
[Fe/H]=$-$0.73$\pm$0.06, consistent with other recent measurements, and they
also confirm that this cluster is enhanced in $\alpha$ elements. Nevertheless,
as a check on this result, we use three tests:

  1. 1.

We have remeasured the near-IR slope of the upper RGB (Cohen et al., 2017, see
their sect. 4.4) from the same 2MASS-calibrated VVV PSF photometry used in
that study, but after applying our differential reddening corrections and
selecting members from Gaia Collaboration et al. (2018), finding a slope of
$-$0.102$\pm$0.003, implying [Fe/H]=$-$0.54$\pm$0.09, although the
slope-[Fe/H] calibration has a scatter of 0.2 dex.

  2. 2.

We measured the difference between the F814W magnitude of the red horizontal
branch (RHB) and the magnitude of the red giant branch bump (RGBB), using a
maximum likelihood fit of an exponential plus single or double Gaussian to the
RGB (e.g. Nataf et al., 2013; Cohen et al., 2018). For NGC 6380, we find
$\Delta I^{\rm RHB}_{\rm RGBB}\approx\Delta F814W^{\rm RHB}_{\rm RGBB}$ =
0.291$\pm$0.021, resulting in [M/H]=$-$0.44$\pm$0.02 for NGC 6380
(corresponding to [Fe/H]=$-$0.73 assuming [$\alpha$/Fe]=+0.4) using eq. 3 of
Nataf et al. (2013), although they estimate their calibration between $\Delta
I^{\rm RHB}_{\rm RGBB}$ and [Fe/H] to have an intrinsic scatter of $\sim$0.05
dex, which they suggest could be due to MWGC age variations.

  3. 3.

We reperformed our relative age dating procedure for NGC 6380 assuming the
higher value of [Fe/H]=$-$0.4 given by C09, finding an age well over the age
of the Universe, although when assuming [Fe/H]=$-$1.0 for NGC 6380 we find an
age of 13.0 Gyr, essentially identical to the result plotted in Fig. 4. This
lower value of [Fe/H] (or a somewhat younger age, by $\sim$0.5 Gyr) would
bring this cluster well within $\sim$1$\sigma$ consistency with the in situ
branch of the MWGC age-metallicity relation, but is at odds with the
photometric metallicity indicators above.

One intriguing possibility to explain the location of NGC 6380 in the age-
metallicity plane could be elemental abundance variations that could mimic an
older age (see Sect. 4.3), but this has been largely ruled out for several
potential culprits (O, Mg, Si) by Fernández-Trincado et al. (2021). However,
Fernández-Trincado et al. (2021) also point out the unusually large extent of
N enhancement in this cluster as well as potential correlations of Ce with N
and Al. High resolution optical spectroscopy and/or blue-ultraviolet
photometry could be particularly beneficial in this case to characterize the
multiple stellar populations in this cluster and constrain any potential
helium enhancement (e.g. Milone et al., 2018).

###  4.2 The Impact of $R_{V}$

We explore the impact of varying the total-to-selective extinction ratio
$R_{V}$ by calculating an alternate set of bolometric corrections assuming
$R_{V}$=2.5 rather than the classical value of $R_{V}$=3.1. Extinction towards
the Galactic bulge is known to be non-standard (see Nataf 2016 for a review),
and such low values of $R_{V}$ have been measured towards the bulge (Sumi,
2004; Nataf et al., 2013), although the parametrization of the extinction law
is still a topic of some debate (Nataf et al., 2016; Schlafly et al., 2016).
Hence, our intention is not to advocate for a particular $R_{V}$ value, but
rather to test if and how a substantial ($\sim$20%) $R_{V}$ variation impacts
derived cluster relative ages.

We assess the impact of a change in $R_{V}$ by examining two hypothetical
MWGCs, each with true values of $E(B-V)$=1.5 and $R_{V}$=2.5, and with [Fe/H]
values of -0.5 and -1.5 (roughly spanning the range of our target clusters).
In each case, we examine isochrones with ages from 10-14 Gyr in steps of 1
Gyr, and register the isochrones following the usual prescription (Sect. 3.3).
The registered isochrones corresponding to the correct values of $R_{V}$ and
$E(B-V)$ are shown in red in Fig. 5, and we now test the impact of the
following two erroneous assumptions:

Figure 5: The impact of fixed bolometric corrections or an incorrect value of
$R_{V}$ on relative ages, assessed using a hypothetical cluster with true
$R_{V}$=2.5, $E(B-V)$=1.5 (at the high end of plausible values for our target
clusters; Table 3), and [Fe/H]=-1.5 (left panel) or [Fe/H]=-0.5 (right panel),
roughly bracketing the range spanned by our target clusters. In each panel,
isochrones from 10-14 Gyr in steps of 1 Gyr are registered following the
prescription in Sect. 3. Isochrones that assume the correct value of
$R_{V}$=2.5 and $E(B-V)$=1.5 are shown in red, isochrones that incorrectly
assume $R_{V}$=3.1 are shown in blue, and isochrones that assume a single,
fixed bolometric correction calculated for zero extinction are shown in black.
In the inset in each panel, we show the gradient of RGB relative color,
measured at a magnitide of (F606W-F606WTO)=$-$2.8 (shown using a grey
horizontal line) versus age.

First, isochrones that incorrectly assume $R_{V}$=3.1 are shown in blue. Note
that by matching the observed MSTO color of the $R_{V}$=2.5 isochrones, these
isochrones correspond to a different value of $E(B-V)$ because of a different
assumed $R_{V}$. Second, isochrones assuming a single fixed zero-extinction
value for the bolometric correction (i.e. not varying as a function of $T_{\rm
eff}$, log g, or total extinction) are shown in black. For each isochrone,
following the usual relative age dating procedure, we record the color of the
RGB relative to the MSTO at a magnitude of (F606W-F606WTOC+0.05)=$-$2.8, shown
as a dashed grey horizontal line. To more clearly illustrate the dependence of
this color (as the relative age indicator) on the above assumptions, the inset
in each panel shows the gradient of RGB relative color with age, with
individual values plotted as filled circles and linear fits as straight lines,
revealing the following:

  * •

If a target cluster is at substantially higher $E(B-V)$ than the reference
cluster being used, the assumption of fixed bolometric corrections via simply
applying horizontal and vertical shifts in their CMDs would result in target
cluster ages that are systematically much too old, by $>$2 Gyr or more
depending on metallicity for $E(B-V)$$\sim$1.5. This effect becomes worse at
higher metallicities and higher extinction.

  * •

When correctly accounting for the variation of bolometric corrections with
stellar parameters and total extinction, an incorrect assumption of $R_{V}$ at
the level of $\sim$20% (assuming $R_{V}$=3.1 for a true $R_{V}$=2.5) has only
a minor ($<$3%) impact on the derived ages, resulting in ages which are
slightly too young when assuming a value of $R_{V}$ that is too high. This
effect is essentially insignificant for more metal-poor clusters ($<$1% at
[Fe/H]=-1.5 for ages $>$10 Gyr) due to their shallower gradient of relative
RGB color versus age. Accordingly:

  * •

Although the values of relative ages are drastically misestimated when using a
single, fixed bolometric correction in each bandpass, the gradient of age as a
function of RGB relative color, shown in the insets of Fig. 5, is quite
insensitive to varying assumptions on $E(B-V)$ or $R_{V}$. Rather, this
gradient depends primarily on metallicity, becoming steeper for more metal-
rich clusters (as pointed out by V13) and thus worsening the systematic impact
of incorrect assumptions on $R_{V}$ and/or $E(B-V)$.

The above results all assume a fixed parametrization of the extinction law
given by Cardelli et al. (1989) and O’Donnell (1994), and further assume that
the single parameter $R_{V}$ is capable of capturing its variation. However,
towards the inner Milky Way there is evidence for both a failure of this
assumed paramerization as well a multi-parameter dependence of the extinction
law (Nataf et al., 2016). While beyond the scope of this study, mono-distance
and mono-metallicity subsamples such as our proper-motion-selected MWGC
members may be ideal to further characterize extinction towards the inner
Milky Way and its variation via well-calibrated photometry over a broad range
of wavelengths.

###  4.3 Helium and Other Chemical Abundance Variations

The relative age dating technique we use here is sensitive to variations in
elements other than iron, including He, $\alpha$\- and light elements such as
O, Mg and Si (see e.g. fig. 17 of V13), between any target and reference
cluster. Regarding helium, there are two reasons why a substantial global
enhancement is unlikely. First, we already find relatively old ages ($\sim$13
Gyr to within uncertainties) for all but the young, metal-rich cluster NGC
6342 (which is spectroscopically verified to be a chemical twin of its
reference cluster) such that any global He enhancement is restricted to fairly
small values ($\Delta$Y$\lesssim$0.04 dex) in order to yield ages younger than
the age of the universe. Second, and relatedly, V13 point out that increasing
the global value of He (unlike some other elements) changes the SGB slope, and
we found no evidence for a difference in SGB slopes between our target
clusters and their respective comparison clusters777Importantly, this
comparison was made after correcting the comparison cluster isochrones to the
color excess values of our target clusters using our fully self-consistent
bolometric corrections, which does affect the shape of the isochrones,
including the SGB.. Moving on from a global enhancement in He, a spread in He
is a common feature of MWGCs. Our target clusters have estimated masses in the
range 4.5$\lesssim$Log M/M☉$\lesssim$5.5 (Baumgardt et al., 2019),
corresponding to He spreads of $\Delta$Y$\lesssim$0.03 dex (Milone et al.,
2018, see their fig. 13). Considering the observational uncertainties on our
photometry and differential reddening corrections, our data are generally
insufficient to disentangle such a modest He spread, which in any case would
not shift our age estimates beyond their uncertainties.

Turning to elements other than He, V13 demonstrate a systematic impact of $>$1
Gyr on relative ages for significant ($\sim$0.4 dex) differences in some light
and $\alpha$ elements. For this reason, we have intentionally chosen
comparison clusters with [$\alpha$/Fe] values similar to those seen for bulge
globulars, and these values are consistent with values of [$\alpha$/Fe] and
[Mg/Fe] found from low-resolution spectroscopy for five of our nine target
clusters (D16). However, more generally, there is also no observational
evidence that our target clusters would show noteworthy differences in
$\alpha$ or light elements with respect to their comparison clusters. In
addition to the minority (three of nine) of our target clusters that have some
of these abundance ratios available from high-resolution optical spectroscopy
(Table 1), a more general comparison of MWGC elemental abundance ratios over a
broad metallicity baseline (Carretta et al., 2009b, c) does not show a
systematic offset from those seen for bulge globulars (Johnson et al., 2016;
Puls et al., 2018) with respect to these elements. The only potential
exception is Si, for which two metal-intermediate inner Milky Way clusters
(NGC 6522 and NGC 6558) show atypically low values close to [Si/Fe]$\sim$0
(Barbuy et al., 2018) that could mimic an older age (by $\lesssim$1 Gyr).
However, without high-quality spectroscopic data for the majority of our
target clusters, we cannot comment further on a case-by-case basis, and self-
consistent multi-element abundance ratios are indeed critical to place
stringent constraints on MWGC ages.

##  5 Summary and Conclusions

We have derived relative ages for nine inner Milky Way globular clusters,
minimizing systematics by using the same filters, evolutionary models,
bolometric corrections, and relative age measurement technique as V13. Two
independent tests of our methodology indicate that the ages we measure using a
classical relative age-dating technique should be directly comparable to those
of V13 at the level of 0.3 Gyr or better. We find that NGC 6342 is a typical
young ($\sim$11.5 Gyr) metal-rich cluster formed in situ, while the remainder
of our sample is uniformly very old, with a mean age of 12.9$\pm$0.4 Gyr
assuming the V13 MWGC age scale. The location of these eight MWGCs in the MWGC
age-metallicity plane suggests that most or all of them formed in situ,
although more secure kinematics-based classifications require distances to be
reliably determined from multi-wavelength measurements of interstellar
extinction towards the target clusters and/or direct measurement of parallaxes
from future astrometric missions. We have explored the impact of both assuming
a fixed zero-extinction bolometric correction, which drastically shifts ages
to higher values, as well as assuming an incorrect value of $R_{V}$, which has
a small ($<$3%) systematic effect on MWGC ages over the range of metallicity
(-1.5$\lesssim$[Fe/H]$\lesssim$-0.5) and color excess ($E(B-V)$$<$1.5) we
explore. Variations in elemental abundances other than [Fe/H] could be
responsible for the unusual location of NGC 6380 in the MWGC age-metallicity
plane, and such measurements from high-resolution spectroscopy on an
internally consistent scale could pose more stringent constraints on MWGC
relative ages.

Support for program HST GO-15065 was provided by NASA through a grant from the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.
LC acknowledges support from the Australian Research Council Future Fellowship
FT160100402.

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# Reentrant phase transition of Born-Infeld-AdS black holes

Amin Dehyadegari  Physics Department and Biruni Observatory, Shiraz
University, Shiraz 71454, Iran  Ahmad Sheykhi<EMAIL_ADDRESS>Physics
Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box
55134-441, Maragha, Iran

###### Abstract

We investigate thermodynamic phase structure and critical behaviour of Born-
Infeld (BI) black holes in an anti-de Sitter (AdS) space, where the charge of
the system can vary and the cosmological constant (pressure) is fixed. We find
that the BI parameter crucially affects the temperature of the black hole when
the horizon radius, $r_{+}$, is small. We observe that depending on the value
of the nonlinear parameter, $\beta$, BI-AdS black hole may be identified as RN
black hole for $Q\geq Q_{m}$, and Schwarzschild-like black hole for $Q<Q_{m}$,
where $Q_{m}=1/\left(8\pi\beta\right)$ is the marginal charge. We analytically
calculate the critical point ($Q_{c},T_{c},r_{+c}$) by solving the cubic
equation and study the critical behaviour of the system. We also explore the
behavior of Gibbs free energy for BI-AdS black hole. We find out that the
phase behaviour of BI-AdS black hole depends on the charge $Q$. For $Q>Q_{c}$,
the Gibbs free energy is single valued and the system is locally stable
($C_{Q}>0$), while for $Q<Q_{c}$, it becomes multivalued and $C_{Q}<0$. In the
range of $Q_{z}<Q<Q_{c}$, a first order phase transition occurs between small
black hole (SBH) and large black hole (LBH). Interestingly enough, in the
range of $Q_{t}\leq Q\leq Q_{z}$, a reentrant phase transition occurs between
intermediate (large) black hole, SBH and LBH in Schwarzschild-type region.
This means that in addition to the first order phase transition which
separates SBH and LBH, a finite jump in Gibbs free energy leads to a zeroth
order phase transition between SBH and intermediate black hole (LBH) where
initiates from $Q=Q_{z}$ and terminates at $Q=Q_{t}$.

###### pacs:

04.70.-s, 05.70.Ce, 04.70.Dy, 04.60.-m

##  I Introduction

Phase transition is a most remarkable property in black hole thermodynamics
which can relate a gravitational system to an accessible experimental system.
The pioneering work on black holes phase transition was done by Hawking and
Page who proved that a first-order phase transition occurs between thermal
radiation and large black hole in the Schwarzschild anti-de-Sitter (AdS)
spacetime HP . After that, Witten showed that this phase transition can be
related to a confinement-deconfinement phase transition in the quark gluon
plasma Witten . Later, it has been claimed that similar to the Van der Waals
phase transition, a charged AdS black hole undergoes a first-order phase
transition between large black hole (LBH) and small black hole (SBH) VDW1 .
While similarity between liquid-gas Van der Waals phase transition and small-
large black hole phase transition in the background of Reissner-Nordstrom-AdS
(RN-AdS) spacetime was provided by extending phase space of black hole
thermodynamics Dolan ; PV . In an extended phase space, the variable
cosmological constant is treated as the pressure, while its conjugate quantity
is the volume of the system. In this approach the mass of black hole is
treated as the enthalpy enthalpy . Critical behaviour of black holes in an
extended phase space have been explored in various setups variousBH ; GBC ;
Hendi ; Sherkat ; Zou ; Kamrani ; Dayyani1 ; Dayyani2 ; Majhi . Recently, this
approach has been utilized to present the emergence of superfluid-like phase
transition for a class of AdS hairy black hole in Lovelock gravity
superfluidBH . Very recently, we disclosed a novel phase transition in dilaton
black holes, in an extended phase space where the charge of the black hole is
regarded as a fixed quantity NAAA .

The reentrant phase transition (RPT) is composed of two (or more) phase
transition, which occurs by a monotonic variation of any thermodynamic
variable so that the initial and final phase of system are macroscopically the
same. This has previously been observed in a nicotine/water mixture Nwater ,
granular superconductors, liquid crystals, binary gases, ferroelectrics and
gels (see RPTreview  and the references therein). Recently, RPT has been
discovered in the four-dimensional BI-AdS black hole and higher-dimensional
singly spinning Kerr-AdS BIRPT ; KerrRPT . These black holes, in certain range
of pressure, undergo a large-small-large black hole phase transition where the
latter “large” state refers to the intermediate black hole (IBH). This RPT is
accompanied by a finite jump in the Gibbs free energy which referred to a
zeroth-order phase transition. The zeroth-order phase transition has been seen
in superfluidity and superconductivity maslov  and recently in the charged
dilaton black hole NAAA . The studies on RPT in an extended phase space for
different black holes have been carried out in MRPT ; quasiRPT ; hgRPT ;
hairyRPT ; massiveRPT ; dSRPT ; DRPT . Microscopic origin of black hole
reentrant phase transitions has been explored in MicroscopicRPT . It was
argued that the type of interaction between small black holes near large/small
transition line, differs for usual and reentrant phase transitions. Indeed,
for usual case, the dominant interaction is repulsive whereas for reentrant
case one may encounter with an attractive interaction MicroscopicRPT . It was
also shown that in reentrant phase transition case, the small black holes
behave like a Bosonic gas whereas in the usual phase transition case, they
behave like a quantum anyon gas MicroscopicRPT . It is worth noting that in
all references mentioned above, the charge of black hole, $Q$, is regarded as
a fixed parameter and the cosmological constant (pressure) is treated as
thermodynamic quantity which can vary.

In a quite different viewpoint, it was argued that one can think of variation
of charge, $Q$, of a black hole and keep the cosmological constant as a fixed
parameter AAA . According to this perspective, the critical behavior occurs in
$Q^{2}$-$\Psi$ plane, where $\Psi=1/2r_{+}$ is the conjugate of $Q^{2}$ AAA .
In this approach, the thermodynamic process is carried out in phase space
where the black hole quantities are varied in a fixed AdS background geometry.
Treating the charge of black hole as a natural external variable affects the
thermodynamic behaviors of gravitational system e.g., it can lead to
interesting critical phenomena AAA . This alternative viewpoint for
description of phase transition and critical behavior, relevant response
function clearly signifies stable and unstable regime. In addition, it leads
to small-large black hole phase transition for charged AdS black holes, and
completes the analogy with van der Waals fluid system AAA .

In this paper, following AAA , we investigate thermodynamic phase structure
and critical behaviour of BI-AdS black holes in a phase space, where the
cosmological constant (pressure) is taken to be constant. It was argued that
the $(3+1)$-dimensional BI-AdS black hole implies a peculiar phase transition
in an extended phase space such as a reentrant phase transition BIRPT . As we
shall see the BI parameter crucially affects the temperature of the black hole
when the horizon radius is small. We shall analytically calculate the critical
point by solving the cubic equation which is missed in the previous studies
(see e.g., Zeng ). Also, we observe the interesting reentrant phase transition
that occurs for certain range of charge of black hole. Our work differs from
previous studies on phase transition in BI-AdS black hole Zeng ; NORPT1 ;
NORPT2 ; NORPT3 ; NORPT4 ; NORPT5 ; NORPT6 ; Dey , in which they do not study
the reentrant phase transition of BI-AdS black holes, while in the present
work we focus on the reentrant phase transition of such system. The
investigation on phase transition of BI-AdS black holes has also been
performed at the fixed electric potential (the grand canonical ensemble) in
Ref. Fernando1 .

The framework of our paper is as follows. In section II, we review the basic
thermodynamic of BI-AdS black hole in four dimensional spacetime. In section
III, we study the critical behavior of BI-AdS black hole in $T-r_{+}$ plane
with fixed cosmological constant. In section IV, we investigate the Gibbs free
energy in order to find the phase transition in the system. We summarize our
results in section V.

##  II Review on BI-AdS black hole

The well known action of four-dimensional Einstein gravity in the presence of
BI nonlinear electrodynamics is Borninfeld

$\mathcal{S}=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\left[\mathcal{R}-2\Lambda+4\beta^{2}\left(1-\sqrt{1+\frac{F_{\mu\nu}F^{\mu\nu}}{2\beta^{2}}}\right)\right],$
(1)

where $\mathcal{R}$ is the Ricci scalar curvature, $\Lambda$ is the
cosmological constant which relates to AdS radius as $\Lambda=-3/L^{2}$ and
the constant $\beta$ is the BI parameter with the dimension of mass that
relates to the string tension as $\beta=1/(2\pi\alpha^{{}^{\prime}})$ ST .
Here, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic tensor field where $A_{\mu}$ is the vector potential. The
static and spherically symmetric metric of spacetime is

$\mathrm{d}s^{2}=-f\left(r\right)\mathrm{d}t^{2}+\frac{1}{f(r)}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},$
(2)

in which, $\mathrm{d}\Omega$ is the metric of an unit $2$-sphere with volume
$\omega=4\pi$ and the metric function $f(r)$ is given by Dey ; Fernando2 ;
Cai1

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1+\frac{r^{2}}{L^{2}}-\frac{m}{r}+\frac{2\beta^{2}r^{2}}{3}\left(1-\sqrt{1+\frac{16\pi^{2}Q^{2}}{\beta^{2}r^{4}}}\right)$
(3) $\displaystyle+\frac{64\pi^{2}Q^{2}}{3r^{2}}\
_{2}\mathcal{F}_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{16\pi^{2}Q^{2}}{\beta^{2}r^{4}}\right],$

where ${}_{2}\mathcal{F}_{1}(a,b,c,z)$ is the hypergeometric function, and $m$
is an integration constant which is related to the mass of the black hole via
$M=m/8\pi$, and $Q$ is the electric charge of the black hole per unit volume
$\omega$. The non-zero component of the gauge potential can be written as

$A_{t}(r)=-\frac{4\pi Q}{r}\
_{2}\mathcal{F}_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{16\pi^{2}Q^{2}}{\beta^{2}r^{4}}\right].$
(4)

In the limit $\beta\rightarrow\infty$, the metric function (3) and the gauge
potential (4) reduce to ‘Reissner-Nordstrom’ (RN)-AdS black holes PV ; AAA .
The Hawking temperature of BI-AdS black hole is given by

$T=\frac{1}{4\pi r_{+}}+\frac{3r_{+}}{4\pi
L^{2}}+\frac{\beta^{2}r_{+}}{2\pi}\left(1-\sqrt{1+\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{4}}}\right),$
(5)

where for large $\beta$, it can be expanded as

$T=\frac{1}{4\pi
r_{+}}\left(1+\frac{3r_{+}^{2}}{L^{2}}-\frac{16\pi^{2}Q^{2}}{r_{+}^{2}}\right)+\frac{16\pi^{3}Q^{4}}{r_{+}^{7}\beta^{2}}+\mathcal{O}\left(\frac{1}{\beta^{4}}\right),$
(6)

where, $r_{+}$ is the event horizon which is defined by the largest positive
real root of $f(r_{+})=0$. Note that the first term in the right hand side of
the above expression is the RN-AdS black hole temperature and the next term is
the leading order BI correction to the temperature. The electric potential at
infinity with respect to the event horizon is given by

$U=-\frac{4\pi Q}{r_{+}}\
_{2}\mathcal{F}_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{4}}\right].$
(7)

The other thermodynamic quantities associated with the Born-Infeld black hole
are BIRPT

$\displaystyle S=\frac{r_{+}^{2}}{4},\quad
P=-\frac{\Lambda}{8\pi}=\frac{3}{8\pi L^{2}},\quad V=\frac{r_{+}^{3}}{3},$ (8)
$\displaystyle\mathfrak{B}$ $\displaystyle=$ $\displaystyle\frac{\beta
r_{+}^{3}}{6\pi}\left(1-\sqrt{1+\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{4}}}\right)+\frac{4\pi
Q^{2}}{3\beta r_{+}}$ (9)
$\displaystyle\times_{2}\mathcal{F}_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{4}}\right],$

where $S$, $P$ and $V$ are entropy, pressure and volume, respectively. Also,
$\mathfrak{B}$ is a quantity conjugate to $\beta$ that is interpreted as the
BI vacuum polarization BIRPT . It is easy to show that the thermodynamic
quantities of BI black hole satisfy the first law of thermodynamic and Smarr
formula, respectively, BIRPT

$dM=TdS+UdQ+VdP+\mathfrak{B}d\beta,$ (10) $M=2TS+UQ-2VP-\mathfrak{B}\beta.$
(11)

It is notable that mass ($M$), charge ($Q$), entropy ($S$) and volume ($V$)
are written per unit volume $\omega$. In the next section, we study the
critical behavior of BI-AdS black hole while the cosmological constant is
fixed and the electric charge of black hole can vary.

##  III Critical behavior of BI-AdS black hole

In this section, we investigate the critical behavior of BI-AdS black hole in
which the charge of black hole can vary whereas the cosmological constant is a
fixed parameter. In order to do this, we can study the behavior of the
specific heat at constant charge

$C_{Q}=T\left(\frac{dS}{dT}\right)_{Q},$ (12)

where, the positive (negative) sign of this quantity indicates the local
thermodynamic stability (instability). Note that Eq. (12) is also calculated
at fixed $L$ (i.e. cosmological constant) and $\beta$. To see the behavior of
$C_{Q}$, we plot the curves of the temperature versus the event horizon for
different values of the charge in Fig. 1. As one can see, for small $r_{+}$,
the behavior of the temperature strongly depends on the charge of the black
holes. Hence, we expand the Hawking temperature for small $r_{+}$ as

$T=\frac{2\beta}{r_{+}}\left(Q_{m}-Q\right)+\frac{r_{+}\left(3+2L^{2}\beta^{2}\right)}{4\pi
L^{2}}-\frac{\beta^{3}r_{+}^{3}}{16\pi^{2}Q}+\mathcal{O}\left(r_{+}^{4}\right),$
(13)

where $Q_{m}=1/\left(8\pi\beta\right)$ is the ‘marginal charge’. Also, the
large $r_{+}$ limit of the temperature is $3r_{+}/\left(4\pi L^{2}\right)$
which is independent of the charge. Hence for large $r_{+}$, the temperature
increases linearly with increasing $r_{+}$. Depending on the value of $\beta$,
BI-AdS black hole may be identified as follows: For $Q\geq Q_{m}$, black hole
is RN type. In this case, we have an extremal black hole because temperature
crosses over from zero with decreasing $r_{+}$. For $Q<Q_{m}$, black hole is
‘Schwarzschild-like’ (S) type. Similar to Schwarzschild solution, black hole
does not exist in the region of low temperature. According to Fig.1, the
rightmost (leftmost) branch of isocharge is related to large black hole which
is locally stable (unstable), i.e. positive (negative) sign of specific heat
at constant charge.

Figure 1: $T-r_{+}$ diagram of BI-AdS black hole. This figure shows the
different behavior depending on the charge of black hole. A negative slope
appears for $Q<Q_{c}$ which indicates the local instability ($C_{Q}<0$). The
critical point is indicated by a black spot. We have set $L=1$ and
$\beta=3.5$.

Now, we turn to obtain the critical point (where continuous phase transition
occurs) with respect to the above mentioned for BI-AdS black hole. In Fig. 1,
isocharge diagrams show that, for constant $L$ and $Q=Q_{c}$, the critical
point is an inflection point which can be characterized by

$\frac{\partial T}{\partial
r_{+}}\Big{|}_{Q_{c}}=0,\quad\quad\quad\frac{\partial^{2}T}{\partial
r_{+}^{2}}\Big{|}_{Q_{c}}=0.$ (14)

Note that for S-type black hole, the critical point must be occurred in the
right branch of $Q_{c}$ which is locally stable. Using the temperature formula
in Eq.(5), the expressions of Eq. (14) are, respectively, written as

$\displaystyle\frac{1}{x}-2x+\left(1+\frac{3}{2\beta^{2}L^{2}}-\frac{1}{2\beta^{2}r_{+c}^{2}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (15) $\displaystyle
x^{3}-\frac{x}{2}-\frac{1}{2x}+\frac{1}{4\beta^{2}r_{+c}^{2}}$
$\displaystyle=$ $\displaystyle 0,$ (16)

where

$x=\left(\sqrt{1+\frac{16\pi^{2}Q_{c}^{2}}{\beta^{2}r_{+c}^{4}}}\right)^{-1}.$
(17)

Since the critical quantities are real positive values, we should have the
following constraint on $x$,

$0\leq x\leq 1.$ (18)

Combining Eqs. (15) and (16) gives the cubic equation

$x^{3}+px+q=0,$ (19)

where

$p=-\frac{3}{2},\quad q=\frac{1}{2}\left(1+\frac{3}{2\beta^{2}L^{2}}\right).$
(20)

Due to fact that $q$ is real and $p<0$, the cubic equation has three or one
real roots. Three roots exist if $4p^{3}+27q^{2}\leq 0$, i.e.

$\beta\geq\beta_{0}=\sqrt{\frac{3}{2}\left(1+\sqrt{2}\right)}/L\approx
1.9098/L.$ (21)

In this case, the roots are written as

$\displaystyle
x_{k}=\sqrt{2}\mathrm{cos}\left(\frac{1}{3}\mathrm{arccos}\left[-\frac{\sqrt{2}}{2}\left(1+\frac{3}{2\beta^{2}L^{2}}\right)\right]-\frac{2\pi
k}{3}\right),$ $\displaystyle k=0,1,2.$ (22)

One can easily verify that the constraint in Eq.(18) is violated by $x_{2}$.
Also, by calculating the critical quantities from $x_{1}$, we find that
$r_{+c}$ occurs in the unstable branch of critical isocharge for S-type black
hole. For $\beta<\beta_{0}$ case, one root is given by

$x_{3}=-\sqrt{2}\mathrm{cosh}\left(\frac{1}{3}\mathrm{arccosh}\left[-\frac{\sqrt{2}}{2}\left(1+\frac{3}{2\beta^{2}L^{2}}\right)\right]\right),$
(23)

which violates the condition in Eq. (18). Consequently, the critical behavior
of BI-AdS black hole can be observed only for $\beta\geq\beta_{0}$. With
$x_{0}$ at hand, the critical quantities are obtained as

$\displaystyle
r_{+c}^{-2}=2\beta^{2}\left(\frac{1}{x_{0}}+x_{0}-2x_{0}^{3}\right),$ (24)
$\displaystyle
Q_{c}^{-2}=64\pi^{2}\beta^{2}\left(1-x_{0}^{2}\right)\left(1+2x_{0}^{2}\right)^{2},$
(25) $\displaystyle
T_{c}=\frac{\sqrt{x_{0}}\left[3+2L^{2}\beta^{2}\left(1-x_{0}\right)\left(1+2x_{0}+2x_{0}^{2}\right)\right]}{4\pi
L^{2}\beta\sqrt{2\left(1-x_{0}^{2}\right)\left(1+2x_{0}^{2}\right)}}$ (26)

The critical charge is larger than $Q_{m}$ when

$\beta>\beta_{1}=\sqrt{\frac{3}{2(\sqrt{6\sqrt{3}-9}-1)}}/L\approx 2.8871/L.$
(27)

Thus for $\beta\geq\beta_{1}$, the critical behavior takes place in the RN-
type of black hole. On the other hand, for $\beta_{0}\leq\beta<\beta_{1}$, the
critical point occurs for S-type black hole at the right branch of $Q=Q_{c}$.

Expanding the critical quantities of BI-AdS black hole for large $\beta$ (RN-
type), lead to

$\displaystyle r_{+c}$ $\displaystyle=$
$\displaystyle\frac{L}{\sqrt{6}}-\frac{7}{24\sqrt{6}\beta^{2}}+\mathcal{O}\left(\frac{1}{\beta^{4}}\right),$
$\displaystyle Q_{c}$ $\displaystyle=$
$\displaystyle\frac{L}{24\pi}+\frac{7}{576\pi
L\beta^{2}}+\mathcal{O}\left(\frac{1}{\beta^{4}}\right),$ $\displaystyle
T_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{L\pi}\sqrt{\frac{2}{3}}-\frac{1}{12\sqrt{6}\pi
L^{3}\beta^{2}}+\mathcal{O}\left(\frac{1}{\beta^{4}}\right).$ (28)

As expected, the first terms on the right-hand side of Eqs.(28) reproduces the
critical values of RN-AdS black hole AAA . The leading BI correction to
critical point is identified by the last terms in above equations. To
determine the possible phase transition in the system, we study the Gibbs free
energy in the next section.

##  IV Gibbs free energy

To find out phase transition and classify its type, we need to explore the
behavior of thermodynamic potential (partition function) of BI-AdS black hole.
This is due to the fact that the thermodynamic potential determines the
globally stable state at equilibrium process. For fixed temperature $T$,
pressure $P$ ($\Lambda$) and charge $Q$, the Gibbs free energy is themodynamic
potential which is computed from the Euclidean action with appropriate
boundary term in the canonical ensemble PV ; BIRPT . In this case, such a
global stable state is corresponding to the lowest Gibbs free energy. Since
the mass of black hole is identified as the enthalpy Eq. (10), one obtains the
Gibbs free energy per unit volume $\omega$ by Legendre transformation BIRPT

$\displaystyle
G\left(T,Q,P\right)=M-TS=\frac{r_{+}}{16\pi}-\frac{r_{+}^{3}}{16\pi
L^{2}}-\frac{r_{+}^{3}\beta^{2}}{24\pi}$
$\displaystyle\times\left(1-\sqrt{1+\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{4}}}\right)+\frac{8\pi
Q^{2}}{3r_{+}}\mathcal{F}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{4}}\right],$
(29)

in which $r_{+}=r_{+}(T,Q,L)$ and $L=L(P)$. In the following, we shall assume
the charge of BI-AdS black hole can vary (parametric changes in the Gibbs free
energy in Eq.(29)), while the pressure do not take different values. It is
worth mentioning that since the pressure and $\beta$ are fixed, one may
consider the above equation as the Helmhotz free energy. However, the results
of this section are not changed.

Figure 2: Gibbs free energy as a function of temperature for different values
of charge and $\beta\geq\beta_{1}$. For $Q\geq Q_{m}$, the phase transition
behavior is reminiscent of RN black hole. Reentrant phase transition
(LBH/SBH/LBH) takes place for $Q_{t}\leq Q\leq Q_{z}$. For $Q<Q_{t}$, the
lower (upper) branch is globally stable (unstable). The positive (negative)
sign of $C_{Q}$ is denoted by the blue solid (dash red) line. Note that curves
are shifted for clarity. We fix $L=1$ and $\beta=3.5$.

Figure 3: The behavior of isocharge $T-r_{+}$ diagram and the corresponding
$G-T$ diagram (inset) of BI-AdS for the case of $L=1$, $Q=0.0101$ and
$\beta=3.5$. As temperature decreases the black hole follows direction of
arrows. The first and zeroth order phase transition are identified by gold and
purple curve, respectively. A large/small/large(intermediate) corresponds to a
reentrant phase transition. The positive (negative) sign of $C_{Q}$ is denoted
by the blue solid (dash red) line.

The behavior of Gibbs free energy for $\beta\approx 3.5$
$\in\left[\beta_{1},\infty\right)$ and $\beta\approx 2.84$
$\in\left[\beta_{0},\beta_{1}\right)$ are illustrated in Figs. 2 and 4,
respectively. As one can see from Figs. 2 and 4, BI-AdS black hole has a
different and complex behavior for various charge $Q$. In Fig. 2, there is a
critical point at $Q=Q_{c}$ in the region of RN-type. For $Q>Q_{c}$, the Gibbs
free energy is single valued and is locally stable ($C_{Q}>0$) every where,
which is shown by the solid blue line. However, for $Q<Q_{c}$, it becomes
multi valued in which negative $C_{Q}$ identified by dash red line. In the
range of $Q_{z}<Q<Q_{c}$, a first order phase transition occurs between small
black hole (SBH) and large black hole (LBH) which is accompanied by a
discontinuity in the slop of $G$ at transition point. Also, for $Q<Q_{m}$,
black hole becomes a S-type where temperature of black hole has a lower bound.
An interesting phenomenon emerges in the range of $Q_{t}<Q<Q_{z}$ where the
reentrant phase transition occurs between intermediate (large) black hole, SBH
and LBH in S-type region. This means that in addition to the first order phase
transition which separates SBH and LBH, a finite jump in Gibbs free energy
leads to a zeroth order phase transition between SBH and intermediate black
hole (LBH) where initiates from $Q=Q_{z}$ and terminates at $Q=Q_{t}$. The
zeroth order phase transition earlier seen in superfluidity and
superconductivity in maslov  and recently has been found in charged dilaton
black hole NAAA . Finally, for $Q<Q_{t}$, only LBH exists that is stable.

Especially, let us consider the case of $Q\approx 0.0101$
$\in\left[Q_{t},Q_{z}\right]$ in Fig. 3. Decreasing the radius of the event
horizon, black hole follows the lower solid blue curve until it joints to the
upper solid blue curve in $G$-$T$ plane (see the inset in Fig. 3). In this
position, black hole enters to the left solid blue curve with a first order
LBH/SBH phase transition which is identified by gold line in $T$-$r_{+}$
plane. If we continue decreasing $r_{+}$, a zeroth order phase transition
occurs between SBH and intermediate black hole (LBH) which is accompanied by a
finite jump in $G$ at the end of solid blue curve. The zeroth order phase
transition is displayed by purple line in Fig. 3. Finally, with decreasing
$r_{+}$, black hole follows the solid blue line until the end. In case of
$\beta\approx 2.84$ $\in\left[\beta_{0},\beta_{1}\right)$, Fig. 4, the
behavior of Gibbs free energy is similar to the mentioned above while all the
phase transition takes place in region $Q<Q_{m}$, i.e. S-type black hole.

Figure 4: Gibbs free energy as a function of temperature for different values
of charge and $\beta$ $\in\left[\beta_{0},\beta_{1}\right)$. For $Q\geq
Q_{m}$, there is no phase transition in the system. Similar to the
schwarzschild-AdS black hole, for $Q_{c}<Q<Q_{m}$, the lower branch (LBH) is
thermodynamically stable, while the upper branch (SBH) is unstable. The first
order phase transition between small and large black holes only occurs in
range of $Q_{z}<Q<Q_{c}$. Same as the Fig.(2) in between $Q_{t}\leq Q\leq
Q_{z}$, there is a reentrant phase transition LBH/SBH/LBH. There is no phase
transition below $Q=Q_{t}$. The positive (negative) sign of $C_{Q}$ is denoted
by the blue solid (dash red) line. Note that curves are shifted for clarity.
We fix $L=1$ and $\beta=2.83$.

Figure 5: Phase diagram corresponding to Fig.(2). Three different transition
are indicated: zeroth-order (purple curve), first-order (gold curve) and a
second-order (black solid circle). The critical point happens in the RN-type
black hole. The inset demonstrates the presence of a large/small/large
(intermediate) black hole reentrant phase transition for a small range of
charge. The marginal charge is shown by black horizontal line. At charges
below the marginal charge, no BH region corresponds to no black hole solution
for region of parameter space.

Figure 6: Phase diagram corresponding to Fig.(4). Three different transition
are indicated: zeroth-order (purple curve), first-order (gold curve) and a
second-order (black solid circle). Phase transition configuration emerges in
the S-type black hole, i.e. below the marginal charge. The inset demonstrates
the presence of a large/small/large (intermediate) black hole reentrant phase
transition for a small range of charge. The marginal charge is shown by black
horizontal line. At charges below the marginal charge, no BH region
corresponds to no black hole solution for region of parameter space.

The corresponding phase diagram of BI-AdS black hole for $\beta\approx 3.5$
$\in\left[\beta_{1},\infty\right)$ and $\beta\approx 2.84$
$\in\left[\beta_{0},\beta_{1}\right)$ are depicted in the $Q$–$T$ diagram in
Figs. 5 and (6), respectively. As is seen from Fig. 5, the critical point is
highlighted by a black spot and the first order phase transition separation
small and large black hole in the region of $Q_{t}<Q<Q_{c}$ which is marked by
the gold curve. For $Q_{t}<Q<Q_{z}$, along with the fist order SBH/LBH, a
zeroth order phase transition separates SBH/LBH (SBH/IBH) which is identified
by purple curve. Hence, we have LBH/SBH/LBH phase transition in the region of
$Q_{t}<Q<Q_{z}$, which is the reentrant phase transition. There is no phase
transition in the charge range $Q>Q_{c}$ as well as $Q<Q_{t}$. Also, no BH
region corresponds to no black hole solution exist. Fig.(6) shows different
phase of BI-AdS black hole for $\beta\approx 2.84$
$\in\left[\beta_{0},\beta_{1}\right)$ which is similar to Fig. 5 while zeroth,
first and second order phase transition occur in S-type of black hole. An
important point can be understood from above discussion: the reentrant phase
transition emerges for the S-type of black hole, i.e. the charge of black hole
is less than marginal charge.

##  V Summary

To summarize, we have investigated charged black holes in the background of
AdS spacetime and in the presence of BI nonlinear electrodynamics. We have
proposed a new viewpoint towards phase space of black holes by assuming that
the cosmological constant, which is regard as the pressure of the system, is a
fixed quantity, while in contrast the charge of the black hole can vary.
Recently, it was argued that such an alternative view towards phase space of
charged AdS black holes can successfully lead to a natural response function
which measures how the size of a black hole ($\Psi\sim r_{+}^{-1}$) changes
with its charge. It was shown that this alternative view more importantly
leads to a natural correspondence of the charged AdS black hole with the Van
der Waals fluid and the associated small-large black hole phase transition
without a need for extended phase space AAA .

First, we explored critical behavior of BI-AdS black holes by studying the
behavior of the specific heat at constant charge, $C_{Q}$, which its sign
indicates the local thermodynamic stability/instability of the system. We find
that for small horizon radius $r_{+}$, the behavior of the temperature
strongly depends on the charge of the black holes. We observed that depending
on the value of the nonlinear parameter, $\beta$, BI-AdS black hole may be
identified as RN black hole for $Q\geq Q_{m}$, and Schwarzschild-like black
hole for $Q<Q_{m}$, where $Q_{m}=1/\left(8\pi\beta\right)$ is the marginal
charge. Besides, similar to Schwarzschild solution, black hole does not exist
in the region of low temperature. We analytically calculated the critical
point ($Q_{c},T_{c},r_{+c}$) by solving the cubic equation and studied the
critical behavior of the system.

We have also explored the behavior of Gibbs free energy for BI-AdS black hole.
We find out that BI-AdS black hole has a different and complex phase behavior
depending on the value of the charge $Q$ of the system. Indeed, there is a
critical point at $Q=Q_{c}$ in the region of RN-type black holes. For
$Q>Q_{c}$, the Gibbs free energy is single valued and the system is locally
stable ($C_{Q}>0$). However, for $Q<Q_{c}$, we have $C_{Q}<0$ and the system
experiences thermally instable phase. We observed that for $Q_{z}<Q<Q_{c}$, a
first order phase transition occurs between SBH and LBH which is accompanied
by a discontinuity in the slop of Gibbs free energy at transition point. An
interesting phenomenon emerges for $Q$ in the range of $Q_{t}<Q<Q_{z}$, where
a reentrant phase transition can be occurred between intermediate (large) BH,
SBH and LBH in Schwarzschild-type region. This implies that in addition to the
first order phase transition which separates SBH and LBH, a finite jump in
Gibbs free energy leads to a zeroth order phase transition between SBH and
intermediate black hole (LBH) where initiates from $Q=Q_{z}$ and terminates at
$Q=Q_{t}$.

Finally, it is important to mention that in the present work we only studied
the phase behavior of BI-AdS black holes. This investigation can also be
extended to other BI-like nonlinear electrodynamics such as Exponential and
Logarithmic nonlinear electrodynamics. Besides, it is worth disclosing the
effects of power parameter of the power-Maxwell electrodynamics on the
reentrant phase transition of black holes in the background of AdS spaces with
variable charge. These issues are now under investigation and the results will
be appeared in the near future.

###### Acknowledgements.

We acknowledge the referees for constructive comments which helped us improve
our paper significantly. We also grateful the Research Council of Shiraz
University. This work has been supported financially by Research Institute for
Astronomy & Astrophysics of Maragha (RIAAM), Iran.

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