# A first Chandra view of the cool core cluster A1668: offset cooling and AGN
feedback cycle

T. Pasini Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029
Hamburg, Germany  M. Gitti Dipartimento di Fisica e Astronomia (DIFA),
Universita‘ di Bologna, via Gobetti 93/2, 40129 Bologna, Italy  Istituto
Nazionale di Astrofisica (INAF) – Istituto di Radioastronomia (IRA), via
Gobetti 101, I-40129 Bologna, Italy  F. Brighenti Dipartimento di Fisica e
Astronomia (DIFA), Universita‘ di Bologna, via Gobetti 93/2, 40129 Bologna,
Italy  E. O’Sullivan Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA02138, USA  F. Gastaldello INAF-IASF Milano, via E.
Bassini 15, I-20133 Milano, Italy  P. Temi Astrophysics Branch, NASA/Ames
Research Center, MS 245-6, Moffett Field, CA 94035  S. L. Hamer Department of
Physics, University of Bath, Claverton Down, BA2 7AY, UK

(Accepted 2021/02/18)

###### Abstract

We present a multi-wavelength analysis of the galaxy cluster A1668, performed
by means of new EVLA and Chandra observations and archival H$\alpha$ data. The
radio images exhibit a small central source ($\sim$14 kpc at 1.4 GHz) with
L${}_{\text{1.4 GHz}}$ $\sim$6 $\cdot$ 1023 W Hz-1. The mean spectral index
between 1.4 GHz and 5 GHz is $\sim$ -1, consistent with the usual indices
found in BCGs. The cooling region extends for 40 kpc, with bolometric X-ray
luminosity L${}_{\text{cool}}=1.9\pm 0.1\cdot$ 1043 erg s-1. We detect an
offset of $\sim$ 6 kpc between the cluster BCG and the X-ray peak, and another
offset of $\sim$ 7.6 kpc between the H$\alpha$ and the X-ray peaks. We discuss
possible causes for these offsets, which suggest that the coolest gas is not
condensing directly from the lowest-entropy gas. In particular, we argue that
the cool ICM was drawn out from the core by sloshing, whereas the H$\alpha$
filaments were pushed aside from the expanding radio galaxy lobes. We detect
two putative X-ray cavities, spatially associated to the west radio lobe
(cavity A) and to the east radio lobe (cavity B). The cavity power and age of
the system are P${}_{\text{cav}}\sim$ 9 $\times$1042 erg s-1 and
t${}_{\text{age}}\sim$5.2 Myr, respectively. Evaluating the position of A1668
in the cooling luminosity-cavity power parameter space, we find that the AGN
energy injection is currently consistent within the scatter of the
relationship, suggesting that offset cooling is likely not breaking the AGN
feedback cycle.

galaxy clusters, AGN, AGN feedback, offset, A1668, cooling flow

††journal: ApJ

##  1 Introduction

In the last two decades, our understanding of the evolution of cool core
galaxy clusters has led to a picture in which the cooling of the Intra-Cluster
Medium (ICM), the cold gas accreting onto the Brightest Cluster Galaxy (BCG),
and the feedback from the central radio source give birth to a tightly-
connected cycle, known as Active Galactic Nuclei (AGN) feedback loop (for
reviews see e.g. McNamara & Nulsen, 2007; Gitti et al., 2012; McNamara &
Nulsen, 2012; Fabian, 2012). Multi-wavelength data provide strong evidences of
this cycle: cavities in the ICM, revealed through deep X-ray observations and
induced by the jets of the central radio galaxy (e.g., McNamara et al., 2000;
Bîrzan et al., 2004; Clarke et al., 2004; Fabian et al., 2006; Gentile et al.,
2007), cold fronts (e.g., Fabian et al., 2006; Markevitch & Vikhlinin, 2007;
Gastaldello et al., 2009; Ghizzardi et al., 2010), optical line emission
(Crawford et al., 1999; McDonald et al., 2010; Hamer et al., 2016) and dust
filaments (Van Dokkum & Franx, 1995; Laine et al., 2003) indicate an extremely
complex and dynamical environment, whose physical processes are still to be
completely understood.
Recently, a number of studies have revealed strong links between the central
BCG, the X-ray core and the cluster dynamics (Sanderson et al., 2009; Hudson
et al., 2010; Rossetti et al., 2016). In particular, spatial offsets between
the BCG, the H$\alpha$ line emission and the X-ray emission peak (e.g. Haarsma
et al., 2010; Hamer et al., 2012, 2016; Barbosa et al., 2018) suggest that ICM
sloshing and offset cooling, together with the AGN, can have a significant
influence on the cluster evolution. Indeed, all these elements affect the
activity of the central Supermassive Black Hole (SMBH) through motions of the
gas, that could be able to regulate the cavity production and, consequently,
the feedback cycle, since the ICM oscillates back and forth with respect to
the central SMBH.
This was recently discussed in Pasini et al. (2019) for the cool core cluster
A2495. Spatial offsets have been observed in this cluster, with the X-ray peak
being separated by $\sim$ 6 kpc from the BCG and $\sim$ 4 kpc from the
H$\alpha$ line emission peak. The analysis presented by the authors on two
putative systems of X-ray cavities, hinted at in the shallow ($\sim 8$ ks)
Chandra observation, suggests that even if cooling is not depositing gas onto
the BCG core, the coupling between the AGN power output and the cooling rate
is still consistent with the observed distribution for cluster samples. In a
forthcoming publication we will present the detailed analysis of the deeper
Chandra observations of A2495, recently allocated ($\sim 130$ ks, P.I.
Gitti111Proposal Number 22800391), which will be key to probe the presence of
two pairs of ICM cavities and test the proposed scenario that the feeding-
feedback cycle is not broken.
A1668 was selected, along with A2495, from the ROSAT Brightest Cluster Sample
(BCS; Ebeling et al. 1998) by choosing objects with X-ray fluxes greater than
10-11 erg cm-2 s-1 and, among these, by selecting those characterized by
logL${}_{\text{H}\alpha}>$ 40 from the catalogue of Crawford et al. (1999). Of
the obtained sample of 13 objects, A2495 and A1668 still lacked Chandra
observations, that were obtained jointly with new VLA data (P.I.
Gitti222Proposal Number 12800143). Pasini et al. (2019) have presented the
results for A2495, making also use of H$\alpha$ line emission data and Hubble
Space Telescope (HST) archival images. In this work we combine the A1668 VLA
and Chandra new observations in order to study the interactions between the
radio source hosted in the BCG and the ICM. As well as for A2495, we included
H$\alpha$ line emission data from Hamer et al. (2016); on the other hand, no
HST data are available for this cluster.
A1668 was previously observed in the radio band by TGSS (TIFR GMRT Sky
Survey), which gives an estimate for the 150 MHz flux density of 1589$\pm$ 159
mJy; Hogan et al. (2015) performed a 5 GHz radio analysis (the data they used
are not the same presented in this work), estimating a flux density of 21.0
$\pm$ 0.1 mJy. A1668 was recently included by Bîrzan et al. (2020) in their
sample of systems observed at 150 MHz by the LOw Frequency ARray (LOFAR, van
Haarlem et al. 2013), showing the presence of large radio lobes, each
extending for more than 50 kpc, and estimating a total flux density of 1.83
$\pm$ 0.44 Jy, consistent with TGSS.

Richness-based estimate of the mass provided values of M${}_{200}\simeq$
1.66$\cdot$10${}^{14}\ \text{M}_{\odot}$ (Andreon, 2016) and M${}_{2500}=$ 3.9
$\pm^{0.8}_{0.7}\cdot$ 10${}^{13}\ \text{M}_{\odot}$ (Pulido et al., 2018).
The cluster’s BCG, IC4130, shows a Star Formation Rate (SFR), estimated from
extinction-corrected H$\alpha$ luminosity obtained from long-slit
observations, of SFR = 2.5 $\pm$ 0.3 $\text{M}_{\odot}$ yr-1 (Pulido et al.,
2018), and extends for $\sim$ 85 kpc (diameter at the isophotal level of 25
mag/arcsec2 in the B-band, Makarov et al. 2014). 333HyperLEDA catalog. Edwards
et al. (2009) also presented IFU observations of the H$\alpha$ emission close
to the BCG, finding a clear velocity gradient from positive values north of
the centre to negative values at the south. They also argued that the line
emitting gas is likely not at rest with respect to the BCG.
In this work, we adopt a $\Lambda$CDM cosmology with H0 = 73 km s-1 Mpc-1,
$\Omega_{M}$= $1-\Omega_{\Lambda}$ = 0.3. The BCG redshift is $z$ = 0.06355
(Hamer et al., 2016) and the luminosity distance is 273.7 Mpc, leading to a
conversion of 1 arcsec = 1.173 kpc.

##  2 Radio analysis

###  2.1 Observations and data reduction

IC4130, the BCG of A1668, was observed with the EVLA on 2011 June 17th in the
1.4 GHz band, and on 2011 March 9th in the 5 GHz band, in A and B
configurations respectively. Details of the observations are shown in Table 1.

Frequency | Number of spw | Channels | Bandwith | Array | Total exposure time
---|---|---|---|---|---
5 GHz (C BAND) | 2 (4832 MHz - 4960 MHz) | 64 | 128 MHz | B | 3h59m21s
1.4 GHz (L BAND) | 2 (1264 MHz - 1392 MHz) | 64 | 128 MHz | A | 2h59m28s
Table 1: Radio observations properties (project code SC0143, P.I. M. Gitti).

The sources J1331+3030 (3C286) and J1327+2210 were used for both the
observations as flux and phase calibrators, respectively. The data reduction
was performed using the NRAO Common Astronomy Software Applications package
(CASA, version 5.3), applying the standard calibration procedure after
carrying out an accurate editing of the visibilities with the CASA task
FLAGDATA. We removed about 6$\%$ of the target visibilities at 5 GHz, whereas
at 1.4 GHz the data were highly contaminated by Radio Frequence Interferences
(RFI), thus producing a visibility loss of $\sim$ 40$\%$.
We applied the standard imaging procedure, making use of the CLEAN task on a
7” $\times$ 7” region centered on the radio source. We took into account the
sky curvature by setting the gridmode=WIDEFIELD parameter and used a two-terms
approximation of the spectral model exploiting the MS-MFFS algorithm (Rau &
Cornwell, 2011).

###  2.2 Results

We produced total intensity radio maps by setting weighting = BRIGGS,
corresponding to ROBUST 0. This baseline weighting provides the best
compromise between angular resolution (determined by long baselines) and
sensitivity to extended emission (provided by short baselines). The
uncertainty on the flux density measurements is 5$\%$, estimated from the
amplitude calibration errors.

Figure 1: 5 GHz VLA map (ROBUST 0) of the radio source hosted in IC4130, the
BCG of A1668. The resolution is 1.14” $\times$ 1.00”, with a rms noise of 6
$\mu$Jy beam-1. Contours are at -3,3,6,12,24,48 $\cdot$ rms. The source flux
density is 19.9 $\pm$ 1.0 mJy. The bottom-left white ellipse represents the
beam.

At 5 GHz (Fig. 1), the radio source exhibits a total flux density of 19.9
$\pm$ 1.0 mJy, consistent with Hogan et al. (2015), that corresponds to a
luminosity of L${}_{\text{5 GHz}}=(1.8\pm 0.1)\cdot 10^{23}$ W Hz-1. The rms
noise is 6 $\mu$Jy beam-1. The source stretches Eastwards for $\sim$ 11 kpc,
with a minor axis of $\sim$ 5.7 kpc. There are no visible hints of larger
emission up to the scale we are sensitive to (60 kpc with the VLA B
configuration at 5 GHz). The equipartition magnetic field was estimated
following the method described in Feretti & Giovannini (2008), finding Heq(5
GHz) = 8.7 $\pm$ 0.1 $\mu$G.

Band | Flux density | rms | beam | Luminosity | Volume | Brightness Temperature | Equipartition Field
---|---|---|---|---|---|---|---
| [mJy] | [$\mu$Jy beam-1] | [arcsec] | [1022 W Hz-1] | [kpc3] | [K] | [$\mu$G]
5 GHz | 19.9 $\pm$ 1.0 | 6 | 1.14x1.00 | 16.8 $\pm$ 0.8 | 185 $\pm$ 22 | 39.6 $\pm$ 10.4 | 8.7 $\pm$ 0.1
1.4 GHz | 70.2 $\pm$ 3.5 | 17 | 1.44x1.08 | 59.1 $\pm$ 2.9 | 359 $\pm$ 30 | 1129.3 $\pm$ 241.2 | 10.3 $\pm$ 0.1
Table 2: Radio properties of A1668 in the two bands observed. The axes of the
radio galaxy are a = 10.9 $\pm$ 1.3, b = 5.7 $\pm$ 1.3 for the 5 GHz map and a
= 14.0 $\pm$ 1.3, b = 7.1 $\pm$ 1.3 for the 1.4 GHz map. The flux density is
estimated within 3$\sigma$ contours, while for the volume we assumed a prolate
elissoid shape. Figure 2: 1.4 GHz VLA map (ROBUST 0) of the radio source
hosted in IC4130. The resolution is 1.44” $\times$ 1.08”, with a rms noise of
17 $\mu$Jy beam-1. Contours are at -3,3,6,12,24,48 $\cdot$ rms. The source
flux density is 70.2 $\pm$ 3.5 mJy. The bottom-left white ellipse represents
the beam.

The 1.4 GHz map (ROBUST 0, Fig. 2) shows no significant differences with
respect to the 5 GHz emission. The source flux density is 70.2 $\pm$ 3.5 mJy
and the rms is $\sim$ 17 $\mu$Jy beam-1. The radio source scale is slightly
larger ($\sim$ 14 kpc for the major axis, $\sim$ 7 kpc for the minor axis),
with a more developed west lobe; again, we did not detect any hint of larger
scale emission up to our sensitivity scale (70 kpc with VLA A configuration at
1.4 GHz). Some cool core clusters show diffuse emission in the form of radio
mini-halos (e.g., Gitti et al., 2004; Govoni et al., 2009; Giacintucci et al.,
2014). Giacintucci et al. (2017) define for mini-halos a minimum radius of 50
kpc since, at smaller radii, diffusion and other transport mechanisms are
plausibly able to spread the relativistic electrons from the central AGN
within their synchrotron radiative cooling time. In Fig. 1 and Fig. 2, the
radio emission is coincident with the optical BCG, and the small scale
suggests that it can all be accounted to the AGN/radio galaxy. It is possible
that diffuse emission larger than our sensitivity scale exists; however, given
the extended double-lobe morphology of the LOFAR 150 MHz image presented in
Bîrzan et al. (2020), the presence of a mini-halo in A1668 looks unlikely. The
equipartition field is Heq(1.4 GHz) = 10.3 $\pm$ 0.1 $\mu$G. Radio properties
can be found in Table 2.
The radio source hosted in the centre of A1668 can be classified as a FRI
galaxy, as demonstrated by both the morphology (asimmetric lobes, no hotspots)
and the 1.4 GHz luminosity (L${}_{\text{1.4 GHz}}=(6.3\pm 0.3)\cdot 10^{23}$ W
Hz-1), that place IC4130 in the 70${}^{\text{th}}$ percentile of the BCG radio
luminosity function presented in Hogan et al. (2015).

####  2.2.1 Spectral index map

The synchrotron spectrum follows a power law $S_{\nu}\propto\nu^{\alpha}$,
where $\alpha$ is the spectral index. The spectral index map (Fig. 3) was
generated using the CASA task IMMATH, combining 1.4 GHz and 5 GHz maps
produced with matched weighting=UNIFORM (to enhance the resolution),
UVRANGE=6.5-152 , and a resolution of 1.4” $\times$ 1.0”. The UVRANGE was set
in order to be sensitive to the same baselines (thus, physical scales) for
both observations.

Figure 3: Spectral index map between 5 GHz and 1.4 GHz of the radio source
hosted in IC4130. Contours are the same as Fig. 2, and typical errors range
from $\Delta\alpha\simeq$ 0.1 for the inner and $\Delta\alpha\simeq$ 0.5 for
the outer regions.

Table 3 lists the peak, the extended and the total radio emission flux
densities at 5 and 1.4 GHz, together with the estimated spectral index between
the two frequencies. The radio core exhibits a flat index ($\alpha\simeq 0$),
as expected from optical thick regions where the radiation is self-absorbed.
Moving towards the outskirt the spectrum becomes steeper, reaching
$\alpha\simeq-2.5$ in the outermost part. The mean index is -0.99$\pm$ 0.06,
consistent with the typical values found in BCGs (Hogan et al., 2015). Table 3
summarizes the spectral index properties.

Region | $S_{C}\pm\Delta S_{C}$ | $S_{L}\pm\Delta S_{L}$ |  $\alpha$ $\pm$ $\Delta\alpha$
---|---|---|---
| [mJy] | [mJy] |
Peak | 7.5 $\pm$ 0.4 | 17.6 $\pm$ 0.9 | -0.67 $\pm$ 0.06
Extended | 12.4$\pm$ 0.6 | 52.6 $\pm$ 2.6 | -1.13 $\pm$0.05
Total | 19.9 $\pm$ 1.0 | 70.2 $\pm$ 3.5 | -0.99 $\pm$ 0.06
Table 3: The first column shows the flux density values at 5 GHz (C band),
while the second displays the 1.4 GHz (L band) values. The third column
presents the corrispondent spectral index values. The extended flux density
was estimated as the difference between the total and the peak fluxes.

##  3 X-ray Analysis

###  3.1 Observation and data reduction

A1668 was observed with the Chandra Advanced CCD Imaging Spectrometer (ACIS),
with the focal point on the S3 CCD, in cycle 12 (ObsID 12877, P.I. Gitti) for
a total exposure of $\sim$ 10 ks. Data were reprocessed with CIAO 4.9
(Fruscione et al., 2006) using CALDB 4.2.1. We ran the Chandra_repro script to
perform the standard calibration process. After background flare removal, we
used the Blanksky template files, filtered and normalized to the count rate of
the source in the hard X-ray band (9-12 keV), in order to subtract the
background. The final exposure time is 9979 s, with roughly $\sim$ 6800 net
counts in a 100$"$ ($\sim$ 120 kpc radius region (0.5-2 keV) centered on the
cluster.
Point sources were identified and removed using the CIAO task WAVDETECT.
Making use of optical catalogues, we found that no astrometry correction was
necessary. Unless otherwise stated, the reported errors are at 68 $\%$
confidence level (1 $\sigma$).

###  3.2 Results

####  3.2.1 Surface Brightness Profile

Figure 4: Chandra image of A1668 in the 0.5-2 keV band, smoothed with a
gaussian filter with a 3 pixel radius.

In Fig. 4 we show the smoothed 0.5-2 keV image of A1668. The ICM exhibits a
roughly circular and regular morphology on large scales ($>$ 30$"$ $\sim$ 35
kpc), while the cluster core shows a region with enhanced emission in the NE-
SW direction. Using the tool SHERPA (Freeman et al., 2001), a surface
brightness profile was produced from a background-subtracted, exposure-
corrected image, making use of 2”-width concentric annuli centered on the
X-ray peak. The profile was then fitted with a single $\beta$-Model (Cavaliere
& Fusco-Femiano, 1976) over the external 30”-100” (35-120 kpc) interval, in
order to exclude the whole core region444The assumption of 30$"$, that was
already justifiable through visual inspection, will be furtherly supported, in
Sec. 3.2.2, by the estimate of the cooling radius. The result of the fit
($\chi^{2}$/DoF $\sim$ 1.71) and its extrapolation to the core region is
represented with the blue line in Fig 5. The best-fit values are: core radius
r0=10.0 $\pm^{0.7}_{0.4}$ arcsec ($\sim$ 11.8 kpc), beta=0.43
$\pm^{0.04}_{0.02}$ and central surface brightness ampl=0.64
$\pm^{0.10}_{0.04}$ counts s-1 cm-2 sr-1.

Figure 5: 0.5-2 keV radial surface brightness profile of A1668. The blue line
represents the single $\beta$-Model fit performed in the external 30”-100”
interval and extrapolated to the center, while the red line is the double
$\beta$-Model fit performed on every radius.

The central brightness excess with respect to the $\beta$-Model is a strong
indication of the presence of a cool core in A1668, as we expected from the
selection criteria described in Sec. 1. This will also be confirmed by the
spectral analysis (see Sec 3.2.2). We therefore fitted the same profile on the
entire radial range with a double $\beta$-Model (Mohr et al., 1999; LaRoque et
al., 2006), represented with the red line in Fig. 5 ($\chi^{2}$/DoF $\sim$
1.49), which provides a better description of the real trend; we found
r01=15.6 $\pm^{4.2}_{2.7}$ arcsec ($\sim$ 18.3 kpc), beta1=0.67
$\pm^{0.14}_{0.08}$ and ampl1= 0.59 $\pm^{0.11}_{0.14}$ counts s-1 cm-2 sr-1
for the first and r02=0.64 $\pm^{0.32}_{0.76}$ arcsec, beta2=0.42
$\pm^{0.02}_{0.01}$ and ampl2=1.78 $\pm^{0.75}_{3.07}$ counts s-1 cm-2 sr-1
for the second $\beta$-Model.

####  3.2.2 Spectral Analysis

Spectra were extracted with the CIAO task specextract in the 0.5-7 keV band;
the extraction was made from a series of concentric rings centered on the
X-ray peak. Each region contains at least $\sim$ 1000 net counts. Background
spectra were also extracted from the Blanksky files of each region. We
individually fitted every spectrum via Xspec (Arnaud, 1996, vv.12.9.1) using a
phabs*apec model, approximating an absorbed, collisionally-ionized diffuse
gas. The redshift was fixed at z=0.06355 and the hydrogen column density was
fixed at NH = 2.20 $\cdot$ 1020 cm-2 (estimated from Kalberla et al. 2005).
The normalization parameter and the temperature kT were left free to vary. The
observation was too shallow to allow us to fit metallicity, which was instead
kept fixed at a value of 0.3 Z⊙555This value was assumed after we tried to
leave the metallicity free to vary. However, errorbars were too large to keep
it thawed.,666The exploited abundance table is from Anders & Grevesse (1989).
Note that these fits do not take in to account projection effects. The best-
fitting parameters are listed in Table 4. The projected temperature profile of
A1668 is shown in blue in Fig 6.

$r_{\text{min}}$-$r_{\text{max}}$ |  $r_{\text{min}}$-$r_{\text{max}}$ | Counts |  kT $\pm\sigma_{\text{kT}}$ |  $\chi^{2}/$DoF
---|---|---|---|---
[arcsec] | [kpc] |  | [keV] |
0 - 12 | 0 - 14 | 1346 (99.6 %) |  1.74 $\pm^{0.15}_{0.08}$ | 45/42
12 - 21 | 14 - 25 | 1274 (99.1 %) |  2.09 $\pm^{0.33}_{0.16}$ | 62/47
21 - 33 | 25 - 35 | 1328 (97.8 %) |  2.84 $\pm^{0.39}_{0.36}$ | 68/52
33 - 45 | 35 - 53 | 1111 (96.9 %) |  3.65 $\pm^{0.66}_{0.51}$ | 60/50
45 - 60 | 53 - 70 | 1101 (93.8 %) |  4.05 $\pm^{0.90}_{0.66}$ | 78/58
60 - 75 | 70 - 88 | 947 (91.2 %) |  3.39 $\pm^{0.81}_{0.54}$ | 123/61
75 - 90 | 88 - 106 | 1006 (89.2 %) |  3.51 $\pm^{0.75}_{0.55}$ | 98/64
Table 4: Fit results for the projected analysis. The first and second columns
show the lower and upper limits of the extraction rings in arcsec and kpc,
while the third column represents the number of source photons coming from
each ring, with the percentage indicating their number compared to the total
photons of the same region. In the last two columns we report the values of
kT, with associated errors, and the $\chi^{2}/$ DoF. Figure 6: Projected
(blue) and deprojected (yellow) temperature profile of A1668. Bars in the
x-axis represent the range of the extraction rings, while in the y-axis are
the errors for the temperature values.

Projection effects were then taken into account extracting spectra from
concentric rings centered on the X-ray peak, containing more than 1500 counts,
and fitting them with a projct*phabs*apec model. Temperature and normalization
were left free to vary, while column density, redshift and abundance were
frozen at the same values of the projected analysis above. Results are listed
in Table 5. The deprojected temperature profile of the cluster is shown in
black in Fig. 6.

$r_{\text{min}}$-$r_{\text{max}}$ |  $r_{\text{min}}$-$r_{\text{max}}$ | Counts | kT |  N(r) (10-4) | Electronic Density | Pressure | Entropy |  tcool
---|---|---|---|---|---|---|---|---
[arcsec] | [kpc] |  | [keV] |  |  [$10^{-2}\ $cm-3] |  [10-11 dy cm-2] |  [keV cm2] | [Gyr]
0 - 20 | 0 - 23.5 | 2506 (99.4 %) |  1.68 $\pm^{0.12}_{0.06}$ |  10.3 $\pm^{0.6}_{0.6}$ |  2.67 $\pm^{0.01}_{0.01}$ |  13.4 $\pm^{1.1}_{0.4}$ |  19.5 $\pm^{0.5}_{0.2}$ |  1.4 $\pm^{0.1}_{0.1}$
20 - 40 | 23.5 - 46.9 | 2162(97.4 %) |  2.51 $\pm^{0.52}_{0.36}$ |  12.5 $\pm^{0.8}_{0.8}$ |  1.07 $\pm^{0.01}_{0.01}$ |  7.7 $\pm^{1.4}_{1.0}$ |  51.8 $\pm^{2.0}_{1.4}$ |  4.3 $\pm^{0.6}_{0.6}$
40 - 65 | 46.9 - 76.2 | 1862 (93.7 %) |  4.18 $\pm^{2.51}_{0.98}$ |  8.1 $\pm^{0.8}_{0.8}$ |  0.44 $\pm^{0.01}_{0.01}$ |  7.5 $\pm^{5.1}_{2.4}$ |  219.9 $\pm^{24.5}_{11.3}$ |  15.8 $\pm^{5.2}_{5.1}$
65 - 90 | 76.2 - 105.5 | 1653 (89.1 %) |  3.36 $\pm^{0.63}_{0.50}$ |  16.9 $\pm^{0.9}_{0.9}$ |  0.44 $\pm^{0.01}_{0.01}$ |  4.2 $\pm^{0.7}_{0.6}$ |  120.7 $\pm^{3.5}_{3.0}$ |  11.7 $\pm^{1.7}_{1.8}$
Table 5: Fit results for the deprojected analysis. The first two columns
report the limits of the annular regions and the number of source photons from
each ring, with the percentage indicating their number compared to the total
photons of the same region. The remaining columns report temperature,
normalization factor, electronic density, pressure, entropy and cooling time.
The fit gives $\chi^{2}/$DoF = 1.46.

Following the same method described in Pasini et al. (2019)777Note the typo in
Eq. 4 of that paper, we estimated the electronic density as :

$\hskip 28.45274ptn_{e}=\sqrt{10^{14}\bigg{(}\dfrac{4\pi\cdot
N(r)\cdot[D_{A}\cdot(1+z)]^{2}}{0.82\cdot V}\bigg{)}}$ (1)

where N(r) is the apec normalization of the deprojected model, V is the shell
volume and DA is the angular distance of the source, estimated as DA =
DL/(1+$z$)2. Table 5 lists the density values for each ring, with the results
showed in Fig. 7.

Making use of the deprojected temperature and density values, we can derive
the cooling time, the pressure and the entropy for each bin. Table 5 presents
the pressure values, calculated as $p=1.83n_{e}kT$, while the entropy, that
was estimated as $S$ = $kTn_{e}^{-2/3}$, is presented in Fig. 8.

Figure 7: Density radial profile of A1668 derived from the deprojected
analysis. Each bin defines an extraction region. Figure 8: Entropy radial
profile of A1668 derived from the deprojected analysis. Each bin defines an
extraction region.

The cooling time is defined as:

$\hskip
28.45274ptt_{\text{cool}}=\dfrac{H}{\Lambda(T)n_{e}n_{p}}=\dfrac{\gamma}{\gamma-1}\dfrac{kT(r)}{\mu
Xn_{e}(r)\Lambda(T)}$ (2)

where $\gamma$=5/3 is the adiabatic index, $H$ is the enthalpy, $\mu\simeq$
0.61 is the molecular weight for a fully ionized plasma, X $\simeq$ 0.71 is
the hydrogen mass fraction and $\Lambda(T)$ is the cooling function
(Sutherland & Dopita, 1993). Results are listed in Table 5, while the cooling
time radial profile is shown in Fig. 9.

Figure 9: Cooling time profile of A1668. Each bin defines an extraction
region. The blue line represents the best-fit function
f(x)=(0.57$\pm$0.99)x0.73±0.43, while the red line is t${}_{\text{age}}$ = 7.7
Gyr.

We thus estimated the cooling radius of the cluster, i.e. the radius within
which the ICM cooling is efficient, assuming t${}_{\text{age}}\sim$ 7.7 Gyr,
corresponding to the look-back time at z=1, as an upper limit for the cluster
age. Consequently, the intersection between the profile best fit (blue line)
and t${}_{\text{age}}$ (red line) defines the cooling radius of A1668, being
$r_{\text{cool}}\approx 34"\approx 40\ \text{kpc}$.
The bolometric X-ray luminosity emitted within this radius was estimated by
extracting a spectrum from an annular region centered on the X-ray peak with r
= $r_{\text{cool}}$. Projection effects were taken into account by using a
second annular region with internal radius coincident with $r_{\text{cool}}$
and external radius $\sim$ 100$"$. By fitting both spectra with a
projct*phabs*apec model, the bolometric luminosity inside the cooling region
results $L_{\text{cool}}=1.9\pm 0.1\cdot 10^{43}\ \text{erg \ s}^{-1}$.
Assuming a steady state cooling flow model, the Mass Deposition Rate of the
cooling flow of A1668 can be estimated as:

${\dot{\text{M}}}\simeq\dfrac{2}{5}\dfrac{\mu m_{p}}{kT}\cdot L_{\text{cool}}$
(3)

In this way, we obtain $\dot{\text{M}}\simeq 29.6\pm 1.6\ \text{M}_{\odot}\
\text{yr}^{-1}$.

As a different approach, we performed a further fit of the spectrum of the
cooling region with a phabs*(apec + mkcflow) model, where the apec component
approximates the ICM emission along the line of sight outside of the cooling
region, while mkcflow is a multiphase component reproducing a cooling flow-
like emission inside the cooling radius. As above, the abundance was fixed at
0.3 Z⊙, while the temperature of the apec model was left free to vary and
bounded to the high temperature parameter of mkcflow. Redshift and absorbing
column density were fixed at the Galactic values (see above), while the low
temperature parameter of mkcflow was fixed at the lowest possible value,
$\sim$ 0.1 keV. The fit gives $\chi^{2}$/DoF = 105/100 and provides an upper
limit of $\dot{\text{M}}<5\ \text{M}_{\odot}\ \text{yr}^{-1}$. The bolometric
luminosity associated to the mkcflow model is L${}_{\text{mkcflow}}=3.2\pm
0.1\cdot 10^{41}\ \text{erg \ s}^{-1}$. The difference between the two
estimates of the mass deposition rate reflects the Cooling Flow (CF) problem:
observed mass deposition rates do not match expectations from the standard CF
model, and heating contribution, likely produced by the central AGN, is
required to balance the ICM radiative losses.

##  4 Discussion

###  4.1 Radio-X-ray combined analysis

In order to investigate the interactions between the cooling ICM and the BCG,
we overlaid the 1.4 GHz radio contours on the X-ray 0.5-2 keV cluster image.
Since we are interested in the core region, in Fig. 10 we show the resulting
image, zoomed in the central $30\times 30$ kpc.

Figure 10: Left Panel: 1.4 GHz radio (green) and H$\alpha$ (black) contours
overlaid on the 0.5-2 keV X-ray image, zoomed towards the cluster centre. The
cyan cross represents the X-ray emission centroid, coincident with the BCG
centre; the red and white crosses are the X-ray and H$\alpha$ peaks,
respectively. Right Panel: From left to right, from top to bottom: 0.5-2 keV,
1.4 GHz, H$\alpha$ and optical (SDSS) images of A1668. All images are centered
on the X-ray peak. The cyan cross represents the BCG.

The cluster X-ray cool core exhibits an elliptical nuclear region. Exploiting
optical catalogues, we found that the radio galaxy is coincident with the BCG
nuclear region, as expected. The emission centroid of the large-scale X-ray
emission ($RA$=13${}^{\text{h}}$03${}^{\text{m}}$43.6${}^{\text{s}}$,
$DEC$=+19$\degree$16${}^{\text{m}}$17.4${}^{\text{s}}$), defined as the center
of the isophotes, lies within this region, too. On the other hand, the X-ray
peak ($RA$=13${}^{\text{h}}$03${}^{\text{m}}$46.6${}^{\text{s}}$,
$DEC$=+19$\degree$16${}^{\text{m}}$12.2${}^{\text{s}}$), is found to the south
of the nucleus of the BCG, exhibiting a significant offset of $\sim$ 5.2$"$,
corresponding to $\sim$ 6 kpc.
In order to check for the possible presence of a field point-source that could
bias the detection of the X-ray peak, we extracted a spectrum from a $\sim$
4$"$ circular region centered on the peak, and fitted it with two models:
phabs*apec and phabs*(apec+powerlaw). The first fit gave
$\chi^{2}/\text{DoF}\sim$ 78/79, while for the second
$\chi^{2}/\text{DoF}\sim$ 71/77; the F-stat method was then applied in order
to check if the addition of the powerlaw component provided a significant
improvement of the fit. We obtained an F-value of 3.4 and p=0.035,
correspondent to a null hypothesis probability of 1-p=0.965. This suggests
that the addition of the point-source emission component is not statistically
significant. As a further check, we looked for possible point-sources in high
energy, optical and infrared catalogues, as well as in a harder band (4-7 keV)
X-ray image; however, we did not detect any point source coincident with the
X-ray peak. We thus conclude that the peak detection is likely not biased, and
therefore the offset is real.
This is analogous with what was found in A2495, that presents a similar-scale
offset between these two components, and with a number of recent works that
found the same feature in other clusters (e.g., Sanderson et al., 2009;
Haarsma et al., 2010; Hudson et al., 2010; Rossetti et al., 2016, and others;
for a brief review of the state-of-the-art literature about BCG/cool core
offsets, see Pasini et al., 2019). We will return on this in Sec. 4.3.

A two-dimensional temperature map is often used in order to further
investigate on the cluster structure and its thermodynamical state, but the
small number of photons prevents us from producing such map. We also estimated
the softness ratio as (S-H)/(S+H), where S and H are the number of counts in
the soft (0.5-2 keV) and hard (2-7 keV) band, respectively. However, the
statistics are still too poor and errors are too large to draw any conclusion
from such analysis.

###  4.2 H$\alpha$ analysis

The presence of optical line emitting nebulae in galaxy clusters is linked to
the thermodynamical conditions of the cluster core; observational studies
(e.g., Cavagnolo et al., 2008; McNamara et al., 2016) have argued that such
warm structures are only found if the central ($\sim$ 10 kpc) entropy falls
below 30 keV cm2 or, alternatively, when t${}_{\text{cool}}$/tff $<$ 10-20
(Voit et al., 2015), where tff = $\sqrt{2R^{3}/GM}$ is the freefall time. The
estimated entropy within the central bin of our spectral analysis ($r<23.5$
kpc, Table 5) is $\sim$ 19.5 keV cm2, thus satisfying the criterion for the
presence of such nebulae in A1668. Hamer et al. (2016) presented VIMOS
observations of the H$\alpha$ line emission of a sample of 73 BCGs, including
A1668, whose image is shown in Fig. 11.

Figure 11: 1.4 GHz radio (green) contours overlaid on the VIMOS H$\alpha$
image (Hamer et al., 2016). The mean seeing is 1.21$"$, and the map units are
in 10-16 erg s-1 cm-2 Å. The white contours are the optical isophotes from
SDSS. The cyan cross represents the radio galaxy centre, coincident with the
BCG core.

The H$\alpha$ structure presents a rather compact shape, extending for $\sim$
9$"$ ($\sim$ 10.6 kpc). Its total luminosity is L${}_{\text{H}\alpha}$ = 3.85
$\pm$ 0.30 $\cdot$ 1040 erg s-1888This estimate differs from Pulido et al.
(2018) since it is not extinction-corrected and is obtained from IFU
observations., and Hamer et al. (2016) classified it as a quiescent object,
showing a simple, centrally-concentrated morphology999The morphology looks
slightly different when compared to Edwards et al. (2009), whose IFU image
shows a somehow better ’resolved’ shape.. To better visualize the interplay of
the three components, in Fig. 10 we also overlay the H$\alpha$ contours on the
X-ray 0.5-2 keV image.
The line emission lies entirely within the BCG, but although it is also within
the cool core as defined from the cooling time profile, it only overlaps one
end of the bright X-ray ridge. This highlights a difference with A2495, in
which the H$\alpha$ structure connects the galaxy with the X-ray peak, and
with other systems (e.g., Bayer-Kim et al., 2002; Hamer et al., 2012, 2016),
in which the line emission seems to be mostly associated to the cooling ICM,
rather than to the BCG.
A significant offset of $\sim$ 6.5$"$ ($\sim$ 7.6 kpc) is present between the
H$\alpha$ ($RA$=13${}^{\text{h}}$03${}^{\text{m}}$46.4${}^{\text{s}}$,
$DEC$=+19$\degree$16${}^{\text{m}}$18.1${}^{\text{s}}$) and the X-ray peaks.
The same feature, albeit smaller, was found in A2495 by Pasini et al. (2019);
offsets between these two peaks were also detected in A1795 (Crawford et al.,
2005) and in a number of systems by Hamer et al. (2016). We will return on
this in Sec. 4.3.

As described in more details in Hamer et al. (2012), the ratio of the
H$\alpha$ plume extent and the total velocity gradient of the warm gas
provides an estimate of the projected offset timescale. Pasini et al. (2019)
showed that, for A2495, such timescale is comparable to the age difference
between two putative cavity pairs, suggesting that, for systems where the BCG
oscillates back and forth through the cooling region, this measure could be a
good indicator for the AGN cycle intermittency. In A1668, the H$\alpha$
structure extends for D’$\sim$ 7.7$"$ from the BCG centre, corresponding to
$\sim$ 9 kpc, and shows a very smooth velocity gradient from East to West. We
measured a velocity difference of $\sim$ +500 km s-1 between the gas at the
BCG centre and that at the tail of the H$\alpha$ structure. This indicates a
projected timescale of T’= D’/V $\sim$ 18 Myr. In order to correct for the
projection effects we assumed a most likely inclination of $\sim$ 60
$\degree$, with a range of 30-75 $\degree$ (Hamer et al., 2012); the corrected
timescale can thus be estimated with T = T’ $\times$ cos(i)/sen(i), with i
being the inclination. We obtained in this way an offset timescale of $\sim$
10.4 Myr, with an upper and lower limit of $\sim$ 31.2 and $\sim$ 4.8 Myr,
respectively; this is consistent with the value of $\sim$ 13 Myr estimated for
A2495, suggesting that the cold gas dynamics in the two systems are similar,
and that the timescale of AGN feedback intermittency is comparable.

One can also estimate the mass of the warm gas by assuming that it is
optically thin:

$M_{\text{H$\alpha$}}\simeq L_{\text{H$\alpha$}}\dfrac{4\mu
m_{p}}{n_{\text{H$\alpha$}}\epsilon_{\text{H$\alpha$}}}$ (4)

where $\epsilon_{\text{H}\alpha}\sim 3.3\cdot 10^{-25}$ erg cm3 s-1 is the
H$\alpha$ line emissivity, while $n_{\text{H$\alpha$}}$ is obtained assuming
pressure equilibrium with the local ICM:

$n_{\text{H$\alpha$}}T_{\text{H$\alpha$}}\simeq n_{\text{ICM}}T_{\text{ICM}}$
(5)

where $T_{\text{H$\alpha$}}\sim 10^{4}$ K, $T_{\text{ICM}}$ is the first value
reported in Tab. 4 ($\sim$ 1.74 keV, since the H$\alpha$ structure is located
within the first spectral bin), while $n_{\text{ICM}}\sim 1.83n_{e}$, where
$n_{e}$ is the electronic density reported in the first row of Tab. 5. In this
way, we obtain $M_{\text{H$\alpha$}}\sim$ (2.4 $\pm$ 0.2) $\cdot$ 106 M⊙.

The [SII]${}_{\lambda}6716$/[SII]${}_{\lambda}6731$ line ratio provides an
independent estimate of the density of the ionised gas and is measured in
Hamer et al. 2016 (Appendix F) as 1.21$\pm$0.2. Assuming case B reionisation
and a temperature of 104 K, this gives an electron density of
$n_{e,\text{H}\alpha}$ =350 $\pm$ 270 cm-3 which corresponds to a total
density of $n_{\text{H}\alpha}$=640 $\pm$ 495 cm-3. These values are
comparable, within the (large) uncertainties, with the value of
$n_{\text{H}\alpha}$=100 $\pm$ ${}^{8}_{3}$ cm-3 derived from Eq. 5. It is
important to consider the impact of the assumed ionised gas temperature
($T_{\text{H}\alpha}$) on the two measurements though. A lower
$T_{\text{H}\alpha}$ would result in a higher $n_{e,\text{H}\alpha}$ from Eq.
5 but a lower $n_{e,\text{H}\alpha}$ from the
[SII]${}_{\lambda}6716$/[SII]${}_{\lambda}6731$ line ratio measurement, while
a higher $T_{\text{H}\alpha}$ would have the opposite effect on both
measurements. The VIMOS data from (Hamer et al., 2016) are not sensitive
enough to provide a reliable estimate of $T_{\text{H}\alpha}$ for Abell 1668,
but deep observations of other objects have found upper limits to
$T_{\text{H}\alpha}$ that are much lower than expected (e.g.
$T_{\text{H}\alpha}<$ 5685 K in the Centaurus cluster, Hamer et al. 2019).
Assuming $T_{\text{H}\alpha}$ = 5000 K, we find $n_{e}$ = 230 $\pm$180 cm-3
and $n_{\text{H}\alpha}$=420 $\pm$ 330 cm-3 from the [SII] ratio and
$n_{\text{H}\alpha}$=200 $\pm$ ${}^{16}_{6}$ cm-3 derived from Eq. 5,
indicating that the two measurements are consistent within the limits of the
available data and assumed values.

###  4.3 Putative cavities and AGN feedback cycle

From the 0.5-2 keV X-ray image (see Fig. 12) it is possible to identify a
number of ICM surface brightness depressions. The present observation is very
shallow ($\sim$ 10 ks), thus the reader shall be warned about the significance
of these deficits, that could possibly be artifacts. However, we focused our
attention on three of these features (showed in Fig. 12) that, due to their
position, could possibly represent real ICM cavities. One of them (A) lies
within the radio galaxy West lobe; another surface brightness depression (B)
is detected in the radio galaxy East lobe, while a symmetrical (with respect
to the X-ray peak), similar-shaped brightness depression (C), is found at the
opposite side of the cluster core, not associated with the radio galaxy. It is
noteworthy to mention that cavity A is also coincident with the H$\alpha$ line
emission peak.

Figure 12: 0.5 - 7 keV image showing three surface brightness depressions: A
and B lie in the west and east lobes of the radio galaxy, respectively, while
C is found at the opposite side of the core. Overlaid are the 1.4 GHz
contours.

To investigate on these features, we estimated their significance as
N${}_{\text{M}}$-N${}_{\text{C}}$/$\sqrt{\text{N}_{\text{M}}+\text{N}_{\text{C}}}$,
where N${}_{\text{M}}$ and N${}_{\text{C}}$ are the number of counts in
regions of equal area close to the candidate cavity and within the cavity,
respectively. The number of counts is different depending on the size of the
elliptical region chosen to cover the putative cavity and, as previously
stated, the current observation requires us to be cautious, since the shape
and extent of the depression observed ’by eye’ can slightly change using
different color scales. For this reason, the upper and lower limits for the
dimensions of these regions were estimated by varying the axes of the ellipse
until reaching a significance of 2$\sigma$ and 3$\sigma$, respectively. The
assumed ’true’ size of the bubble is the mean between these two limits. Cavity
A exhibits a circular shape, with a diameter of 4.8 $\pm$ 0.6 kpc, while
cavity B is more elliptical, with a major axis of 5.2 $\pm$ 0.4 kpc and a
minor axis of 3.0 $\pm$ 0.3 kpc. On the other hand, the brightness deficit
related to candidate cavity C is less enhanced and, in order to reach the
desired significances, requires the size to be larger than the observed
depression. Therefore, in the following analysis we will only discuss
candidate cavities A and B, while cavity C will not be considered. Following
the method described in Bîrzan et al. (2004), we determined the cavity power:

$P_{\text{cav}}=\dfrac{E_{\text{cav}}}{t_{\text{cav}}}=\dfrac{4pV}{t_{\text{cav}}}$
(6)

where $t_{\text{cav}}$ is the age of the cavity, calculated as
$t_{\text{cav}}=R/c_{s}$, with R being the cavity distance from the BCG
center, $p$ is the pressure at the distance of the cavity, $V$ is the cavity
volume and $c_{s}$ is the sound velocity. The volume was estimated assuming an
oblate elipsoidal shape for both cavities, while for pressure and temperature
we assumed the values listed in Table 5 corresponding to the annular bin the
cavities lie in.
We obtained the same age for both cavities: t${}_{\text{cav}}$ = 5.2 $\pm 0.7$
Myr. This leads to P${}_{\text{cav, A}}$ = 5.1 $\pm 2.6\ \cdot$ 1042 erg s-1
for cavity A, and P${}_{\text{cav, B}}$ = 3.8 $\pm 1.4\ \cdot$ 1042 erg s-1
for cavity B. The age is consistent with the offset timescale estimated in
Sec. 4.2. Finally, we compared the estimated values of P${}_{\text{cav}}$ (in
the hypothesis that the cavities are real) and L${}_{\text{cool}}$ with the
typical distribution observed for cool core clusters (Bîrzan et al., 2017).
The result is presented in Fig. 13.

Figure 13: Blue points are the data from Bîrzan et al. (2017), the orange ones
are the values for A2495 (Pasini et al., 2019), while red represents the
cavity system detected in A1668. Dashed lines represent, from left to right,
P${}_{\text{cav}}=$ L${}_{\text{cool}}$ assuming $pV$, $4pV$ or $16pV$ as the
deposited energy.

The L${}_{\text{cool}}$-P${}_{\text{cav}}$ relationship in A1668 is consistent
within the scatter of the expected distribution, despite being on its lower
edge. The detected offsets, therefore, do not seem to affect the feeding-
feedback cycle, that is still maintained. The same result was found in A2495,
where the two cavity systems (i.e., the AGN) have enough energy to balance the
radiative losses within the cooling region. We then argue that small offsets
are not able to break the AGN feeding-feedback cycle.

###  4.4 Offsets, cooling and H$\alpha$ emission: are sloshing and AGN
activity shaping the core of A1668?

As described in the previous sections, the core of A1668 contains a complex
set of structures whose origin is not immediately clear. It is worth
reiterating that the available Chandra data is only a snapshot, providing
somewhat limited information on the ICM. It should also be remembered that all
of the structures we observe are contained within the central $\sim$ 20 kpc,
inside the stellar body of the BCG.

The offsets between the peaks of the radio, X-ray and H$\alpha$ emission raise
the question of how these components came to be separated. The peak of
emission from the hot ICM lies not in the BCG nucleus, but $\sim$6 kpc to the
south. The radio emission is reasonably evenly distributed Eastwards and
Westwards with respect to the nucleus, but the H$\alpha$ peaks in the region
of the west radio jet/lobe, extending around the western half of the radio
structure and overlapping the optical centroid of the BCG.

We suggest a qualitative scenario which might explain the relative
morphologies of the different components. At some point in the past, A1668 may
have been a relaxed cluster with a cool core. In that core, centred on the
BCG, gas had begun to cool and condense out of the ICM, forming the kind of
filamentary H$\alpha$ nebula observed in other cool core cluster. The small
velocity gradient in the H$\alpha$ emission is consistent with such an origin.
At that stage, A1668 underwent a minor merger, which caused the core to begin
sloshing, oscillating around the centre of the cluster gravitational
potential. Sloshing motions in the plane of the sky are typically visible as a
spiral pattern in the ICM, but if the plane of motion is aligned along the
line of sight, the motions produce pairs of nested cold fronts, and the cool
ICM gas drawn out from the core can appear as a tail to one side of the BCG.
Our Chandra observation is too short for fronts to be visible, but we do see
the tail: the ridge structure.

At this stage, cooling (and H$\alpha$ emission) would still have been centred
in the core of the BCG. About 5 Myr ago ($\sim$ cavity age), sufficient cooled
material reached the central SMBH to trigger an outburst. This produced the
radio jets/lobes we observe, and as these expanded they pushed aside the pre-
existing H$\alpha$ filaments and hotter ICM gas. This produced a correlated
H$\alpha$/radio morphology, with much of the H$\alpha$ wrapped around the west
jet/lobe. It also disrupted the centre of the cool core, reducing the X-ray
surface brightness as a large part of the volume in the core of the BCG was
filled by the radio lobes, producing the apparent cavities in the ICM. This
brings us to the current situation, where the brightest X-ray emission is in
the tail to the south of the BCG nucleus.

The expansion timescale of the radio lobes is only a few 106 yr. This is very
short compared to typical sloshing timescales. The hot ICM oscillates with
sloshing timescale given by $t_{\rm slosh}=2\pi/\omega_{\rm BV}$ where
$\omega_{\rm BV}=\Omega_{K}\left(\frac{1}{\gamma}\frac{d\ln S}{d\ln
r}\right)^{1/2}$. Here $d$ls$S$/$d$ln$r$ is the logarithmic entropy gradient,
$\Omega_{K}$=$\sqrt{GM/r^{3}}$ and $\gamma$=5/3 for the ionized ICM plasma. We
also know that the free-fall time in the cluster is $t_{\rm ff}=\sqrt{2r/g}$.
Our Chandra data are not sufficient to accurately model the mass profile, but
we know that the stellar velocity dispersion, $\sigma_{*}$, in the inner
regions of the BCG will follow the gravitational potential, so that $t_{\rm
ff}\simeq r/\sigma_{*}$ (Voit et al., 2015). We can therefore approximate
$\omega_{\rm BV}$ as $\frac{1}{t_{ff}}\sqrt{\frac{6}{5}\frac{d\ln S}{d\ln
r}}$.

Our data do not allow the calculation of the entropy profile on scales
$r\lesssim 25$ kpc (Fig. 8), thus we use the average slope $d{\rm ln}S/d{\rm
ln}r=0.67$ given by Hogan et al. (2017b) (see also Panagoulia et al. 2014).
This returns $t_{\rm slosh}\sim 7t_{\rm ff}$. We do not know the scale of the
sloshing, but it must be greater than the length of the X-ray tail ($\sim$16
kpc). Based on the measured $\sigma_{*}$=226$\pm$7 km s-1 (Pulido et al.,
2018), at 16 kpc $t_{\rm ff}$=70 Myr, and thus $t_{\rm slosh}$ may be as much
as $\sim$ 490 Myr. As expected this is considerably longer than the AGN
expansion timescale, confirming that if sloshing is occurring, it cannot yet
have affected the structure of the radio lobes.

As argued by Olivares et al. (2019), the fact that filamentary nebulae in cool
core clusters generally lack a significant velocity gradient indicates that
the cool H$\alpha$ or CO-emitting gas they contain is at least partially tied
to the surrounding ICM. It is unclear how the different phases are connected,
but it has been suggested that the denser material may be enveloped by many
diffuse layers of warmer gas (Li et al., 2018), or threaded through by
magnetic fields (McCourt et al., 2015), either of which could increase drag
forces. We would thus expect the H$\alpha$ emission to trace the regions in
which gas has most recently cooled from the ICM, after modification by the
expanding radio lobes. If the inflation of cavities has disrupted the cooling
region, we might expect the locus of any future cooling to be at the new X-ray
peak of surface brightness, source of the BCG nucleus. However, the lack of
H$\alpha$ emission at that location suggests that cooling there is slower than
it was near the BCG core, and that no reservoir of cooled material has yet
built up at that location.

This scenario is of course speculative, given the constraints available from
the data. Several aspects are uncertain. All, part or none of the H$\alpha$
emitting gas might have formed as a result of the AGN outburst, with the
expanding lobes triggering condensation (e.g., Qiu et al., 2020). If the
cluster is sloshing, we cannot know the scale or alignment of the motion
without deeper data. However, our scenario explains several basic facts: the
radio and H$\alpha$ emission are correlated because the radio has at least
partly determined the morphology of the H$\alpha$-emitting gas. The X-ray
offset is the result of sloshing, which has not affected the radio sources or
H$\alpha$ because the radio source expansion timescale is short compared to
the sloshing timescale. The BCG is no longer the centre of ICM cooling because
the AGN has pushed aside the dense gas which fuelled the outburst.

The scenario also makes testable predictions. If the cluster is sloshing, we
should expect deeper Chandra data to reveal nested cold fronts, and the X-ray
ridge should contain relatively cool, high abundance gas. Deeper imaging
should also allow us to more accurately measure the morphology of any
cavities, which should be correlated with the rado jets and lobes. Higher
resolution radio data may be required to make this comparison. Lastly, we
might expect higher resolution H$\alpha$ imaging to reveal complex structure
in the cooled material, consistent with a filamentary nebula disturbed by an
AGN outburst.

###  4.5 Alternative explanations for the origin of the H$\alpha$ emission and
spatial offsets

An alternative hypothesis that could explain the observed displacement of the
H$\alpha$ emission is cooling in situ, perhaps stimulated by the same AGN
outburst which originated the X-ray cavities. Inhomogeneous cooling scenarios
in clusters have been the object of a long, lively debate (see early reviews
by Fabian et al. 1991, 1994, and references therein). Nowadays, many lines of
evidence suggest that hot gas cools at a (mean) low rate and in a spatially
distributed fashion, when the ISM/ICM conditions are appropriate (see Hogan et
al. 2017b; Pulido et al. 2018; Lakhchaura et al. 2018 for a quantitative
discussion). The primary trigger of this localized cooling (that is, the
origin of thermally unstable perturbations) might be turbulence (e.g. Chaotic
Cold Accretion, CCA, see Gaspari et al., 2012, 2013; Voit et al., 2015),
lifting of low entropy gas by X-ray cavities (Revaz et al., 2008; Brighenti et
al., 2015) or the sloshing itself.

CCA implies the trigger of thermal instabilities, that is favoured by a
central ($\leq$ 10 kpc) cooling time $t_{\rm cool}\sim 1$ Gyr (Hogan et al.,
2017a; Pulido et al., 2018), or by $t_{\rm cool}/t_{ff}\leq 10-20$ (Voit et
al., 2015). It is easier to ensue at the position of the X-ray peak, but it
can be triggered wherever these conditions are respected. Therefore, the
displacement observed for the H$\alpha$ gas could be the result of CCA
detached from the emission peak. CCA at the current position of the peak could
still be happening, but it could have not built yet enough material to be
detected in H$\alpha$. However, given the available X-ray observation, we are
not in a position to accurately estimate the cooling time in the central
region of the cluster.

We can explore in some more detail the scenario where the warm gas derives
from a cool component of the ICM, originally located close to the nucleus of
the BCG, uplifted by the cavities and then cooled to $10^{4}$ K. Following
Archimedes’ principle, cavities can lift an amount of gas equal to their
displacement, though simulations suggest that the maximum amount is only
$\sim$50% of this value (Pope et al., 2010). This corresponds to
M${}_{\text{uplift}}$ = 9 $\cdot$ 106 M⊙ for cavity A, and
M${}_{\text{uplift}}$ = 3 $\cdot$ 106 M⊙ for cavity B. The mass of the
H$\alpha$ plume is lower ($M_{\text{H$\alpha$}}\sim$ 2.4 $\cdot$ 106 M⊙, see
Sec. 4.2). However, given the state-of-the-art correlation between
L${}_{\text{H}\alpha}$ and molecular gas mass M${}_{\text{mol}}$ (see e.g.,
Edge, 2001; Salomé & Combes, 2003; Pulido et al., 2018), that is usually found
to be co-spatial with H$\alpha$, we would expect to have M${}_{\text{mol}}$
$\sim$ 109 M⊙ (lower than the upper limit $1.5\times 10^{9}$ M⊙ quoted by
Salomé & Combes 2003). The total amount of gas would therefore be too large
for the cavities to uplift and this, along with the radio/H$\alpha$
morphology, would imply the need for an earlier cycle of AGN jet activity if
uplift is responsible.

The observed H$\alpha$ line emission could also be the remnant of the ISM of a
gas-rich galaxy which merged with the BCG. To test this hypothesis, we
examined SDSS and DSS optical images and catalogs in order to check whether a
member galaxy could be interacting with the BCG. However, the closest system
lies more than 40 kpc away from the BCG, not showing any hint of interplay.
Therefore, the merging hypothesis looks unlikely with the current data.

Finally, the warm gas could originate from the stellar mass loss in the BCG
(Mathews, 1990; Li et al., 2019). With a total B-band luminosity $L_{B}\sim
1.3\times 10^{11}$ $L_{B,\odot}$ (Makarov et al., 2014) and a stellar mass to
ratio for an old population $(M/L_{B})\sim 7$ (e.g., Maraston, 2005), the
expected mass loss rate is $\dot{M}_{*}\sim 1.35$ M⊙/yr (Mathews, 1989). Thus,
the observed amount of emission line gas can be accumulated in less than 2
Myr. However, the displacement with respect to the BCG center and its
filamentary and disturbed distribution (Edwards et al., 2009; Hamer et al.,
2016) are not easily accounted for by this scenario.

##  5 Conclusions

We performed a multi-wavelength analysis of the cool core cluster A1668, by
means of new radio (EVLA) and X-ray (Chandra) observations and of H$\alpha$
line emission data from Hamer et al. (2016). The results can be summarized as
follows:

  * •

The radio analysis at 1.4 (L${}_{\text{1.4 GHz}}\sim 6\cdot 10^{23}$ W Hz-1)
and 5 GHz (L${}_{\text{5 GHz}}\sim 2\cdot 10^{23}$ W Hz-1) shows a small
($\sim$ 11 -14 kpc) and elongated FRI radio galaxy, with no hints of larger
scale emission at these frequencies. The mean spectral index is
$\alpha=0.99\pm 0.06$, consistent with the usual values found in BCGs.

  * •

The X-ray analysis confirms the classification of A1668 as a cool core
cluster, with a cooling radius of $\sim$ 40 kpc inside which we estimate a
bolometric luminosity $L_{\rm cool}\sim$ 1.9 $\cdot$ 1043 erg s-1.

  * •

The multi-wavelength analysis reveals two spatial offsets, with the first of
$\sim$ 6 kpc being between the BCG nucleus and the X-ray peak, while the
second of $\sim$ 7.6 kpc between the H$\alpha$ and the X-ray peaks. This is
similar to what was found in another similar cluster, A2495, with two offsets
of 6 and 4 kpc, respectively (Pasini et al., 2019). The compact H$\alpha$
emission structure extends for $\sim$ 11 kpc and is mostly co-spatial with the
BCG, unlike A2495, where the line emission seems to be linked to the cluster
cool core, rather than to the central galaxy.

  * •

We identify three X-ray surface brightness depressions, one of them (A)
coincident with the west radio lobe and with the H$\alpha$ peak, another one
(B) lying within the east radio lobe, while the third one (C) being more
uncertain. For the system of cavities A and B we determine an age of $\sim$
5.2 Myr. The L${}_{\text{cool}}$-P${}_{\text{cav}}$ estimates for A1668 are in
agreement with the relationship observed for other systems (e.g., Bîrzan et
al., 2017), suggesting that the detected offsets are not able to break the AGN
feeding-feedback cycle.

  * •

Finally, we discuss possible explanations for the multiphase gas and for the
displacements observed in the core of A1668. We propose that, initially, all
the components were spatially coincident in the cluster cool core. Sloshing
was likely triggered by a minor merger, causing some of the cool gas around
the BCG to be drawn out into a tail that we now observe as an X-ray ridge
structure. On the other hand, the densest, most rapidly cooling gas, still in
and around the BCG core, condensed out to form the H$\alpha$ nebula. About 5
million years ago, the condensed material fuelled the central SMBH, triggering
the outburst that produced the observed radio jets/lobes. The expansion of the
lobes finally pushed aside the H$\alpha$ nebula and the hot ICM, disrupting
the cool core centre. Alternative explanations for the misplacement of the
H$\alpha$ emission include cooling in situ through thermal instabilities,
uplift from the cavities, reminiscence from a past merger with a gas-rich
galaxy, or stellar mass loss from the BCG, although the last three look
unlikely (see Sec. 4.5).

We thank the referee for thoughtful comments and suggestions, which have
significantly improved the presentation of our results. TP is supported by the
BMBF Verbundforschung under grant number 50OR1906. E.O’S. ackonwledges support
for this work from the National Aeronautics and Space Administration through
Chandra Award Number G07-18162X issued by the Chandra X-ray Center, which is
operated by the Smithsonian Astrophysical Observatory for and on behalf of the
National Aeronautics Space Administration under contract NAS8-03060. Based on
observations made with ESO Telescopes at the La Silla or Paranal Observatories
under programme ID 080.A-0224 and 082.B-0671. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc.

## References

  * Anders & Grevesse (1989) Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197, doi: 10.1016/0016-7037(89)90286-X
  * Andreon (2016) Andreon, S. 2016, A&A, 587, A158, doi: 10.1051/0004-6361/201526852
  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Barbosa et al. (2018) Barbosa, C. E., Arnaboldi, M., Coccato, L., et al. 2018, A&A, 609, A78, doi: 10.1051/0004-6361/201731834
  * Bayer-Kim et al. (2002) Bayer-Kim, C. M., Crawford, C. S., Allen, S. W., Edge, A. C., & Fabian, A. C. 2002, MNRAS, 337, 938, doi: 10.1046/j.1365-8711.2002.05969.x
  * Bîrzan et al. (2017) Bîrzan, L., Rafferty, D. A., Brüggen, M., & Intema, H. T. 2017, MNRAS, 471, 1766, doi: 10.1093/mnras/stx1505
  * Bîrzan et al. (2004) Bîrzan, L., Rafferty, D. A., McNamara, B. R., Wise, M. W., & Nulsen, P. E. J. 2004, ApJ, 607, 800, doi: 10.1086/383519
  * Bîrzan et al. (2020) Bîrzan, L., Rafferty, D. A., Brüggen, M., et al. 2020, MNRAS, doi: 10.1093/mnras/staa1594
  * Brighenti et al. (2015) Brighenti, F., Mathews, W. G., & Temi, P. 2015, ApJ, 802, 118, doi: 10.1088/0004-637X/802/2/118
  * Cavagnolo et al. (2008) Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2008, ApJ, 683, L107, doi: 10.1086/591665
  * Cavaliere & Fusco-Femiano (1976) Cavaliere, A., & Fusco-Femiano, R. 1976, A&A, 49, 137
  * Clarke et al. (2004) Clarke, T. E., Blanton, E. L., & Sarazin, C. L. 2004, ApJ, 616, 178, doi: 10.1086/424911
  * Crawford et al. (1999) Crawford, C. S., Allen, S. W., Ebeling, H., Edge, A. C., & Fabian, A. C. 1999, MNRAS, 306, 857, doi: 10.1046/j.1365-8711.1999.02583.x
  * Crawford et al. (2005) Crawford, C. S., Sanders, J. S., & Fabian, A. C. 2005, MNRAS, 361, 17, doi: 10.1111/j.1365-2966.2005.09149.x
  * Ebeling et al. (1998) Ebeling, H., Edge, A. C., Bohringer, H., et al. 1998, MNRAS, 301, 881, doi: 10.1046/j.1365-8711.1998.01949.x
  * Edge (2001) Edge, A. C. 2001, MNRAS, 328, 762, doi: 10.1046/j.1365-8711.2001.04802.x
  * Edwards et al. (2009) Edwards, L. O. V., Robert, C., Mollá, M., & McGee, S. L. 2009, MNRAS, 396, 1953, doi: 10.1111/j.1365-2966.2009.14897.x
  * Fabian (2012) Fabian, A. C. 2012, ARA&A, 50, 455, doi: 10.1146/annurev-astro-081811-125521
  * Fabian et al. (1994) Fabian, A. C., Canizares, C. R., & Boehringer, H. 1994, ApJ, 425, 40, doi: 10.1086/173959
  * Fabian et al. (1991) Fabian, A. C., Nulsen, P. E. J., & Canizares, C. R. 1991, A&A Rev., 2, 191, doi: 10.1007/BF00872767
  * Fabian et al. (2006) Fabian, A. C., Sanders, J. S., Taylor, G. B., et al. 2006, MNRAS, 366, 417, doi: 10.1111/j.1365-2966.2005.09896.x
  * Feretti & Giovannini (2008) Feretti, L., & Giovannini, G. 2008, Clusters of Galaxies in the Radio: Relativistic Plasma and ICM/Radio Galaxy Interaction Processes, Vol. 740, 24, doi: 10.1007/978-1-4020-6941-3_5
  * Freeman et al. (2001) Freeman, P., Doe, S., & Siemiginowska, A. 2001, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4477, Proc. SPIE, ed. J.-L. Starck & F. D. Murtagh, 76–87, doi: 10.1117/12.447161
  * Fruscione et al. (2006) Fruscione, A., McDowell, J. C., Allen, G. E., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6270, Proc. SPIE, 62701V, doi: 10.1117/12.671760
  * Gaspari et al. (2013) Gaspari, M., Ruszkowski, M., & Oh, S. P. 2013, MNRAS, 432, 3401, doi: 10.1093/mnras/stt692
  * Gaspari et al. (2012) Gaspari, M., Ruszkowski, M., & Sharma, P. 2012, ApJ, 746, 94, doi: 10.1088/0004-637X/746/1/94
  * Gastaldello et al. (2009) Gastaldello, F., Buote, D. A., Temi, P., et al. 2009, ApJ, 693, 43, doi: 10.1088/0004-637X/693/1/43
  * Gentile et al. (2007) Gentile, G., Rodríguez, C., Taylor, G. B., et al. 2007, ApJ, 659, 225, doi: 10.1086/512479
  * Ghizzardi et al. (2010) Ghizzardi, S., Rossetti, M., & Molendi, S. 2010, A&A, 516, A32, doi: 10.1051/0004-6361/200912496
  * Giacintucci et al. (2014) Giacintucci, S., Markevitch, M., Brunetti, G., et al. 2014, ApJ, 795, 73, doi: 10.1088/0004-637X/795/1/73
  * Giacintucci et al. (2017) Giacintucci, S., Markevitch, M., Cassano, R., et al. 2017, ApJ, 841, 71, doi: 10.3847/1538-4357/aa7069
  * Gitti et al. (2012) Gitti, M., Brighenti, F., & McNamara, B. R. 2012, Advances in Astronomy, 2012, doi: 10.1155/2012/950641
  * Gitti et al. (2004) Gitti, M., Brunetti, G., Feretti, L., & Setti, G. 2004, A&A, 417, 1, doi: 10.1051/0004-6361:20031750
  * Govoni et al. (2009) Govoni, F., Murgia, M., Markevitch, M., et al. 2009, A&A, 499, 371, doi: 10.1051/0004-6361/200811180
  * Haarsma et al. (2010) Haarsma, D. B., Leisman, L., Donahue, M., et al. 2010, ApJ, 713, 1037, doi: 10.1088/0004-637X/713/2/1037
  * Hamer et al. (2012) Hamer, S. L., Edge, A. C., Swinbank, A. M., et al. 2012, MNRAS, 421, 3409, doi: 10.1111/j.1365-2966.2012.20566.x
  * Hamer et al. (2016) —. 2016, MNRAS, 460, 1758, doi: 10.1093/mnras/stw1054
  * Hamer et al. (2019) Hamer, S. L., Fabian, A. C., Russell, H. R., et al. 2019, MNRAS, 483, 4984, doi: 10.1093/mnras/sty3459
  * Hogan et al. (2017a) Hogan, M. T., McNamara, B. R., Pulido, F., et al. 2017a, ApJ, 837, 51, doi: 10.3847/1538-4357/aa5f56
  * Hogan et al. (2015) Hogan, M. T., Edge, A. C., Hlavacek-Larrondo, J., et al. 2015, MNRAS, 453, 1201, doi: 10.1093/mnras/stv1517
  * Hogan et al. (2017b) Hogan, M. T., McNamara, B. R., Pulido, F. A., et al. 2017b, ApJ, 851, 66, doi: 10.3847/1538-4357/aa9af3
  * Hudson et al. (2010) Hudson, D. S., Mittal, R., Reiprich, T. H., et al. 2010, A&A, 513, A37, doi: 10.1051/0004-6361/200912377
  * Kalberla et al. (2005) Kalberla, P. M. W., Burton, W. B., Hartmann, D., et al. 2005, A&A, 440, 775, doi: 10.1051/0004-6361:20041864
  * Laine et al. (2003) Laine, S., van der Marel, R. P., Lauer, T. R., et al. 2003, AJ, 125, 478, doi: 10.1086/345823
  * Lakhchaura et al. (2018) Lakhchaura, K., Werner, N., Sun, M., et al. 2018, MNRAS, 481, 4472, doi: 10.1093/mnras/sty2565
  * LaRoque et al. (2006) LaRoque, S. J., Bonamente, M., Carlstrom, J. E., et al. 2006, ApJ, 652, 917, doi: 10.1086/508139
  * Li et al. (2019) Li, Y., Bryan, G. L., & Quataert, E. 2019, ApJ, 887, 41, doi: 10.3847/1538-4357/ab4bca
  * Li et al. (2018) Li, Y., Ruszkowski, M., & Tremblay, G. 2018, 854, 91, doi: 10.3847/1538-4357/aaa843
  * Makarov et al. (2014) Makarov, D., Prugniel, P., Terekhova, N., Courtois, H., & Vauglin, I. 2014, A&A, 570, A13, doi: 10.1051/0004-6361/201423496
  * Maraston (2005) Maraston, C. 2005, MNRAS, 362, 799, doi: 10.1111/j.1365-2966.2005.09270.x
  * Markevitch & Vikhlinin (2007) Markevitch, M., & Vikhlinin, A. 2007, Phys. Rep., 443, 1, doi: 10.1016/j.physrep.2007.01.001
  * Mathews (1989) Mathews, W. G. 1989, AJ, 97, 42, doi: 10.1086/114955
  * Mathews (1990) —. 1990, The Ins and Outs of Gas Flows in Elliptical Galaxies, ed. G. Fabbiano, J. S. Gallagher, & A. Renzini, Vol. 160, 265, doi: 10.1007/978-94-009-0543-6_33
  * McCourt et al. (2015) McCourt, M., O’Leary, R. M., Madigan, A.-M., & Quataert, E. 2015, 449, 2, doi: 10.1093/mnras/stv355
  * McDonald et al. (2010) McDonald, M., Veilleux, S., Rupke, D. S. N., & Mushotzky, R. 2010, ApJ, 721, 1262, doi: 10.1088/0004-637X/721/2/1262
  * McNamara & Nulsen (2007) McNamara, B. R., & Nulsen, P. E. J. 2007, ARA&A, 45, 117, doi: 10.1146/annurev.astro.45.051806.110625
  * McNamara & Nulsen (2012) —. 2012, New Journal of Physics, 14, 055023, doi: 10.1088/1367-2630/14/5/055023
  * McNamara et al. (2016) McNamara, B. R., Russell, H. R., Nulsen, P. E. J., et al. 2016, ApJ, 830, 79, doi: 10.3847/0004-637X/830/2/79
  * McNamara et al. (2000) McNamara, B. R., Wise, M., Nulsen, P. E. J., et al. 2000, ApJ, 534, L135, doi: 10.1086/312662
  * Mohr et al. (1999) Mohr, J. J., Mathiesen, B., & Evrard, A. E. 1999, ApJ, 517, 627, doi: 10.1086/307227
  * Olivares et al. (2019) Olivares, V., Salomé, P., Combes, F., et al. 2019, 631, A22, doi: 10.1051/0004-6361/201935350
  * Panagoulia et al. (2014) Panagoulia, E. K., Fabian, A. C., & Sanders, J. S. 2014, MNRAS, 438, 2341, doi: 10.1093/mnras/stt2349
  * Pasini et al. (2019) Pasini, T., Gitti, M., Brighenti, F., et al. 2019, ApJ, 885, 111, doi: 10.3847/1538-4357/ab4808
  * Pope et al. (2010) Pope, E. C. D., Babul, A., Pavlovski, G., Bower, R. G., & Dotter, A. 2010, MNRAS, 406, 2023, doi: 10.1111/j.1365-2966.2010.16816.x
  * Pulido et al. (2018) Pulido, F. A., McNamara, B. R., Edge, A. C., et al. 2018, ApJ, 853, 177, doi: 10.3847/1538-4357/aaa54b
  * Qiu et al. (2020) Qiu, Y., Bogdanović, T., Li, Y., McDonald, M., & McNamara, B. R. 2020, Nature Astronomy, 4, 900, doi: 10.1038/s41550-020-1090-7
  * Rau & Cornwell (2011) Rau, U., & Cornwell, T. J. 2011, A&A, 532, A71, doi: 10.1051/0004-6361/201117104
  * Revaz et al. (2008) Revaz, Y., Combes, F., & Salomé, P. 2008, A&A, 477, L33, doi: 10.1051/0004-6361:20078915
  * Rossetti et al. (2016) Rossetti, M., Gastaldello, F., Ferioli, G., et al. 2016, MNRAS, 457, 4515, doi: 10.1093/mnras/stw265
  * Salomé & Combes (2003) Salomé, P., & Combes, F. 2003, A&A, 412, 657, doi: 10.1051/0004-6361:20031438
  * Sanderson et al. (2009) Sanderson, A. J. R., Edge, A. C., & Smith, G. P. 2009, MNRAS, 398, 1698, doi: 10.1111/j.1365-2966.2009.15214.x
  * Sutherland & Dopita (1993) Sutherland, R. S., & Dopita, M. A. 1993, ApJS, 88, 253, doi: 10.1086/191823
  * Van Dokkum & Franx (1995) Van Dokkum, P. G., & Franx, M. 1995, AJ, 110, 2027, doi: 10.1086/117667
  * van Haarlem et al. (2013) van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2, doi: 10.1051/0004-6361/201220873
  * Voit et al. (2015) Voit, G. M., Donahue, M., Bryan, G. L., & McDonald, M. 2015, Nature, 519, 203, doi: 10.1038/nature14167


# Searches for long-duration gravitational wave transients in LIGO and Virgo
data

A. Macquet1 M.A. Bizouard1 N. Christensen1 and M. Coughlin2 1Artemis,
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Nice 06300,
France  2School of Physics and Astronomy, University of Minnesota,
Minneapolis, Minnesota 55455, USA

###### Abstract

Long-lived gravitational wave (GW) transients have received interest in the
last decade, as the sensitivitity of LIGO and Virgo. Such signals, lasting
between $10^{1}-10^{3}$s, can come from a variety of sources, including
accretion disk instabilities around black holes, binary neutron stars post-
merger, core-collapse supernovae, non-axisymmetric deformations in isolated
neutron stars, and magnetar giant flares. Given the large parameter space and
the lack of precisely modeled waveforms, searches must rely on robust
detection algorithms, which make few or no assumption on the nature of the
signal. Here we present a new data analysis pipeline to search for long-lived
transient gravitational wave signals, based on an excess cross-power statistic
computed over a network of detectors. It uses a hierarchical strategy that
allows to estimate the background quickly and implements several features
aimed to increase detection sensitivity by $\sim 30\%$ for a wide range of
signal morphology compared to an older implementation. We also report upper
limits on the GW energy emitted from a search conducted with the pipeline for
GW emission around a sample of nearby magnetar giant flares, and discuss
detection potential of such sources with second and third generation
detectors.

##  1 Presentation of an unmodeled GW search pipeline

When the GW signal searched is precisely modelled, e.g for compact binary
mergers or cosmic strings, searches usually rely on matched filter techniques,
which have optimal sensitivity. However, detection of GW transients which are
poorly modelled, often refered as GW ”bursts”, require a more agnostic
approach. Most of the searches rely on finding zones of excess power in time-
frequency representation of the data. This is the case of the STAMP
pipeline[1], for which we propose here an enhanced version, PySTAMPAS,
designed to perform all-sky / all-time GW searches at a reduced computational
cost and increased detection efficiency[2].

###  1.1 Coherent detection statistic

We consider a pair $(I,J)$ of GW detectors, and their strain time-series
$s_{I}(t)$ and $s_{J}(t)$. A simple time-frequency representation of the data
can be obtained by taking the Fourier transform of short segments of duration
$\Delta T$. We note $\tilde{s}_{I}(t;f)$ such a representation, where $t;$
denotes the start time of each segment. Following [1], an estimator for the
coherent GW power signal-to-noise ratio (SNR) in a time-frequency pixel
$(t,f)$ is

$\rm{SNR}(t;f,\hat{\Omega})=Re\left[\frac{2\,\tilde{s}_{I}^{\star}(t;f)\tilde{s}_{J}(t;f)}{\sqrt{\frac{1}{2}P_{I}(t;f)P_{J}(t;f)}}\leavevmode\nobreak\
e^{2\pi if\hat{\Omega}\cdot\Delta\vec{x}_{IJ}/c}\right],$ (1)

where $P_{I}(t;f)$ is the power spectral density of the detector’s noise and
$\Delta\vec{x}_{IJ}$ the distance between the two detectors. The GW signal is
assumed to originate from a point-like source whose direction is given by the
unit vector $\hat{\Omega}$. The phase term comes from the delay between the
two detectors of the arrival of the GW signal.

When doing an all-sky search, the direction of the source is not known a
priori, introducing a degeneracy in the computation of
$\textrm{SNR}(t;f,\hat{\Omega})$. However, a long-lived GW can spread over
several pixels, forming a cluster $\Gamma$. This breaks the degeneracy as
$\hat{\Omega}$ remains identical for each pixel. By testing a sufficient
number of sky positions, we maximize the summed SNR of the cluster

$\textrm{SNR}_{\Gamma}(\hat{\Omega})\equiv\sum\limits_{(t;f)\in\Gamma}\textrm{SNR}(t;f,\hat{\Omega}).$
(2)

Noise fluctuations in detectors can form clusters that mimic the shape of a GW
signal. To better discriminate signal from noise, we assign each cluster a
ranking statistic $p_{\Lambda}$ that reflects its significance, based on the
coherent and incoherent components of the SNR of the pixels[2].

###  1.2 Implementation of the detection algorithm

Because of the large parameter space and the amount of data to process,
unmodeled searches are computationaly expensive, and compromises have to be
made between detection efficiency and computational cost, leading to
suboptimal sensitivity. PySTAMPAS implements a hierarchical method [3] to
address computational cost issues while preserving detection sensitivity. The
different steps of these methods are described below.

#### Time-frequency maps

The data stream from each detector is split into windows of $\sim 500$s
duration, and for each window, time-frequency maps ($tf$-maps) of the
detectors’ strain data are built using short segments of duration $\Delta T$.
The segments are hann-windowed and overlap by $50\%$. In order to better
reconstruct fast frequency evoluting signals, different values of $\Delta T$
are used and the resulting maps are combined to form a multi-resolution $tf-
map$. The maps are whitened by the one-sided amplitude spectral density
$\sqrt{P_{I}(t;f)}$, which is estimated by taking the median of
$|\tilde{s}_{I}(t;f)|^{2}$ over frequency neighbouring pixels.

#### Clustering

A statistically significant GW signal will generate a track of excess power
pixels in a $tf$\- map. To identify and extract those tracks, a clustering
algorithm is run over the maps. Several types of clustering algorithms have
been implemented for un-modelled GW searches, which are adapted for different
classes of signal. To remain as agnostic as possible on the signal morphology,
we opt for a seed-based clustering algorithm. The burstegard algorithm[4]
identifies pixels of single detector $tf$-maps with autopower above a given
threshold and groups the neighbouring pixels together to form clusters (see
Fig. 1). Cluster which contain a sufficient number of pixels are kept to be
analyzed coherently at the next step of the analysis.

Figure 1: Left : multi-resolution time-frequency map of a long-duration GW
signal injected in Gaussian data. Right : cluster extracted from this $tf$-map
by the clustering algorithm.

#### Coincident analysis

Clusters extracted from one detector are matched with corresponding pixels
from the other detector’s $tf$-map. Assuming a sky direction $\hat{\Omega}$,
the coherent SNR of each pixel is computed following eq.1. Because cross-
correlation is only computed over a reduced number of pixels, it is possible
to test many sky positions (up to several hundreds), and therefore maximize
the summed quantity $\textrm{SNR}_{\Gamma}$ while keeping computational cost
affordable. The coincident triggers extracted are ranked by significance
following their coherent statistic $p_{\Lambda}$.

#### Background estimation

To assess a false-alarm probability to triggers found in coincidence, one has
to estimate the distribution of $p_{\Lambda}$ for triggers generated by
detectors’ noise (background noise). Like a lot of unmodeled searches,
PySTAMPAS uses the time-slide method to estimate background noise
distribution. The principle is to shift the data stream from one detector with
respect to the other in order to remove any coherent GW signal from the data,
while preserving any non Gaussian or non-stationary feature that can be
present in real GW detector’s data. By repeating the process for multiple
values of time-shift, it is possible to simulate a large amount of background
noise and therefore assess precisely the significance of coincident triggers.
With the hierarchical method implemented, PySTAMPAS is able to estimate the
significance of triggers up to $5\,\sigma$ for a year-long observing run.

###  1.3 Performances

We estimate the performances of the pipeline by injecting simulated signals in
the data at different amplitudes and try to recover them. To simulate a
realistic all-sky search, detectors’ response to each signal is computed
assuming a random sky localization and source’s plane inclination. We
considered a set of a dozen waveforms that span the parameter space, with
duration and frequency between $8-240$ s and $10-1800$ Hz respectively, and
diverse spectral morphology. Results on Gaussian simulated data and real data
from Advanced LIGO’s second observing run show a detection sensitivity
increased by $\sim 30\%$ in average for the set of waveforms tested, as
compared to the previous all-sky implementation of the STAMP pipeline.

##  2 Search for long-duration GW emission around Magnetar Giant Flares

###  2.1 Motivation

Magnetar Giant Flares (MGF) are highly energetic events that occur in
magnetars, young neutron stars with very large magnetic field (up to $10^{14}$
G). They usually consist in a bright millisecond flare of gamma-rays of energy
$10^{44-47}$ erg followed by pulsating tails at $\sim 10^{44}$ erg that can
last several hundred of seconds.

The recent observation of GRB 200415a, a short gamma-ray burst (GRB) in NGC
253 and its possible magnetar giant flare origin[5], along with $3$ other
short GRBs from nearby galaxies, has motivated us to perform a search for a GW
counterparts around those events with PySTAMPAS. Indeed, the large energy
emission at relatively close distance ($<5$ Mpc) could make them observable
with GW, and the lack of knowledge about the coupling between oscillation
modes and GW emission require an un-modelled search algorithm.

###  2.2 Data sample and search methodology

We considered $3$ short GRBs associated with MGF that happened between 2005
and 2007 for which data from initial LIGO detectors at Hanford (H1, H2) and
Livingston (L1) were available111LIGO and Virgo detectors were off at the time
of GRB 200415a. Since a host galaxy has been identified for each of these
events, PySTAMPAS pipeline was used in a targeted configuration, with only one
sky position tested per search. Searches were conducted over a
$[-500\,\rm{s},+500\,\rm{s}]$ around each event to account for the potential
delay between EM and possibly long-duration GW emission.

###  2.3 Results and future prospects

No significant GW candidates were found for any of the three events searched.
To place upper limits on the GW energy emitted we injected sinusoidal
waveforms damped with an exponential enveloppe with decay times between
$0.2-10$ s and central frequency between $50-500$ Hz. For two of the events,
GRB 051103 and GRB 070222, we derive upper limits at $100$ Hz of $6\times
10^{51}$ erg and $3\times 10^{50}$ erg respectively. These are within the same
order of magnitude as upper limits previously published in [6], with a factor
2 improvement. We also place an upper limit of $10^{52}$ erg for GRB 070222,
which had only been recently associated with a potential MGF. Detailed results
can be found in [7].

We also studied the prospectives of future detections using simulated data of
Advanced LIGO at design sensitivity (expected during O5) and of Einstein
Telescope. We report the results in Fig. 2 that shows in particular that
galactic MGF should be observable with Advanced LIGO if they emit up to $1\%$
of their EM energy in GW. Furthermore, Third generation detectors could push
these boundaries further and make even less energetic MGFs (potentially
associated with SGR and FRB) within detection range.

Figure 2: Estimated sensitivity of PySTAMPAS to GW energy emitted vs distance
to the source for Advanced LIGO at design sensitivity (grey band) and Einstein
Telescope (yellow band). The yellow band shows the typical electromagnetic
energy emitted by MGFs for comparison.

## References

## References

  * [1] E. Thrane et al.  Long gravitational-wave transients and associated detection strategies for a network of terrestrial interferometers.
  * [2] A. Macquet et al.  in preparation, 2021.
  * [3] E. Thrane and M. Coughlin.  Detecting Gravitation-Wave Transients at 5$\sigma$: A Hierarchical Approach.
  * [4] T. Prestegard.  Unmodeled search for long-lasting gravitational-wave signals with LIGO and studies of underground seismic noise for future gravitational-wave detectors. PhD thesis, University of Minnesota, 2016.
  * [5] E. Burns et al.  Identification of a Local Sample of Gamma-Ray Bursts Consistent with a Magnetar Giant Flare Origin.  Astrophys.J.Lett., 907(2):L28, February 2021.
  * [6] J Abadie et al.  Implications for the origin of grb 051103 from ligo observations.  Astrophys. J., 755(1):2, 2012.
  * [7] Adrian Macquet, Marie-Anne Bizouard, Eric Burns, Nelson Christensen, Michael Coughlin, Zorawar Wadiasingh, and George Younes.  Search for Long-duration Gravitational-wave Signals Associated with Magnetar Giant Flares.  arXiv e-prints, page arXiv:2105.02086, May 2021.