# Main sequence of star formation and colour bimodality considering galaxy
environment

Pius Privatus  0000-0002-6981-717X<EMAIL_ADDRESS>Department of
Physics, Dibrugarh University, Dibrugarh 786004, Assam, India  Department of
Natural Sciences, Mbeya University of Science and Technology, Iyunga, 53119,
Mbeya, Tanzania  Umananda Dev Goswami  0000-0003-0012-7549<EMAIL_ADDRESS>Department of Physics, Dibrugarh University, Dibrugarh 786004, Assam, India

###### Abstract

This study involves the use of Sloan Digital Sky Survey Data Release 12 (SDSS
DR12) also referred to as the Legacy Survey to investigate the influence of
the galaxy environment on the main sequence of star formation, colour
bimodality and the quenching of star formation rate. We classify the galaxies
according to the ratio of their emission lines and based on their environment
(isolated and non-isolated). We find that for $z\lesssim 0.09$, the fraction
of non-isolated galaxies is greater than isolated galaxies, whereas for
$z>0.09$ the opposite result is observed. Quenching is observed to be
influenced by the environment at $M_{\star}<10^{10.7}M_{\odot}$ (mostly for
the star-forming and composite galaxies), while for $M_{\star}\geq
10^{10.7}M_{\odot}$ (mostly for Seyfert galaxies and low-ionization nuclear
emission-line regions), the effect of the environment is very weak. We observe
the decrease in the slope of the star formation main sequence by $\sim\\!0.02$
dex and the intercept by $\sim\\!0.17$ dex for non-isolated galaxies in
comparison to isolated galaxies. We also find that star-forming, composite,
Seyfert galaxies and low-ionization nuclear emission-line regions form the
evolutionary pathways, where most star-forming galaxies ($\sim\\!60\%$) are
found in the blue cloud, both composite ($\sim\\!50\%$) and Seyfert
($\sim\\!49\%$) galaxies in the green valley and low-ionization nuclear
emission-line regions ($\sim 60\%$) in the red sequence. The study concludes
that the environment in which the galaxies reside influences the shape of the
star formation main sequence, quenching and hence colour bimodality especially
for star-forming and composite galaxies while for Seyfert and low-ionization
nuclear emission-line region galaxies, there is a mild impact.

Main sequence; Colour bimodality; Green valley; Galaxy environment.

##  I Introduction

The star-forming (SF) main sequence (MS) refers to a close relationship
between the rates at which stars are formed and the changes in the masses of
stars ($M_{\star}$s) in the majority of the galaxy population undergoing star
formation. The baseline of the correlation between these two parameters
changes over time, indicating that the intensity of star formation changes
significantly with time. There is considerable uncertainty regarding the slope
and dispersion of this correlation. The literature presents different values
of slope between $0.2$ and $1.2$, and the dispersion, ranging from $0.3$ to
$0.6$ [1]. The predominant source of uncertainty arises from the biases
introduced through various factors. These may include the criteria employed
for selecting the SF galaxy population, the indicators used for the star
formation rates (SFRs), the fitting method applied (such as mean, median, or
mode), and the non-linearity of the relation, where the slope varies across
different mass scales.

A comprehensive understanding of how the MS evolves is crucial for capturing
fundamental insights into how galaxies grow. In this regard, the slope of the
correlation between SFR and $M_{\star}$ provides information on how star
formation activity changes across diverse stellar masses, while the dispersion
of this correlation unveils the degree of unpredictability in the history of
gas accretion and the efficiency of star formation [2, 3, 4, 5, 6, 7].
Therefore, comprehending the trajectory of galaxies over time in the SFR
versus $M_{\star}$ plane serves as a potential tool for identifying the
processes responsible for the gradual decline in star formation activity as
time progresses.

The fitting parameters of the MS provide information about the nature of
stellar mass buildup and star formation where the slope is related to the
slope of low mass SF galaxies and the normalization of the MS evolution is
thought to be due to the evolution of gas density with redshift [10, 1, 9, 8].
Refs. [11, 12] revealed that the MS requires a more complex stellar mass fit.
By selecting the SF galaxies from the seventh release of SDSS data with the
range of redshift $0.02<z<0.085$ Ref. [13] obtained the absence of deviation
from the simple power-law MS relation in the local galaxies. This study
revealed that the deviation originates from the inconsistency in SF galaxies’
selection, however in this study the active galactic nucleus (AGN) galaxies
were excluded from the analysis and only the SF galaxies were discussed.

Moreover, the galaxy colour versus stellar mass which serves as a potential
tool for identifying the mechanism deriving this bimodality [14], is another
very important subject to undertake. Different studies tried to include the
AGNs in the colour versus stellar mass diagram and obtained that these
galaxies populate the red sequence and the state between blue cloud and red
sequence which is known as green valley. For example, Ref. [15] obtained that
the inclusion of AGNs in these diagrams enable to obtain the role of AGN’s
feedback in the quenching of SFR of early type galaxies. Ref. [16] obtained
that the AGNs are populated on a distinct region of the colour versus stellar
mass diagram referred to as the red sequence or top of the blue cloud,
believing that a key stage in the massive galaxies’ evolution is the quenching
of SFR at which there is the migration of blue cloud to the red sequence. Ref.
[16] outlined that AGNs may be the cause or they maintain the quenching
process. Furthermore, the ongoing star formation is not a necessary condition
for the AGN activity since the black holes’ accretions are observed when the
star formation has been terminated.

In a simplified representation, the process guiding these transformation can
be broadly categorized as ‘internal’ and ‘external’. The ongoing debate
revolves around the extent to which each of these scenarios influences galaxy
properties. The internal process involves the co-evolution of a galaxy and its
black hole. The majority of galaxy formation models observe AGN feedback to be
a primary mechanism affecting the main sequence [17]. Although this mechanism
may have an impact in the case of massive galaxies exceeding $10^{11}$ solar
masses, there is a lack of observed feedback effects for the majority of the
SF galaxies [18]. On the other hand, external processes involve processes
associated with interactions either among more than one galaxy or with the
surrounding environment. Both internal and external processes have undergone
extensive studies in the literature [20, 21, 22, 19, 23, 24, 25, 26, 27, 28].

Ref. [26] observed morphology quenching to be a reason for the reduction of
star formation activities in galaxies mainly as a bulge formation which
stabilizes the gas disc against the gravitational instabilities. Ref. [27]
obtained that morphology and environment have a combined role in slowing down
the star formation activities in galaxies. Furthermore, they observed that a
long-time scale environmental effect appears at low redshift. Ref. [28]
suggested that the quenching of SFR is mainly due to internal and linked with
bulge growth. However, the existence of the relation between morphology and
density may provide a different turn in the relation of SFR, stellar mass and
the environment where a particular galaxy resides. These studies aim to
discern the mechanism or combination of mechanisms that lead to the quenching
of star formation processes in galaxies, thereby influencing their evolution.
Despite the fact that to build a global picture of the galaxy evolution
involves understanding of how the evolution of galaxy and environment are
related. Currently, there is still a deficiency of comprehensive studies
relating the positioning of galaxies with respect to the MS for different
environments.

In this paper, our primary goal is to investigate whether the positioning of
galaxies on the star formation MS depends on the galaxy environment by using
isolated and non-isolated galaxy samples. Additionally, we aim to quantify the
evolutionary trajectory of composite, Seyfert (Seyfert 1 not included), and
low-ionization nuclear emission-line region (LINER) galaxies with respect to
the colour-stellar mass diagram, seeking insights into whether galaxy
environment plays a role in shaping galaxy colour bimodality. The width of the
MS is defined as $\pm\,0.3$ dex ranging from the best fit line of SFR–M⋆
plane. This approach enables us to assess the linearity of the MS and explore
the dispersion of the MS.

The structure of this paper is as follows: In the next section, we explain the
source of data and the method of getting samples. Section III is dedicated to
presenting the results. In Section IV the results are discussed. Section V
presents the summary and conclusion. Cosmological constants are adopted from
Ref. [29], wherein the dark energy density parameter $\Omega_{\Lambda}=0.692$,
Hubble constant of $H_{0}=67.8$ km s-1 Mpc-1 and the matter density parameter
$\Omega_{m}=0.308$ are recorded.

##  II Data and Analysis

In this research work the catalog data extracted from the twelveth release of
Sloan Digital Sky Survey (SDSS DR12) as detailed in Refs. [30, 31]
specifically, the value added catalog detailed in Refs. [33, 32] was employed.
This catalog contains $584\,449$ galaxies which are either in groups, clusters
or isolated up to $z=0.2$, all brighter than Petrosian r-band magnitude of
$17.77$. We performed $1$ arcsec cross-correlation between the data obtained
from the methods outlined in Ref. [32] and the Max Planck Institute for
Astrophysics and Johns Hopkins University (MPA-JHU) to obtain a total of
$571\,459$ galaxies that satisfy the requisite cut-off condition. We applied a
signal-to-noise condition of $3$ for [OIII]$\lambda$5007, [NII]$\lambda$6584
spectral lines, and the Balmer recombination lines including H$\alpha$ and
H$\beta$ resulting in a total of $274\,517$ galaxies.

The stellar masses obtained from the MPA-JHU team were calculated from the
Bayesian approach as detailed in Ref. [34]. Stellar mass calculation within
the SDSS spectroscopic fiber aperture relies on the fiber magnitudes, whereas
the total stellar mass is determined using the model magnitudes. The MPA-JHU
total SFR used in this study was derived from the MPA-JHU database and
estimated using the methods of Refs. [35, 36] with adjustments made for non SF
galaxies. The MPA-JHU team uses the H$\alpha$ calibration [37] to determine
the SFR for galaxies classed as SF. In contrast to the approach taken by Ref.
[35], the MPA-JHU team applied aperture corrections for SFR by fitting the
photometric data from the outer regions of the galaxies. Specifically, for SFR
computation, Ref. [35] outlined the calculation within the galaxy fiber
aperture. It is important to keep in mind that the region beyond the fiber SFR
is estimated using the methods of Ref. [38]. Furthermore, for the case of AGN
and weak emission line galaxies, SFR was determined using photometry. Although
the output for stellar mass and SFR are presented in probability distribution
functions at $2.5\%,16\%,50\%,84\%,$ and $97.5\%$ values, in this study we use
the median estimates ($50\%$) of the probability distributions. The galaxies
having no $M_{\star}$ and SFR and the specific star formtaion rate (SSFR)
measurements were removed from the sample making a total of $268\,954$
galaxies.

Figure 1: The diagnostic diagram for all the classified galaxies with the
scatter plot (left panel) and the contour plot (right panel).

We classify the galaxies using the $N\\!I\\!I$ diagnostic diagram as detailed
in Ref. [41, 40, 39, 15], which is based on the following equations:

$\displaystyle\log_{10}\left({\frac{[{\it OIII}]}{{\it H\beta}}}\right)$
$\displaystyle<0.61\left(\log_{10}\left({\frac{[{\it NII}]}{{\it
H\alpha}}}\right)-0.05\right)^{-1}+1.3,$ (1)
$\displaystyle\log_{10}\left({\frac{[{\it OIII}]}{{\it H\beta}}}\right)$
$\displaystyle<0.61\,\left(\log_{10}\left({\frac{[{\it NII}]}{{\it
H\alpha}}}\right)-0.47\mbox{}\right)^{-1}+1.19,$ (2)
$\displaystyle\log_{10}\left({\frac{[{\it OIII}]}{{\it H\beta}}}\right)$
$\displaystyle>0.61\,\left(\log_{10}\left({\frac{[{\it NII}]}{{\it
H\alpha}}}\right)-0.47\mbox{}\right)^{-1}+1.19,$ (3)
$\displaystyle\log_{10}\left({\frac{[{\it OIII}]}{{\it H\beta}}}\right)$
$\displaystyle-0.45<1.05\left(\log_{10}\left({\frac{[{\it NII}]}{{\it
H\alpha}}}\right)\mbox{}\right).$ (4)

Here, SF galaxies were obtained by Eq. (1), composite galaxies by Eq. (2), AGN
galaxies by Eq. (3), while the Seyfert and LINER galaxies were discerned from
AGNs by using Eq. (4). After application of all stated procedures, the sample
contains $170\,797$ ($63\%$) SF, $58\,763$ ($22\%$) composite, $12\,536$
($5\%$) Seyfert and $26\,858$ ($10\%$) LINERs. The distribution of galaxies is
shown in the diagnostic diagram in Fig. 1.

Figure 2: Step plot showing redshift distribution for isolated galaxies (iso)
and non-isolated galaxies (nis) for the star-forming (blue top-left panel),
composite (green top-right panel), Seyfert (red bottom-left panel) and LINER
(orange bottom-right panel) galaxies.

The galaxies with no neighbour (with $N\\!gal=1$ flag) are isolated, while the
galaxies with more than one neighbour (with $N\\!gal\geq 2$ flag) are non-
isolated. The final dataset contains $93\,727$ ($55\%$) isolated, $77\,070$
($45\%$) non-isolated SF galaxies; $30\,652$ ($52\%$) isolated, $28\,111$
($48\%$) non-isolated composite galaxies; $6\,643$ ($53\%$) isolated, $5\,893$
($47\%$) non-isolated Seyfert galaxies; and $11\,355$ ($42\%$) isolated,
$15\,503$ ($58\%$) non-isolated LINER galaxies. The distribution of galaxies
according to redshift in the final sample for isolated galaxies and non-
isolated galaxies is shown in Fig. 2. From this figure, it is observed that
the number of non-isolated galaxies in the lower redshift range ($z<0.09$) is
larger than the isolated, while in the higher redshift range ($z\geq 0.09$)
the number of isolated galaxies is higher than the number of non-isolated
galaxies. This indicates that isolated galaxies are preferentially found at
higher redshift while non-isolated galaxies are found at lower redshift.

##  III Results

###  III.1 The Main Sequence (MS)

The relationship between the SFR and stellar mass $M_{\star}$ serves as the
tracer on how the stars are formed in relation to the stellar mass within a
galaxy. The MS is generally characterized by such a relation as given by

$\log_{10}(\mbox{SFR})=\beta\,\log_{10}(M_{\star})+\alpha,$ (5)

where $\beta$ and $\alpha$ are slope and intercept, respectively. Aiming at
studying how the environment affects the SFR relative to the $M_{\star}$ for
the SF, composite, Seyfert and LINER classified sub-samples of galaxies, we
generate the equation of the best-fitted line of the SF galaxies (the equation
of MS based on Eq. (5)). This best-fitted line is used as a reference to
understand the behaviour of SFR relative to $M_{\star}$ for all other sub-
samples of galaxies residing in different environments. Figs. 3, 4, 5 and 6
show the distributions of SFR with respect to $M_{\star}$ of the SF,
composite, Seyfert and LINER galaxies for isolated and non-isolated cases
along with the corresponding best-fitted MS line of the SF galaxies. The width
of the MS of $\pm 0.3$ dex (dashed lines) in these figures is adopted from
Refs. [14, 19, 9, 42]. The figures show the positions of galaxies of all sub-
samples with respect to the MS and the corresponding numbers of galaxies with
their percentages are shown in Table 1. Furthermore, the galaxies are divided
into the redshift slices containing a binning size of $0.01$ aiming to
decrease the Malmquist bias as detailed in Ref. [43]. Table 2 shows the
Kolmogorov-Smirnov (KS) test for all redshift slices, a test that uses
statistical probability to assess the degree of similarity between the two
independent distributions. The larger probability value (P $>0.05$) indicates
that there is no significant difference between the distributions, which
implies that the two distributions are the same, while a small probability
value (P $<0.05$) indicates the existence of a significant difference between
the distributions, which implies that the two distributions are different
[44].

(a) isolated

(b) non-isolated

Figure 3: Scatter plots showing SFR as the function of stellar mass for
isolated (left plot) and non-isolated (right plot) star-forming galaxies. The
violet, indigo, green, yellow and red colours in the left plot respectively
show the $0\%$, $25\%$, $50\%$, $75\%$ and $100\%$ of the probability
distributions for isolated star-forming galaxies, while the blue, cyan, green,
yellow and red colours in right plot respectively show the $0\%$, $25\%$,
$50\%$, $75\%$ and $100\%$ of the probability distributions for non-isolated
star-forming galaxies. The right-bottom cross bars on each diagram correspond
to $2\sigma$ Poissonian error in the calculated values of stellar mass and the
SFR.

The general equations of the best-fitted line for the isolated and non-
isolated galaxies for our data are respectively given by

$\displaystyle\log_{10}(\mbox{SFR})$
$\displaystyle=0.72\,\log_{10}(M_{\star})-7.06,$ (6)
$\displaystyle\log_{10}(\mbox{SFR})$
$\displaystyle=0.70\,\log_{10}(M_{\star})-6.89.$ (7)

Table 3 indicates the percentage change between isolated and non-isolated
galaxies above MS, within MS and below MS for SF (column 2), composite (column
3), Seyfert (column 4) and LINER (column 5) galaxies. This table is used as
the tracer on how the positioning of galaxies in the plane of SFR versus
$M_{\star}$ and hence the SFR are affected by changing the environment in
which the galaxies reside.

(a) isolated

(b) non-isolated

Figure 4: Scatter plots showing SFR as the function of stellar mass for
isolated (left plot) and non-isolated (right plot) composite galaxies. The
violet, indigo, green, yellow and red colours in the left plot respectively
show the $0\%$, $25\%$, $50\%$, $75\%$ and $100\%$ of the probability
distributions for isolated composite galaxies, while the blue, cyan, green,
yellow and red colours in the right plot show respectively the $0\%$, $25\%$,
$50\%$, $75\%$ and $100\%$ of the probability distributions for non-isolated
composite galaxies. The right bottom cross bars on each diagram correspond to
$2\sigma$ Poissonian error in the calculated values of stellar mass and the
SFR.

(a) isolated

(b) non-isolated

Figure 5: Scatter plots showing SFR as the function of stellar mass for
isolated (left plot) and non-isolated (right plot) Seyfert galaxies. The
violet, indigo, green, yellow and red colours in the left plot respectively
show the $0\%$, $25\%$, $50\%$, $75\%$ and $100\%$ of the probability
distributions for isolated Seyfert galaxies, while the blue, cyan, green,
yellow and red colours in the right plot show respectively the $0\%$, $25\%$,
$50\%$, $75\%$ and $100\%$ of the probability distributions for non-isolated
Seyfert galaxies. The right bottom cross bars on each diagram correspond to
$2\sigma$ Poissonian error in the calculated values of stellar mass and the
SFR.

(a) isolated

(b) non-isolated

Figure 6: Scatter plots showing SFR as the function of stellar mass for isolated (left plot) and non-isolated (right plot) LINERs The violet, indigo, green, yellow and red colours in the left plot respectively show the $0\%$, $25\%$, $50\%$, $75\%$ and $100\%$ of the probability distributions for isolated LINERs, while the blue, cyan, green, yellow and red colours in the right plot show respectively the $0\%$, $25\%$, $50\%$, $75\%$ and $100\%$ of the probability distributions for non-isolated LINERs. The right bottom cross bars on each diagram correspond to $2\sigma$ Poissonian error in the calculated values of stellar mass and the SFR. Table 1: The number of galaxies within (MS), above (Above MS), and below (Below MS) the star-forming main sequence for isolated (iso) and non-isolated (nis) environments. | Star-forming (%) | Composite (%) | Seyfert (%) | LINERs (%)
---|---|---|---|---
Position | iso | nis | iso | nis | iso | nis | iso | nis
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
MS | $65164(69.53)$ | $50614(65.67)$ | $8418(27.46)$ | $6630(23.59)$ | $1376(20.71)$ | $1123(19.06)$ | $433(3.81)$ | $391(2.52)$
Above MS | $16720(17.84)$ | $12915(16.76)$ | $304(0.99)$ | $275(0.98)$ | $41(0.62)$ | $36(0.61)$ | $20(0.18)$ | $14(0.09)$
Below MS | $11843(12.64)$ | $12541(17.57)$ | $21930(71.55)$ | $21206(75.44)$ | $5226(78.67)$ | $4734(80.33)$ | $10902(96.01)$ | $15098(97.39)$
Total | $93727(100)$ | $77070(100)$ | $30652(100)$ | $28111(100)$ | $6643(100)$ | $5893(100)$ | $11355(100)$ | $15503(100)$
Table 2: The Kolmogorov-Smirnov probability values for stellar mass P
($M_{\star}$) and star formation rate P (SFR) within the redshift bin size of
$0.01$.

| P ($M_{\star}$) | P (SFR)
---|---|---
Redshift | Star-forming | Composite | Seyfert | LINERs | Star-forming | Composite | Seyfert | LINERs
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
$z<0.04$ | $2.3\text{\times}{10}^{-22}$ | $2.16\text{\times}{10}^{-10}$ | $0.0109$ | $9.54\text{\times}{10}^{-16}$ | $3.91\text{\times}{10}^{-07}$ | $5.54\text{\times}{10}^{-18}$ | 0.324 | $9.416\text{\times}{10}^{-06}$
$0.04\leq z<0.05$ | $5.07\text{\times}{10}^{-09}$ | $5.16\text{\times}{10}^{-07}$ | $0.055$ | $6.6\text{\times}{10}^{-10}$ | $3.34\text{\times}{10}^{-08}$ | $8.12\text{\times}{10}^{-08}$ | $0.774$ | $0.8403$
$0.05\leq z<0.06$ | $6.48\text{\times}{10}^{-07}$ | $0.000122$ | $0.0139$ | $4.71\text{\times}{10}^{-10}$ | $0.000485$ | $1.83\text{\times}{10}^{-10}$ | $0.277$ | $0.08203$
$0.06\leq z<0.07$ | $2.84\text{\times}{10}^{-11}$ | $2.75\text{\times}{10}^{-07}$ | $0.629$ | $1.82\text{\times}{10}^{-15}$ | $7.39\text{\times}{10}^{-05}$ | $6.26\text{\times}{10}^{-09}$ | $1.7\text{\times}{10}^{-05}$ | $0.0263$
$0.07\leq z<0.08$ | $2.35\text{\times}{10}^{-12}$ | $1.23\text{\times}{10}^{-13}$ | $0.0158$ | $6.66\text{\times}{10}^{-10}$ | $4.93\text{\times}{10}^{-05}$ | $5.22\text{\times}{10}^{-08}$ | $0.0192$ | $0.1355$
$0.08\leq z<0.09$ | $2.17\text{\times}{10}^{-08}$ | $4.48\text{\times}{10}^{-09}$ | $0.153$ | $2.43\text{\times}{10}^{-08}$ | $0.00233$ | $4.7\text{\times}{10}^{-05}$ | $0.401$ | $0.1036$
$0.09\leq z<0.10$ | $7.9\text{\times}{10}^{-09}$ | $2.67\text{\times}{10}^{-16}$ | $0.00465$ | $3.64\text{\times}{10}^{-09}$ | $0.00988$ | $0.000148$ | $0.742$ | $0.04011$
$0.10\leq z<0.11$ | $0.00027$ | $1.22\text{\times}{10}^{-10}$ | $0.518$ | $6.36\text{\times}{10}^{-09}$ | $0.0067$ | $0.0162$ | $0.412$ | $0.00207$
$0.11\leq z<0.12$ | $0.000166$ | $6.98\text{\times}{10}^{-06}$ | $0.000509$ | $0.000901$ | $0.0107$ | $0.000455$ | $0.0254$ | $0.8636$
$0.12\leq z<0.13$ | $8.13\text{\times}{10}^{-05}$ | $3.07\text{\times}{10}^{-05}$ | $0.173$ | $1.55\text{\times}{10}^{-09}$ | $0.006$ | $5.65\text{\times}{10}^{-05}$ | $0.00857$ | $0.02497$
$0.13\leq z<0.14$ | $0.000142$ | $4.5\text{\times}{10}^{-11}$ | $0.12$ | $7.32\text{\times}{10}^{-08}$ | $0.002$ | $2.66\text{\times}{10}^{-05}$ | $0.599$ | $0.01283$
$0.14\leq z<0.15$ | $0.0492$ | $0.000808$ | $0.00162$ | $0.00602$ | $0.0047$ | $0.0237$ | $0.19$ | $0.04616$
$0.15\leq z<0.16$ | $0.0111$ | $6.93\text{\times}{10}^{-05}$ | $0.129$ | $0.0225$ | $0.395$ | $0.0508$ | $0.216$ | $0.02163$
$0.16\leq z<0.17$ | $0.184$ | $0.000254$ | $0.731$ | $0.0206$ | $0.913$ | $0.541$ | $0.0246$ | $0.003163$
$z\geq 0.17$ | $0.2$ | $0.000803$ | $0.717$ | $0.000157$ | $0.177$ | $0.0744$ | $0.2$ | $0.04614$

Table 3: The percentage change ($\bigtriangleup$ (%)) of isolated and non-
isolated galaxies within MS, above MS and below MS.

|  $\bigtriangleup$ (%)
---|---
Position | Star-forming | Composite | Seyfert | LINERs
(1) | (2) | (3) | (4) | (5)
MS | $3.86$ | $3.87$ | $1.65$ | $1.29$
Above MS | $1.08$ | $0.01$ | $0.01$ | $0.09$
below MS | $4.93$ | $3.89$ | $1.66$ | $1.38$
Total | $9.88\sim 10$ | $7.77\sim 8$ | $3.32\sim 3$ | $2.76\sim 3$

###  III.2 Quenching and Colour Bimodality

Galaxies can be categorized into two groups: those actively forming stars,
appearing blue, and those lacking significant star formation, appearing red.
The galaxies initially fall into blue sub-category and then they change
gradually to red [45, 46]. It is clear to say that evolution from one category
to another must involve processes which quench their rate of forming new stars
from the blue cloud passing the intermediate stage (green valley) to the red
sequence [47, 48]. The factors for this transformation may be due to internal
mechanisms like negative feedback from the AGNs and the galaxy environment. To
study the effect of the environment on quenching, Table 4 shows the number of
galaxies with respect to the green valley (GV) for isolated and non-isolated
galaxies respectively, and the position of galaxies on the colour against the
stellar mass diagrams are shown in Figs. 7(a), 7(b), 7(c) and 7(d) for the SF,
composite, Seyfert and LINER galaxies respectively. The width of the GV is
derived following the Ref. [49], which is obtained from equations,

$\displaystyle u-r$ $\displaystyle=-0.24+0.25\times M_{\star},$ (8)
$\displaystyle u-r$ $\displaystyle=-0.75+0.25\times M_{\star}.$ (9)

Here $u$ and $r$ magnitudes were derived from the SDSS database with
extinction corrected, stellar mass is in the unit of $M_{\odot}$. Table 5
indicates the percentage change between isolated and non-isolated galaxies
above GV, within GV and below GV for the SF (column 2), composite (column 3),
Seyfert (column 4) and LINERs (column 5). This table trace how the positioning
of galaxies in the colour against stellar mass diagram and hence quenching are
affected by changing the galaxies’ environment.

(a) star-forming

(b) composite

(c) Seyfert

(d) LINERs

Figure 7: Distribution of galaxy’s rest-frame colour against stellar mass for star-forming (top-left plot), composite (top-right plot), Seyfert (bottom-left plot) and LINER (bottom-right plot) galaxies. The filled contour plots are for the isolated galaxies while the non-filled contour plots are for the non-isolated galaxies. Contours are drawn at $25\%$, $50\%$, $75\%$ and $100\%$ of the maximum density of each classified galaxy. The cross bars correspond to $2\sigma$ Poissonian error in the calculated values of stellar mass and rest-frame colour. Table 4: The number of galaxies within the green valley (GV), above the green valley (Above GV), and below the green valley (Below GV) for isolated (iso) and non-isolated (nis) galaxies. | Star-forming (%) | Composite (%) | Seyfert (%) | LINERs (%)
---|---|---|---|---
Position | iso | nis | iso | nis | iso | nis | iso | nis
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
GV | $28401(30.30)$ | $26810(34.79)$ | $15754(51.40)$ | $13643(48.53)$ | $3284(49.44)$ | $2857(48.48)$ | $4234(37.29)$ | $5198(33.53)$
Above GV | $5745(6.13)$ | $6394(8.29)$ | $8411(27.44)$ | $10206(36.31)$ | $2228(33.53)$ | $2331(39.56)$ | $6524(57.45)$ | $9756(62.93)$
Below GV | $59581(63.57)$ | $43866(56.92)$ | $6487(21.16)$ | $4262(15.16)$ | $1131(17.03)$ | $705(11.96)$ | $597(5.26)$ | $549(3.54)$
Total | $93727(100)$ | $77070(100)$ | $30652(100)$ | $28111(100)$ | $6643(100)$ | $5893(100)$ | $11355(100)$ | $15503(100)$
Table 5: The percentage change ($\bigtriangleup$ (%)) of isolated and non-
isolated galaxies within the green valley (GV), above the green valley (Above
GV) and below the green valley (below GV).

|  $\bigtriangleup$ (%)
---|---
Position | Star-forming | Composite | Seyfert | LINERs
(1) | (2) | (3) | (4) | (5)
GV | $4.49$ | $2.87$ | $0.96$ | $3.76$
Above GV | $2.26$ | $8.87$ | $2.03$ | $4.48$
below GV | $8.85$ | $6.00$ | $5.07$ | $0.76$
Total | $15.60\sim 16$ | $17.74\sim 18$ | $8.06\sim 8$ | $9.00\sim 9$

Table 6: The Kolmogorov-Smirnov probabilities for colour P ($u-r$) within the
redshift binning size of $0.01$ involving star-forming, composite, Seyfert and
LINER galaxies.

| P ($u-r$)
---|---
Redshift | Star-forming | Composite | Seyfert | LINERs
(1) | (2) | (3) | (4) | (5)
$z<0.04$ | $2.25\text{\times}{10}^{-51}$ | $3.16\text{\times}{10}^{-18}$ | $0.00151$ | $6.06\text{\times}{10}^{-04}$
$0.04\leq z<0.05$ | $3.42\text{\times}{10}^{-16}$ | $9.45\text{\times}{10}^{-11}$ | $0.661$ | $0.536$
$0.05\leq z<0.06$ | $3.68\text{\times}{10}^{-17}$ | $1.39\text{\times}{10}^{-07}$ | $0.179$ | $4.65\text{\times}{10}^{-05}$
$0.06\leq z<0.07$ | $2.84\text{\times}{10}^{-11}$ | $9.42\text{\times}{10}^{-13}$ | $3.54\text{\times}{10}^{-07}$ | $0.611$
$0.07\leq z<0.08$ | $2.52\text{\times}{10}^{-20}$ | $3.68\text{\times}{10}^{-12}$ | $0.0294$ | $0.467$
$0.08\leq z<0.09$ | $2.54\text{\times}{10}^{-05}$ | $4.47\text{\times}{10}^{-11}$ | $0.966$ | $0.116$
$0.09\leq z<0.10$ | $1.04\text{\times}{10}^{-07}$ | $5.9\text{\times}{10}^{-11}$ | $0.0487$ | $0.314$
$0.10\leq z<0.11$ | $0.00859$ | $4.17\text{\times}{10}^{-06}$ | $0.224$ | $0.475$
$0.11\leq z<0.12$ | $0.00142$ | $2.13\text{\times}{10}^{-06}$ | $0.053$ | $0.0087$
$0.12\leq z<0.13$ | $0.000105$ | $6.25\text{\times}{10}^{-06}$ | $0.013$ | $0.481$
$0.13\leq z<0.14$ | $6.84\text{\times}{10}^{-05}$ | $3.77\text{\times}{10}^{-06}$ | $0.446$ | $0.011$
$0.14\leq z<0.15$ | $0.00135$ | $0.000201$ | $0.0421$ | $0.0081$
$0.15\leq z<0.16$ | $0.0297$ | $0.00409$ | $0.00346$ | $0.0821$
$0.16\leq z<0.17$ | $0.0788$ | $0.00759$ | $0.724$ | $0.0161$
$z\geq 0.17$ | $0.0547$ | $0.019$ | $0.878$ | $0.0504$

’

Figure 8: Variation of median star formation rate versus the reddening-
corrected rest-frame colour (left panel), and median stellar mass versus the
reddening-corrected rest-frame colour (right panel) for star-forming (blue
points), composite (green points), Seyfert (red points) and LINER (indigo
points) galaxies. The error bar corresponds to $1\sigma$ Poissonian error in
SFR and stellar mass. The solid lines are used to connect the median values of
all these subclasses of galaxies residing in isolated and non-isolated
environments respectively.

##  IV Discussion

From columns (2), (3) and (5) of Table 2 it is observed that the SF, composite
and LINER stellar masses depend on the environment resulting from having
probability values less than $0.05$ that is $\sim\\!0.02$, $\sim\\!2.95\times
10^{-4}$, and $\sim\\!0.02$ on average, respectively. The relationships of
their masses with the environment decrease with the increase in redshift. From
columns (6) and (7) of Table 2 it is observed that the SF and composite SFR
depend on the environment for low redshift ($z\lesssim 0.15$) although at
higher redshift ($z>0.15$) they are only mildly influenced by the environment.
From column (9), we observe that there is a weak dependence of LINER SFR on
the environment across all redshift ranges. This result is supported by the
findings from simulations based on the density approach reported in Ref. [50].
From columns (4) and column (8) of this table it is observed that the Seyfert
galaxies’ stellar mass and SFR weakly depend on the environment, this results
from having probability value of $\sim\\!0.2$ and $\sim\\!0.3$ on average for
$M_{\star}$ and SFR respectively, which are much larger than $0.05$ (the
standard probability value in statistics). This implies that there is no
significant difference in measurements of $M_{\star}$ and SFR between the two
corresponding distributions.

In summary, composite galaxies have lower $M_{\star}$ and higher SFR than
Seyfert which has lower $M_{\star}$ and higher SFR than LINERs. The stellar
mass of SF, composite, and LINERs depends on the environment while the stellar
mass of Seyfert weakly depends on the environment. The SFR of SF and composite
galaxies depends on the environment at lower redshift (in this case $z\lesssim
0.15$), for higher redshift (in this case $z>0.15$) the dependence of SFR on
the environment for SF and composite galaxies is very weak. The SFR of Seyfert
and LINERs weakly depend on the environment following the evidence from the KS
probability values in Table 2. These results support the findings in Refs.
[20, 21, 22, 19, 23, 24] indicating that the SF galaxies in dense regions of
the universe have suppressed SFRs. However, these results disagree with
findings by Ref. [25] using the density approach which observed a strong
dependence of AGNs stellar mass and SFR on environment. On the other hand, the
study agrees with Ref. [25] on one side that the fraction of low-density AGNs
is larger than the sample of higher-density AGNs. This study revealed that the
fact is valid for isolated and non-isolated galaxies at lower redshift (in
this case $z>0.09$) as shown in Fig. 2, the fraction of non-isolated galaxies
is higher than isolated galaxies. For the case of $z\lesssim 0.09$, the
fraction of isolated Seyfert and LINERs (AGNs) is larger than the sample of
non-isolated AGNs. This relation is not only valid for AGNs but also SF and
composite galaxies. The study revealed that non-isolated galaxies reside
mostly at the lower redshift while the isolated galaxies reside at higher
redshift for all the classified galaxies.

The equations of the main sequence for SF galaxies in Fig. 3 show a tight
relationship between the $\log(\mbox{SFR})$ and $\log(M_{\star})$ as explained
in Section III. From Fig. 4, 5, 6 it is observed that the loci of SFR versus
stellar mass for composite, Seyfert and LINER galaxies respectively, are
almost perpendicular to the SF MS. Composite galaxies having extended interval
in SFR, $+1$ dex above to $-2$ dex below the MS, Seyfert lie below the MS
between the SFR range $0.5$ to $-2.5$ dex, while LINERs lie more below the MS
between the SFR range $0$ to $-2.5$ dex. These results are consistent with the
findings by Refs. [19, 1, 14, 51]. It is observed that isolated SF MS in Fig.
3(a) deviate from the non-isolated SF MS in Fig. 3(b) by $0.02$ dex and $0.17$
dex in slope and intercept, respectively. This indicates that the SFR
decreases for non-isolated galaxies when compared to isolated galaxies. From
Fig. 4 it is evident that composite galaxies possess stellar masses greater
than $10^{10}$$M_{\odot}$, unlike the SF galaxies, the SFR of composite
galaxies shows almost no relation with $M_{\star}$. The shapes of composite
galaxies distribution stretch beyond the highest point of the SF MS, this area
with high SFRs may indicate galaxies experiencing vigorous bursts of star
formation, showing consistent results with Refs. [54, 52, 53, 10]. The
composite galaxies span from the higher mass part of the main sequence to a
minimal active star formation region. Composite galaxies have a greater
average stellar mass compared to purely SF galaxies, yet their average SFR is
lower means the onset of star-formation suppression may have begun during the
composite stage, this is consistent with the results from Refs. [55, 14]. From
Fig. 6 it is evident that most LINERs fall below the SF MS, this indicates
that they have very little current star formation activities compared to
Seyfert galaxies as shown by Fig. 5, which is consistent with the results from
Refs. [39, 14] that LINERs are mostly observed in the older stellar
population.

Observations from Table 1 indicate that the non-isolated environment increases
the rate of quenching, this is indicated by the higher percentage of galaxies
below the MS when compared with the isolated environment. Furthermore, the
lower percentage of the number of galaxies within and above the MS for the
non-isolated environment indicates the decrease of SFR for this environment
compared to the isolated one. From Table 3 it is observed that the total
percentage change of galaxies in isolated and non-isolated environments for SF
galaxies is $\sim\\!10\%$, for composite is $\sim\\!8\%$, for Seyfert and
LINERs is only $\sim\\!3\%$ indicating that the quenching of SF and composite
galaxies depends on the environment while the quenching of the AGNs (Seyfert
and LINERs) have very weak dependence on the environment. This result of the
weak dependance of SFR for Seyfert and LINERs on the environment is also
indicated in Table 2 showing higher KS probabilities on these galaxies.

The u-r colour against $M_{\star}$ diagrams in Figs. 7(a), 7(b), 7(c), and
7(d) reinforce the relation with respect to the MS in Figs. 3, 4, 5 and 6,
respectively. The composite galaxies are more redder and massive than the SF,
the Seyfert galaxies are more massive and redder than composite, and LINERs
have the highest stellar mass and most redder, this observations are in line
with the finding by Ref. [14]. Table 4, indicates that the non-isolated
environment increases the rate of quenching, this is indicated by the higher
percentage of galaxies above the green valley (Above GV) and within the green
valley (GV) when compared with the isolated environment. Furthermore the lower
percentage of number of galaxies below the green valley for non-isolated
galaxies when compared to isolated galaxies indicates the decrease of SFR for
non-isolated galaxies. From Table 5 it is observed that the total percentage
change of galaxies in isolated and non-isolated environments for SF galaxies
is $\sim\\!16\%$, for composite is $\sim\\!18\%$, indicating that quenching of
SF and composite galaxies depends on the environment supported by Fig. 7(a)
and 7(b) whereby the contour plot for the non-isolated galaxies (non-filled
contour) are seen to diverge from the isolated SF galaxies (filled contour).
For Seyfert and LINERs, the percentage change is only $\sim\\!8\%$ and
$\sim\\!9\%$ respectively, indicating that quenching of AGNs (Seyfert and
LINERs) has a weak dependence on the environment. From Figs. 7(c) and 7(d) it
is observed that the colour against stellar mass relationships for Seyfert and
LINERs show deviation for isolated (filled contour) and non-isolated (non-
filled contour), but the deviation is insignificant and less than SF and
composite galaxies as observed in Table 5, indicating weak dependence of
Seyfert and LINERs on the environment. Again these results arise from the weak
dependence of SFR for Seyfert and LINERs on the environment supporting the
evidence derived from columns (8) and (9) of Table 2 and columns (4) and (5)
of Table 3. Furthermore, all these results are supported by the KS probability
values for u-r colour in Table 6 where columns (2) and (3) indicate that SF
and composite galaxies’ u-r colour depend on the environment by having average
probability values of $\sim\\!0.01$ and $\sim\\!$4.51\text{\times}{10}^{-03}$$
respectively, which are smaller than the standard probability value in
statistics ($0.05$), while columns (4) and (5) indicate that u-r colour of
Seyfert and LINERs weakly depend on the environment by having average
probability values of $\sim\\!0.3$ and $\sim\\!0.2$ respectively, which are
much larger than the standard probability value. Again from Table 6 it is seen
that the relationship of u-r colour with the environment decreases with the
increase in redshift.

It is observed that the SF galaxies in Fig. 7(a) form a blue cloud since most
of the galaxies are below the green valley ($\sim\\!60\%$) as shown in Table 4
while the composite galaxies in Fig. 7(b) form a green valley as most of the
galaxies lie within demarcation lines ($\sim\\!50\%$) on average. Moreover,
the evidence from Table 4 indicates that most of Seyfert galaxies are within
the green valley ($\sim\\!49\%$) on average and LINERs are above the green
valley ($\sim 60\%$) on average, these support the observations from Figs.
7(c) and 7(d). Therefore from Table 4 and Figs. 7 it is clear to say that
composite, Seyfert and LINERs populate the green valley and red-sequence while
SF galaxies populate the blue cloud, which also was observed by Refs. [16, 15,
56, 57, 58].

The observation from the right panel of Fig. 8 indicates that the average
stellar mass increases in the sequence of SF, composite, Seyfert and LINERs
while the left panel of Fig. 8 indicates that the average SFR decreases in the
same sequence, agreeing with the findings by Ref. [59] which observed that
galaxies hosting AGNs (in this case Seyfert and LINERs) are preferentially
more massive and have low SFR. Fig. 8 also indicates that quenching of SFR
increases from SF to composite, Seyfert and LINERs galaxies in which the
dependence on the environment is seen to decrease at SFR $\sim\\!-0.2$, which
is equivalent to $M_{\star}=10^{10.7}M_{\odot}$ as shown by Fig. 8. This
implies that other mechanisms, for example, the AGNs feedback are responsible
for quenching of galaxies with $M_{\star}\geq 10^{10.7}M_{\odot}$.

##  V Summary and Conclusion

For the first time, we use the catalogue of groups and clusters derived by
Ref. [32] from Sloan Digital Sky Survey as detailed in Refs. [30, 31], and the
MPA-JHU data obtained using the methods outlined in Ref. [35, 36], to include
the composite, Seyfert and LINERs on the local star formation MS then to study
the influence of isolated and non-isolated environments on the star formation
MS, colour bimodality and hence quenching of SFR. We apply the signal-to-noise
criteria $3$, to obtain a total of $274\,517$ galaxies. Again by removing the
galaxies with no stellar mass and SFR measurements a total of $268\,954$
galaxies was obtained. The galaxies are classified based on emission line
ratio using the NII diagnostic diagram obtaining $170\,797\,(63\%)$ SF,
$58\,763\,(22\%)$ composite, $12\,536\,(5\%)$ Seyfert and $26\,858\,(10\%)$
LINERs. Using the catalog of groups and clusters by Ref. [32] we divide
galaxies with no neighbour ($N\\!gal=1$) as isolated, while the galaxies with
more than one neighbour ($N\\!gal\geq 2$) as non-isolated. The final dataset
contains $93\,727\,(55\%)$, $77\,070\,(45\%)$ SF galaxies, $30\,652\,(52\%)$,
$28\,111(48\%)$ composite galaxies, $6\,643\,(53\%)$, $5\,893\,(47\%)$ Seyfert
galaxies, and $11\,355\,(42\%)$, $15\,503\,(58\%)$ LINER galaxies with
isolated and non-isolated environments respectively. Our results suggest that:

  * 1.

Non-isolated galaxies preferentially reside at lower redshift regions (in this
case $z\lesssim 0.09$) while the isolated galaxies reside at higher redshift
regions (in this case $z>0.09$) of the universe.

  * 2.

SFR and stellar mass of SF and composite galaxies depend on the environment
where high-mass galaxies are found in the isolated environment while the low-
mass galaxies is a characteristic of the non-isolated environment.
Furthermore, the isolated environment contains galaxies with high SFR while
the non-isolated environment contains galaxies with low SFR. The dependence of
$M_{\star}$ and SFR on the environment decreases as the redshift increases.

  * 3.

Stellar masses of LINERs depend on the environment like SF and composite
galaxies but its SFR weakly depends on the environment. Whereas the dependence
of stellar mass and SFR of Seyfert galaxies on the environment is very weak.
This implies that emission from AGNs (in this case Seyfert and LINERs) weakens
the dependance of SFR on the environment.

  * 4.

The decrease of the slope for the MS of SF galaxies by $0.02$ dex and
intercept by $0.17$ dex are observed between isolated and non-isolated
galaxies implying that the environment is among the factors shaping the star
formation main sequence.

  * 5.

There is a significant difference in the measurement of u-r colour for the SF
and composite galaxies between the isolated and non-isolated environments
while this difference for Seyfert and LINERs is insignificant. This implies
that there is a weak dependence of u-r colour on the environment for Seyfert
and LINERs.

  * 6.

SF, composite, Seyfert and LINER galaxies form evolutionary pathways where
most SF galaxies are found in the blue cloud ($\sim\\!60\%$), both composite
($\sim\\!50\%$) and Seyfert ($\sim\\!49\%$) in the green valley, and LINERs
($\sim\\!60\%$) in the red sequence.

  * 7.

The dependence of quenching on the environment decreases at
$M_{\star}=10^{10.7}M_{\odot}$ indicating that for $M_{\star}\geq
10^{10.7}M_{\odot}$ other mechanisms (including the AGNs feedback) dominate
the quenching process.

In the future, as the follow-up of this work, we will classify galaxies based
on morphology, aiming to investigate what factor takes the reading on the
dependance of galaxy properties on environment by comparing the isolated,
group and cluster galaxies.

## Acknowledgements

PP acknowledges support from The Government of Tanzania through the India
Embassy, Mbeya University of Science and Technology (MUST) for Funding and
SDSS for providing data. UDG is thankful to the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India for the Visiting Associateship
of the institute.

## References

  * [1] J. Speagle, C. Steinhardt, P. Capak, J. Silverman et al. _A Highly Consistent Framework for the Evolution of the Star-Forming” Main Sequence” from $z\sim 0-6$_, ApJS 214, 15 (2014) [arXiv:1405.2041].
  * [2] P. Behroozi, R. Wechsler, C. Conroy, _The average star formation histories of galaxies in dark matter halos from $z\sim 0-8$_, ApJ 770, 57 (2013) [arXiv:1207.6105].
  * [3] K. Noeske, B. Weiner, S. Faber, C. Papovich et al. _Star formation in aegis field galaxies since z= 1.1: The dominance of gradually declining star formation, and the main sequence of star-forming galaxies_ , ApJ 660, L43 (2007) [arXiv:astro-ph/0701924].
  * [4] G. Rodighiero, E. Daddi, I. Baronchelli, A. Cimatti et al. _The lesser role of starbursts in star formation at $z=2$_, ApJL 739, L40 (2011) [arXiv:astro-ph/0701924].
  * [5] N. Reddy, M. Pettini, C. Steidel, A. Shapley et al. _The characteristic star formation histories of galaxies at redshifts $z\sim 2-7$_, ApJ 754, 25 (2012) [arXiv:1205.0555].
  * [6] B. Salmon, C. Papovich, S. Finkelstein, V. Tilviy et al. _The relation between Star Formation Rate and Stellar Mass for Galaxies at $3.5\leq z\leq 6.5$ in CANDELS_, ApJ 799, 183 (2015) .
  * [7] P. Santini, A. Fontana, M. Castellano, M. Criscienzo et al. _The star formation main sequence in the Hubble Space Telescope Frontier Fields_ , ApJ 847, 76 (2017) [arXiv:1706.07059].
  * [8] G. Magdis, E. Daddi, M. Béthermin, M. Sargent et al. _The evolving interstellar medium of star-forming galaxies since $z=2$ as probed by their infrared spectral energy distributions_, ApJ 760, 6 (2012) [arXiv:1210.1035].
  * [9] K. Whitaker, P. Dokkum, G. Brammer, M. Franx _The star formation mass sequence out to z= 2.5_ , ApJL 754, L29 (2015) [arXiv:1205.0547].
  * [10] Y. Peng, S. Lilly, K. Kovač, M. Bolzonella et al. _Mass and Environment as Drivers of Galaxy Evolution in SDSS and zCOSMOS and the Origin of the Schechter Function_ , ApJ 721, 193 (2010) [arXiv:1003.4747].
  * [11] N. Drory, M. Alvarez _The contribution of star formation and merging to stellar mass buildup in galaxies_ , ApJ 680, 41 (2008)[arXiv:0803.1489].
  * [12] K. Whitaker, M. Franx, J. Leja, P. Dokkum et al. _Constraining the Low-mass Slope of the Star Formation Sequence at $0.5<z<2.5$ _, ApJ 795, 104 (2008)[arXiv:1407.1843].
  * [13] A. Renzini, Y. Peng, _An objective definition for the main sequence of star-forming galaxies_ , ApJ 801, L29 (2008)[arXiv:1502.01027].
  * [14] S. Leslie, L. Kewley, D. Sanders, N. Lee _Quenching star formation: insights from the local main sequence_ , MNRAS 455, L82 (2015) [arXiv:1509.03632].
  * [15] K. Schawinski, D. Thomas, M. Sarzi, C. Maraston et al. _Observational evidence for AGN feedback in early-type galaxies_ , MNRAS 382, 1415 (2007) [arXiv:0709.3015].
  * [16] K. Nandra, A. Georgakakis, C. Willmer, M. Cooper et al. _AEGIS: The color-magnitude relation for X-ray-selected active galactic nuclei_ , ApJ 660, L11 (2007).
  * [17] D. Croton, V. Springel, S. White, G. Lucia et al. _The many lives of active galactic nuclei: cooling flows, black holes and the luminosities and colours of galaxies_ , MNRAS 365, 11 (2013) [arXiv:astro-ph/0508046].
  * [18] D. Rosario, P. Santini, D. Lutz, L. Shao et al. _The mean star formation rate of X-ray selected active galaxies and its evolution from $z\sim 2.5$: results from PEP-Herschel_, A & A 545, A45 (2012) [arXiv:1203.6069].
  * [19] D. Elbaz, E. Daddi, D. Borgne, M. Dickinson et al. _The reversal of the star formation-density relation in the distant universe_ , A&A 468, 33 (2007) [arXiv:astro-ph/0703653].
  * [20] I. Lewis, M. Balogh, R. Propris, W. Couch, R. Bower et al. _The 2dF Galaxy Redshift Survey: the environmental dependence of galaxy star formation rates near clusters_ , MNRAS 334, 673 (2002) [arXiv:astro-ph/0203336
  * [21] P. Gomez, R. Nichol, _Galaxy Star-Formation as a Function of Environment in the Early Data Release of the Sloan Digital Sky Survey_ , A& A 584, 210 (2003) [arXiv:astro-ph/0210193].
  * [22] M. Tanaka, T. Goto, _The Environmental Dependence of Galaxy Properties in the Local Universe: Dependence on Luminosity, Local Density, and System Richness_ , AJ 128, 2677 (2004) [arXiv:astro-ph/0411132].
  * [23] M. Cooper, J. Newman, _The DEEP2 Galaxy Redshift Survey: The Role of Galaxy Environment in the Cosmic Star-Formation History_ , MNRAS 383, 1058 (2008) [arXiv:0706.4089].
  * [24] S. Patel, B. Holden, _The Dependence of Star Formation Rates on Stellar Mass and Environment at $z\sim 0.8$_, ApJ 705, L67 (2007) [arXiv:0910.0837].
  * [25] X. Deng, P. Wu, X. Qian, C. Luo, _Environmental Dependence of Stellar Mass, Star Formation Rate, Specific Star Formation Rate, and AGN Activity for an Apparent Magnitude Limited Main Galaxy Sample of the SDSS DR7_ , PASJ 64, 93 (2012).
  * [26] M. Martig, F. Bournaud, R. Teyssier, A. Dekel et al. _Morphological quenching of star formation: making early-type galaxies red_ , ApJ 707, 250 (2009)[arXiv:0905.4669].
  * [27] G. Erfanianfar, P. Popesso, A. Finoguenov, D. Wilman et al. _Non-linearity and environmental dependence of the star-forming galaxies main sequence_ , MNRAS 455, 2839 (2016)[arXiv:1511.01899].
  * [28] P. Lang, S. Wuyts, R. Somerville, N. Schreiber, R. Genzel et al. _Bulge Growth and Quenching since $z=2.5$ in CANDELS/3D-HST _, ApJ 788, 788 (2014)[arXiv:1402.0866].
  * [29] P. Ade, N. Aghanim, _Planck 2015 results. XXIII. The thermal Sunyaev-Zeldovich effect–cosmic infrared background correlation_ , A& A 594, A23 (2016) [arXiv:1509.06555].
  * [30] D. Eisenstein, D. Weinberg, _SDSS-III: Massive spectroscopic surveys of the distant universe, the Milky Way, and extra-solar planetary systems_ , AJ 142, 72 (2011) [arXiv:1101.1529].
  * [31] S. Alam, F. Albareti, C. Prieto, _The Eleventh and Twelfth Data Releases of the Sloan Digital Sky Survey: Final Data from SDSS-III_ , ApJ 219, 12 (2015) [arXiv:1501.00963].
  * [32] E. Tempel, T. Tuvikene, _Merging groups and clusters of galaxies from the SDSS data. The catalogue of groups and potentially merging systems_ , A & A 602, A100 (2017) [arXiv:1704.04477]
  * [33] E. Tempel, A. Tamm, M. Gramann, _Flux- and volume-limited groups/clusters for the SDSS galaxies: catalogues and mass estimation_ , A& A 566, A1 (2014) [arXiv:1402.1350].
  * [34] G. Kauffmann, S. White, _The Environmental Dependence of the Relations between Stellar Mass, Structure, Star Formation and Nuclear Activity in Galaxies_ , MNRAS 353, 713 (2004) [arXiv:astro-ph/0402030].
  * [35] J. Brinchmann, S. Charlot, _The physical properties of star forming galaxies in the low redshift universe_ , MNRAS 351, 1151 (2004) [arXiv:astro-ph/0311060].
  * [36] C. Tremonti, T. Heckman, _The Origin of the Mass–Metallicity Relation: Insights from 53,000 Star-Forming Galaxies in the SDSS_ , ApJ 613, 898 (2004) [arXiv:astro-ph/0405537].
  * [37] C. Kennicutt, _Star Formation in Galaxies Along the Hubble Sequence_ , A&A 39, 189 (1998) [arXiv:astro-ph/9807187].
  * [38] S. Salim, J. Lee, R. Davé, _On the Mass-Metallicity-Star Formation Rate Relation for Galaxies at $z\sim 2$_, ApJ 808, 14pp (2015) [arXiv:1506.03080].
  * [39] L. Kewley, B. Groves, G. Kauffmann, T. Heckman et al. _The host galaxies and classification of active galactic nuclei_ , MNRAS 372, 961 (2006) [arXiv:astro-ph/0605681].
  * [40] G. Kauffmann, T. Heckman, C. Tremonti, J. Brinchmann et al. _The host galaxies of active galactic nuclei_ , MNRAS 346, 1055 (2003) [arXiv:astro-ph/0304239].
  * [41] L. Kewley, M. Dopita, R. Sutherland, C. Heisler et al. _Theoretical modeling of starburst galaxies_ , ApJ 556, 121 (2001) [arXiv:astro-ph/0106324].
  * [42] T. Shimizu, R. Mushotzky, M. Meléndez, M. Koss et al. _Decreased specific star formation rates in AGN host galaxies_ , MNRAS 452, 1841 (2007) [arXiv:1506.07039].
  * [43] P. Teerikorp, _Eddington-Malmquist bias in a cosmological context_ , A & A 576, A75 (2015) [arXiv:1503.02812].
  * [44] D. Harari, S. Mollerach, _Kolmogorov-Smirnov test as a tool to study the distribution of ultra-high energy cosmic ray sources_ , A& A 394, 916 (2009) [arXiv:0811.0008].
  * [45] T. Gonçalves, D. Martin, K. Menéndez-Delmestre, T. Wyder et al. _Quenching star formation at intermediate redshifts: downsizing of the mass flux density in the green valley_ , ApJ 759, 67 (2012) [arXiv:1209.4084].
  * [46] J. Moustakas, A. Coil, J. Aird, M. Blanton, R. Cool et al. _Quenching star formation at intermediate redshifts: downsizing of the mass flux density in the green valley_ , ApJ 767, 50 (2013) [arXiv:1301.1688].
  * [47] S. Faber, C. Willmer, C. Wolf, D. Koo, B. Weiner, J. Newman et al. _Galaxy Luminosity Functions to z $\sim$1: DEEP2 vs. COMBO-17 and Implications for Red Galaxy Formation_, ApJ 665, 265 (2007) [arXiv:astro-ph/0506044].
  * [48] R. Hickox, J. Mullaney, D. Alexander, C. Chen, F. Civanoet al. _Black hole variability and the star formation–active galactic nucleus connection: do all star-forming galaxies host an active galactic nucleus_ , ApJ 782, 9 (2014) [arXiv:1306.3218].
  * [49] K. Schawinski, C. Urry, B. Simmons, L. Fortson, S. Kaviraj et al. _The green valley is a red herring: Galaxy Zoo reveals two evolutionary pathways towards quenching of star formation in early-and late-type galaxies_ , MNRAS 440, 889 (2014).
  * [50] G. Rihtaršič, V. Biffi, D. Fabjan, K. Dolag et al. _Environmental dependence of AGN activity and star formation in galaxy clusters from Magneticum simulations_ , A&A 683, A57 (2024) [arXiv:2307.06374]
  * [51] E. Daddi, M. Dickinson, G. Morrison, R. Chary et al. _Multiwavelength study of massive galaxies at $z\sim 2$. I. Star formation and galaxy growth_, ApJ 670, 156 (2007). [arXiv:0705.2831].
  * [52] J. Rich, L. Kewley, M. Dopita, _Galaxy-Wide Shocks in Late-Merger Stage Luminous Infrared Galaxies_ , ApJ 782, 9 (2014) [arXiv:1104.1177].
  * [53] S. Leslie, J. Rich, L. Kewley, M. Dopita, _The energy source and dynamics of infrared luminous galaxy ESO 148-IG002_ , MNRAS 444, 1842 (2014) [arXiv:1408.0789].
  * [54] T. Yuan, L. Kewley, D. Sanders, _The Role of Starburst-Active Galactic Nucleus Composites in Luminous Infrared Galaxy Mergers: Insights from the New Optical Classification Scheme_ , ApJ 709, 884 (2010)
  * [55] D. Schiminovich, T. Wyder, D. Martin, B. Johnson et al. _The UV-Optical Color Magnitude Diagram II: Physical Properties and Morphological Evolution On and Off of a Star-Forming Sequence_ , ApJS 173, 315 (2007) [arXiv:0711.4823].
  * [56] V. Smolčić, G. Zamorani, E. Schinnerer, S. Bardelli et al. _Cosmic evolution of radio selected active galactic nuclei in the cosmos field_ , ApJ 696, 24 (2009) [arXiv:0901.3372].
  * [57] K. Alatalo, K. Nyland, G. Graves, S. Deustua et al. _NGC 1266 as a local candidate for rapid cessation of star formation_ , ApJ 780, 186 (2014) [arXiv:1311.6469].
  * [58] K. Alatalo, S. Cales, P. Appleton, L. Kewley et al. _Catching Quenching Galaxies: The Nature of the WISE Infrared Transition Zone_ , ApJL 794, L13 (2014) [arXiv:1409.2489].
  * [59] X. Deng, Y. Xin, P. Wu, P Jiang et al. _Some Properties of Active Galactic Nuclei in the Volume-limited Main Galaxy Samples of SDSS DR8_ , ApJ 754, 82 (2012).


††thanks: Corresponding author<EMAIL_ADDRESS>
# High-precision measurement of the atomic mass of 84Sr and implications to
isotope shift studies

Z. Ge  University of Jyväskylä, Department of Physics, Accelerator laboratory,
P.O. Box 35, FI-40014, Jyväskylä, Finland  S. W. Bai  School of Physics and
State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871, China  T. Eronen  University of Jyväskylä, Department of
Physics, Accelerator laboratory, P.O. Box 35, FI-40014, Jyväskylä, Finland  A.
Jokinen  University of Jyväskylä, Department of Physics, Accelerator
laboratory, P.O. Box 35, FI-40014, Jyväskylä, Finland  A. Kankainen
University of Jyväskylä, Department of Physics, Accelerator laboratory, P.O.
Box 35, FI-40014, Jyväskylä, Finland  S. Kujanpää  University of Jyväskylä,
Department of Physics, Accelerator laboratory, P.O. Box 35, FI-40014,
Jyväskylä, Finland  I. D. Moore  University of Jyväskylä, Department of
Physics, Accelerator laboratory, P.O. Box 35, FI-40014, Jyväskylä, Finland  D.
A. Nesterenko  University of Jyväskylä, Department of Physics, Accelerator
laboratory, P.O. Box 35, FI-40014, Jyväskylä, Finland  M. Reponen  University
of Jyväskylä, Department of Physics, Accelerator laboratory, P.O. Box 35,
FI-40014, Jyväskylä, Finland

###### Abstract

The absolute mass of 84Sr was determined using the phase-imaging ion-
cyclotron-resonance technique with the JYFLTRAP double Penning trap mass
spectrometer. A more precise value for the mass of 84Sr is essential for
providing potential indications of physics beyond the Standard Model through
high-precision isotope shift measurements of Sr atomic transition frequencies.
The mass excess of 84Sr was refined to be -80649.229(37) keV/c2 from high-
precision cyclotron-frequency-ratio measurements with a relative precision of
4.8$\times$10-10. The obtained mass-excess value is in agreement with the
adopted value in the Atomic Mass Evaluation 2020, but is 30 times more
precise. With this new value, we confirm the previously observed nonlinearity
in the study of the isotope shift of strontium. Moreover, the double-beta
($2\beta^{+}$) decay $Q$ value of 84Sr was directly determined to be
1790.115(37) keV, and the precision was improved by a factor of 30.

##  I Introduction

Isotope shifts in atomic transition frequencies arise from differences in
neutron numbers among isotopes sharing the same atomic number. The isotopic
shifts in the frequency of an atomic transition demonstrate an approximately
linear correlation with the isotope shift observed in a second transition.
These shifts reveal contributions from field and mass shifts king1984isotope,
originating from the differing nuclear masses of isotopes and variations in
their nuclear charge distribution. Isotopic shifts can be systematically
studied using a King plot analysis. In this analysis, the isotope shifts in
two sets of electronic transitions within the same isotopes are correlated. To
perform the King plot analysis, one measures the energies of two transitions
for three or more isotopes of a specific element. The King plot is expected to
exhibit linearity, with the experimentally determined slope serving as a
reliable benchmark for theoretical predictions flam18. Deviations from
linearity are crucial for refining atomic structure calculations
dammalapati2009; shi2017; naze2015. Recent theoretical proposals suggest that
the linearity in King plots could be leveraged to impose constraints on
higher-order effects on isotope shifts or on physics beyond the Standard Model
(SM) frugiuele2017; berengut2018. Motivated by this, a recent surge in efforts
miyake2019; manovitz2019; knollmann2019; counts2020; solaro2020;
NESTERENKO2020-2; Counts20; rehbehn2021; muller21; hur2022; figueroa2022;
ono2022 has significantly advanced the precision of isotope shift
measurements. Essential to placing constraints on proposed electron-neutron
interactions and other novel physics through King’s linearity is a two-fold
requirement. First, experimental data in the form of precision optical
spectroscopy and nuclear mass measurements are needed to empirically limit the
potential size of the nonlinearity. Second, if a nonlinearity is observed,
precise atomic and nuclear theory is necessary to calculate beyond-first-order
SM sources of nonlinearity. A natural inquiry emerges regarding the potential
for the King plot to maintain its linearity at an enhanced level of
experimental precision. Recent experiments conducted with strontium and
ytterbium ions miyake2019; Counts20 have provided initial indications that
this linearity is, in fact, disrupted at a magnitude of several standard
deviations. Strontium presents favorable properties for studying isotope
shifts, boasting an abundance of stable isotopes and very narrow optical
transitions stellmer2014. Earlier theoretical work has also proposed the
measurement of strontium isotope shifts as a promising probe for new physics
frugiuele2017; berengut2018. Strontium features four stable isotopes: three
bosons (84,86,88Sr) and one fermion (87Sr). Precise and accurate determination
of the atomic mass of the stable Sr isotopes is crucial for probing potential
causes for such nonlinearities.

Atomic masses of 86-88Sr have been measured at the FSU Penning trap rana2012
with high precision (uncertainty of $\leq$ 6 eV/c2) as adopted in the most
recent Atomic Mass Evaluation 2020 (AME2020) Wang2021; Kondev2021. The atomic
mass uncertainty for 84Sr, the least abundant naturally occurring isotope of
strontium, is however notably higher at 1.2 keV/c2. For investigating
fundamental physics and exploring phenomena beyond the SM via high-precision
King-plot tests with strontium, it is critical to measure the atomic mass of
84Sr directly with high precision. In this article, we report on the most
precise absolute mass value of 84Sr to date, achieved via high-precision
cyclotron-frequency-ratio measurements with the JYFLTRAP double Penning trap
mass spectrometer (PTMS). We employ this refined mass value in an updated King
plot analysis for two transitions of strontium.

Figure 1: (Color online). (a) Diagram depicting the ion production and mass
measurement process using the PI-ICR technique. Stable 84Sr+ and 84Kr+ ions
were concurrently generated in an offline glow-discharge ion source. Selection
of the final ion species occurred through buffer-gas cooling in the
preparation trap (b), and the cyclotron frequency was determined using the
phase-imaging technique at the measurement trap (c). (d) The position-
sensitive MCP detector captured the projected images representing different
radial-motion phases (1-3). (e) Illustration depicting the radial-motion (1:
"magnetron", 2: "cyclotron", and 3: "center") projection of 84Sr+ ions onto
the position-sensitive MCP detector. Panel 1 shows the magnetron phase spot,
panel 2 the cyclotron phase spot, and panel 3 the center spot. The angle
difference between magnetron and cyclotron motion spots relative to the center
spot was used to deduce the cyclotron frequency of the measured ion. Colors in
each pixel indicate the number of ions.

##  II Experimental method

The experiment was performed at the Ion Guide Isotope Separator On-Line
facility (IGISOL) using the JYFLTRAP double PTMS Eronen2012a; Eronen2016, at
the University of Jyväskylä Moore2013. Stable 84Sr+ ions were generated using
an offline glow-discharge ion source. For a precise mass measurement of 84Sr,
reference ions of 84Kr+ with a well-known mass value (uncertainty of 4 eV/c2
Wang2021) were concurrently produced from the same ion source. As illustrated
in Fig. 1(a), the gas cell within the glow-discharge ion source comprises two
sharp electrodes, with one composed of naturally abundant strontium. By
introducing a gas containing natural krypton, it facilitates the simultaneous
generation of stable ions for both strontium and krypton.

The generated ions of 84Sr+ and 84Kr+ were extracted using helium gas flow and
electric fields facilitated by a sextupole ion guide (SPIG) Karvonen2008.
Following acceleration over an electric potential of $\approx$30 kV, the ions
with mass number of $A$ = 84 were mass-separated using a 55∘ dipole magnet
with a mass resolving power of $M/\Delta{M}$ $\approx$ 500\. Post isobaric
separation, the ions were directed to a radiofrequency quadrupole cooler-
buncher (RFQ) Nieminen2001, where they were accumulated, cooled, and bunched.

The ion bunches from the RFQ were injected to the JYFLTRAP double PTMS,
consisting of two cylindrical Penning traps equipped with a 7-T
superconducting solenoid. The first trap, functioning as a purification trap,
is filled with buffer gas and is employed for isobaric purification using the
sideband buffer gas cooling technique Savard1991. This technique alone
achieves mass purification with a resolving power of approximately $10^{5}$ by
selectively converting ion motion from magnetron to reduced cyclotron motion.

In the purification trap, all cooled and centered ions (84Sr+ and 84Kr+) were
initially excited to a large orbit of revolution by applying a dipole
excitation at the magnetron motion frequency $\nu_{-}$ for approximately 11
ms. Subsequently, a quadrupole excitation was executed to center the ions of
interest (only 84Sr+ or 84Kr+) through collisions with the buffer gas for
about 100 ms.

In the precision trap, the phase-imaging ion-cyclotron-resonance (PI-ICR)
method Nesterenko2018; nesterenko2021 was employed to measure the actual
cyclotron frequency, $\nu_{c}={qB}/{(2\pi m)}$, where $B$ is the magnetic
field strength, $q$ is the charge state, and $m$ is the mass of the stored
ion. The scheme of the PI-ICR technique nesterenko2021; Nesterenko2018;
Eliseev2014; Eliseev2013 at JYFLTRAP relies on direct measurements of the
cyclotron motion and magnetron motion simultaneously by projecting their
radial ion motion onto a position-sensitive MCP detector.

To determine the phases of the radial motions, the center has to be determined
for the ion spots on the detector. This is done by storing the ions for a few
milliseconds without exciting their cyclotron motion, and the ions are
directly extracted from the trap and projected onto the MCP detector.

Two patterns, as detailed in nesterenko2021; Nesterenko2018, are utilized to
measure the magnetron or cyclotron motion phases, respectively. The angle
between two phase images of the projected radial motions with respect to the
center spot is $\alpha_{c}=\alpha_{+}-\alpha_{-}$, where $\alpha_{+}$ and
$\alpha_{-}$ are the polar angles of the cyclotron and magnetron motion
phases. The cyclotron frequency $\nu_{c}$ is derived from:

$\nu_{c}=\frac{\alpha_{c}+2\pi n_{c}}{2\pi{t_{acc}}},$ (1)

where $n_{c}$ is the full number of revolutions of the measured ions during
the phase accumulation time $t_{acc}$. A few different accumulation times for
84Sr+ and 84Kr+ were used to confirm the unambiguity of the cyclotron
frequency. A fixed accumulation time of 400 ms was employed for the actual
measurements to determine the final $\nu_{c}$. A measurement with "cyclotron"
and "magnetron" phase spots collected with respect to the center spot is shown
in Fig.1(b).

The atomic mass $M_{{}^{84}Sr}$ was derived from the measured cyclotron
frequency ratio ($R$ = ${\nu_{c}(^{84}Kr^{+}})/{\nu_{c}}(^{84}Sr^{+})$)
measurements of singly charged ions of the decay pair 84Sr-84Kr:

$M{(^{84}Sr)}=R(M{(^{84}Kr)}-qm_{e})+qm_{e}+(R\cdot
B{(^{84}Kr)}-B{(^{84}Sr}))/c^{2},$ (2)

Here, $M{(^{84}Sr)}$ and $M{(^{84}Kr)}$ represent the respective masses of the
parent and daughter atoms, and $q$ denotes the charge state for singly charged
ions ($q=1$). $m_{e}$ and $c$ correspond to the mass of an electron and the
speed of light in vacuum. The electron binding energies, $B{(^{84}Sr)}$ and
$B{(^{84}Kr)}$, are 5.69486745(12) eV and 13.9996055(20) eV, respectively, as
obtained from NIST_ASD.

The $Q$ value for the double-beta decay of 84Sr can be determined from the
mass difference: $Q_{2\beta^{+}}=(M{(^{84}Sr)}-M{(^{84}Kr)})c^{2}$.

Table 1: The resulting $Q_{2\beta^{+}}$ and mass-excess values of 84Sr determined in this work based on the weighted mean of the cyclotron frequency ratio $\overline{R}$. The frequency ratio $\overline{R}$, $Q_{2\beta^{+}}$ values (in keV), and the mass excess (ME, in keV/c2) of the parent, as determined in this work, are provided alongside the corresponding values from AME2020 Wang2021 for comparison. | $\overline{R}$ | $Q_{2\beta^{+}}$ | ME
---|---|---|---
This Work | 1.000 022 902 36(48) | 1790.115(37) | -80649.229(37)
AME2020 |  | 1789.8(12) | -80649.6(12)

##  III Results and discussion

###  III.1 Mass and $Q$-value determination

A full scanning measurement (one cycle) of the magnetron phase, cyclotron
phase, and center spot in sequence was completed in less than 3 minutes for
each ion species of 84Kr+ and 84Sr+. In the analysis, the position of each
spot was fit using the maximum likelihood method. A few cycles were summed to
ensure reasonable counts for fitting before determining the position of each
spot. The phase angles were calculated accordingly based on the determined
positions of the phases to deduce the cyclotron frequencies of each ion
species. The cyclotron frequency $\nu_{c}$ of the daughter ion 84Kr+ was used
as a reference and was linearly interpolated to the time of the measurement of
the parent 84Sr+ (ion of interest) to deduce the cyclotron frequency ratio
$R$. Bunches with less than five detected ions per bunch were considered in
the data analysis to reduce a possible cyclotron frequency shift due to ion-
ion interactions Kellerbauer2003; Roux2013. Up to 5 detected ions per bunch
were taken into acount for the analysis, and no count-rate related frequency
shifts were observed in the analysis. The temporal fluctuation of the magnetic
field for JYFLTRAP was measured to be $\delta_{B}(\nu_{c})/\nu_{c}=\Delta
t\times 2.01(25)\times 10^{-12}$/min nesterenko2021, where $\Delta t$ is the
time interval between two consecutive reference measurements. The contribution
of temporal fluctuations of the magnetic field to the final frequency ratio
uncertainty was less than $10^{-10}$ since the parent-daughter measurements
were interleaved with $\Delta t<10$ minutes. To minimize the systematic
uncertainty arising from the conversion of cyclotron motion to magnetron
motion and potential distortion of the ion-motion projection onto the
detector, the positions of the magnetron-motion and cyclotron-motion phase
spots were deliberately chosen. The angle $\alpha_{c}$ between them was kept
to a few degrees or less, effectively reducing this uncertainty to a level
well below 10-10 Eliseev2014. Moreover, the commencement of the initial
dipolar excitation with frequency $\nu_{+}$ was systematically scanned across
one magnetron period (6 points), and the extraction was scanned over one
cyclotron period (6 points) to mitigate any lingering effects of residual
magnetron and cyclotron motion that might influence the distinct spots. The
measurements were conducted in eight separate time slots, each lasting around
4 hours. For each slot of measurement, a weighted mean ratio $R_{4h}$ was
calculated, and the maximum of internal and external errors Birge1932 was
selected. The final ratio $\overline{R}$ was then obtained as a weighted mean
ratio of all $R_{4h}$ sets, taking into account the maximum of internal and
external errors. The systematic uncertainties and experimental statistical
uncertainty were added in quadrature to the final result. The final frequency
ratio $\overline{R}$ with its uncertainty, as well as the corresponding
$Q_{2\beta^{+}}$ and mass-excess values, are 1.000 022 902 36(48),
1790.115(37) keV, and -80649.229(37) keV/c2 respectively.

In Fig.2, the analysis results, which include all data sets, are compared to
literature values. A comparison of these results to the literature values are
also tabulated in Table. 1.

Figure 2: (Color online). The deviation (left axis) of the individually
measured cyclotron frequency ratios $R_{4h}$
($\nu_{c}$(84Kr+)/$\nu_{c}$(84Sr+)) from the final ratio value $\overline{R}$
and (right axis)) mass-excess values in this work compared to values adopted
from AME2020 Huang2021; Wang2021. The red points, accompanied by
uncertainties, represent individual data obtained with the PI-ICR method in
eight distinct time slots. The solid red line illustrates the weighted average
value from this work, $\overline{R}$ = 1.000 022 902 36(48), and its 1$\sigma$
uncertainty band is shaded in red. The dashed blue line represents the
difference between the new value in this work and the one referred to in
AME2020, with its 1$\sigma$ uncertainty area shaded in blue.

The mass excess (-80649.229(37) keV/c2) and $Q_{2\beta^{+}}$ (1789.8(12) keV)
from this work are both a factor of $\approx$30 more precise than those
derived from the evaluated masses in AME2020 Huang2021; Wang2021, but both
agree well with the values documented in AME2020. The mass-excess value in
AME2020 is derived primarily from two PTMS experiments Keller07; Emma11 with
an influence of 88.8%. A slight contribution of 6.8% is from endpoint energy
measurements of 84Rb($\beta^{-}$)84Sr BOOIJ1971 and the smallest influence of
2.1% is related to a nuclear reaction experiment 84Sr(d,p)85Sr MORTON1971.

Figure 3: (Color online). King plot of the strontium isotope shifts in
$\alpha$ (689 nm) and $\beta$ (698 nm) transitions as reported in miyake2019;
Takano_2017. The implication of the mass value of 84Sr from this work compared
to values adopted from AME2020 Huang2021; Wang2021 is evident. Black data
points and their accompanying error bars in black are obtained using the
literature mass values, while the red data point with error bars is obtained
with the new mass value from this work. Error bars and the difference between
the 84Sr-88Sr points derived from our measurement and from miyake2019;
Takano_2017; Huang2021; Wang2021 are not visible at the scale being used due
to a good agreement of this work with the AME2020. Linear fits were conducted
for the set with literature data (black) and the new mass value (red), with a
zoomed-in inset demonstrating a rather small change in the linear fit and no
significant change in nonlinearity.

###  III.2 King plot analysis

The isotope shift between two isotopes with mass numbers $A_{1}$ and $A_{2}$,
($\nu^{A_{1}A_{2}}_{i}$) is defined as the difference in transition
frequencies: $\nu^{A_{1}A_{2}}_{i}=\nu^{A_{1}}_{i}-\nu^{A_{2}}_{i}$. The
primary contributions to the isotope shift arise from the mass shift (MS) and
field shift (FS). The MS results from the mass difference of isotopes $A_{1}$
and $A_{2}$ and is expressed as an electronic coefficient $k_{i}$ multiplied
by the isotope-dependent reduced mass, given by:
$\mu_{A_{1}A_{2}}=1/m_{A_{1}}-1/m_{A_{2}}$. The FS originates from the
differing volumes of the two isotopes, factorized into an electronic, isotope-
independent coefficient $F_{i}$ and the charge radius variance $\delta\langle
r^{2}\rangle_{A_{1}A_{2}}=\langle r^{2}_{A_{1}}\rangle-\langle
r^{2}_{A_{2}}\rangle$. The isotope shift composition in terms of electronic
and nuclear quantities factorizes into the total isotope shift equation:
$\nu^{A_{1}A_{2}}_{i}=k_{i}\mu_{A_{1}A_{2}}+F_{i}\delta\langle
r^{2}\rangle_{A_{1}A_{2}}+...$, where the first term represents the MS, and
the second term represents the FS king63; king1984isotope, with the dots
denoting potential higher-order corrections and new physics contributions. The
frequency shifts are commonly normalized by the reduced mass
$\mu_{A_{1}A_{2}}$ to obtain the modified isotope shift,
$\nu^{A_{1}A_{2}}_{i}/\mu_{A_{1}A_{2}}$. Consequently, the MS (the sum of the
normal MS and specific MS) reduces to the electronic factor $k_{i}$, while the
FS factor $F_{i}$ is multiplied by the modified charge radius variance,
$\delta\langle r^{2}\rangle_{A_{1}A_{2}}/\mu_{A_{1}A_{2}}$, establishing a
linear dependence between the two sets of modified frequency shifts known as
the King linearity king63. To quantify the observed linearity, a measure of
nonlinearity is defined flam18; Berengut18. A King plot analysis can be used
to systematically quantify and visually examine isotope shifts in various
atomic transitions referenced to the same isotope.

The refined mass-excess value obtained for 84Sr was employed to carry out an
updated King plot analysis for two transitions at 589 nm and 598 nm, as
detailed in miyake2019; Takano_2017. Figure 3 displays the King plot featuring
both the literature mass values from AME2020 and our updated mass value. The
main uncertainties in the plot are from the frequency shifts measurements in
the transitions between 84Sr, 86Sr, and 87Sr relative to 88Sr as adopted from
miyake2019. The transition of 84Sr-88Sr at 598 nm is from Takano_2017 which
has a better precision than that of miyake2019. The observed nonlinearity from
miyake2019, achieved through our new mass value for 84Sr is evident, and the
uncertainty is no longer constrained by the mass uncertainty. Given that the
uncertainty in the mass contributes to the calculation of the modified isotope
shifts in similar ways, the impact on the linearity of the King plot is
preserved due to the agreement of our result with the literature value.

A linear orthogonal distance regression analysis ODR2004 that provides unified
standard error estimates for the uncertainties implanted in $x$ and $y$
directions, was conducted to determine the slope and intercept. The resulting
slope and intercept from the fit to the AME2020 mass values and the updated
mass value are 0.98148(51) and 8578(326) MHz$\cdot$amu, respectively. Our
results align well with the fitting values of 0.981(5) and 8560(3450)
MHz$\cdot$amu, reported in miyake2019, offering smaller uncertainties. These
results confirm the nonlinearity announced in miyake2019, using the
nonlinearity measure defined in flam18; Berengut18. The uncertainty from the
mass values now allows for a potential future precision measurement of optical
spectroscopy to reach a relative precision by more than three orders of
magnitude, thus potentially leading to an unprecedented sensitivity for new
physics. This will reach the spectroscopic precision necessary to attain
state-of-the-art bounds on a spin-independent fifth force, as current
calculations place the required uncertainty at < 1 Hz Ludlow15; Berengut18.
This would essentially demand measuring optical atomic transitions with a
relative precision of 10-15 or lower, a challenging accomplishment feasible
only for a select few leading optical clock laboratories Ludlow15.

##  IV Conclusion and outlook

In summary, we have performed direct high-precision mass and double-beta decay
$Q$ value measurements of 84Sr using the PI-ICR technique with the JYFLTRAP
double PTMS. The mass and $Q$ values were determined with a precision
approximately 30 times higher than those adopted in AME2020, aligning well
with the literature. The new mass value was utilized to conduct an updated
King plot analysis for transitions at 689 nm and 698 nm for 84,86,87Sr in
relation to 88Sr. Our results showcase an enhanced nonlinear King plot, in
good agreement with previously reported values, offering smaller
uncertainties. The current contribution of the mass uncertainty to the King
plot analysis enables future precision enhancements by several orders of
magnitude in optical spectroscopy, mitigating statistical and systematic
errors in both transitions. This potential advancement could lead to
unparalleled sensitivity to new physics. More precise isotope shift
measurements of Sr isotopes, providing stringent experimental constraints on
King linearity, are highly required. Performing such a measurement with a
radioactive isotope e.g., 90Sr, to avoid complications due to hyperfine
structure, is feasible at the IGISOL facility. In conjunction with additional
measurements of isotope shifts in elements such as calcium, this will
contribute to enhancing precision in atomic structure calculations and
imposing constraints on new physics.

###### Acknowledgements.

We acknowledge the staff of the Accelerator Laboratory of University of
Jyväskylä (JYFL-ACCLAB) for providing stable online beam. We thank the support
by the Academy of Finland under the Finnish Centre of Excellence Programme
2012-2017 (Nuclear and Accelerator Based Physics Research at JYFL) and
projects No. 306980, No. 312544, No. 275389, No. 284516, No. 295207, No.
314733, No. 315179, No. 327629, No. 320062, No. 354589 and 345869. The support
by the EU Horizon 2020 research and innovation program under grant No. 771036
(ERC CoG MAIDEN) is acknowledged. We express gratitude for the productive
results stemming from conversations with R. de Groote and X. F. Yang.

## References

  * King (1984) W. H. King,  _Isotope Shifts in Atomic Spectra_ (Plenum, New York, 1984).
  * Flambaum  _et al._ (2018) V. V. Flambaum, A. J. Geddes, and A. V. Viatkina, Phys. Rev. A 97, 032510 (2018).
  * Dammalapati  _et al._ (2009) U. Dammalapati, S. De, K. Jungmann, _et al._ , Eur. Phys. J. D 53, 1 (2009).
  * Shi  _et al._ (2017) C. Shi, F. Gebert, C. Gorges, _et al._ , Appl. Phys. B 123, 2 (2017).
  * Nazé  _et al._ (2015) C. Nazé, J. G. Li, and M. Godefroid, Phys. Rev. A 91, 032511 (2015).
  * Frugiuele  _et al._ (2017) C. Frugiuele, E. Fuchs, G. Perez, and M. Schlaffer, Phys. Rev. D 96, 015011 (2017).
  * Berengut  _et al._ (2018a) J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Phys. Rev. Lett. 120, 091801 (2018a).
  * Miyake  _et al._ (2019) H. Miyake, N. C. Pisenti, P. K. Elgee, _et al._ , Phys. Rev. Research 1, 033113 (2019).
  * Manovitz  _et al._ (2019) T. Manovitz, R. Shaniv, Y. Shapira, R. Ozeri, and N. Akerman, Phys. Rev. Lett. 123, 203001 (2019).
  * Knollmann  _et al._ (2019) F. W. Knollmann, A. N. Patel, and S. C. Doret, Phys. Rev. A 100, 022514 (2019).
  * Counts  _et al._ (2020a) I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung, J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V. Vuletić, Phys. Rev. Lett. 125, 123002 (2020a).
  * Solaro  _et al._ (2020) C. Solaro, S. Meyer, K. Fisher, J. C. Berengut, E. Fuchs, and M. Drewsen, Phys. Rev. Lett. 125, 123003 (2020).
  * Nesterenko  _et al._ (2020) D. Nesterenko, R. de Groote, T. Eronen, Z. Ge, M. Hukkanen, A. Jokinen, and A. Kankainen, International Journal of Mass Spectrometry 458, 116435 (2020).
  * Counts  _et al._ (2020b) I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung, J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V. Vuletić, Phys. Rev. Lett. 125, 123002 (2020b).
  * Rehbehn  _et al._ (2021) N.-H. Rehbehn, M. K. Rosner, H. Bekker, _et al._ , Phys. Rev. A 103, L040801 (2021).
  * Müller  _et al._ (2021) R. A. Müller, V. A. Yerokhin, A. N. Artemyev, and A. Surzhykov, Phys. Rev. A 104, L020802 (2021).
  * Hur  _et al._ (2022) J. Hur, D. P. L. A. Craik, I. Counts, _et al._ , Phys. Rev. Lett. 128, 163201 (2022).
  * Figueroa  _et al._ (2022) N. L. Figueroa, J. C. Berengut, V. A. Dzuba, _et al._ , Phys. Rev. Lett. 128, 073001 (2022).
  * Ono  _et al._ (2022) K. Ono, Y. Saito, T. Ishiyama, _et al._ , Phys. Rev. X 12, 021003 (2022).
  * Stellmer  _et al._ (2014) S. Stellmer, F. Schreck, and T. C. Killian, in  _Annual Review of Cold Atoms and Molecules_, Vol. 2, edited by K. W. Madison, K. Bongs, L. D. Carr, A. M. Rey, and H. Zhai (World Scientific, Singapore, 2014) Chap. 1.
  * Rana  _et al._ (2012) R. Rana, M. Höcker, and E. G. Myers, Phys. Rev. A 86, 050502 (2012).
  * Wang  _et al._ (2021) M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, Chinese Physics C 45, 030003 (2021).
  * Kondev  _et al._ (2021) F. Kondev, M. Wang, W. Huang, S. Naimi, and G. Audi, Chinese Physics C 45, 030001 (2021).
  * Eronen  _et al._ (2012) T. Eronen, V. S. Kolhinen, V. V. Elomaa, D. Gorelov, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, P. Karvonen, S. Kopecky, I. D. Moore, H. Penttilä, S. Rahaman, S. Rinta-Antila, J. Rissanen, A. Saastamoinen, J. Szerypo, C. Weber, and J. Äystö, European Physical Journal A 48, 1 (2012).
  * Eronen  _et al._ (2016) T. Eronen, A. Kankainen, and J. Äystö, Progress in Particle and Nuclear Physics 91, 259 (2016).
  * Moore  _et al._ (2013) I. D. Moore, T. Eronen, D. Gorelov, J. Hakala, A. Jokinen, A. Kankainen, V. S. Kolhinen, J. Koponen, H. Penttilä, I. Pohjalainen, M. Reponen, J. Rissanen, A. Saastamoinen, S. Rinta-Antila, V. Sonnenschein, and J. Äystö, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms 317, 208 (2013).
  * Karvonen  _et al._ (2008) P. Karvonen, I. D. Moore, T. Sonoda, T. Kessler, H. Penttilä, K. Peräjärvi, P. Ronkanen, and J. Äystö, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms 266, 4794 (2008).
  * Nieminen  _et al._ (2001) A. Nieminen, J. Huikari, A. Jokinen, J. Äystö, P. Campbell, and E. C. Cochrane, Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 469, 244 (2001).
  * Savard  _et al._ (1991) G. Savard, S. Becker, G. Bollen, H. J. Kluge, R. B. Moore, T. Otto, L. Schweikhard, H. Stolzenberg, and U. Wiess, Physics Letters A 158, 247 (1991).
  * Nesterenko  _et al._ (2018) D. A. Nesterenko, T. Eronen, A. Kankainen, L. Canete, A. Jokinen, I. D. Moore, H. Penttilä, S. Rinta-Antila, A. de Roubin, and M. Vilen, European Physical Journal A 54, 0 (2018).
  * Nesterenko  _et al._ (2021) D. A. Nesterenko, T. Eronen, Z. Ge, A. Kankainen, and M. Vilen, Eur. Phys. J. A 57, 302 (2021).
  * Eliseev  _et al._ (2014) S. Eliseev, K. Blaum, M. Block, A. Dörr, C. Droese, T. Eronen, M. Goncharov, M. Höcker, J. Ketter, E. M. Ramirez, D. A. Nesterenko, Y. N. Novikov, and L. Schweikhard, Applied Physics B: Lasers and Optics 114, 107 (2014).
  * Eliseev  _et al._ (2013) S. Eliseev, K. Blaum, M. Block, C. Droese, M. Goncharov, E. Minaya Ramirez, D. A. Nesterenko, Y. N. Novikov, and L. Schweikhard, Physical Review Letters 110, 82501 (2013).
  * Kramida  _et al._ (2020) A. Kramida, Yu. Ralchenko, J. Reader, and and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.8), [Online]. Available: https://physics.nist.gov/asd [2021, January 19]. National Institute of Standards and Technology, Gaithersburg, MD. (2020).
  * Kellerbauer  _et al._ (2003) A. Kellerbauer, K. Blaum, G. Bollen, F. Herfurth, H. J. Kluge, M. Kuckein, E. Sauvan, C. Scheidenberger, and L. Schweikhard, European Physical Journal D 22, 53 (2003).
  * Roux  _et al._ (2013) C. Roux, K. Blaum, M. Block, C. Droese, S. Eliseev, M. Goncharov, F. Herfurth, E. M. Ramirez, D. A. Nesterenko, Y. N. Novikov, and L. Schweikhard, The European Physical Journal D 67, 1 (2013).
  * Birge (1932) R. T. Birge, Physical Review 40, 207 (1932).
  * Huang  _et al._ (2021) W. Huang, M. Wang, F. Kondev, G. Audi, and S. Naimi, Chinese Physics C 45, 030002 (2021).
  * Kellerbauer  _et al._ (2007) A. Kellerbauer, G. Audi, D. Beck, K. Blaum, G. Bollen, C. Guénaut, F. Herfurth, A. Herlert, H.-J. Kluge, D. Lunney, S. Schwarz, L. Schweikhard, C. Weber, and C. Yazidjian, Phys. Rev. C 76, 045504 (2007).
  * Haettner  _et al._ (2011) E. Haettner, D. Ackermann, G. Audi, K. Blaum, M. Block, S. Eliseev, T. Fleckenstein, F. Herfurth, F. P. Heßberger, S. Hofmann, J. Ketelaer, J. Ketter, H.-J. Kluge, G. Marx, M. Mazzocco, Y. N. Novikov, W. R. Plaß, S. Rahaman, T. Rauscher, D. Rodríguez, H. Schatz, C. Scheidenberger, L. Schweikhard, B. Sun, P. G. Thirolf, G. Vorobjev, M. Wang, and C. Weber, Phys. Rev. Lett. 106, 122501 (2011).
  * Booij  _et al._ (1971) H. Booij, E. Van Hoek, H. Van Der Molen, W. Slot, and J. Blok, Nuclear Physics A 160, 337 (1971).
  * Morton  _et al._ (1971) J. Morton, W. Davies, W. Mclatchie, D. W., and J. Kitching, Nuclear Physics A 161, 228 (1971).
  * Takano  _et al._ (2017) T. Takano, R. Mizushima, and H. Katori, Applied Physics Express 10, 072801 (2017).
  * King (1963) W. H. King, J. Opt. Soc. Am. 53, 638 (1963).
  * Berengut  _et al._ (2018b) J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Phys. Rev. Lett. 120, 091801 (2018b).
  * York  _et al._ (2004) D. York, N. M. Evensen, M. L. Martınez, and J. De Basabe Delgado, American Journal of Physics 72, 367 (2004).
  * Ludlow  _et al._ (2015) A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Rev. Mod. Phys. 87, 637 (2015).


# Blind Deep-Learning-Based Image Watermarking Robust Against Geometric
Transformations ††thanks: This work was funded in part by the Research
Foundation – Flanders (FWO), IDLab (Ghent University – imec), Flanders
Innovation & Entrepreneurship (VLAIO), the Flemish Government, and the
European Union. The computational resources (STEVIN Supercomputer
Infrastructure) and services used in this work were kindly provided by Ghent
University, the Flemish Supercomputer Center (VSC), the Hercules Foundation
and the Flemish Government department EWI.

Hannes Mareen, Lucas Antchougov, Glenn Van Wallendael, and Peter Lambert
Ghent University – imec, IDLab, Department of Electronics and Information
Systems, Ghent, Belgium,
<EMAIL_ADDRESS>https://media.idlab.ugent.be

###### Abstract

Digital watermarking enables protection against copyright infringement of
images. Although existing methods embed watermarks imperceptibly and
demonstrate robustness against attacks, they typically lack resilience against
geometric transformations. Therefore, this paper proposes a new watermarking
method that is robust against geometric attacks. The proposed method is based
on the existing HiDDeN architecture that uses deep learning for watermark
encoding and decoding. We add new noise layers to this architecture, namely
for a differentiable JPEG estimation, rotation, rescaling, translation,
shearing and mirroring. We demonstrate that our method outperforms the state
of the art when it comes to geometric robustness. In conclusion, the proposed
method can be used to protect images when viewed on consumers’ devices.

###### Index Terms:

Watermarking, Image Forensics.

##  I Introduction

Image watermarking has applications such as copyright protection,
authentication, and fingerprinting or traitor tracing [1, 2]. It is typically
important for the watermark to be robust. For example, the watermark should
survive attacks such as compression and geometric transformations.
Additionally, to not hinder the consumers’ viewing experience, it is desirable
for the watermark to be imperceptible. Moreover, for the detection process to
be applicable on a broad scale (such as on consumer devices), it is also
important for the watermark to be blind, meaning the watermarking can be
detected without requiring the original image nor additional information.

HiDDeN (Hiding Data with Deep Networks) is an advanced, deep-learning-based
image watermarking technique [3]. This technique involves the integration of
multiple networks into one end-to-end Convolutional Neural Network (CNN). A
first network aims to embeds a watermark imperceptibly into the image, and
second network aims to accurately decode the embedded watermark. The decoding
does not require the original content, making it a blind watermarking
technique. Although HiDDeN demonstrated good performance, it is not robust
against geometric transformations.

This paper proposes a novel watermarking method robust against geometric
attacks, by extending the HiDDeN architecture [3] with noise layers that
simulate geometric attacks.

##  II Proposed Method

Figure 1: Architecture of HiDDeN [3], consisting of an encoder and decoder,
separated by noise layers. We include noise layers that simulate geometric
transformations to provide robustness against geometric attacks.

We propose to extend the HiDDeN architecture [3] in order to make it robust
against gemoetric distortions. The HiDDeN architecture is visualized in Fig.
1, and consists of three networks to achieve this task: an encoder, decoder,
and adversary. The encoder network takes an image and a bit message as input
and produces an output image with the message embedded into it. The decoder
network then takes the encoded image as input and returns a decoded message.
The decoder is blind, i.e., it does not take any additional information as
input. The adversary network plays a crucial role by being trained to
differentiate between watermarked and non-watermarked images, essentially
acting as a discriminator. This dynamic enables the encoder and decoder
networks to learn how to ”outsmart” the adversary network, resulting in a
substantial improvement in imperceptibility for the embedded watermark. Using
the adversary network results in a substantial improvement in imperceptibility
[3].

Between the encoder and the decoder networks, multiple so-called noise layers
can be inserted. These layers simulate attacks on the watermark. The
originally proposed HiDDeN model [3] used the following noise layers: Crop,
Gaussian Blur, Cropout, Dropout, and JPEG compression through JPEG-Mask &
JPEG-Drop. We opted to not use Cropout and Dropout, as these use information
from the unwatermarked image and are therefore not realistic attacks.
Additionally, we replaced the JPEG-Mask and JPEG-Drop layers because we could
not achieve good robustness against JPEG compression with them. Instead, we
propose to use a JPEG noise layer on a differentiable approximation of JPEG
compression called JPEGdiff [4] (instead of the JPEG-Mask and JPEG-Crop noise
layers that the original method used). JPEGdiff works in a similar way as
JPEG, with the difference that the rounding operation after quantization uses
a rounding approximation instead. That is because the rounding operation
$\lfloor I\rceil$ has a derivative of 0 nearly everywhere. The differentiable
approximate rounding is done in the following way: $\lfloor
I\rceil_{approx}=\lfloor I\rceil+(I-\lfloor I\rceil)^{3}$. This network is
trained separately to simulate JPEG and is then used as a noise layer.

To provide robustness against geometric transformations, we add the following
noise layers: Rescale (factor between 50% and 200%), Translate (5% to 50% of
the image width and height), Rotate (angles between +/- 10 and 60 degrees),
Shear (angles between +-/ 10 and 45 degrees), and Mirror. These operations are
all differentiable. Additionally, we deleted the Crop noise layer, as we found
that the base model (without noise layers) already provided sufficient
robustness against cropping (see Section III and Fig. 2).

TABLE I: Imperceptibility results Identity | Rescale | Translate | Rotate | Shear
---|---|---|---|---
0.975 | 0.960 | 0.977 | 0.963 | 0.966
Mirror | Gaussian Blur | JPEGdiff | Combined | RivaGAN
0.980 | 0.963 | 0.963 | 0.930 | 0.977

##  III Evaluation

###  III-A Experimental Setup

To create the training, validation, and testing, we selected 10000, 1000, and
1000 images from the COCO dataset [5], respectively. During training, random
crops of $128\times 128$ pixels are taken, whereas testing was done on
rescaled images of $512\times 512$ pixels (to evaluate imperceptibility) and
$256\times 256$ pixels (to evaluate robustness). Additionally, we embedded
randomly-generated 30-bit messages. We train and evaluate an Identity model
(no noise layers), 7 specialized models (single noise layer), and a Combined
model (6 noise layers, selected at random during training). The Shear noise
layer was not added in the Combined model, because the Rotate layer already
sufficient provided robustness against shearing. We additionally evaluate and
compare against the state-of-the-art RivaGAN method [6].

###  III-B Imperceptibility

To measure the imperceptibility, we calculate the structured similarity index
measure (SSIM) between the original image and its watermarked version. A
higher value signifies a less perceptible watermark. Table I gives average
SSIM values for all methods. Although the combined models scores the lowest
SSIM score (0.930), it remains an imperceptible result. Furthermore, RivaGAN
exhibits an SSIM score of 0.977, performing similarly to most specialized
models. We manually inspected the watermarked images and confirm that it is
very hard (if not impossible) to spot artefacts at native resolution. Example
images are available on our website:
https://media.idlab.ugent.be/watermarking-blind-icce.

###  III-C Robustness

To measure the robustness, we evaluate the bit accuracy of the decoded
messages for a range of attacks. It can be observed that the identity model
(blue line) performs relatively poorly against most attacks. The surprising
exception is cropping, for which it maintains a bit accuracy close to 100% for
crops with ratio $p\geq 0.2$. The specialized models (green lines) never fall
below 65% bit accuracy for the considered attacks. The combined model shows
significantly improved robustness compared to the identity model.
Unsurprisingly, the combined model has a lower resilience than the specialized
models for their targeted attacks, though. This is especially the case for
JPEG compression and Gaussian Blurring. Future work should investigate how to
provide additional robustness in these cases.

In contrast, RivaGAN shows very good performance against JPEG compression and
Gaussian blurring. This demonstrates RivaGAN’s ability to extract low-level
features of images that can survive certain attacks without an explicit need
for noise layers [6]. However, most notably, our specialized and combined
methods outperform RivaGAN when exposed to geometric attacks.

Figure 2: Robustness results.

##  IV Conclusion

We presented a blind deep-learning-based watermarking method based on the
HiDDeN architecture [3], which uses noise layers to provide robustness against
specific attacks. We demonstrate that the embedded watermarks are
imperceptible and, most notably, we outperform state-of-the-art methods in
robustness against geometric attacks. As such, the watermarking method can be
used to protect copyright of images viewed on consumer electronic devices.

## References

  * [1] M. Asikuzzaman and M. R. Pickering, “An overview of digital video watermarking,” _Proc. IEEE Int. Symposium Circuits Syst. (ISCAS)_ , vol. 28, no. 9, pp. 2131–2153, 2018.
  * [2] H. Mareen, J. De Praeter, G. Van Wallendael, and P. Lambert, “A novel video watermarking approach based on implicit distortions,” _IEEE Trans. Consum. Electron._ , vol. 64, no. 3, pp. 250–258, 2018.
  * [3] J. Zhu, R. Kaplan, J. Johnson, and L. Fei-Fei, “HiDDeN: Hiding data with deep networks,” _CoRR_ , vol. abs/1807.09937, 2018.
  * [4] R. Shin and D. Song, “JPEG-resistant adversarial images,” in _NIPS Workshop Mach. Learning Computer Security,_ , vol. 1, 2017, p. 8.
  * [5] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick, “Microsoft COCO: Common objects in context,” in _Proc. Eur. Conf. Computer Vis._ Springer, 2014, pp. 740–755.
  * [6] K. A. Zhang, L. Xu, A. Cuesta-Infante, and K. Veeramachaneni, “Robust invisible video watermarking with attention,” _arXiv preprint arXiv:1909.01285_ , 2019.