###### keywords:

neutron stars, superdense matter, equation of state.

Ofengeim et al.]<EMAIL_ADDRESS>D. Ofengeim1,2, P. S. Shternin2, T.
Piran1

[Maximally Massive Neutron Stars]Maximal Mass Neutron Star as a Key to
Superdense Matter Physics

The Herbew University of Jerusalem, Jerusalem, Israel Ioffe Institute, Saint
Petersburg, Russia

We propose a universal approximation of the equation of state of superdense
matter in neutron star (NS) interiors. It contains only two parameters, the
pressure and the density at the center of the maximally massive neutron star.
We demonstrate the validity of this approximation for a wide range of
different types of equations of state, including both baryonic and hybrid
models. Combined with recently discovered correlations of internal (density,
pressure, and speed of sound at the center) and external (mass, radius)
properties of a maximally massive neutron star, this approximation turns out
to be an effective tool for determining the equation of state of superdense
matter using astrophysical observations.

10.31857/S…

##  1 Introduction

Determining the equation of state of superdense matter is one of the central
problems of neutron star astrophysics (Haensel et al. 2007). A natural
approach to this problem is to infer observational constraints on the NS
physical characteristics like mass $M$, radius $R$, moment of inertia $I$,
etc., and then compare these with the predictions of NS structure theory
(Lattimer, 2021). On this path, one usually takes a certain microphysical
model, which allows one to calculate the equation of state, and by comparing
the results of modeling the NS structure with observations, one then
constrains the parameters of the original model.

However, the properties of matter at densities significantly larger than the
saturated nuclear matter density ($\rho_{0}=2.8\times 10^{14}$ g/cm3) are
practically impossible to study in terrestrial laboratories. When constructing
models of such matter, one has to rely on extrapolations of various methods
and approaches that have proven themselves under less extreme conditions.
These extrapolations are based on a variety of ideas about the microphysics of
superdense nuclear matter. As a result, the “astrophysical market” offers a
huge number of different equations of state, which at first glance are
completely different from each other (see section 2 below).

It is, therefore, interesting to identify universal relations between NS
parameters that weakly depend on specific microphysical models (for example,
Lattimer and Prakash 2001; Bejger and Haensel 2002; Yagi and Yunes 2013a,b;
Zhang and Yagi 2020; Ofengeim 2020; Cai et al., 2023). Such relations are
intended to describe in a unified way the combinations of NS characteristics
encountered within the framework of various approaches to modeling superdense
matter. They are convenient to use for interpreting observations as
restrictions on possible combinations of stellar parameters become largely
model-independent. This work is devoted to the development of such an
approach.

Many observational properties of NSs ($M$, $R$, etc.) are determined by the
solution of the Tolman-Oppenheimer-Volkoff equations (Tolman, 1939;
Oppenheimer and Volkoff, 1939). To close this system of equations, it is
necessary to specify the relation between pressure and density, $P=P(\rho)$.
This relation, the equation of state (Haensel et al. 2007), is in the one-to-
one correspondence to the $M-R$ curve of NSs (Lindblom 1992). That is if we
manage to accurately determine the $M-R$ curve, the problem of finding the
equation of state $P(\rho)$ will also be solved. In reality, we can only
measure masses and radii of individual NSs, and this only with a finite
accuracy, which significantly complicates the situation.

Whatever the equation of state is, the $M-R$ curve always have a global
maximum, $M_{{}_{\text{TOV}}}$, along the mass axis (e.g. Shapiro, Teukolsky,
1985). However, the properties of the maximally massive neutron star (MMNS)
are different for each equation of state model. If the latter predicts
$M_{{}_{\text{TOV}}}$ less than the mass of any of the observed NS masses,
then the model is incorrect. Therefore, any observation of a sufficiently
massive NS imposes significant constraints on the equation of state.

The value of $M_{{}_{\text{TOV}}}$ specifies the natural scale of NS masses
for a given equation of state. Similar characteristics are the radius of the
MMNS, $R_{{}_{\text{TOV}}}$, the density at its center,
$\rho_{{}_{\text{TOV}}}$, and the corresponding pressure,
$P_{{}_{\text{TOV}}}$. Note that $\rho_{{}_{\text{TOV}}}$ and
$P_{{}_{\text{TOV}}}$ (for the true equation of state realized in nature) are
the maximum possible density and pressure of matter in a stationary object in
the contemporary Universe. In what follows, we will also need the speed of
sound at the center of the MMNS, $c_{\text{s}_{\text{TOV}}}$.

Ofengeim (2020) showed that among the parameters characterizing the MMNS
($M_{{}_{\text{TOV}}}$, $R_{{}_{\text{TOV}}}$, $\rho_{{}_{\text{TOV}}}$,
$P_{{}_{\text{TOV}}}$, $c_{\text{s}_{\text{TOV}}}$), only two are independent.
This was confirmed on a set of 50 equations of state for nucleonic and
hyperonic compositions. Recently, a possible explanation for these findings
was given in Cai et al. (2023) based on a perturbative analysis of the
dimensionless Tolman-Oppenheimer-Volkoff equations. In the present work, the
existence of correlations between these parameters is confirmed on an extended
sample of 162 equations of state, including nucleon, hyperon, and hybrid
models (i.e., with a quark inner core), and new compact approximation formulas
are proposed to describe these correlations.

In addition, we constructed a universal approximation of the $P(\rho)$
relations for $\rho\gtrsim 3\rho_{0}$ which is based on only two parameters,
namely $\rho_{{}_{\text{TOV}}}$ and $P_{{}_{\text{TOV}}}$. Taking into account
the one-to-one correspondence between the pair
$\rho_{{}_{\text{TOV}}},P_{{}_{\text{TOV}}}$ and the pair
$M_{{}_{\text{TOV}}},R_{{}_{\text{TOV}}}$, this provides a model-independent
tool for a direct transformation of observational constrains on the properties
of MMNS into constraints on the equation of state of superdense matter at
densities that are most difficult to reach in laboratory research.

##  2 The Zoo of equations of state

Figure 1: Distributions of (a) NS maximal mass, (b) speed of sound in the
center of the star, (c) $R_{1.4}/R_{{}_{\text{TOV}}}$ ratio for the sample of
equations of state used. Different hatching patterns corresponds to different
model types (nucleon, hyperon, hybrid).

We consider 162 NS equations of state. Among them, 97 have nycleon
composition, 32 allow for the appearance of hyperons and $\Delta$-isobars (but
do not allow for free quarks) in NS interiors, and 33 equations of state
predict the presence of internal quark core of the star. Main sources for
equations of state are the CompOSE database111https://compose.obspm.fr. (Typel
et al., 2015), the collection of models from Read et al. (2009) and models
used earlier by Ofengeim (2020). A complete list of the equations of state
used containing heir main characteristics and references is given in Appendix.

The considered models are based on a variety of approaches to modeling nuclear
interactions and the microphysics of superdense matter. These include obsolete
models of free degenerate neutron gas and free $npe$ gas; models based on
effective energy density functionals, including purely phenomenological (PAL,
PAPAL, BGN families), non-relativistic Skyrme-type (SLy, BSk, SkI, etc.) or
Gogni (D1M*) and others, numerous relativistic mean field functionals; models
derived from microscopic baryon interaction potentials using many-body methods
(APR, WFF, BBB); and also some other models. Among the hybrid equations of
state, both those in which a first-order phase transition occurs between
hadronic and quark matter (for example, CMF, VQCD) and those in which there is
a quark-hadron crossover (QHC) between these phases are present.

Figure 1 shows how this sample of models is distributed over
$M_{{}_{\text{TOV}}}$ (panel a), over $c_{\text{s}_{\text{TOV}}}$ (panel b),
and over the ratio of the radius $R_{1.4}$ of a “canonical” NS with a mass of
$1.4$$M_{\odot}$ to the radius of the most massive star, $R_{{}_{\text{TOV}}}$
(panel c). Notice that all 162 equations of state satisfy the condition
$R_{{}_{\text{TOV}}}<R_{1.4}$.

Realistic equations of state, apparently, should satisfy the causality
condition $c_{\text{s}_{\text{TOV}}}<c$, where $c$ is the speed of light in
vacuum (Haensel et al. 2007), and explain present observations of massive
radio pulsars (Demorest et al., 2010; Antoniadis et al., 2013; Fonseca et al.,
2021). About one third of the equations of state of our sample violate the
second condition, and about 10% violate the first one. We do believe, however,
that it is important to include both realistic and non-realistic models in
order to comprehensively explore the generality of the detected correlations
and the constructed approximations.

We also note, that the speed of sound is not always a monotonic function of
the density (especially for hybrid models), and $c_{\text{s}_{\text{TOV}}}$ is
not necessarily the highest possible speed of sound within a star. For
example, for the JJ(VQCD)soft and JJ(VQCD)intermediate models,
$\rho_{{}_{\text{TOV}}}$ falls into the region of a first-order phase
transition, and $c_{\text{s}_{\text{TOV}}}$ is formally equal to zero (these
models are not shown in figure 1(b)). However, it turns out that the more
stringent condition, $c_{\text{s}}<c$ in the entire volume of the star, is
violated by the same number of equations of state as the condition
$c_{\text{s}_{\text{TOV}}}<c$.

Figure 2: Correlations of MMNS characteristics for different equations of
state. Different symbols cortespond to different models
($\boldsymbol{\bigcirc}$ to nucleonic models, $\boldsymbol{\triangle}$ to
hyperonic and isobar models, $\boldsymbol{\Box}$ to hybrid models,
respectively). The solid lines show The approximations given in equations (1).
The quantity $\mathcal{R}$ is defined in equation (1d), $R_{\rm
TOV}^{\text{fit}}$ is given by equation (1b). The exponents $p_{i}$ and
$q_{i}$, with $i=M,R,c$, are given in table 1. Relative errors of the fits are
shown in lower panels.

##  3 Correlations of MMNS characteristics

There are strong correlations between the values of $M_{{}_{\text{TOV}}}$,
$R_{{}_{\text{TOV}}}$, $\rho_{{}_{\text{TOV}}}$, $P_{{}_{\text{TOV}}}$, and
$c_{\text{s}_{\text{TOV}}}$ calculated within different models (Ofengeim,
2020; Cai et al., 2023), for which we propose here new fitting formulae:

$M_{{}_{\text{TOV}}}=\frac{\rho_{{}_{\text{TOV}}}\mathcal{R}^{3}}{f_{M}(P_{{}_{\text{TOV}}},\rho_{{}_{\text{TOV}}})},$
(1a)
$R_{{}_{\text{TOV}}}=\frac{\mathcal{R}}{f_{R}(P_{{}_{\text{TOV}}},\rho_{{}_{\text{TOV}}})},$
(1b) $c_{\text{s}_{\text{TOV}}}=\sqrt{G\rho_{{}_{\text{TOV}}}}\ \mathcal{R}\
\frac{f_{c}(P_{{}_{\text{TOV}}},\rho_{{}_{\text{TOV}}})}{f_{R}(P_{{}_{\text{TOV}}},\rho_{{}_{\text{TOV}}})}.$
(1c) These formulas differ from those proposed earlier by Ofengeim (2020) and
Cai et al. (2023) and are based on dimensional analysis. The key to their
construction is the introduction of the characteristic “Jeans” scale for the
radius
$\mathcal{R}=\sqrt{\frac{P_{{}_{\text{TOV}}}}{G\rho_{{}_{\text{TOV}}}^{2}}}$
(1d) and the corresponding scale for the mass
$\rho_{{}_{\text{TOV}}}\mathcal{R}^{3}$. The dimensionless functions $f_{M}$,
$f_{R}$ and $f_{c}$ have the same structure
$f_{i}=c_{i}\left(\frac{P_{{}_{\text{TOV}}}}{\rho_{{}_{\text{TOV}}}c^{2}}\right)^{p_{i}}\left(\frac{\rho_{{}_{\text{TOV}}}}{\rho_{0}}\right)^{q_{i}}+d_{i},$
(1e)

where the optimal values of the fitting parameters $c_{i}$, $d_{i}$, $p_{i}$
and $q_{i}$ are given in table 1. Figure 2 clearly demonstrates the accuracy
of the obtained approximations. It can be seen that the correlations of
$M_{{}_{\text{TOV}}}$ and $R_{{}_{\text{TOV}}}$ with $P_{{}_{\text{TOV}}}$,
$\rho_{{}_{\text{TOV}}}$ are described with good accuracy, while the
approximation formula for $c_{\text{s}_{\text{TOV}}}$ shows a larger scatter.

Table 1: Parameters of the approximations given in equation (1). The last two columns show the relative root mean squared fit error (rrms) and the maximal relative fit error (max). Models-outliers JJ(VQCD)soft and JJ(VQCD)intermediate are not included in the calculation of these errors. $i$ | $p_{i}$ | $q_{i}$ | $c_{i}$ | $d_{i}$ | rrms | max
---|---|---|---|---|---|---
$M$ | $1.41$ | $0.0177$ | $5.86$ | $0.273$ | 0.86% | 5.4%
$R$ | $0.518$ | $-0.0755$ | $2.74$ | $-0.0741$ | 2.0% | 9.3%
$c$ | $0.76$ | $-0.016$ | $3.27$ | $0.19$ | 4.9% | 22%

Among the 162 models considered, there are two that stand out significantly
from these correlations. These are the JJ(VQCD)soft and JJ(VQCD)intermediate
models, for which the center of the most massive NS falls in the region of the
first-order phase transition. In this case, the maximum of the $M-R$ curve
turns out to be non-smooth, which is apparently crucial for the correlation
$M_{{}_{\text{TOV}}}(\rho_{{}_{\text{TOV}}},P_{{}_{\text{TOV}}})$ to be
fulfilled. The values of $M_{{}_{\text{TOV}}}$ for these equations of state
differ by $\sim 20\%$ from those predicted by the universal trend (1a).
Likewise, $c_{\text{s}_{\text{TOV}}}\to 0$ at the center of these stars, which
differs from the predictions of (1c) by $\sim 100\%$. These models are not
shown in Figure 2(c) and the bottom panel of Figure 2(a). On the other hand,
they satisfy the correlations
$R_{{}_{\text{TOV}}}(\rho_{{}_{\text{TOV}}},P_{{}_{\text{TOV}}})$ along with
the others. In all other cases, when the maximum of the $M-R$ curve is smooth,
no significant deviations from correlations (1) are observed.

##  4 The universal approximation $P(\rho)$

If we express the dependencies $P(\rho)$ in dimensionless variables
$P/P_{{}_{\text{TOV}}}$, $\rho/\rho_{{}_{\text{TOV}}}$, the behavior of all
162 models will be very similar, especially when approaching the center of the
MMNS. A good universal approximation at $\rho>3\rho_{0}$ is:

$P=P_{{}_{\text{TOV}}}\
g_{P}\left(\rho/\rho_{{}_{\text{TOV}}},c_{\text{s}_{\text{TOV}}}^{\text{(fit)}}/c,\gamma_{\text{max}}^{\text{(fit)}}\right)\
,$ (2a) where $c_{\text{s}_{\text{TOV}}}^{\text{(fit)}}$ is given by equation
(1c), the value
$\gamma_{\text{max}}^{\text{(fit)}}=\left(c_{\text{s}_{\text{TOV}}}^{\text{(fit)}}\right)^{2}\rho_{{}_{\text{TOV}}}/P_{{}_{\text{TOV}}}$
plays the role of the adiabatic index222$\gamma_{\text{max}}^{\text{(fit)}}$
may differ significantly from the actual value of the adiabatic index at the
point $\rho=\rho_{{}_{\text{TOV}}}$. Although the fitting error (1c) for
$c_{\text{s}_{\text{TOV}}}$ is generally not too large, but the error of
$c_{\text{s}_{\text{TOV}}}^{2}$ in a few worst cases exceeds $50\%$.  in the
MMNS center, and the function $g_{P}$ is $g_{P}(x,\zeta,\gamma)=x^{\gamma-
a_{0}-a_{1}\zeta}\Bigl{[}1\>+\\\
+(a_{0}+a_{1}\zeta)(1-x)+(b_{0}+b_{1}\zeta)(1-x)^{p}\Bigr{]}^{-1}$ (2b) with
$a_{0}=-0.6268$, $a_{1}=-0.1294$, $b_{0}=-0.5588$, $b_{1}=1.023$ and
$p=2.494$.

Figure 3: Typical examples of the application of the approximation (2) (lines)
to nucleonic (BSk24, circles), hyperonic (CMF-1,traingles) and hybrid
(QHC21_A, squares) equations of state (symbols). Each curve is plotted in the
region $3\rho_{0}\leqslant\rho\leqslant\rho_{{}_{\text{TOV}}}$. The lower
panel shows the relative approximation errors.

On average, the approximation error (2) does not exceed a few percent. Figure
3 shows three typical examples for the nucleon, hyperon, and hybrid equations
of state. It can be seen that the fit works best at high densities, and is
worst at lower ones.

Figure 4: The distribution of the errors of the fit (2) for the whole sample
of equations of state.

Figure 4 shows the distributions of the root-mean-square and maximal
deviations from the approximation (2) at $\rho>3\rho_{0}$ for the sample of
equations of state (see also the table LABEL:tab:zooNuc in Appendix). For
nucleonic models, even the largest error does not exceed 10%. The error can be
significantly higher for models with phase transitions (either to hyperonic or
quark matter). About 10 equations of state give a maximal error of more than
$30-60\%$, but for most of them the standard deviation is still $\lesssim
10\%$. This is a manifestation of the heteroscedasticity of the fitting error,
already demonstrated in figure 3. The standard deviation is dominated by the
region of large densities, where the approximation works well, while the
largest error, on the contrary, characterizes the fit quality at low
densities.

Three models give a large, $\sim 20-40\%$, root mean square error, i.e. they
are unsatisfactorily described by proposed approximations at any density. Two
of them are the already mentioned JJ(VQCD)intermediate and JJ(VQCD)soft models
that show strong density jumps during first-order phase transitions. The third
one, RSGMT(QMC700) model, has extreme values even for those properties that
are subject to laboratory testing, and therefore seems unrealistic.

Figure 5: MMNSs positions on $M,R$ (a) and $P,\rho$ (b) planes. Filled symbols
show subluminal equations of state ($c_{\text{s}_{\text{TOV}}}<c$), open
symbols correspond to superluminal ones ($c_{\text{s}_{\text{TOV}}}\geqslant
c$). Solid lines show the boundaries $c_{\text{s}_{\text{TOV}}}=c$, obtained
using approximations (1). Long-dashed lines show the constraint from Kandel
and Romani (2023), $M_{{}_{\text{TOV}}}>2.19$$M_{\odot}$, short-dashed lines
show the constraint $M_{{}_{\text{TOV}}}<2.33$$M_{\odot}$ from Rezzolla et al.
(2018). Dash-dotted line shows the constraint
$R_{{}_{\text{TOV}}}<R_{1.4}<13.6\,$km (Annala et al., 2018). Regions which
fulfill all these constraints are doubly hatched. The single-hatched region in
panel (b) shows the region occupied by all $P(\rho)$ curves, whose MMNS points
$(\rho_{{}_{\text{TOV}}},P_{{}_{\text{TOV}}})$ are anywhere in the allowed
(double-hatched) region. This region describes the allowed equations of state
under the constraints imposed here. Note that for readability the scale of (b)
was chosen such that some of the models are out of the boundaries of the
figure.

##  5 Applications to observational constraints on MMNSs

We turn now to demonstrate how the proposed universal correlations enable us
to impose model-independent constraints on the equation of state of superdense
matter based on observational data.

Figure 5(a) show MMNS positions on the $M,R$ plane for all considered
equations of state (i.e., each symbol is a point
$(R_{{}_{\text{TOV}}},\,M_{{}_{\text{TOV}}})$) corresponding to a specific
equation of state. Different types of symbols correspond to different types of
equations of state (nucleonic, hyperonic, hybrid). In addition, open symbols
indicate “superluminal” models for which $c_{\text{s}_{\text{TOV}}}\geqslant
c$. The shaded area in the upper right corner shows the so-called maximum
compactness limit for possible values of NS masses and radii (for example,
Lattimer and Prakash, 2016). Figure 5(b) shows the same data, but on the
$P,\rho$ plane (each symbol is a point
$(\rho_{{}_{\text{TOV}}},\,P_{{}_{\text{TOV}}})$), however for a readability,
a scale was chosen at which some of the models fell out of the boundaries of
the figure.

There are various constraints on $M_{{}_{\text{TOV}}}$ in the literature with
varying degrees of certainty. The maximal mass is limited from below by
observations of the most massive NSs. As an example of such a restriction, we
chose the condition from Kandel and Romani (2023), namely
$M_{{}_{\text{TOV}}}>2.19M_{\odot}$ at the 2$\sigma$ significance level. It is
based on measurements of the masses of pulsars in binary systems. This limit
is shown in Figure 5(a) with long-dashed line. In addition, Rezzolla et al.
(2018), based on an analysis of the neutron star merger event GW170817 (Abbott
et al., 2017), proposed an upper limit on the mass of the MMNS,
$M_{{}_{\text{TOV}}}<2.33M_{\odot}$ at the 2$\sigma$ significance level. This
limit is shown in Figure 5(a) with short-dashed line. In the following we use
these constraints for illustrative purposes and demonstrate how observational
constraints can be used to put limits on the equation of state.

Direct inference of the MMNS radius from observations is difficult. We will
resort to the following considerations. As shown in Figure 1(c), for all 162
equations of state $R_{{}_{\text{TOV}}}$ turns out to be less than the radius
$R_{1.4}$ of the “canonical” NS with $M=1.4$$M_{\odot}$. Inferences of the
constraints on on the radii of medium-mass NSs are simpler due to their
greater accessibility for various types of observations (for example, Degenaar
and Suleimanov, 2018). Conservatively, we employ the results of the analysis
of the gravitational wave event GW170817 from the work of Annala et al.
(2018), in which $R_{1.4}<13.6\,$km was obtained at the $90\%$ significance
level333Stronger limits can be obtained from other observations. We set the
same condition on $R_{{}_{\text{TOV}}}$, which is shown in Figure 5(a) with
the dash-dotted line.

These three constraints can be mapped onto the $P,\rho$ plane by considering
the fits (1a) and (1b) as equations for determining $\rho_{{}_{\text{TOV}}}$
and $P_{{}_{\text{TOV}}}$. Their solutions are depicted in Figure 5(b) with
lines of the same styles as the corresponding constraints in Figure 5(a).

Finally, realistic equations of state must satisfy the theoretical restriction
$c_{\text{s}_{\text{TOV}}}<c$. Using formulas (1), this boundary is shown on
both planes of figure 5 with solid lines. The true MMNS should be located
below these lines. Notice that due to the approximate nature of the fitting
formulas, a small number of superluminal models (empty symbols) fall below,
and a small number of subluminal (filled symbols) fall above the solid lines.
This uncertainty, as well as the uncertainties in transferring observational
constraints from $M_{{}_{\text{TOV}}}$ and $R_{{}_{\text{TOV}}}$ to
$P_{{}_{\text{TOV}}}$ and $\rho_{{}_{\text{TOV}}}$, must be taken into account
in a thorough analysis, but here we neglect them in our illustrative analysis.

Combining these constraints, we obtain the allowed regions of
$(M_{{}_{\text{TOV}}},\,R_{{}_{\text{TOV}}})$ and
$(P_{{}_{\text{TOV}}},\,\rho_{{}_{\text{TOV}}})$ which are shown in figure 5
as a double hatch region. Using approximation (2) from each point of this
region one can draw a curve towards lower densities. The set of these curves
will then occupy the area shown in Figure 5(b) as a single hatch region. This
area represents the final constraint on the equation of state that is
constructed using this method. It appears to be at least as stringent as
limits proposed in other recent work (Raaijmakers et al., 2021; Zhang et al.,
2023). But note that this result is purely illustrative. A more thorough
analysis requires taking into account uncertainties in the approximations (1)
and (2) and analyzing the systematic errors of the employed observations, as
well as accounting for other observations that we haven’t used here.

##  6 Conclusions

We elaborated and improved the method for determining the equation of state of
superdense matter from the properties of MMNS, previously proposed by Ofengeim
(2020). The correlations between the quantities $M_{{}_{\text{TOV}}}$,
$R_{{}_{\text{TOV}}}$, $P_{{}_{\text{TOV}}}$, $\rho_{{}_{\text{TOV}}}$ and
$c_{\text{s}_{\text{TOV}}}$ discovered in that work are confirmed on a much
wider set of equations of state, which now includes both baryonic and hybrid
models. Using dimensional analysis we present new approximate formulae (1)
that describe theses correlations.

In addition, it is shown that the $P(\rho)$ relations for a variety of the
equation of state models can be described by a single approximation (2), which
has only two parameters, namely pressure $P_{{}_{\text{TOV}}}$ and density
$\rho_{{}_{\text{TOV}}}$ at the center of MMNS. This fit has a low accuracy at
small densities $\rho\lesssim 3\rho_{0}$, but it works well in the more
interesting denser region.

By applying formulas (1) to a number of theoretical and observational
constraints on the characteristics of MMNS, we limit significantly the
possible values of $M_{{}_{\text{TOV}}}$, $R_{{}_{\text{TOV}}}$,
$P_{{}_{\text{TOV}}}$ and $\rho_{{}_{\text{TOV}}}$. Using the formula (2), the
constraints on the last pair are converted into constraints on the entire
equation of state in the region $\rho\gtrsim 3\rho_{0}$.

The nature of the universal properties studied here remains unclear. In the
work by Ofengeim (2020), the existence of correlations between the properties
of MMNS was interpreted as indirect confirmation of the approximate two-
parametric nature of realistic equations of state identified by Lindblom
(2010). The universality of the $P(\rho)$ curves reported in the present work,
at first glance, directly extends Lindblom’s findings to almost all existing
models and proposes a pair of quantities $P_{{}_{\text{TOV}}}$ and
$\rho_{{}_{\text{TOV}}}$ (or, taking into account correlations (1),
$M_{{}_{\text{TOV}}}$ and $R_{{}_{\text{TOV}}}$) as these two parameters.
However, Lindblom’s two-parametric description had an “anchor” point at the
crust-core boundary, i.e. at a density of $\sim 0.5\rho_{0}$, while the
expression (2) works above $3\rho_{0}$. Therefore, the parametrization of the
equations of state with two parameters used in this work is somewhat different
from that discovered by Lindblom (2010). In addition, both the Lindblom
parametrization and the parametrizations of the $P(\rho)$ curves proposed here
are completely phenomenological and have no microscopic justification.
Finally, Cai et al. (2023) recently proposed an explanation for the
correlations between the characteristics of the MMNS, which does not rely at
all on the properties of the equation of state. So perhaps the correlations
(1) and the approximations (2) have different physical origins.

This work can be extended in three natural directions. First, one can
extrapolate the approximation of $P(\rho)$ curves to the region
$\rho<3\rho_{0}$. In order to do so, it probably will be necessary to increase
the number of independent parameters in the fitting formula. Second, due to
the one-to-one correspondence between the relations $P(\rho)$ and $M,R$
(Lindblom, 1992), the latter set of curves should also be describable by a
small number of real parameters. Accordingly, one can try to propose a
universal approximation for the $M,R$ curves. Such approximation will be
useful in almost all studies related to observations of NS masses and radii.
Finally, it is necessary to determine which of the explanations for the
correlations of MMNS properties, those of Ofengeim (2020), Cai et al. (2023)
or some other is correct. Regardless of these open questions, we expect that
the method presented here can be used to infer new limits from observations on
the equation of state of matter at very high densities.

The work was supported by RSF # 19-12-00133 (PS) and by an Advanced ERC grant
MultiJets (DO, TP).

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## Supplementary Materials: Equations of state zoo catalogue

The selection of equations of state for this work was carried out as follows.
From the array of CompOSE models (Typel et al., 2015) (as of November 2022),
the category Cold Neutron Star EoS was taken. Only the tables covering both
the NS crust and core were used. Models from Read et al. (2009) were added to
this sample (this collection of equations of state was available for some time
through the link provided in the review by Özel and Freire, 2016; an archived
copy is available now), and several dozen models more which are available to
the authors (see Ofengeim, 2020). Then duplicates and equations of state
tables of which ended at $\rho<\rho_{{}_{\text{TOV}}}$ were discarded. As a
result, we selecteed a sample of 162 models, presented in the table
LABEL:tab:zooNuc.

The nomenclature used here follows the CompOSE tradition and is based on the
following principle: <acronym of the original paper>(<model designation>).
Tables for each equation of state, as well as its detailed description
(nuances of choosing microphysical parameters, crustal models, etc.) can be
found in the references indicated in the “source” column. There one can also
find a decoding of the acronym of the original work where this particular
model was proposed (which rarely coincides with the source). If neither the
source nor the references given there describe a model for a crust, it means
that the BSk24 crust was used (Pearson et al., 2018), stitched to the core by
requiring the continuity of pressure at the boundary. Model designations in
the table LABEL:tab:zooNuc may undergo reasonable abbreviations in comparison
with those used in the referenced sources.

Additionally, for each equation of state the maximum mass of the NS, its
radius, pressure, density and speed of sound at the center of such a star, as
well as the radius of the <<canonical>> NS $R_{1.4}$ are indicated in the
table LABEL:tab:zooNuc. The root mean square and maximum relative fitting
errors of the $P(\rho)$ fit (2) are also indicated for the range
$\rho>3\rho_{0}$.

Table S1:  Neutron star equations of state  Name | Reference | $M_{{}_{\text{TOV}}}$ | $R_{{}_{\text{TOV}}}$ | $R_{1.4}$ | $\rho_{{}_{\text{TOV}}}$ | $P_{{}_{\text{TOV}}}$ | $c_{\text{s}_{\text{TOV}}}$ |  Fit $P(\rho)$ [%]
---|---|---|---|---|---|---|---|---
|  |  [$M_{\odot}$] | [km] | [km] | $[\rho_{0}]$ | $[\rho_{0}c^{2}]$ | $[s]$ | $\sqrt{\langle\delta P^{2}\rangle}$ | $\max|\delta P|$
nycleon models
APR(APR1) | R+09 | 1.68 | 8.28 | 9.34 | 14.9 | 7.42 | 0.839 | 8.6 | 12
APR(APR2) | R+09 | 1.81 | 8.72 | 10.16 | 13.4 | 7.63 | 1.03 | 5.4 | 9.8
APR(APR3) | R+09 | 2.39 | 10.76 | 12.06 | 8.35 | 5.78 | 1.13 | 1.9 | 7.5
ZBL(BPAL12) | R+09 | 1.45 | 9.01 | 10.04 | 14.2 | 4.29 | 0.704 | 2.8 | 9.3
EOHJBO(ENG) | R+09 | 2.24 | 10.41 | 12.01 | 9.19 | 5.68 | 1.00 | 3.5 | 6.7
FP(FPS) | R+09 | 1.80 | 9.28 | 10.84 | 12.1 | 5.40 | 0.870 | 2.7 | 5.4
PAL(PAL6) | R+09 | 1.48 | 9.27 | 10.54 | 13.5 | 3.93 | 0.692 | 1.7 | 5.9
WFF(WFF1) | R+09 | 2.13 | 9.41 | 10.39 | 10.8 | 8.04 | 1.18 | 6.2 | 16
WFF(WFF2) | R+09 | 2.20 | 9.81 | 11.14 | 10.1 | 7.37 | 1.15 | 2.9 | 4.9
WFF(WFF3) | R+09 | 1.84 | 9.50 | 10.90 | 11.3 | 4.86 | 0.836 | 2.8 | 4.6
MPA(MPA1) | R+09 | 2.46 | 11.32 | 12.44 | 7.55 | 4.56 | 0.991 | 2.6 | 7.0
MS(MS1) | R+09 | 2.77 | 13.33 | 14.84 | 5.53 | 2.80 | 0.892 | 0.94 | 1.4
MS(MS1b) | R+09 | 2.78 | 13.29 | 14.52 | 5.51 | 2.80 | 0.894 | 0.95 | 1.4
MS(MS2) | R+09 | 1.81 | 11.68 | 13.72 | 7.79 | 1.89 | 0.582 | 2.3 | 7.2
ABHT(QMC-RMF1) | CompOSE | 1.95 | 10.25 | 11.85 | 9.82 | 3.96 | 0.806 | 1.5 | 4.6
ABHT(QMC-RMF2) | CompOSE | 2.04 | 10.49 | 12.02 | 9.26 | 3.93 | 0.828 | 1.6 | 5.0
ABHT(QMC-RMF3) | CompOSE | 2.14 | 10.71 | 12.24 | 8.81 | 4.11 | 0.862 | 2.5 | 7.2
ABHT(QMC-RMF4) | CompOSE | 2.21 | 11.03 | 12.33 | 8.15 | 3.69 | 0.852 | 1.6 | 5.2
APR(APR) | CompOSE | 2.19 | 9.93 | 11.33 | 9.93 | 6.87 | 1.15 | 1.8 | 6.0
BG(BGN1) | HPY07 | 2.17 | 10.91 | 13.00 | 8.66 | 4.24 | 0.939 | 2.4 | 4.1
BG(BGN2) | HPY07 | 2.48 | 11.74 | 13.53 | 7.21 | 4.23 | 1.04 | 3.3 | 6.4
BBB(BBB1) | HPY07 | 1.79 | 9.67 | 11.03 | 11.0 | 4.05 | 0.765 | 2.9 | 6.4
BBB(BHF-BBB2) | CompOSE | 1.92 | 9.50 | 11.12 | 11.4 | 5.75 | 0.915 | 2.4 | 3.6
BL(chiral) | CompOSE | 2.08 | 10.28 | 12.31 | 9.78 | 5.13 | 0.985 | 1.2 | 1.7
GDTB(DDH$\delta$) | CompOSE | 2.15 | 11.23 | 12.65 | 7.92 | 3.13 | 0.790 | 1.3 | 4.9
GKYG(APR-I) | G+05 | 1.92 | 10.31 | 12.14 | 9.85 | 3.86 | 0.801 | 0.83 | 3.7
GKYG(APR-II) | G+05 | 1.92 | 10.27 | 12.07 | 9.91 | 3.89 | 0.802 | 0.81 | 3.7
GKYG(APR-III) | G+05 | 1.93 | 10.38 | 12.26 | 9.75 | 3.80 | 0.799 | 0.92 | 3.8
GM(GM1) | CompOSE | 2.38 | 12.16 | 14.19 | 6.91 | 3.07 | 0.848 | 2.2 | 6.3
GMSR(BSk14) | CompOSE | 1.93 | 9.29 | 11.17 | 11.9 | 7.10 | 1.13 | 4.9 | 14
GMSR(H7) | CompOSE | 2.48 | 11.23 | 12.68 | 7.64 | 5.64 | 1.38 | 10 | 15
GMSR(LNS5) | CompOSE | 1.97 | 9.92 | 11.48 | 10.4 | 4.84 | 0.890 | 2.2 | 5.8
GMSR(SLy5) | CompOSE | 2.10 | 10.06 | 11.79 | 10.0 | 5.67 | 1.00 | 0.81 | 2.0
GPPVA(DD2) | CompOSE | 2.42 | 11.88 | 13.19 | 6.99 | 3.30 | 0.866 | 1.3 | 3.9
GPPVA(DDME2) | CompOSE | 2.48 | 12.07 | 13.23 | 6.71 | 3.24 | 0.873 | 1.4 | 2.9
GPPVA(FSU2) | CompOSE | 2.07 | 12.06 | 13.89 | 7.11 | 2.08 | 0.632 | 1.7 | 4.3
GPPVA(FSU2H) | CompOSE | 2.38 | 12.36 | 13.30 | 6.36 | 2.33 | 0.703 | 3.5 | 5.0
GPPVA(FSU2R) | CompOSE | 2.05 | 11.65 | 12.95 | 7.35 | 2.18 | 0.632 | 2.5 | 3.7
GPPVA(NL3$\omega\rho$L55) | CompOSE | 2.75 | 13.00 | 13.73 | 5.66 | 2.92 | 0.894 | 1.8 | 2.6
GPPVA(TM1e) | CompOSE | 2.12 | 11.84 | 13.17 | 7.12 | 2.24 | 0.654 | 2.3 | 3.3
GPPVA(TW) | CompOSE | 2.08 | 10.64 | 12.33 | 9.05 | 3.95 | 0.839 | 1.8 | 6.3
KKPY(APR-IV) | K+14 | 2.16 | 10.86 | 12.72 | 8.68 | 4.12 | 0.904 | 1.6 | 4.8
free $n$ | O20 | 0.709 | 9.73 | — | 15.0 | 1.44 | 0.369 | 0.58 | 1.2
free $npe$ | O20 | 0.697 | 9.82 | — | 14.7 | 1.36 | 0.362 | 0.69 | 1.5
PCGS(PCSB0) | CompOSE | 2.53 | 12.13 | 13.30 | 6.63 | 3.35 | 0.888 | 2.2 | 3.2
PFCPG(BSk19) | Po+13 | 1.86 | 9.10 | 10.73 | 12.4 | 6.76 | 1.01 | 3.9 | 8.7
PFCPG(BSk20) | Po+13 | 2.16 | 10.17 | 11.74 | 9.61 | 5.79 | 1.06 | 0.53 | 2.6
PFCPG(BSk21) | Po+13 | 2.27 | 11.04 | 12.59 | 8.17 | 4.24 | 0.963 | 1.6 | 4.4
PCPFDDG(BSk22) | Pe+18 | 2.26 | 11.20 | 13.04 | 8.07 | 4.02 | 0.951 | 2.6 | 4.6
PCPFDDG(BSk24) | Pe+18 | 2.28 | 11.07 | 12.57 | 8.10 | 4.15 | 0.955 | 1.4 | 4.8
PCPFDDG(BSk25) | Pe+18 | 2.22 | 11.04 | 12.36 | 8.11 | 3.76 | 0.896 | 0.90 | 4.1
PCPFDDG(BSk26) | Pe+18 | 2.17 | 10.19 | 11.76 | 9.57 | 5.76 | 1.06 | 0.65 | 3.1
RG(KDE0v) | CompOSE | 1.96 | 9.66 | 11.43 | 11.0 | 5.82 | 0.984 | 1.4 | 2.9
RG(KDE0v1) | CompOSE | 1.97 | 9.79 | 11.64 | 10.8 | 5.53 | 0.965 | 0.92 | 1.9
RG(Rs) | CompOSE | 2.12 | 10.76 | 12.95 | 8.96 | 4.23 | 0.922 | 2.8 | 3.7
RG(Sk255) | CompOSE | 2.14 | 10.85 | 13.16 | 8.88 | 4.32 | 0.937 | 2.6 | 4.1
RG(Sk272) | CompOSE | 2.23 | 11.09 | 13.33 | 8.40 | 4.30 | 0.965 | 2.8 | 5.0
RG(Ska) | CompOSE | 2.21 | 10.89 | 12.92 | 8.63 | 4.46 | 0.970 | 2.4 | 4.4
RG(Skb) | CompOSE | 2.19 | 10.61 | 12.22 | 8.85 | 4.64 | 0.974 | 1.6 | 3.8
RG(SkI2) | CompOSE | 2.16 | 11.12 | 13.50 | 8.45 | 3.89 | 0.914 | 3.9 | 5.3
RG(SkI3) | CompOSE | 2.24 | 11.31 | 13.57 | 8.08 | 3.89 | 0.937 | 3.8 | 5.5
RG(SkI4) | CompOSE | 2.17 | 10.67 | 12.39 | 8.86 | 4.47 | 0.949 | 1.5 | 4.0
RG(SkI5) | CompOSE | 2.24 | 11.47 | 14.10 | 7.96 | 3.76 | 0.932 | 5.0 | 7.0
RG(SkI6) | CompOSE | 2.19 | 10.76 | 12.50 | 8.71 | 4.42 | 0.951 | 1.6 | 4.3
RG(SkMp) | CompOSE | 2.11 | 10.53 | 12.51 | 9.26 | 4.58 | 0.946 | 1.9 | 2.8
RG(SkOp) | CompOSE | 1.97 | 10.13 | 12.14 | 10.2 | 4.72 | 0.906 | 0.87 | 1.3
RG(SLy2) | CompOSE | 2.05 | 10.05 | 11.80 | 10.1 | 5.34 | 0.977 | 0.39 | 1.5
RG(SLy4) | CompOSE | 2.05 | 9.99 | 11.72 | 10.2 | 5.46 | 0.985 | 0.46 | 1.3
RG(SLy9) | CompOSE | 2.16 | 10.63 | 12.48 | 9.01 | 4.59 | 0.952 | 1.3 | 3.9
RG(SLy230a) | CompOSE | 2.10 | 10.25 | 11.85 | 9.60 | 4.95 | 0.948 | 1.3 | 4.7
VGBCMR(D1M*) | CompOSE | 2.00 | 10.21 | 11.72 | 9.72 | 4.28 | 0.855 | 1.8 | 5.6
XMLSLZ(DD-LZ1) | CompOSE | 2.56 | 12.28 | 13.14 | 6.38 | 3.12 | 0.875 | 1.6 | 2.3
XMLSLZ(DDME-X) | CompOSE | 2.56 | 12.35 | 13.36 | 6.36 | 3.10 | 0.876 | 1.2 | 2.0
XMLSLZ(GM1) | CompOSE | 2.36 | 11.94 | 13.75 | 7.08 | 3.17 | 0.851 | 2.1 | 6.3
XMLSLZ(MTVTC) | CompOSE | 2.02 | 10.90 | 13.09 | 8.87 | 3.46 | 0.803 | 2.6 | 5.6
XMLSLZ(NL3) | CompOSE | 2.77 | 13.28 | 14.58 | 5.53 | 2.81 | 0.890 | 1.2 | 1.7
XMLSLZ(PK1) | CompOSE | 2.31 | 12.66 | 14.36 | 6.33 | 2.14 | 0.682 | 1.7 | 2.5
XMLSLZ(PKDD) | CompOSE | 2.33 | 11.77 | 13.62 | 7.33 | 3.30 | 0.852 | 1.6 | 5.7
XMLSLZ(TM1) | CompOSE | 2.18 | 12.37 | 14.27 | 6.72 | 2.11 | 0.659 | 1.4 | 4.3
XMLSLZ(TW99) | CompOSE | 2.08 | 10.62 | 12.26 | 9.06 | 3.96 | 0.839 | 1.8 | 6.2
YHSHP(PAL1_120) | Y+11 | 1.47 | 9.18 | 10.37 | 13.8 | 4.09 | 0.699 | 2.1 | 7.4
YHSHP(PAL1_180) | Y+11 | 1.74 | 9.93 | 12.04 | 11.2 | 3.97 | 0.771 | 1.1 | 1.6
YHSHP(PAL1_240) | Y+11 | 1.95 | 10.60 | 12.77 | 9.50 | 3.70 | 0.811 | 1.9 | 3.7
YHSHP(PAL2_120) | Y+11 | 1.48 | 9.72 | 11.33 | 12.6 | 3.47 | 0.677 | 2.8 | 4.0
YHSHP(PAL2_180) | Y+11 | 1.75 | 10.35 | 12.93 | 10.5 | 3.52 | 0.753 | 3.2 | 4.4
YHSHP(PAL2_240) | Y+11 | 1.97 | 10.97 | 13.55 | 9.03 | 3.35 | 0.795 | 3.5 | 5.4
YHSHP(PAL2_400_0) | Y+11 | 2.85 | 12.69 | 14.39 | 6.00 | 4.49 | 1.23 | 4.5 | 8.4
YHSHP(PAL3_120) | Y+11 | 1.42 | 8.45 | 8.95 | 15.9 | 5.14 | 0.729 | 5.8 | 14
YHSHP(PAL3_180) | Y+11 | 1.69 | 9.36 | 11.07 | 12.3 | 4.66 | 0.799 | 1.8 | 3.7
YHSHP(PAL3_240) | Y+11 | 1.91 | 10.12 | 11.94 | 10.2 | 4.18 | 0.828 | 0.45 | 2.2
YHSHP(PAL3_260_01) | Y+11 | 2.06 | 10.41 | 12.20 | 9.47 | 4.46 | 0.907 | 0.86 | 3.0
YHSHP(PAL3_300) | Y+11 | 1.83 | 10.51 | 12.22 | 9.47 | 3.02 | 0.709 | 1.0 | 4.0
YHSHP(PAL3_300_0) | Y+11 | 2.34 | 11.08 | 12.71 | 8.07 | 4.69 | 1.04 | 2.1 | 4.9
YHSHP(PAL3_400) | Y+11 | 2.83 | 12.32 | 13.30 | 6.20 | 4.92 | 1.26 | 3.0 | 6.7
YHSHP(PAPAL_120) | Y+11 | 1.44 | 8.74 | 9.55 | 15.0 | 4.70 | 0.717 | 4.2 | 12
YHSHP(PAPAL_180) | Y+11 | 1.71 | 9.59 | 11.47 | 11.9 | 4.37 | 0.786 | 1.0 | 2.4
YHSHP(PAPAL_240) | Y+11 | 1.93 | 10.32 | 12.29 | 9.92 | 3.98 | 0.822 | 0.94 |
${}^{\text{a}}$ $\rho_{{}_{\text{TOV}}}$ falls into the first-order phase
transition region, References: R+09 — Read et al. (2009), Özel and Freire
(2016), available through the archive url
https://web.archive.org/web/20200830055909/http://xtreme.as.arizona.edu/neutronstars/;
HPY07 — Haensel et al. (2007); G+05 — Gusakov et al. (2005); K+14 — Kaminker
et al. (2014); O20 — Ofengeim (2020); Po+13 — Potekhin et al. (2013); Pe+18 —
Pearson et al. (2018); Y+11 — Yakovlev et al. (2011); O+19 — Ofengeim al.
(2019); GHK14 — Gusakov et al. (2014); CompOSE — CompStar Online Supernova
Equations of State database, https://compose.obspm.fr (Typel et al., 2015).

Models with hyperons and $\Delta$-isobars G(GNH3) R+09 1.96 11.39 14.18 8.49
2.85 0.747 4.7 8.3 LNO(H1) R+09 1.55 10.97 12.85 9.16 1.95 0.541 4.7 7.3
LNO(H2) R+09 1.67 11.51 13.47 8.02 1.71 0.564 4.5 6.8 LNO(H3) R+09 1.79 11.85
13.82 7.52 1.68 0.518 3.0 6.3 LNO(H4) R+09 2.03 11.76 13.74 7.51 2.30 0.653
3.4 6.7 LNO(H5) R+09 1.73 11.31 13.23 8.19 1.92 0.596 4.4 6.2 LNO(H7) R+09
1.68 10.85 12.92 9.18 2.32 0.630 4.4 6.3 RSGMT(QMC700) OF16 1.98 12.36 12.84
5.99 1.17 0.404 21 34 BG(BGN1H1) R+09 1.63 9.33 12.89 13.6 5.48 0.876 11 55
BG(BGN1H2) HPY07 1.57 9.04 12.52 14.6 5.95 0.897 14 62 BG(BGN2H1) HPY07 1.82
9.57 13.52 12.4 6.36 1.01 7.4 31 BG(BGN2H2) HPY07 1.74 9.12 13.15 13.9 7.45
1.05 11 49 DNS(CMF) CompOSE 2.07 11.88 13.59 7.19 2.19 0.686 2.1 4.8 DS(CMF)-1
CompOSE 2.07 11.88 13.59 7.20 2.19 0.686 2.1 4.8 DS(CMF)-2 CompOSE 2.13 12.00
13.71 7.05 2.25 0.692 2.0 5.6 DS(CMF)-3 CompOSE 2.00 11.54 13.19 7.65 2.32
0.685 1.5 4.4 DS(CMF)-4 CompOSE 2.05 11.59 13.28 7.59 2.41 0.689 1.6 5.3
DS(CMF)-5 CompOSE 2.07 11.48 13.23 7.74 2.65 0.753 2.4 4.9 DS(CMF)-6 CompOSE
2.11 11.57 13.32 7.62 2.65 0.744 2.1 5.5 DS(CMF)-7 CompOSE 2.07 11.49 13.23
7.73 2.64 0.752 2.4 4.9 DS(CMF)-8 CompOSE 2.09 11.58 13.32 7.60 2.58 0.744 2.4
4.9 GHK(GM1A) O+19 1.99 11.94 13.72 7.18 2.02 0.686 5.7 7.9 GHK(TM1C) O+19
2.05 12.47 14.31 6.58 1.77 0.644 3.0 4.1 GHK(GM1’B) GHK14 2.01 11.46 13.76
8.13 2.78 0.748 5.0 7.4 OGHF(FSU2H) O+19 1.99 11.98 13.31 6.90 1.83 0.637 3.4
4.8 OGHF(NL3$\omega\rho$) O+19 2.71 12.94 13.73 5.73 2.85 0.889 0.67 0.96
OPGR(GM1Y4) CompOSE 1.79 13.01 13.77 5.23 0.823 0.424 1.5 3.3 OPGR(GM1Y5)
CompOSE 2.12 12.30 13.77 6.61 1.91 0.667 3.2 5.2 OPGR(GM1Y6) CompOSE 2.29
12.12 13.77 6.84 2.64 0.808 3.0 4.1 R(DD2Y$\Delta$)1.1-1.1 CompOSE 2.04 11.21
13.00 8.22 2.89 0.741 1.5 6.2 R(DD2Y$\Delta$)1.2-1.1 CompOSE 2.05 10.98 12.31
8.39 3.05 0.731 3.4 5.9 R(DD2Y$\Delta$)1.2-1.3 CompOSE 2.03 11.44 13.26 7.89
2.63 0.739 3.8 5.3 Hybrid models ABPR(ALF1) R+09 1.49 9.21 9.89 11.9 2.95
0.565 8.9 26 ABPR(ALF2) R+09 2.09 11.96 13.18 6.94 1.95 0.553 4.3 6.5
ABPR(ALF3) R+09 1.47 9.50 10.31 11.6 2.75 0.565 5.4 15 ABPR(ALF4) R+09 1.94
10.88 11.66 8.28 2.33 0.506 13 20 PCL(PCL2) R+09 1.48 10.18 11.73 10.8 2.54
0.599 5.2 8.5 BFH(QHC19-A) CompOSE 1.93 10.23 11.55 9.63 3.66 0.781 2.1 5.3
BFH(QHC19-B) CompOSE 2.07 10.55 11.59 8.82 3.59 0.799 4.5 11 BFH(QHC19-C)
CompOSE 2.18 10.74 11.59 8.37 3.70 0.815 7.7 18 BFH(QHC19-D) CompOSE 2.28
10.86 11.59 8.12 3.92 0.833 11 23 BFH(QHC18) CompOSE 2.04 10.38 11.48 9.22
3.88 0.804 7.2 21 DS(CMF)-1 CompOSE 1.97 11.19 13.59 8.54 2.98 0.784 9.9 20
DS(CMF)-2 CompOSE 1.96 11.14 13.71 8.70 3.08 0.787 10. 22 DS(CMF)-3 CompOSE
1.99 11.24 13.19 8.21 2.78 0.777 6.7 14 DS(CMF)-4 CompOSE 1.98 11.24 13.28
8.27 2.82 0.778 8.2 17 DS(CMF)-5 CompOSE 2.02 11.88 13.24 6.92 1.93 0.727 4.7
9.8 DS(CMF)-6 CompOSE 2.01 11.99 13.32 6.76 1.84 0.728 6.6 14 DS(CMF)-7
CompOSE 2.02 11.89 13.24 6.92 1.92 0.727 4.9 11 DS(CMF)-8 CompOSE 2.02 11.89
13.32 6.92 1.92 0.727 6.0 13 JJ(VQCD)intermediate CompOSE 2.14 11.79 12.41
7.83 1.95 —${}^{\text{a}}$ 33 41 JJ(VQCD)soft CompOSE 2.02 11.84 12.33 7.90
1.51 —${}^{\text{a}}$ 45 63 JJ(VQCD)stiff CompOSE 2.33 11.86 12.54 6.81 2.68
0.763 2.4 3.6 KBH(QHC21_A) CompOSE 2.18 11.24 12.37 7.83 3.15 0.798 1.2 3.8
KBH(QHC21_AT) CompOSE 2.13 10.87 11.84 8.25 3.35 0.798 2.0 4.0 KBH(QHC21_B)
CompOSE 2.25 11.41 12.36 7.49 3.08 0.803 2.3 5.9 KBH(QHC21_BT) CompOSE 2.20
11.07 11.85 7.86 3.25 0.803 3.0 6.3 KBH(QHC21_C) CompOSE 2.31 11.56 12.36 7.22
3.03 0.808 3.6 8.3 KBH(QHC21_CT) CompOSE 2.26 11.13 11.74 7.68 3.31 0.812 5.5
10 KBH(QHC21_D) CompOSE 2.36 11.69 12.34 6.99 3.01 0.813 5.0 11 KBH(QHC21_DT)
CompOSE 2.32 11.28 11.75 7.41 3.27 0.817 6.8 13 OOS(DD2_FRG)(2) CompOSE 2.05
12.53 13.22 5.92 1.36 0.551 5.2 16 OOS(DD2_FRG)(2+1) CompOSE 1.84 12.75 13.22
5.33 0.897 0.466 4.9 15 OOS(DD2-FRG)vec(2) CompOSE 2.12 12.69 13.22 5.67 1.32
0.559 12 27 OOS(DD2-FRG)vec(2+1) CompOSE 1.93 12.98 13.22 4.99 0.844 0.467 14
34


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# Adversarial Attacks on Human Vision

Victor A. Mateescu and Ivan V. Bajić
School of Engineering Science, Simon Fraser University, Canada

###### Abstract

This article presents an introduction to visual attention retargeting, its
connection to visual saliency, the challenges associated with it, and ideas
for how it can be approached. The difficulty of attention retargeting as a
saliency inversion problem lies in the lack of one-to-one mapping between
saliency and the image domain, in addition to the possible negative impact of
saliency alterations on image aesthetics. A few approaches from recent
literature to solve this challenging problem are reviewed, and several
suggestions for future development are presented.

##  I Visual Saliency

Visual attention retargeting is a field of research in which the content of an
image or video is altered in an effort to guide viewer’s attention from where
it normally would be, towards another, desired location.111More recently, when
applied to machine vision rather than human vision, these kinds of methods
have been referred to as adversarial attacks. The origins of attention
retargeting are closely tied to visual saliency, a measure of propensity for
drawing visual attention. A considerable amount of research has been done on
saliency computation [1]; we can confidently predict the areas of an image
that are likely to draw our immediate attention. The problem of going in the
reverse direction, i.e., dictating where we should look and finding the image
that satisfies this requirement, has been relatively unexplored. Since
attention retargeting builds upon principles of early attention, we begin our
overview with a discussion of visual saliency.

A full, detailed analysis of every object within view at any single point in
time is an incredibly complex task, far beyond the capabilities of human
visual processing [2]. And yet, most would agree that the simple act of seeing
is rather effortless; watching television is not often thought of as a taxing
activity. To cope with the vast amount of input obtained through our eyes [3],
our attentional mechanisms will focus on a few key areas for in-depth analysis
using a two-stage process [4].

(a)

(b)

Figure 1: Illustration of visual saliency in terms of (a) orientation and (b)
color, from [5].

The initial processing narrows down the list of potentially relevant areas
across our entire visual field [3] by identifying objects that appear to pop
out in terms of basic visual features, such as color, orientation, and motion
[6]. An interesting or conspicuous object (in the context of these visual
primitives) can be perceived within 25 - 50 ms [3] in an effortless manner,
regardless of clutter from uninteresting objects [4]. The second stage of
attention involves a more complex, detailed processing out of the viewer’s own
volition. Although the deliberate nature of this mechanism can override the
attention given to the objects that popped out in the first stage, it cannot
be deployed at such a fast rate. This means that if the scene suddenly
changes, we involuntarily draw our gaze toward objects that are sufficiently
salient in the pre-attentive stage regardless of any intent to avoid them. For
example, you’ll likely be distracted for a moment by the backlit mobile phone
screen in a dark movie theater the instant someone in front of you takes it
out to start texting. Soon afterwards however, you resolve to ignore it and
return your focus onto the movie.

Figure 2: A popular model for visual saliency computation [7].

Visual saliency primarily refers to the initial stage of visual attention. A
region of interest (ROI) is salient if it is perceptually different from its
local surroundings in terms of basic visual features (Fig. 1). A popular
architecture for visual saliency computation [7] is illustrated in Fig. 2. A
set of visual features that drive early attention is extracted first from the
input stimuli. In the subsequent stage, conspicuities are identified within
each feature channel by computing local dissimilarity among features
corresponding to each region of the input. For example, image luminance can be
divided into non-overlapping square patches, each of which is compared against
their adjacent patches. This operation, frequently referred to as “center-
surround,” produces a conspicuity map for each feature channel, which details
the particular feature’s influence on the overall saliency. Each conspicuity
map is normalized to ensure that their values fall within a common range and
to punish uniformity, and then combined with other maps.

The final result is a grayscale saliency map, as illustrated in Fig. 2
(bottom). Highly salient regions that are predicted to draw viewer’s attention
are denoted by bright pixels, whereas non-salient regions that are likely to
be neglected are indicated by dark pixels. The above architecture can be
applied to predict saliency in videos as well. The input would be a sequence
of images, and temporal features, such as flicker and motion, would be
utilized in addition to spatial features to produce a saliency map for each
image in the sequence.

The past few decades of research demonstrate a fairly decent grasp of
predicting visual saliency [1]. This is significant because the involuntary
nature of visual saliency makes it a powerful tool for orienting attention. We
may then pose an interesting problem: if we can reasonably accurately estimate
visual saliency, how can we alter it to manipulate viewers’ attention? In the
following section, we describe attention retargeting as a saliency inversion
problem and specify the main challenges involved. Various approaches to this
problem are provided with examples from existing work. We finish our overview
with a discussion of subliminal attention guiding and provide some suggestions
for future work.

##  II Attention Retargeting

Attention retargeting refers to modifying an image or video in an effort to
alter viewer’s gaze patterns in a desired way. This can be thought of as a
saliency inversion problem, as illustrated in Fig. 3. Given an image and a
(target) map of desired saliency, we want to obtain the modified image whose
saliency matches the target saliency map. Unfortunately, this is an ill-posed
problem since there is no one-to-one mapping between saliency and images. This
is largely due to the fact that saliency can stem from various different
features. As illustrated in Fig. 3, the desired change in saliency can be
achieved through the manipulation of features like intensity, color, or
spatial frequency, among others. This problem can persist even when the
retargeting is constrained to a single feature, since saliency is based on
context. For example, the saliency of a ROI can be altered by raising the
intensity of the ROI, or decreasing the intensity of its surroundings, or a
combination of the two.

The fact that a desired change in saliency can be obtained in many ways leads
to the impression that modifying saliency is easy. This is a difficult point
to contest; a drastic change in the intensity of a region can easily make it
stand out, and an application of Gaussian blurring to reduce visual
conspicuities may serve to conceal it. Although modifying saliency may be
simple in itself, doing so in a manner that preserves the aesthetics of the
original image is not. The concept of aesthetics is fairly subjective and
difficult to measure.

Figure 3: Image and corresponding saliency (top). Target saliency map and
possible retargeted images (bottom).

Attention retargeting is relatively unexplored and lacks a unified framework.
In the remainder of this section, we present three broad classes of methods
that have been developed for this purpose. Examples and ideas for improvement
are provided with a brief overview of existing work in the field.

###  II-A Iterative Black-Box Approach

The simplest approach to attention retargeting is to use saliency computation
as a black box in an iterative optimization procedure, as shown in Fig. 4
(excluding the the signal drawn in red). At each iteration, the saliency $S$
of the image $I$ is obtained after it has been modified with respect to a
chosen set of features. These features can be different from those used to
compute saliency. The goal is to minimize the error between the saliency $S$
and the target saliency $T$, given by $e=T-S$. In order to maintain the
naturalness or aesthetics of the input image, a set of constraints can be
introduced to prevent overmodification. The process ends when $e$ becomes
small enough, or when the modifications can no longer produce an appreciable
change in saliency towards the intended goal.

Figure 4: Attention Retargeting as an optimization procedure. The signal shown
in red indicates that the model uses intermediate outputs of saliency
computation (feature and conspicuity maps) to help determine the correct
modification at each step.

An example of this approach can be found in the work of Wong and Low [8]. In
their model, the user segments an image into $N$ segments and enumerates them
in order of importance. A target importance value $T_{i}$ is assigned to each
segment, where $i=1$ is the most important segment and $i=N$ is the least
important. Modifications are made to the intensity, color saturation, and
sharpness of the original image until the average saliency within each segment
matches the target importance value for the corresponding segment.
Specifically, the saliency error is defined as

$e=\sum\limits_{i=1}^{N}|\mathcal{N}(T_{i})-\mathcal{N}(S_{i})|,$ (1)

where $S_{i}$ is the average saliency of the $i$th segment, and
$\mathcal{N}(\cdot)$ is a normalization operator defined as
$\mathcal{N}(X_{i})=X_{i}/\sum_{j=1}^{N}X_{j}$.

The main disadvantage of this approach is the limited insight into why the
images are modified the way they are. This limitation stems from the use of
saliency computation as a black-box function, which only lets us observe the
effect of a modification after the change to the image has been made. If
$\mathcal{N}(T_{i})-\mathcal{N}(S_{i})$ is small for some $i$, no
modifications are necessary in that region. If
$\mathcal{N}(T_{i})-\mathcal{N}(S_{i})$ is large and positive then we have to
increase saliency, and if $\mathcal{N}(T_{i})-\mathcal{N}(S_{i})$ is large and
negative then we have to decrease saliency in order to match the target
saliency. All we can do is modify the image where
$|\mathcal{N}(T_{i})-\mathcal{N}(S_{i})|$ is large, recompute the saliency,
and see whether the cost function is actually minimized. Since the
modification may not necessarily alter saliency in the right direction, many
iterations may be needed.

This can be remedied if the saliency computation block contains feature
channels that are pertinent to the modifications being made. For example, let
us consider a saliency detector that extracts spatial frequency information
from blocks of $I$ to produce a feature map $\beta_{f}$. Center-surround
operations are applied on $\beta_{f}$ to produce the conspicuity map
$\beta_{c}$, describing the saliency of $I$ that originates from spatial
frequency. The modification in this example is either blurring or sharpening
(e.g., via Gaussian or Laplacian filters) applied to specific regions of $I$,
which decrease or increase their spatial frequency, respectively, by varying
degrees.

Consider the case where saliency must be reduced in a particular region. If
$\beta_{c}$ is small in this region, then the feature associated with it does
not contribute to saliency and does not need to be modified. Conversely, if
$\beta_{c}$ is large then the feature in $\beta_{f}$ must be altered in the
opposite direction. In our example, if $\beta_{f}$ indicates high spatial
frequency in that region of $I$ then we need to blur it; otherwise, we need to
sharpen it. Similar logic is applied in the case where saliency must be
increased in a particular region. A large $\beta_{c}$ in this region indicates
that it is already salient with respect to its corresponding feature, hence no
modification is needed. If $\beta_{c}$ is small in this region, then the
corresponding region of $I$ must be made salient with a modification that
alters the feature in $\beta_{f}$ in the opposite direction, as before. Thus,
the information extracted during saliency computation
$\beta=\\{\beta_{f},\beta_{c}\\}$ can be used to guide the direction and
possibly even the magnitude of the modifications at each step. This improved
algorithm, shown by the addition of the red signal in Fig. 4, can be thought
of as a steepest descent algorithm for attention retargeting.

Hagiwara et al. apply this methodology in [9]. They reverse engineer a
simplified version of the well-known saliency model by Itti et al. [7], which
decomposes an RGB image into a set of four color features $r=R-(G+B)/2$,
$g=G-(R+B)/2$, $b=B-(R+G)/2$, and $y=(R+G)/2-|R-G|/2-B$, and an additional
intensity feature $v=(R+G+B)/3$. Conspicuity maps are then obtained for each
feature and combined to compute saliency. Hagiwara et al. first select a ROI
whose saliency is to be maximized. They determine the change in the image
channels $\Delta R$, $\Delta G$, $\Delta B$ needed to increment saliency at
each pixel as a function of the features and their corresponding conspicuity
maps. Since the saliency of the ROI is to be maximized relative to the rest of
the image, each pixel within the ROI is modified by $\Delta\alpha$ (calculated
individually for each pixel), and pixels outside are modified by their
respective $-\Delta\alpha$, where $\alpha\in\\{R,G,B\\}$.

###  II-B Direct Mapping

The optimization procedure in Fig. 4 uses feedback from saliency computation
to estimate how the image is to be modified in a series of steps. Mapping an
additive or multiplicative change to the saliency map (or conspicuity map of a
particular feature) directly onto the image domain is a highly non-trivial
task. However, one of the earlier works on attention retargeting by Su et al.
[10], which de-emphasizes distracting textures in images, demonstrates that
this is indeed possible. Since there currently are no generalizations of this
approach, we illustrate the concept of direct mapping by summarizing the
methodology of Su et al..

Figure 5: Computation of visual saliency from texture using steerable filter
banks [10].

The texture-based saliency of an image is computed using steerable filter
banks, as shown in Fig. 5. The local frequency content $s_{Ln}$ is computed
for each subband by rectification (since $s_{n}$ is band-limited) followed by
local averaging with a Gaussian filter. Conspicuities within each subband
$s_{Hn}$ can be found by applying a high pass filter to the local frequency
content. Regions of the image with highly salient textures are identified by
large values in the conspicuity maps $s_{Hn}$.

Minimizing the saliency of these regions can be thought of as removing any
conspicuities from the local frequency content, i.e., subtracting $s_{Hn}$
from $s_{Ln}$. To ensure that the values of $s_{Ln}$ remain positive after any
modification, the procedure is revised so that any subsequent processing of
$s_{Ln}$ is done in the log domain. Thus, $s_{Hn}$ is subtracted from
$\ln(s_{Ln})$ instead, which is equivalent to a scaling of the original
coefficients

$s_{n}^{\prime}=s_{n}e^{-s_{Hn}}.$ (2)

The modified image can be reconstructed as a linear combination of the
modified subbands.

###  II-C ROI-Based Retargeting

Often, it is only desirable to draw attention to a specific ROI, or perhaps a
few ROIs. For example, the target saliency map in the bottom-left of Fig. 3
indicates that we want the flamingo in the top right of the image to draw more
attention than the rest of the image. To this end, we may supply a simple
binary mask of the desired ROI with the objective of making this region as
salient as possible relative to the rest of the image.

Kim and Varshney apply this concept in a straightforward manner to guide
attention towards selected regions in visualizations of volumetric datasets
[11] and 3-D meshes [12]. They model saliency as the convolution of a center-
surround kernel, e.g., difference of Gaussians, with a feature map of an
image. This rudimentary saliency computation can be written as a matrix
multiplication $\mathbf{Cx}=\mathbf{s}$, where $\mathbf{C}$ is the center-
surround operation matrix, while $\mathbf{x}$ and $\mathbf{s}$ are,
respectively, the vectorized feature and saliency maps. Suppose instead that
we supply $\mathbf{s}$ as a binary mask of the ROI, with the goal of solving
for an unknown $\mathbf{x}$. If $\mathbf{C}$ is well-conditioned then
$\mathbf{C}^{-1}\mathbf{s}$ is a vector of scaling factors that can be used to
modulate feature channels (e.g., intensity and color saturation in [11]) to
increase the relative saliency of the ROI.

The absence of the target saliency as an input highlights a big redundancy—the
lack of need for full-scale saliency computation—in the approaches outlined in
Section II-A. Saliency computation typically requires center-surround
differences over the whole image. With the simplified goal of either
increasing or decreasing saliency within a single region, it suffices to
perform this operation on the ROI and its surroundings alone, rather than the
entire image.

A model proposed in [13], which estimates the relative changes in saliency of
a ROI under rotation, demonstrates what can be accomplished by avoiding this
inefficiency. Upon selecting a ROI, e.g., the tile in Fig. 6(b), edge
distributions within this ROI and its surroundings are obtained using
statistical Hough Transform. The solid red curve in Fig. 6(a) shows that the
ROI in Fig. 6(b) primarily consists of horizontal
($\theta=0^{\circ}/180^{\circ}$) and vertical ($\theta=90^{\circ}$) edges,
with a similar composition for the surrounding region, shown by the blue
curve. If we were to rotate the ROI by $45^{\circ}$, then the edges of the
tile would be primarily oriented at $45^{\circ}$ and $135^{\circ}$. Therefore,
a counter-clockwise rotation of the ROI can be represented as a leftward
circular shift of its corresponding distribution, as shown in Fig. 6(a) by the
dashed red curve.

(a)

(b)

(c)

Figure 6: Illustration of the method used to predict relative saliency of the
ROI

(a)

(b)

(c)

(d)

Figure 7: Example of attention retargeting by orientation change [13] in
action: (a) original image; (b) altered image - one of the cables in the
bottom left has been rotated to increase saliency; (c) human gaze heat map for
the original image; (d) human gaze heat map for the modified image - more
attention paid to the bottom left

The dissimilarity between the ROI and its surroundings, i.e., the center-
surround operation that estimates the saliency of the ROI, is measured in [13]
as the symmetric Kullack-Leibler divergence between their respective edge
distributions. We can rapidly obtain this measure for all possible rotation
angles $\phi$ of the ROI due to the concise representation of this
modification as a circular shift of a 1-D distribution. The result, shown in
Fig. 6(c), is a complete inverse mapping of the ROI’s relative saliency to
rotation angle (note that a one-to-one mapping between saliency and images
still does not exist, despite only a single feature being modified). The
approach taken here chooses features in a manner that allows a simpler
representation of the modification being made, and utilizes a center-surround
operation that avoids redundancy by only focusing on the ROI. These two
aspects grant the ability to rapidly predict the outcome of all possible
modifications of the particular feature, in this case edge orientation.
Another example of the orientation change by the method from [13] is shown in
Fig. 7. A similar approach has been applied to hue in [5] for color-based
attention retargeting.

Aside from benefits to computational complexity, this simplification may also
allow more freedom in the problem formulation, particularly in the objective
function used in the optimization. For example, a recently proposed model by
Nguyen et al. used a graph-based optimization for attention retargeting [14].
The input image is segmented into patches, and the set of patches $i$
comprising the ROI undergo color transfer from a set of candidate patches
$x_{i}$. The candidate patches are mined from a large image dataset and
correspond to the same objects found in the ROI so that color transfer only
occurs between similar objects in an effort to maintain naturalness. Candidate
patches for color transfer to the ROI are found by minimizing

$E(x)=\sum\limits_{i\;\in\;\text{ROI}}E_{d}(x_{i})+\lambda\sum\limits_{i}\sum\limits_{j\;\in\;N(i)}E_{s}(x_{i},x_{j}).$
(3)

The data cost $E_{d}$ is designed to consider global center-surround
differences, rather than local ones. It demands that the ideal candidate patch
be highly dissimilar from the entire image content outside of the ROI. The
smoothness energy $E_{s}$ is designed to punish dissimilarity between
neighboring patches that are of a similar color, e.g., belonging to the same
object, and encourage it otherwise. In contrast to the data cost $E_{d}$, this
considers local center-surround differences to further enhance saliency. At
the same time, it ensures that different objects in the ROI remain
individually consistent in appearance.

###  II-D Summary

A summary of attention retargeting methods—what they aim to achieve, their
methodology, and any constraints imposed—is presented in Table I. Most
existing methods are concerned with guiding attention towards a region by
increasing its saliency [5, 8, 9, 11, 12, 13, 14]. The problem of repelling
attention by reducing saliency is less explored ([10]). From the analysis
above, it is reasonable to conclude that [8] and [9] are the most
computationally demanding since they use full-scale saliency computation.
Although setting hard limits on the modifications may be the simplest way to
preserve aesthetics ([8, 10, 11, 12] and to a lesser extent [10]), they may
not match the versatility of adaptive constraints [14]. Since an objective
comparison of these methods currently does not exist, we will refrain from
speculating any further on their strengths and weaknesses.

TABLE I: Summary of attention retargeting methods. Method |  Goal |  Features and Approach |  Aesthetics Constraints
---|---|---|---
Wong and Low [8] |  Change saliency of each image segment according to specified order of importance. |  • Intensity, chromaticity, sharpness.  • Iterative adjustment of each segment of the image.  • Uses saliency computation as a black box. |  Upper and lower limits for the amount that each segment can be changed with respect to each feature.
Hagiwara et al. [9] |  Make ROI the most salient part of the image. |  • Intensity and color.  • Iterative adjustment of each pixel in the entire image.  • Uses feedback from saliency computation. |  None.
Su et al. [10] |  Decrease saliency of textures in ROI. |  • Spatial frequency.  • Image is decomposed with steerable pyramids.  • Saliency subtraction is mapped directly as a scaling of steerable coefficients. |  • Scaling factors with extreme values are clamped.  • Histogram of reconstructed output image is matched to that of the original input.
Kim and Varshney [11, 12] |  Increase the relative saliency of the ROI. |  • Intensity, chromaticity [11] and curvature [12].  • Express saliency as a matrix multiplication; invert the operation with a binary map of the ROI.  • Scale features at each pixel/voxel within the ROI and local surroundings. |  Scaling factors are normalized to a certain range.
Mateescu and Bajić [5, 13] |  Map relative ROI saliency to all possible modifications of the ROI. |  • Hue in [5], edge orientation in [13].  • Efficiently represent the modification.  • Measure dissimilarity between ROI and surround for all possible modifications. |  • No direct aesthetics constraints.  • Shows which modifications have a lighter impact on saliency, and hence more subtle.
Nguyen et al. [14] |  Increase the relative saliency of the ROI. |  • Intensity and color.  • Graph-based optimization that maximizes global and local dissimilarity of ROI by changing the intensity and color of each object within. |  • New color of objects in ROI can only come from similar objects in an image dataset.  • Graph-based optimization maintains the color-consistency of each object in ROI.

##  III Subliminal Attention Guiding

Subtlety is an important aspect of attention retargeting. Modifications that
degrade an image so severely that they detract from the viewing experience are
of limited practical use, regardless of whether they guide viewers’ attention
as intended. It would be of great value to be able to guide attention in the
least intrusive manner possible. Ideally, the modifications wouldn’t be
perceivable at all, thus guiding attention subliminally. Retargeting attention
without any perceivable alteration to the original stimuli may seem like an
implausible concept. However, it is a topic of research in neuroscience and
psychology that has received a fair bit of interest.

Experiments on subliminal orienting of attention generally consist of
reaction-time target detection tasks. In a typical experiment, each trial
begins with the participant fixating at the center of a blank screen. After
some time, a subliminal cue, i.e., a visually unperceivable stimulus
(experimentally confirmed to be unperceivable), is presented on the screen at
a random location from a set of predetermined possible locations. Sometime
later, a target, i.e., a visible stimulus, is presented randomly at one of
these locations. Prior to the experiment, the participant is told to respond
as quickly as possible to the target by issuing a key-press corresponding to
the target’s location. If the subliminal cue is truly capable of orienting the
viewer’s attention, reaction times to the target are expected to be shorter in
trials where the cue is presented in the same location as the target. In this
situation, the cue would serve as a subconscious hint to the location of the
upcoming target. However, reaction times are expected to increase in trials
where the cue and the target appear in different locations. Here, the cue
would subconsciously distract the viewer.

Figure 8: Experimental setup used in [15] to investigate the effects of
subliminal flicker on attention.

A study by Cheadle et al. investigates subliminal flicker as a cue for
orienting attention [15]. Their display consists of three Gabor patterns,
equally spaced on an invisible circle around the black fixation cross at the
center, as shown in Fig. 8. The subliminal cue in their experiments consists
of alternating the contrast of one of the three Gabor patches from maximum to
minimum at a frequency of 50 Hz (corresponding to a 100 Hz refresh rate). The
flicker is no longer discernable at such a high frequency and the flickering
Gabor patch appears identical to the other two. The target that follows the
subliminal cue is a change in spatial frequency of one of the three Gabor
patches. Participants’ reactions to the target were found to be, on average,
15 ms faster in cases where the cue is presented in the same location as the
target, which suggests that subliminal flicker is capable of drawing
attention.

A temporally asynchronous cue was shown to subliminally draw attention by
Mulckhuyse et al. [16]. In their experiment, they displayed three grey
circular discs arranged horizontally, one of which, either the left or the
right, appeared a short instant (16 ms) before the other two. In their study,
they determined that participants could not accurately locate the disc that
appeared earlier, suggesting that all three discs were perceived to appear
simultaneously. The target consisted of a black dot that appeared inside
either the left or right circular discs. Participants were told to locate the
target as fast as possible. Reaction times were 11 ms shorter, on average, in
cases where the target was in the same location as the cue.

Although these studies demonstrate that it may be possible to guide attention
without awareness, they all use synthetic stimuli, and they used reaction time
as a proxy for attention. It is uncertain whether such effects on attention
would be observed if similar cues were to be applied in natural images and
videos. Since reaction-time target detection tasks are less suitable for
natural stimuli, investigation of subliminal orienting in natural images and
videos would require eye-tracking as a more direct measure of visual
attention. One of the two existing studies to have done this is the work by
Huang et al. [17], who displayed a circular blob (radius of 50 pixels and
Weber contrast of 20%) for 50 ms prior to displaying an image in an attempt to
subconsciously draw attention toward the blob’s location. The other study [18]
investigated the effect of subliminal flicker in natural images, where flicker
was introduced to selected regions alternating the contrast in these regions
from high to low at a frequency of 50 Hz. A comparison of eye-tracking data
between participants who viewed the flickering images against those who viewed
the original images suggests that subliminal flicker may, on average, repel
attention rather than attract it [18]. However, a more detailed analysis is
required to assess statistical significance of these findings.222A more
detailed study of this effect, subsequently termed flicker observer effect,
has been presented in [19].

##  IV Conclusions and Future Work

As a relatively unexplored topic, attention retargeting presents a wide array
of possibilities for improvement. For measurable progress, it is necessary to
establish an objective framework for comparison of various approaches.
Changing saliency is easy, but doing so in an aesthetically responsible manner
is not. A reliable measure of aesthetics is of critical importance in this
regard. Reverse-engineering existing saliency models may provide valuable
insight and could pave the way to tractable models for predicting how
simultaneous changes in multiple features affect saliency, which would be an
important step forward. Subliminal attention retargeting may be considered an
ultimate goal of this research direction, but further work is needed to
investigate its plausibility in natural images and video. We hope that this
article sparks interest in attention retargeting and motivates others to
contribute.

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