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MNRAS 000,1–10 (2018) Preprint 13 December 2018 Compiled using MNRAS L
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Revealing the nonlinear behaviour of the lensed quasar
Q0957+561
A. Bewketu Belete,1?B. L. Canto Martins,1I. C. Le˜ao,1J. R. De Medeiros1
1Departamento de F´ısica Te´orica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN 59078-970, Brazil
Accepted 2018 December 10. Received 2018 December 10; in original form 2018 October 3.
ABSTRACT
Knowledge about how the nonlinear behaviour of the intrinsic signal from lensed back-
ground sources changes on its path to the observer provides much information, par-
ticularly about the matter distribution in lensing galaxies and the physical properties
of the current universe, in general. Here, we analyse the multifractal (nonlinear) be-
haviour of the optical observations of A and B images of Q0957+561 in the rand
gbands. AIMS: To verify the presence, or absence, of extrinsic variations in the ob-
served signals of the quasar images and investigate whether extrinsic variations affect
the multifractal behaviour of their intrinsic signals. METHOD: We apply a wavelet
transform modulus maxima-based multifractality analysis approach. RESULTS: We
detect strong multifractal (nonlinear) signatures in the light curves of the quasar im-
ages. The degree of multifractality for both images in the rband changes over time in
a non-monotonic way, possibly indicating the presence of extrinsic variabilities in the
light curves of the images, i.e., the signals of the quasar images are a combination of
both intrinsic and extrinsic signals. Additionally, in the r band, in periods of quiescent
microlensing activity, we find that the degree of multifractality (nonlinearity) of im-
age A is stronger than that of B, while B has a larger multifractal strength in recent
epochs (from day 5564 to day 7527) when it appears to be affected by microlensing.
Finally, comparing the optical bands in a period of quiescent microlensing activity,
we find that the degree of multifractality is stronger in the rband for both quasar
images. In the absence of microlensing, the observed excesses of nonlinearity are most
likely generated when the broad-line region (BLR) reprocesses the radiation from the
compact sources.
Key words: methods: statistical – galaxies: quasar: individual: Q0957+561
1 INTRODUCTION
Though quasars, in general, are known by their extremely
high luminosities, gravitationally lensed quasars (hereinafter
GLQs) are brighter than their unlensed counterparts and
produce different image components Agnello et al. (2015).
Study of GLQs provides significant information, mainly
about background source quasar variability mechanisms and
accretion disk structure (Jim´enez-Vicente et al. 2015;Poo-
ley et al. 2007), the mass distribution in lensing galaxies
(Bate et al. 2008) and, in general, information to constrain
physical properties of the Universe (Gil-Merino et al. 2018;
Kostrzewa-Rutkowska et al. 2018). It has been understood
that the time delay caused by gravitational lensing is di-
rectly related to the current expansion rate of the Universe
(the Hubble constant) and the mean surface mass density of
?E-mail: asnakew@fisica.ufrn.br
the lensing galaxy (e.g., Kochanek & Schechter 2004;Refsdal
1964). In the study of GLQs, great attention has been de-
voted to determining time delays between images, constrain-
ing the Hubble constant, disentangling intrinsic and extrin-
sic signals and identifying their variability mechanisms.
The quasar Q0957+561, at z = 1.41, is the first identified
GLQ (Walsh et al. 1979) and one of the most studied lensed
quasars (e.g., Hainline et al. 2011;Nakajima et al. 2009;
Cuevas-Tello et al. 2006;Schild 1996;Rhee 1991;Gond-
halekar & Wilson 1980); its lensing galaxy, which is known
to be part of a cluster of galaxies, is located at z = 0.36
(Keeton et al. 2000;Rhee 1991). The doubly imaged quasar
Q0957+561 has been observed since its discovery in 1979
(Walsh et al. 1979) in different electromagnetic bands, in-
cluding the r(read arm) and g(blue arm) bands, both
within the optical ranges. It has been said that lensing is
due to matter inhomogeneity at some point between the
source and observer, and intervening matter (mainly lensing
©2018 The Authors
arXiv:1812.04635v1 [astro-ph.GA] 11 Dec 2018
2A. Bewketu et al.
galaxies) is known to split light from background sources
(Wambsganss 1998). The travel times between image com-
ponents of background source do not agree with each other
due to the difference in their ray paths. As a result, this
difference introduces a time delay between components of
the same source, which in turn provides extremely valuable
astrophysical information.
Shalyapin et al. (2008) studied the light curves of
Q0957+561 in the gand rpassbands, determined a time
delay ∆tBA = 417 ±2 d in the g band (1σ, A is leading),
and used data for image A to measure the delay between
a large event in the g band and the corresponding event in
the r band. The time delay between both optical bands was
∆trg = 4 ±2 d (1σ, g band event is leading). In the same pa-
per, they demonstrated the absence of extrinsic variability
in the quasar light curves and identified that reverberation
within the gas disc around the supermassive black hole is a
possible mechanism for the observed intrinsic variability (see
also Gil-Merino et al. 2012). From other monitoring data of
Q0957+561 in several optical bands, time delays between
the two images A and B are found to be mainly in the range
from 417-425 d (A is leading; e.g., Colley et al. 2003;Ovald-
sen et al. 2003;Oscoz et al. 2001). A more recent study of
Q0957+561 has shown that ∆trg ∼1 d for image A and
∆trg ∼4 d for image B, which agrees with ∆tB A ∼420 d in
the rband (Shalyapin et al. 2012). This extra delay of ap-
proximately 3 d in the rband was also favoured in a previous
study of the lens system (Kundi´c et al. 1997) and is roughly
consistent with several estimates in red bands (e.g., Ovald-
sen et al. 2003;Serra-Ricart et al. 1999). Though Shalyapin
et al. (2012) considered a dense cloud within the cD lensing
galaxy along the line of sight to the image A as a possible
cause for a 3 d lag between optical bands, this lag relies
on standard cross-correlation techniques that could lead to
biased results, so the true time delay ∆tBA is most likely
achromatic (Gil-Merino et al. 2018). Even though no mi-
crolensing effects have been detected in the light curves of
the lensed quasar Q0957+561 over several decades, strong
evidence for extrinsic variability was found in the initial and
last years of monitoring, as reported in Pelt et al. (1998) and
Gil-Merino et al. (2018).
Here, we study the multifractal (nonlinear) behaviour of
the optical observations of the lensed quasar Q0957+561
in the rand gbands separately. Our questions are the fol-
lowing: Is there any difference in nonlinearity between the
signals of the images A and B of Q0957+561? What can
we learn about the intrinsic variability of the quasar and
the extrinsic mechanisms distorting it? To address these
questions, we analyse the multifractal (nonlinear) behaviour
of the light curves of images A and B of the gravitation-
ally lensed quasar Q0957+561 in the rand gbands us-
ing a wavelet transform-based multifractality analysis ap-
proach called Wavelet Transform Modulus Maxima (here-
inafter WTMM). Belete et al. (2018b) have shown that radio
quasars are intrinsically multifractal and that the multifrac-
tality (or nonlinearity) strength is different across electro-
magnetic spectrum. In addition, Belete et al. (2018a) verified
that redshift correction does not affect quasars’ multifractal-
ity behaviour, which is the same in both the observed and
rest frames of quasars. Multifractality analysis has been ap-
plied in different science cases (e.g., Maruyama et al. 2017;
Maruyama 2016a;Kasde et al. 2016;Agarwal et al. 2016;de
Freitas et al. 2016;Aliouane & Ouadfeul 2013;Jagtap et al.
2012;Ouahabi & Femmam 2011;Nurujjaman et al. 2009;
Lin & Sharif 2007;Degaudenzi & Arizmendi 1998).
Our work is organized as follows. In Section 2, we present the
light curves, method, and procedures. We discuss the results
in Section 3. The summary and conclusions are included in
Section 4.
2 DATA COLLECTION, METHOD AND
PROCEDURES
2.1 Data collection: Light curves
We use the the long-term optical light curves of Q0957+561
(1996-2016) for images A and B in the rband (Fig. 1, top
panel) and the 5.5-year (2007-2010) optical observational
data of Q0957+561 in the rand gbands (Fig.1, middle and
bottom panels, respectively) from the GLQ database, which
can be accessed via the GLENDAMA website1. In the re-
sults section of this website, the long-term light curves in the
rband (those spanning from 1996 to 2016) are given in mag-
nitudes. Since our aim is to analyse multifractal signatures
in the flux observations (if any), we have converted the mag-
nitudes of the signals to their equivalent fluxes according to
the relation:
Fl ux =(3631 Jy) × 10(−0.4×magnitu de)(1)
where Jystands for Jansky. For more information about
the light curves used in this study, refer to Gil-Merino et al.
(2018), Shalyapin et al. (2012), and Goicoechea et al. (2008).
2.2 Method and Procedures
The continuous wavelet transform is an excellent tool for
mapping the changing properties of nonstationary signals.
Because of its capability of decomposing a signal into small
fractions that are well localized in time and frequency and
detecting local irregularities of a signal (areas of the signal
where a particular derivative is not continuous) such as non-
stationarity, oscillatory behaviour, breakdown, discontinuity
in higher derivatives, the presence of long-range dependence,
and other trends (Maruyama 2016b;Puckovs & Matvejevs
2012), wavelet analysis remains one of the most preferred
signal analysis techniques to date. Additionally, there is a
claim that wavelet transforms are suitable for multifractal
analysis and allow reliable multifractal analysis to be per-
formed (Muzy et al. 1991). Here, we apply WTMM-based
multifractality analysis, which was originally introduced by
Muzy et al. (1991). We follow the procedures discussed by
Puckovs & Matvejevs (2012):
1. We calculate wavelet coefficients of the signal X(t) using
the following mathematical relation:
W(s,a)=
1
√s∫T
0Ψt−a
s.X(t)dt (2)
where Wis the wavelet coefficients; Ψ(s,a,t) is the mother
wavelet function; sis the scaling parameter; ais the shift
parameter; Xis the signal; tis the time at which the signal
1http://grupos.unican.es/glendama/database/
MNRAS 000,1–10 (2018)
Nonlinear difference between images of Q0957+561 3
Figure 1. Top: Light curves of images A and B – red and blue, respectively – of Q0957+561. Middle: Light curves of image A in the r
band (red) and gband (blue) of Q0957+561. Bottom: Light curves of image B in the rband (red) and gband (blue) of Q0957+561.
is recorded; and Tis maximal time value or signal length.
The analysing wavelet Ψ(t) is generally chosen to be well
localized in space and frequency. At lower scales s≈0, the
number of local maxima lines (hereinafter LcMx) tends to
infinity. Though it has been suggested that the scaling pa-
rameter sused in the WTMM approach should be limited
to s≤[1, 28], it should be in the interval [1, T
2]and also
can be in the interval [1, T
4], mainly to reduce the compu-
tation time. Since it is the maxima line that points towards
regularities or carries information about any singularity or
nonlinearity in the signal X(t), we only calculate wavelet co-
efficients that correspond to local maxima lines; therefore, it
is unnecessary to calculate wavelet coefficients that do not
contain maxima lines. Hence, determining the limit of the
most informative maximal scales is dependent on the pres-
ence and absence of local maxima lines. In our case, knowing
that it is not important to calculate wavelet coefficients at
large scales that do not correspond to maxima lines (rather,
it is only a waste of time), our scale parameter is determined
to be in the interval [1, T
4], which is still informative (Puck-
ovs & Matvejevs 2012). The shifting parameter acannot be
greater than the signal length T, and therefore, a≤T.
Usually, Ψ(t) is only required to be of zero mean, but in ad-
dition to these requirements, for the particular purpose of
multifractal analysis, Ψ(t) is also required to be orthogonal
to some low-order polynomials, up to the degree n-1 (i.e., to
have nvanishing moments) Enescu et al. (2006):
∫+∞
−∞
tmΨ(t)dt =0,∀m,0≤m≤n(3)
The Mexican Hat (MHAT) wavelet (second-order Gauss
wavelet) has been chosen to be the analysing wavelet. This
wavelet is one of the wavelets that has been applied for
WTMM-based analysis and is represented by the relation
MNRAS 000,1–10 (2018)
4A. Bewketu et al.
Ψ(t)=(1−t2).e−t2
2(4)
where Ψ(t) is the analysing wavelet function; tis the time
at which the signal is recorded.
2. We represent the calculated absolute wavelet coefficients
in matrix form as
Wsq
s,a=(W(s,a))2|(s,a∈N)∧(s∈ [1,smax ]) ∧ (a∈ [1,T]),(5)
where Wsq is the squared wavelet coefficients matrix; smax
is the maximal scaling parameter; sis the scaling parameter;
ais the shifting parameter; and Tis the signal length.
3. We calculate the skeleton function from the squared
wavelet coefficient matrix and express it in matrix form as
Lc M xs,a=Lc M x(s,a)(6)
under the conditions (s,a∈N)∧(s∈ [1,smax ]) ∧(a∈ [1,T]),
where Lc M x is the wavelet skeleton function; W(s,a) are the
wavelet coefficients; sis the scaling parameter; sma x is the
maximal scaling parameter; ais the shift parameter; and T
is the signal length. The skeleton function is a collection of
local maxima lines at each scale, i.e., it is the scope of all
local maxima lines that exist on each scale s. In other word,
skeleton matrix construction is a technique of excluding co-
efficients from the absolute wavelet coefficients matrix that
are not maximal. As a result, in the skeleton matrix, only
absolute wavelet coefficients that correspond to local max-
ima lines exist. The need to collect all the maxima lines at
each scale together in matrix form, the skeleton function, is
because it is the maxima lines that carry valuable informa-
tion about the signals, i.e., maxima lines point towards reg-
ularities in the signal. Since wavelet coefficients on corners
provide minimal or no information, and consequently local
maxima lines (LML) on corners also provide no significant
information about the singularity in the signal, we therefore
take the edge effect into consideration by removing the lo-
cal maxima lines on corners using the formula provided by
Puckovs & Matvejevs (2012). We fix broken lines, gaps, and
single points in the LcMx matrix by applying an algorithm
called supremum algorithm, which consists of seven steps,
as explained by Puckovs & Matvejevs (2012).
4. Using the collected local maxima lines, we calculate the
thermodynamic partition function, a function that connects
the wavelet transform and multifractality analysis part as
follows:
Zq(s)=
T−1
Õ
a=1(C(s).WT M M )q|(Lc M xs,a=1),(7)
where Zq(s)is the thermodynamic partition function;
WT M M is the wavelet modulus maxima coefficients; C(s) is
a constant depending on scaling parameter s;sis the scal-
ing parameter; qis the moment, which takes any interval
with zero mean – in our case, q∈[-5, 5]; and L cM x is the
wavelet skeleton function (aggregate of local maxima lines
in matrix form). The thermodynamic partition function is
a function of two arguments - the scaling parameter sand
power argument q. The moment qdiscovers different regions
of singularity measurement in the signal, i.e., it indicates the
presence of wavelet modulus maxima coefficients of different
values. The condition Lc M xs,a= 1 is to inform that only
modulus maxima coefficients are used. What is important
here is the relationship between Zq(s) and s, which deter-
mines the scalability of the signal under consideration. In the
WTMM approach, the wavelet transform maxima are used
to define a partition function, whose power-law behaviour
is used for an estimation of the local exponents. On small
scales, the following relation is expected:
Zq(s) ∼ sτ(q),(8)
where τ(q)is the scaling exponent function, which is the
slope of the linear fitted line on the log-log plot of Z q(s) and
sfor each q.
5. We determine the scaling exponent function τ(q) using
the following relation:
τ(q)=lim
s→0
ln(Zq(s))
ln(a),(9)
where Zq(s) is the thermodynamic partition function; τis
the local scaling exponent; sis the scaling parameter; and q
is the moment. The condition τ(q= 0) +1 = 0 is important
for multifractal spectrum calculation. The scaling exponent
function τ(q) is a function of one argument qand is deter-
mined from the slope of the line fitted line on the log-log
plot of Z q(s) against the logarithm of time scale sfor each
q, which means that the behaviour of the scaling function
τ(q) is completely dependent on the nature of the thermo-
dynamic partition function. We define monofractal and mul-
tifractal as follows: a time series is said to be monofractal
if τ(q) is linear with respect to q; if τ(q) is nonlinear with
respect to q, then the time series considered is classified as
multifractal (Frish and Parisi, 1985).
6. At last, once we determine the scaling exponent τ(q), it
is necessary to estimate the multifractal spectrum f(α) to
be able to fully draw conclusions about the multifractal or
nonlinear behaviour of a considered signal. We estimate the
multifractal spectrum function f(α) via the Legendre trans-
formation as follows Halsey et al. (1986):
α=α(q)=
∂τ(q)
∂q,(10)
where αis the singularity exponent or Holder exponent.
f(α)=q.α −τ(q),(11)
where f(α) is the multifractal spectrum function. Smaller
values of ∆α(i.e., ∆αnears zero) indicate the monofractal
limit, whereas larger values indicate the strength of the mul-
tifractal behaviour in the signal (Telesca et al. 2004;Ashke-
nazy et al. 2003;Shimizu et al. 2002). When the multifractal
structure is sensitive to the small-scale fluctuation with large
magnitudes, the spectrum will be found with right trunca-
tion, whereas the multifractal spectrum will be found with
left-side truncation when the time series has a multifractal
structure that is sensitive to the local fluctuations with small
magnitudes.
3 RESULTS AND DISCUSSION
Here we analyse, and discuss, the multifractal behaviour of
the light curves given in Fig. 1. By using Eqs. 1through 3,
we compute the absolute wavelet coefficients and construct
the corresponding skeleton functions by collecting absolute
wavelet coefficients that only hold local maxima lines for all
MNRAS 000,1–10 (2018)
Nonlinear difference between images of Q0957+561 5
the light curves considered. The multifractality analysis for
each light curve in Fig. 1is discussed below.
3.1 Analysis of the light curves of images A and B
in the rband
Using the skeleton functions, the collected local maxima
lines, constructed for the two images in the rband (top
panel in Fig. 1) we determine the relationship between the
logarithm of the thermodynamic partition function Z q(s)
and the scale sfor images A and B as shown in Figs. 2
and 3, respectively. As one can see, the thermodynamic par-
tition functions fluctuate in a nonlinear manner, revealing
the nonlinear functionality between Zq(s) and s. Though we
have information about how Zq(s) changes against sat this
level, it is usual to determine the scaling exponent functions
τ(q), the slope of the log-log plots of the thermodynamic
partition function Z q(s) and the scale s, for both images to
confirm the true relationship between the thermodynamic
partition function and the scale. A nonlinear relationship
between the thermodynamic partition function Z q(s) and
the scale sobserved, based on the thermodynamic partition
function against the scale splots in (Figs. 2and 3), is fur-
ther confirmed by the scaling exponent function τ(q) versus
the moment qplots for images A and B. The nonlinearity
between τ(q) and q, which is the slope of log(Z q(s)) against
log(s), clearly indicates the presence of multifractal (nonlin-
ear) structure in the light curves of the two images. From
the scaling exponent functions τ(q) versus q(Figs. 2and 3),
the difference in the degree of nonlinearity between images
A and B is visible. Once we are sure about the presence
of multifractal (nonlinear) signature in the considered light
curves based on the nonlinear scaling exponent function,
the next step is determining the degree of multifractality
or nonlinearity (i.e., how strong is the observed multifractal
signature?) for the signals of both images. To that end, we
estimate, and plot, the multifractal spectrum functions f(α)
and calculate the width (∆α=αmax -αmi n) using equations
Eqs. 10 and 11. The calculated width ∆αvalues for images
A and B are ∆αA= 1.6030 and ∆αB= 1.1567, respectively.
Wider width values confirm the presence of strong multi-
fractal signatures and the intermittent nature of the light
curves. The difference in nonlinearity observed between im-
ages A and B in the scaling exponent function plots, that
the nonlinearity strength of image A is stronger than image
B, is further confirmed by the calculated width values of the
multifractal spectrum function (Figs. 2and 3).
To verify whether there is a change in the degree of multi-
fractality in the long-term light curves of images A and B in
the rband, top panel in Fig. 1, we divide their light curves
into three time segments containing equal data points: from
day 117 to day 1827 (segment 1), from day 1835 to day 4525
(segment 2), and from day 4526 to day 7506 (segment 3). We
perform multifractality analysis for each time segment sepa-
rately. Following the same procedures applied in the previous
subsections, we have computed absolute wavelet coefficients,
constructed skeleton functions, and determined the thermo-
dynamic partition functions Z q(s) using the collected abso-
lute wavelet coefficients that only hold local maxima lines.
The slope calculated from the log-log plots of the thermo-
dynamic partition function Z q(s) against the scale sis rep-
resented by the scaling exponent τ(q) (Fig. 4). In addition,
the corresponding multifractal spectrum function f(α) is es-
timated (Fig. 4). The calculated width ∆αfor each time seg-
ment of the quasar images A and B are 1.3150/1.6293/1.3375
and 0.9888/0.9416/1.6943 (segment 1/segment 2/segment
3), respectively. The nonlinear behaviour of the scaling expo-
nent and the corresponding width values reveal the presence
of strong multifractal (nonlinear) signature in each time seg-
ment of the light curves. Comparing the degree of multifrac-
tality (nonlinearity) between the corresponding light curves
of the images, the multifractality (nonlinearity) strength of
image A is found to be stronger than that of image B in the
first two time segments, from day 117 to day 1827 and from
day 1835 to day 4525. In contrast, image B is stronger in
the last time segment (segment 3). For both images, A and
B, the order of time segments from the highest to the low-
est in the degree of multifractality (nonlinearity) is segment
2, segment 3, and segment 1 (for image A) and segment
3, segment 1, and segment 2 (for image B). Though mi-
crolensing effects were detected in the initial years of the
quasar monitoring (Pelt et al. 1998), strong evidence of mi-
crolensing was also detected in the last years of monitoring
Gil-Merino et al. (2018). Of the two quasar images, image B
is most likely affected by microlensing since it is the closest
to the centre of the galaxy and thus crosses an internal re-
gion of the galaxy’s luminous halo. To investigate the effect
of microlensing on image B, particularly in the last years
of monitoring, we divide the light curve of the image (top
panel in Fig. 1) from day 4936 to day 7527 into two equal
time segments, though the number of data points is small –
from day 4936 to day 5372 (segment 1) and from day 5564
to day 7527 (segment 2) – and perform the same multifrac-
tality analysis as in the previous cases. The results obtained
– namely, the scaling exponent and multifractality spectrum
function, which are given in Fig. 5– clearly show the exis-
tence of multifractal behaviour. The calculated width values
are 0.8791 and 2.4548 in segment 1 and segment 2, respec-
tively. The degree of multifractality detected in segment 2
is stronger than the one detected in segment 1. Considering
the results obtained here for segment 2 and the one obtained
in the previous analysis for segment 3 (from day 4526 to day
7506), we can see that the degree of multifractality of parts
of the light curve in image B (specifically, parts that include
data points in the range of day 5564 to day 7500) is stronger
than those that did not include this range.
3.2 Analysis of the light curves of image A in the
rand gbands
Here, we analysed the light curve of image A in the rand
gbands (Fig. 1, middle panel). By using the collected local
maxima lines (the skeleton functions) for the light curves,
we determine the corresponding thermodynamic partition
functions Zq(s) against the scale sas presented in Fig. 6.
The slope of the log-log plots of the thermodynamic parti-
tion functions Z q(s) versus the scale sin the rand gbands
is represented by the scaling exponent τ(q) in Fig. 6, re-
vealing the nonlinear relationship between Zq(s) and s. To
further confirm the observed nonlinear relationship between
the thermodynamic partition function Z q(s) and the scale s,
and to determine the strength of the detected multifractal
(nonlinear) signature, we estimate the multifractal spectrum
functions for the signal of image A in the rand gbands. The
MNRAS 000,1–10 (2018)
6A. Bewketu et al.
Figure 2. The thermodynamic partition function Zq(s) (left), the scaling exponent function τ(q) (middle), and the multifractal spectrum
function f(α) (right) of image A of Q0957+561 in the rband.
Figure 3. The thermodynamic partition function Zq(s) (left), the scaling exponent function τ(q) (middle), and the multifractal spectrum
function f(α) (right) of image B of Q0957+561 in the rband.
Figure 4. Top panel: The scaling exponent functions τ(q) (left) and the multifractal spectrum functions f(α) (right) for all time segments
of image A of Q0957+561 in the rband. Bottom panel: The same as the upper panels, but for image B of Q0957+561 in the rband.
For both images: red (segment 1, from day 117 to day 1827), black (segment 2, from day 1835 to day 4525), and blue (segment 3, from
day 4526 to day 7506).
MNRAS 000,1–10 (2018)
Nonlinear difference between images of Q0957+561 7
Figure 5. The scaling exponent functions τ(q) (left) and the corresponding multifractal spectrum functions f(α) (right) of image B in
two different time segments, red (segment 1, from day 4936 to day 5372) and blue (segment 2, from day 5564 to day 7527).
Figure 6. Upper panel: The thermodynamic partition function Zq(s) (left), the scaling exponent function τ(q) (middle), and the
multifractal spectrum function f(α) (right) of image A of Q0957+561 in the rband. Bottom panel: The same as the upper panels, but
for image A of Q0957+561 in the gband.
estimated multifractal spectrum functions f(α) of image A
in the rand gbands are shown in Fig. 6. The calculated
width values in rand gbands are ∆αAr= 1.5319 and ∆αAg
= 1.22, respectively, proving the multifractal behaviour of
image A of Q0957+561 in the rand gbands. Here also,
there is a clear difference in the degree of nonlinearity be-
tween intrabands: the nonlinearity in the rband is observed
to be stronger than the one detected in the gband.
3.3 Analysis of the light curves of image B in the
rand gbands
Similarly, following the same procedures as discussed in the
last subsections, we have analysed the light curves of im-
age B of Q0957+561 in the rand gbands ( Fig. 1, bottom
panel). The thermodynamic partition functions Z q(s) calcu-
lated by using the skeleton functions for the light curves of
image B in the rand gbands, which in turn were constructed
from the absolute wavelet coefficients of the corresponding
light curves, are given in Fig. 7. The scaling exponent func-
tions τ(q), i.e., the slope of the log-log plot of Z q(s) against
s, is also presented in Fig. 7. The τ(q) versus qplot clearly
shows the presence of nonlinear signatures in the light curves
of image B in the rand gbands. The nonlinearity or multi-
fractality detected in the light curves is further strengthened
by the estimated multifractal spectrum function in Fig. 7.
The calculated width (∆α=αmax -αmi n) values are ∆αBr
= 1.0481 and ∆αBg= 0.7739 for the rand gbands of im-
age B of Q0957+561, respectively, confirming the conclusion
based on the scaling exponent function. As discussed in sub-
sections 3.2 and 3.3, for both quasar images, the degree of
multifractality is stronger in the rband.
MNRAS 000,1–10 (2018)
8A. Bewketu et al.
Figure 7. Upper panel: The thermodynamic partition function Zq(s) (left), the scaling exponent function τ(q) (middle), and the
multifractal spectrum function f(α) (right) of image B of Q0957+561 in the rband. Bottom panel: The same as the upper panels, but
for image B of Q0957+561 in the g band.
4 SUMMARY AND CONCLUSIONS
We analyse the presence of multifractal (nonlinear) signa-
tures in the light curves of images A and B of the lensed
quasar Q0957+561 in the optical rand gbands using a
wavelet transform modulus maxima-based multifractality
analysis technique. In addition, we divide the long-term
monitoring light curves of the quasar images in the rband
into different time segments and repeat the same multirac-
tality analysis for each time segment separately. First, we
calculate the absolute wavelet coefficients using the contin-
uous wavelet transform approach and form a matrix of lo-
cal maxima lines (construct skeleton functions) by aggregat-
ing the absolute wavelet coefficients that only hold maxima
lines. Second, using the constructed skeleton function, we
determine the thermodynamics partition function for all the
light curves considered. Third, we estimate the slope of the
log-log plots of the thermodynamic partition function Z q(s)
and the scale s. The estimated behaviours of the slopes are
quantified by the scaling exponent function τ(q) versus the
moment qplots. Finally, we estimate the multifractal spec-
trum at each frequency for all the light curves and calculate
the degree of multifractality from the width ∆αof the spec-
trum. Our main findings are the following: (i) we observed
strong multifractal behaviour in all the light curves anal-
ysed; (ii) the degree of multifractality for both images in
the rband changes over time in a non-monotonic way; (iii)
in the r band, in periods of quiescent microlensing activity,
we found that the degree of multifractality (nonlinearity) of
image A is stronger than that of B, while B has the larger
multifractal strength in recent epochs (from day 5564 to
7527), when it appears to be affected by microlensing; and
(iv) in a period of quiescent microlensing activity in the g
and rbands, the degree of multifractality is stronger in the
rband for both quasar images.
The detection of a multifractal signature in the quasar light
curves is in agreement with our previous results that quasars
are intrinsically multifractal or nonlinear systems (Belete
et al. 2018a,b). The observed multifractality could be due
to different physical mechanisms. It is the variation in flux
that results in a multifractal signature in a light curve. It has
been identified that reverberation within the gas disc around
the supermassive black hole is responsible for most observed
variations in Q0957+561 (Shalyapin et al. 2008); therefore,
most likely it is this physical mechanism that causes mul-
tifractal (nonlinearity) signatures in the light curves con-
sidered. Though no extrinsic signals or microlensing effects
were detected for decades in the light curves of the quasar
Q0957+571 (Shalyapin et al. 2008), recently, strong evidence
for the presence of microlensing effects in the light-curves of
Q0957+561 has been found (Gil-Merino et al. 2018). It is
most likely that quasar images in the same band, in the
optical in our case, have similar (if not the same) radia-
tion mechanisms and regions and are consequently expected
to have similar nonlinear behaviours unless otherwise con-
taminated in a way that changes the intrinsically nonlin-
ear structure. This effect is because for intrinsically variable
quasars, the fluxes measured from the images are expected
to have similar light curves, except for certain lags – time
delays – and an overall offset in magnitude (Wambsganss
1998); as a result, the light curves are expected to be sim-
ilarly nonlinear or multifractal. If there were no contribu-
tion from extrinsic variation, mainly due to microlensing,
the nonlinear signature of the signals would not differ that
much. Also it has been indicated, at a given epoch, intrin-
sic variability should affect images of a lensed quasar in the
same way, whereas microlensing may induce differences be-
tween the spectra of different images. Therefore, assuming
MNRAS 000,1–10 (2018)
Nonlinear difference between images of Q0957+561 9
that all signals of the images in the rband have similar
(if not the same) radiation mechanisms and regions (from
the accretion disk or continuum compact source), or both
signals are intrinsically similarly nonlinear, any difference
in their nonlinearity strength would most likely be due to
the existence of extrinsic variabilities of a different nature
in the observed light curves of the images or due to mi-
crolensing by stars in the lensing galaxies affecting image B,
since microlensing affects the light curves of quasar images
(Kostrzewa-Rutkowska et al. 2018). Therefore, the change in
the degree of multifractality over time in a non-monotonic
way provides us physically important information about the
existence of extrinsic variations in the observed light curves
of the quasar images. In other words, the observed difference
in the degree of multifractality between the time segments
of the light curves indicates that the images are affected
by different physical phenomena along their paths to the
observer, and consequently, the degree of multifractality is
different between them. In particular, in the case of image
B, the increase in the degree of multifractality in the last
years of monitoring could be taken as an additional evidence
for the presence of extrinsic variability due to microlensing
effects since it is the one presumably playing a more rele-
vant role in the image B. The nonlinearity in both quasar
images (leaving aside microlensing, in addition to the ”ex-
cesses” in A and r) should have an intrinsic origin (within
the source quasar). The observed nonlinearity may be pro-
duced in the central high-energy source that irradiates the
accretion disk (AD), when its central flares are reprocessed
in the AD and/or when these flux variations from the AD are
again reprocessed in the broad-line region (BLR). Moreover,
in the absence of microlensing, both observed ”excesses” of
nonlinearity are most likely associated with the presence of
a compact dusty region in the lensing galaxy. However, it re-
mains unclear how dust extinction works to generate these
”excesses”. A significant ”excess” of nonlinearity is generated
when the BLR reprocesses the radiation from the compact
sources. The resulting diffuse light plays a more important
role in A because its direct light is partially extinguished by
the compact dusty cloud in the lensing galaxy. This ratio-
nale can explain the ”excess” in A (in relation to B). The
diffuse contribution could also be more important in the r
band (in relation to g), which would explain the ”excess” in
rin both images. It has been indicated that measuring the
abundance and strength of nonlinear potential fluctuations
along sightlines to high redshift provides a powerful test of
cosmic structure formation scenarios (Cen et al. 1994). We
believe that our results provide significant information to
better understand the physical properties of the interven-
ing medium and construct a model to better understand the
matter distribution in foreground lensing galaxies. Further-
more, multifractality (nonlinearity) analysis could be used
to check whether signals from background sources are only
intrinsic or a combination of both intrinsic and extrinsic sig-
nals, in addition to techniques already developed to do so.
To the best of our knowledge, this is the first time that this
nonlinearity analysis technique has been applied to extra-
galactic lensed sources, and we consider that this approach
could be used to study the relationship between the change
in the nonlinear structure of intrinsic signals from different
background sources at different redshifts, and mass, of lens-
ing galaxies based on the change in the multifractal (non-
linear) behaviour of the signals, from which we can learn
much about the nature of the matter distribution in lensing
galaxies and background quasi-stellar objects.
ACKNOWLEDGEMENTS
This publication makes use of data from the GLQ
database in the GLENDAMA archive available at
http://grupos.unican.es
/glendama/database/. The research activities of the Ob-
servational Astronomy Board of the Federal University of
Rio Grande do Norte (UFRN) are supported by contin-
uous grants from CNPq, CAPES and FAPERN Brazilian
agencies. We also acknowledge financial support from INCT
INEspa¸co/CNPq/MCT. AB acknowledges a CAPES PhD
fellowship. ICL acknowledges a CNPq/PDE fellowship. We
warmly thank the anonymous reviewer for the fruitful com-
ments and suggestions that greatly improved this work.
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