Constraining $\gamma$-ray dissipation site in gravitationally lensed quasar -- PKS 1830$-$211

Abstract

Variable γ\gamma-ray flares upto minute timescales reflect extreme particle acceleration sites. However, for high-redshift blazars, the detection of such rapid variations remains limited by current telescope sensitivities. Gravitationally lensed blazars serve as powerful tools to probe γ\gamma-ray production zones in distant sources, with time delays between lensed signals providing crucial insights into the spatial distribution of emission regions relative to the lens's mass-weighted center. We have utilized 15 years of Fermi-LAT γ\gamma-ray data from direction of PKS 1830−-211 to understand the origin of flaring high-energy production zone at varying flux states. To efficiently estimate the (lensed) time delay, we used a machine learning-based tool - the Gaussian Process regression algorithm, in addition to - Autocorrelation function and Double power spectrum. We found a consistent time delay across all flaring activity states, indicating a similar location for the γ\gamma-ray emission zone, possibly within the radio core. The estimated time delay of approximately 20 days for the five flaring epochs was significantly shorter than previously estimated radio delays. This suggests that the γ\gamma-ray emission zone is closer to the central engine, in contrast to the radio emission zone, which is expected to be much farther away. A linear relationship between lag and magnification has been observed in the identified source and echo flares. Our results suggest that the γ\gamma-ray emission zone originates from similar regions away from the site of radio dissipation.

Full text

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Constraining 𝛾-ray dissipation site in gravitationally lensed quasar -
PKS 1830−211
Sushmita Agarwal1★, Amit Shukla1, Pranjali Sharma1,2
1Department of Astronomy, Astrophysics and Space Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore, 453552, India
2Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Variable 𝛾-ray flares upto minute timescales reflect extreme particle acceleration sites. However, for high-redshift blazars,
the detection of such rapid variations remains limited by current telescope sensitivities. Gravitationally lensed blazars serve as
powerful tools to probe 𝛾-ray production zones in distant sources, with time delays between lensed signals providing crucial
insights into the spatial distribution of emission regions relative to the lens’s mass-weighted center. We have utilized 15 years
of Fermi-LAT 𝛾-ray data from direction of PKS 1830−211 to understand the origin of flaring high-energy production zone at
varying flux states. To efficiently estimate the (lensed) time delay, we used a machine learning-based tool - the Gaussian Process
regression algorithm, in addition to - Autocorrelation function and Double power spectrum. We found a consistent time delay
across all flaring activity states, indicating a similar location for the 𝛾-ray emission zone, possibly within the radio core. The
estimated time delay of approximately 20 days for the five flaring epochs was significantly shorter than previously estimated radio
delays. This suggests that the 𝛾-ray emission zone is closer to the central engine, in contrast to the radio emission zone, which
is expected to be much farther away. A linear relationship between lag and magnification has been observed in the identified
source and echo flares. Our results suggest that the 𝛾-ray emission zone originates from similar regions away from the site of
radio dissipation.
Key words: gravitational lensing: strong - methods: statistical - galaxies: active - galaxies: high-redshift - galaxies: jets -
gamma-rays: galaxies
1 INTRODUCTION
The accretion of matter onto supermassive black holes powers Active
Galactic Nuclei (AGN). The interplay of magnetic field and rotation
either of the black hole (Blandford & Znajek 1977) or of the accretion
disk (Blandford & Payne 1982) are believed to generate collimated
plasma jets that extend from the central engine to large distances.
A subset of these jetted AGNs, known as Blazars, are aligned with
our line of sight and exhibit extremely luminous and highly variable
emissions across a broad electromagnetic spectrum. Notably, Multi-
frequency variability of these point-jetted AGNs provides insights
into the size of the emission region in the jet (Madejski & Sikora
2016). The detection of extremely rapid variability, on timescales
comparable to the light-crossing time of the black hole, in both
high-energy (HE) and very high-energy (VHE) emissions suggests
that the emission zone is compact and located close to the black
hole, within a few tens of gravitational radii (Agarwal et al. 2023;
Shukla et al. 2018;Ackermann et al. 2016;Aleksić et al. 2011).
However, HE and VHE 𝛾-ray photons originating near the black
hole are highly susceptible to attenuation from photon-photon pair
production due to the dense field of ultraviolet (UV) and optical seed
★E-mail: sush.agarwal16@gmail.com , phd1901221002@iiti.ac.in
photon (Liu & Bai 2006). This suggests that 𝛾-ray emission must
occur farther from regions dominated by such external seed photons.
Nonetheless, the lack of sufficient seed photons at larger distances
from central engine complicates the understanding of where high-
energy dissipation occurs.
Variable radio emissions on parsec (pc) to megaparsec (Mpc)
scales often correlate with 𝛾-ray emissions, suggesting co-spatial
origins within the jet (Ghirlanda et al. 2011;Marscher et al. 2008a).
However, the absence of rapid variability in radio bands and potential
synchrotron self-absorption up to hundreds of GHz limit the detec-
tion of radio emissions in smaller structures (Rybicki & Lightman
1979). Observations across radio to X-ray wavelengths indicate ra-
diation regions spanning subparsec to megaparsec scales (Marscher
et al. 2008b;Fuentes et al. 2023;Harris & Krawczynski 2006;Tavec-
chio et al. 2007). The limited resolution of high-energy telescopes
further complicates identification of 𝛾-ray emission regions, leaving
fast variability as the primary probe for high-energy processes. This
limitation constrains our understanding of the link between radio and
𝛾-ray variability (Jorstad et al. 2001;Blandford & Levinson 1995).
The consistency of radiation sources across energy levels remains un-
certain, particularly in high-redshift blazars, where rapid variability
detection is hindered by current telescope sensitivities.
Another probe of the 𝛾-ray production regions is gravitational
©2025 The Authors
arXiv:2501.04775v1 [astro-ph.HE] 8 Jan 2025

2S. Agarwal et. al
lensing. One of the earliest predictions of Einstein’s general relativity
was the deflection of light by the Sun (Einstein 1936). Later, Zwicky
(1937b) and Zwicky (1937a) proposed that galaxies, like stars, could
also act as gravitational lenses. In such a lensed system, photons from
a background galaxy are bent around a foreground lensing galaxy,
creating a magnified and distorted images of the background source.
Radiation from the same point on the source follows different paths,
causing time-variable sources to show similar variability patterns
in different lensed images, but with time delays and magnification.
These time delays and magnifications depend on the geometry of the
source-lens-observer system.
Time delays and magnifications across energy ranges directly re-
flect the size of the emission region and the distribution of the emis-
sion zone around the mass-weighted center of the lens (Barnacka
et al. 2014). Distribution of such time delays provide an alternative
approach to understanding the origin of 𝛾-rays in blazars. Continuous
observations by telescopes like Fermi enable long-term monitoring
of distant sources, providing insights into the evolution of flux vari-
ability over time. Among known gravitationally lensed quasars, two
have been detected at 𝛾-ray energies: PKS 1830-211 (Abdo et al.
2015) and QSO B0218+357 (Cheung et al. 2014).
PKS 1830-21 was first identified as a gravitationally lensed system
by the Very Large Array Radio Telescope, revealing two compact
components in the northeast and southwest (Subrahmanyan et al.
1990). The Australian Telescope Compact Array later showed these
double components separated by 0.98” and connected by an elliptical
Einstein ring (Jauncey et al. 1991;Nair et al. 1993). The flat spectrum
radio quasar PKS 1830-211 (𝑧=2.507), which would typically
appear as a point source, exhibits a double radio structure indicative
of an intervening lens galaxy at redshift 𝑧=0.89 (Wiklind & Combes
1996;Winn et al. 2002;Koopmans & de Bruyn 2005). Evidence also
suggests a second intervening galaxy at 𝑧=0.19, with H I and OH
absorption, though its effect on lensing is expected to be negligible
(Lovell et al. 1996;Muller et al. 2020;Winn et al. 2002;Nair et al.
1993).
Radio observations with the Australia Telescope Compact Array at
8.6 GHz measured radio time delays of 26+4
−5days (Lovell et al. 1998).
Within two years of Fermi/LAT observations, the first gravitational
time delay in 𝛾-rays during its quiescent state was measured at 27.1±
0.6days (Barnacka et al. 2011). The consistency of time delays in
𝛾-rays and radio bands may suggests a co-spatial origin during low
states of 𝛾-ray activity (Barnacka et al. 2014). Later searches in
Fermi-LAT during active states revealed shorter time delays of 23 ±
0.5days and 19.7±1.2days (Barnacka et al. 2015). An independent
study using molecular absorption lines derived a differential time
delay of 24+5
−4days, with the north-east component leading (Wiklind
& Combes 2001).
The search for time delays relies on the length of the light curve.
Understanding the origin of 𝛾-ray flares can be enhanced by study-
ing various flaring states at different flux levels. PKS 1830-211 has
shown significant activity over the past decade, with multiple flares
detected in the Fermi-LAT light curve. We estimated the time delays
during such high flaring periods using the autocorrelation function
and double power spectrum. We also used a machine-learning tech-
nique called Gaussian Process regression to estimate time delays in
different flux states. A comprehensive 15-year search for time delays
was conducted, focusing on the active states of the source. The paper
is structured as follows: Section 2describes the data analysis and the
tools and techniques used for time delay estimation. Sections 3and 4
present the results and discussion, respectively. The summary of our
results is provided in Section 5.
2 METHODS AND TECHNIQUES
2.1 Data Reduction - Fermi-LAT
Despite being a lensed quasar, PKS 1830−211 appears as a point
source due to the limited spatial resolution of the Fermi-LAT tele-
scope. Regardless of this limitation, the Fermi-LAT telescope proves
to be useful. It allows us to utilize the combined flux from the two
anticipated lensed images, which have now coalesced into a signal
exhibiting a distinct time delay. These delays are expected to provide
valuable constraints on the emission size of the source.
Fermi-LAT is a pair-conversion 𝛾-ray telescope, sensitive to pho-
tons in the energy range of 20 MeV - 300 GeV (Atwood et al. 2009).
We selected data from 15.5 years of Fermi observations, covering the
period from MJD 54683 to MJD 60373, within 10◦of the location of
PKS 1830−211. This source, located approximately ∼5◦from the
galactic center, is prone to galactic contamination. To minimize this,
we selected photons with energies within 0.2 - 300 GeV.
To extract the photon statistics, the standard analysis procedure
suggested by the Fermi Science Tools and the open-source Fermipy
package (Wood et al. 2017) was used in the energy range between
0.2−300 GeV, employing the latest instrument response function
P8R3_SOURCE_V3. A zenith angle cut of 90◦, a GTMKTIME cut of
DATA_QUAL >0 && LAT_CONFIG==1, and evtype=3 were used in
the analysis. Only those events highly probable of being photons
were considered for further analysis by applying a GTSELECT cut on
event class to account for SOURCE class events using evclass=128.
A source model was prepared by including the source at RA =
278.413 and Dec = -21.075 and considering all the 4FGL catalog
sources within 20◦around the region of interest. The source is mod-
elled with log-parabola model, parametrized as:
𝑑𝑁
𝑑𝐸
=𝑁◦𝐸
𝐸𝑏−(𝛼+𝛽(log(𝐸/𝐸𝑏))) (1)
where scale parameter 𝐸𝑏was fixed to 4FGL catalog value of
645.56 MeV, 𝛼is specral index , 𝛽is curvature parameter and 𝑁𝑜is
the Normalization. Spectral parameters for the sources within 5◦of
the region of interest were allowed to vary. Sources outside 5◦were
fixed to 4FGL catalog values. Additionally, the background modeled
with the diffuse galactic emission model (gll_iem_v07) and the ex-
tragalactic isotropic diffuse emission model for point source analysis
(iso_P8R3_SOURCE_V3_v1) was allowed to vary.
We performed a binned likelihood analysis using GTLIKE to evalu-
ate the best-fit model parameters, including the source’s spectrum and
intensity at different epochs. The significance of detection is quan-
tified using the Test Statistic (TS), defined as TS =−2 ln(L0/L1),
where L0and L1are the likelihood values without and with the
point source at the position of interest, respectively. Only significant
epochs with TS >9, predicted photons >3, and bins with flux greater
than its uncertainty (𝐹𝑡> 𝜎𝑡) are considered for further analysis.
2.2 Analysis Tools and Techniques
2.2.1 Bayesian Block and HOP algorithm
To enable the detection and characterization of localized variability
structures over time, we represent flux points and their uncertainties
as step-functions using Bayesian Block (BB; Scargle et al. 2013).
Each point of change in the block in the BB representation highlights
a 3𝜎variation from the previous block. The BB output is then pro-
cessed by the HOP algorithm, which is based on a watershed concept
derived from topological data analysis (Eisenstein & Hut 1998). HOP
algorithm by itself identifies flaring states or periods of higher flux
MNRAS 000,1–11 (2025)

Lensing delay in high states of PKS 1830−211 3
55000 56000 57000 58000 59000 60000
Time (MJD)
0
50
100
150
Flux × 10 7
(ph cm 2 sec 1)
F1 F2 F3 F4 F5
BB
Mean Flux
10 day binned
UL (TS<9)
0
100
200
300
Significance (
TS
)
Figure 1. 10-day binned light curve of ≈15.5 years of Fermi-LAT observation of gravitationally lensed FSRQ PKS 1830−211. The grey region represents the
high-flux state, and the white region represents the low-flux state. The light curve is divided into flaring epochs identified using HOP groups, marked by grey
patches. HOP groups separated by less than 50 days are combined into flaring states, labeled as F1 to F5 (indicated by horizontal lines). The secondary y-axis
(right) shows the detection significance as √TS. Periods with TS <9 are represented by upper limits.
by clustering data points from neighboring regions where the flux
exceeds a threshold. The combination of BB and HOP identifies a
block higher than those before and after it as a peak. This approach
then traces down from the peak in both directions, stopping when
each subsequent block is lower than the previous one. Here, we use
the mean flux as a lower threshold.
This technique divides the light curve into flaring and quiescent
epochs, with consecutive connected BBs above the mean flux base-
line referred to as a HOP group. The flare identification code de-
veloped by Wagner et al. (2021) differentiates between various HOP
groups, leading to the identification of nine HOP groups, represented
by grey patches for our source in Fig. 1. The maximum time delay
between the source and its lensed counterpart is expected to be ap-
proximately 70 days, as suggested by Barnacka et al. (2015). Out of
the nine HOP groups represented in Fig. 1, some are less than 70 days
apart. These close intervals suggest probable pairs of the source and
its echo flare, arising from lensing. Therefore, we group HOP groups
that are separated by less than 70 days, leading to five flaring states:
F1 (MJD 55450 - 55600), F2 (MJD 56063, 56173), F3 (MJD 58363
- 58963), F4 (MJD 59063 - 59153), and F5 (MJD 59683 - 59943) as
shown in Fig. 1. The lag and magnification from these flaring groups
are further discussed.
2.2.2 Power Spectrum
To identify the intrinsic temporal behavior of the time series, we
evaluate the power spectral density (PSD) of high-energy 𝛾-ray light
curves. For a stochastic time series, the power distribution at each
frequency typically follows a power-law (PL) (𝑃(𝑓) ∝ 𝑓−𝑘) across
various wavelengths and timescales, with an index ranging from
approximately 1 to 3 (Sobolewska et al. 2014;Finke & Becker 2014;
Nakagawa & Mori 2013). The average slope for the 𝛾-ray PSD of the
brightest 22 flat-spectrum radio quasars and 6 BL Lac objects is 1.5
and 1.7, respectively (Abdo et al. 2010). During the quiescent state,
blazars typically exhibit temporal variability characterized by pink
noise behavior, with 𝛼∼1.
We compute the power-law variability index for the 1d and 12-
hour binned Fermi-LAT light curve (LC) for flares F1 - F5 to quan-
tify the temporal variability during the observed period, using the
PSRESP method described in Max-Moerbeck et al. (2014), based
on Uttley et al. (2002). The obtained PSD is fitted with a PL model
of the form PSD(𝜈) ∝ 𝜈−𝑘, where 𝑘and 𝜈are the spectral index
and frequency, respectively. We simulate 1000 LCs with similar flux
distribution and statistical variability as the observed LC using Con-
nolly (2015). We have accounted for red noise leakage and aliasing
effects as described in Goyal et al. (2022).
2.3 Estimating Time delay
Gravitational lensing is often used as a promising tools for deter-
mining cosmological distances (Refsdal 1964;Schechter et al. 1997;
Blandford & Narayan 1992). Additionally, the time delay and mag-
nification ratio derived at any wavelngth can explicate the location
of the emission region relative to the central black hole (Barnacka
et al. 2014). Atwood (2007) predicted that LAT could detect delayed
emission from bright lensed objects. High-energy observations of
blazars exhibit significant variability due to the small emission re-
gion. The lensing-induced delay in photon arrival is expected to
alter the intrinsic flux pattern of the source. Unlike radio and opti-
cal telescopes, which can resolve magnified, multiple images of a
lensed source, high-energy observations are often limited by poor
spatial resolution. Consequently, the composite flux from source and
its echo image appears as a point source to Fermi. This results in a
repeated flux pattern spaced by adays and demagnified by a factor
of bin the time domain. The total observed flux can be expressed as:
𝑆obs =𝑠(𝑡) + 𝑠(𝑡+𝑎)/𝑏(2)
Thus, the total flux from the two images is integrated into the
combined light curve when observed by high energy telescopes like
Fermi. Thus, an added challenge is disentangling the repeated flares
imprinted in the combined light curve from the apparent point source.
Cheung et al. (2014) attempted to separate these flares and identified
a leading and trailing component in the lensed blazar B0218+537.
In this work, we have used three techniques to estimate the lags in
data using (1) autocorrelation Function, (2) double power spectrum,
and (3) gaussian process regression. Fig. 1represents the 5 flaring
epochs that have been explored in further work.
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4S. Agarwal et. al
Table 1. PSD results
Flux state1Time period2𝑇𝑜𝑏𝑠3𝑁𝑇 𝑆>9/𝑁𝑡 𝑜𝑡 4Δ𝑇𝑚𝑖𝑛5Δ𝑇𝑚𝑎𝑥 6𝑇𝑚𝑒𝑎 𝑛7𝑘±𝑘𝑒𝑟 𝑟 8𝑝𝛽9𝐹𝑣𝑎𝑟 ±Δ𝐹𝑣𝑎𝑟 10
[day] [day] [day] [day] [day]
F1 MJD 55450 - 55600 150 123/150 1 5.0 1.22 0.91 ±0.39 0.85 0.66±0.02
168/360 0.55.0 0.77 0.96 ±0.24 0.60 0.60±0.02
F2 MJD 56063 - 56173 110 101/110 1 3.0 1.08 0.72 ±0.33 0.05 0.46±0.03
153/220 0.54.0 0.71 1.15 ±0.24 0.70 0.37±0.03
F3 MJD 58363 - 58963 600 517/600 1 21.0 1.18 1.36 ±0.24 0.83 0.88±0.01
941/1200 0.521.0 0.64 1.37 ±0.22 0.54 0.84±0.01
F4 MJD 59063 - 59153 90 75/90 1 4.0 1.19 0.88 ±0.47 0.14 0.19±0.05
105/180 0.55.0 0.86 0.50 ±0.27 0.34 0.12±0.08
F5 MJD 59683 - 59943 260 213/260 1 22.0 1.22 1.27±0.37 0.72 0.53±0.02
355/320 0.521.5 0.73 1.23 ±0.20 0.08 0.45±0.02
Note: (1) Flux states extracted from use of BB and HOP algorithm (2) Period of the flaring states (3) Total exposure time (4) Fraction of points
having TS >9 (5) Minimum sampling time in observed LC (6) Maximum sampling time in observed LC (7) Mean Sampling time i.e. total
observation time over a number of data points in that interval (8) The power law index for the power law model of PSD analysis using PSRESP
method (9) p-value corresponding to the power law model. The power law model is considered a bad fit if 𝑝𝛽≤0.1as the rejection confidence
for such model is >90% (10) Fractional variability
Figure 2. (Left) Kernel visualization using covariance between each sample location and zeroth point for RBF, Periodic and RBF ×Periodic. (Right) Covariance
matrix of the sample space for RBF ×Periodic kernel where warmer colors indicate higher correlations.
2.3.1 Auto-Correlation Function (ACF)
The Autocorrelation function (ACF) is a standard statistical tool for
assessing the similarity of a time series with a delayed copy of itself.
ACF allows the identification of periodicity or repeated patterns in a
signal, making it suitable for estimating lags in data.
For a noise-dominated signal, variable structures are inherently
present in power-law noise with an index greater than 1. Barnacka
et al. (2015) described the role of ACF in deciphering the lag in
noisy lensed signals and found that time delay detection is easier for
a source with large variability index (𝑘). Thus, a steep spectrum with
spurious peaks improves the chances of confident detection.
The significance of the estimated lags is evaluated using the Monte
Carlo simulation described in Section 2.3.4. To make the time series
continuous, the epochs of no or less significant observation are inter-
polated with zeros as in Barnacka et al. (2015). We then performed
autocorrelation on 1-day and 12-hour binned time series for flares F1
to F5. The results are discussed in Section 3.
2.3.2 Double Power Spectrum (DPS)
A lensed time series is represented in equation 2. Given the long,
continuous, and evenly spaced nature of Fermi-LAT light curves, it is
feasible to extract the lag from the data using the Fourier transform, as
described by Barnacka et al. (2011), Barnacka (2013), and Barnacka
et al. (2015). This method was first used for lag estimation in lensed
light curves by Barnacka et al. (2011). The idea is to take a Fourier
transform of the first power spectrum of Equation 2. The Fourier
transform of the first component, 𝑠(𝑡), is ˜𝑠(𝑓), and for the second
component, it is ˜𝑠(𝑓)𝑒−2𝜋 𝑖 𝑓 𝑎. Therefore, the observed signal in the
frequency domain is transformed into:
F(𝑆obs)=˜𝑠(𝑓)(1+𝑏−1𝑒−2𝜋𝑖 𝑓 𝑎)(3)
The first power spectrum (FPS) is the square modulus of Fourier
transform of 𝑆obs, i.e. :
|˜
𝑆(𝑓)|2=|˜𝑠(𝑓)|2(1+𝑏−2+2𝑏−1𝑐𝑜𝑠2𝜋 𝑓 𝑎)(4)
MNRAS 000,1–11 (2025)

Lensing delay in high states of PKS 1830−211 5
0 20 40 60 80 100 120 140
MJD [ 55450 + ]
0
1
2
3
4
5
Flux × 10 6
[ph cm 2 s 1]
F1
Mean prediction
68% confidence interval
10 20 30 40 50 60
Lags [days]
0.2
0.0
0.2
0.4
0.6
0.8
ACF
10 20 30 40 50 60
Lags [days]
0.0
0.1
0.2
0.3
0.4
0.5
DPS
1 2 3 4
Figure 3. (Top panel) 1-day binned (black) and 12hr binned (red) light curve
of flaring epochs F1 [MJD 55453 - 55583] (marked in Figure. 1) of FSRQ
PKS 1830−211. In blue is the GPR predictions on 1 day binned data for the
paramaters with largest marginal likelihood (Middle panel) ACF on 1-day
binned light curve of F1 period (Bottom panel) DPS on 1-day binned light
curve.
If an intrinsic lag is imprinted onto the signal, it should be encoded
into the periodicity of the FPS with a time period inversely propor-
tional to the lag. Therefore, a power spectrum of the FPS is expected
to show a large amplitude of the time delay signal. Barnacka et al.
(2011) found that DPS is 90% efficient in detecting the encoded lag
in the signal, significantly improving over the 10
To correct for the smearing effect caused by the finite length of
the signal and sampling in the data, the signal must undergo specific
processing steps. We use the method described in Barnacka et al.
(2015), based on Brault & White (1971), to extract the time delay
present in the signal. This method can accurately extract time delays
regardless of whether the light curve is white noise or red noise
dominated and eliminates fake time delay peaks expected in red
noise signals.
0 20 40 60 80 100
MJD [ 56063 + ]
0.0
0.5
1.0
1.5
2.0
2.5
Flux × 10 6
[ph cm 2 s 1]
F2
Mean prediction
68% confidence interval
10 20 30 40 50 60
Lags [days]
0.2
0.0
0.2
0.4
0.6
0.8
ACF
10 20 30 40 50 60
Lags [days]
0.0
0.1
0.2
0.3
0.4
0.5
DPS
1 2 3 4
Figure 4. (Top panel) 1-day binned (black) and 12hr binned (red) light curve
of flaring epochs F2 [MJD 56063 - 56173] (marked in Figure. 1) of FSRQ
PKS 1830−211. In blue is the GPR predictions on 1-day binned data for the
paramaters with largest marginal likelihood (Middle panel) ACF on 1-day
binned light curve of F2 period (Bottom panel) DPS on 1-day binned light
curve of F2 period
2.3.3 Gaussian Process Regression (GPR)
A Gaussian Process (GP) is a random process where any point 𝑥in
the real domain is a random variable 𝑓(𝑥), and the joint distribution
of a finite number of these variables follows a Gaussian distribution.
Mathematically, for a set of inputs 𝑥1,𝑥2,...,𝑥𝑛with corresponding
outputs 𝑦1,𝑦2,...,𝑦𝑛, wherein y = f(x), the function values f(x)
follow a joint Gaussian distribution. GP can be seen as a general-
ization of the infinite-dimensional multivariate Gaussian distribu-
tion. In a finite-dimensional Gaussian distribution, the correlation
between random variables is defined using a covariance matrix. For
an infinite-dimensional Gaussian distribution, this matrix is replaced
by a "covariance function", known as a Kernel (Rasmussen 2004).
The mean function of the infinite-dimensional Gaussian is typically
set to zero for easier computation, but the mean of the observational
data is later added to obtain predictions on the original scale, leverag-
ing the scaling property of Gaussian distributions. Standardization,
which involves subtracting the mean and dividing by the standard
MNRAS 000,1–11 (2025)

6S. Agarwal et. al
0 100 200 300 400 500 600
MJD [ 58361 + ]
0
2
4
6
8
10
12
Flux × 10 6
[ph cm 2 s 1]
F3
Mean prediction
68% confidence interval
10 20 30 40 50 60 70
Lags [days]
0.2
0.0
0.2
0.4
0.6
0.8
ACF
10 20 30 40 50 60 70
Lags [days]
0.0
0.1
0.2
0.3
0.4
0.5
DPS
1 2 3 4
Figure 5. (Top panel) 1-day binned (black) and 12hr binned (red) light curve
of flaring epochs F3 [MJD 58363 - 58963] (marked in Fig. 1) of FSRQ
PKS 1830−211. In blue is the GPR predictions on 1-day binned data for the
paramaters with largest marginal likelihood (Middle panel) ACF on 1-day
binned light curve of F3 period (Bottom pane;) DPS on 1-day binned light
curve of F3 period
deviation of the data before fitting the GP, is a common practice.
Kernel selection: The choice of kernel requires some prior informa-
tion about the data. In our analysis, we incorporate the prior knowl-
edge that the data exhibits a lag effect. This leads to the selection of
the following kernel:
𝜅(𝑥, 𝑥 ′)=exp −|𝑥−𝑥′|2
2𝑙2×exp −2
𝑙2sin2𝜋|𝑥−𝑥′|
𝑝 (5)
where 𝑙is the length scale, and 𝑝is the distance between repe-
titions. The first multiplicative element in this kernel corresponds
to a Gaussian-shaped correlation function, while the second multi-
plicative element represents a periodic correlation. Fig. 2depicts the
shape of the chosen kernel after multiplication. In this context, the
periodicity parameter 𝑝effectively functions as the lag.
Hyper-parameter estimation : The likelihood function can serve
as an objective function for a non-linear optimization algorithm to
obtain the maximum likelihood parameter values. In GPR, this is
replaced by the log marginal likelihood, which incorporates both
0 20 40 60 80
MJD [ 59063 + ]
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Flux × 10 6
[ph cm 2 s 1]
F4
Mean prediction
68% confidence interval
10 20 30 40 50 60
Lags [days]
0.2
0.0
0.2
0.4
0.6
0.8
ACF
10 20 30 40 50 60
Lags [days]
0.0
0.1
0.2
0.3
0.4
0.5
DPS
1 2 3 4
Figure 6. (Top panel) 1-day binned (black) and 12hr binned (red) light curve
of flaring epochs F4 [MJD 59063 - 59153] (marked in Fig. 1) of FSRQ
PKS 1830−211. In blue is the GPR predictions on 1-day binned data for the
paramaters with largest marginal likelihood (Middle panel) ACF on 1-day
binned light curve of F4 period (Bottom panel) DPS on 1-day binned light
curve of F4 period
the data fit term and a penalty term to prevent overfitting. The log
marginal likelihood consists of three terms added together: The first
term, −1
2y𝑇(K(X,X′)+𝜎2
𝑛I)−1y, quantifies the quality of the fit. The
second term, −1
2log det(K(X,X′) + 𝜎2
𝑛I), helps helps avoid over-
fitting. The last term, −𝑛
2log(2𝜋)is a normalization term to ensure
a valid probability distribution, where K(X,X′)is the covariance
matrix, Iis the identity matrix, and 𝑛is the number of data points.
We optimize the hyperparameters of the kernel K. This study
utilizes the scikit-learn GPR module (Pedregosa et al. 2011) for eas-
ier computation. We optimize the length scale hyperparameter for
various fixed periodicity hyperparameter values and obtain the log
marginal likelihood profile across different lag values, as shown in
Fig. 8. We select a grid of lag values from 1 to 70 days in steps of 1
day. The upper limit for the lag is chosen based on prior information
about the gravitationally lensed source (Zhang et al. 2008). Since the
marginal likelihood can exhibit very similar values across different
lags, we introduce a metric for better comparison, termed the "likeli-
hood metric". Given that the marginal likelihood can be negative, we
multiply it by -1 and subtract the maximum of this value from each
MNRAS 000,1–11 (2025)

Lensing delay in high states of PKS 1830−211 7
0 50 100 150 200 250
MJD [ 59683 + ]
0
1
2
3
4
5
Flux × 10 6
[ph cm 2 s 1]
F5
Mean prediction
68% confidence interval
10 20 30 40 50 60
Lags [days]
0.2
0.0
0.2
0.4
0.6
0.8
ACF
10 20 30 40 50 60
Lags [days]
0.0
0.1
0.2
0.3
0.4
0.5
DPS
1 2 3 4
Figure 7. (Top panel) 1-day binned (black) and 12hr binned (red) light curve
of flaring epochs F5 [MJD 59683 - 59943] (marked in Fig. 1) of FSRQ
PKS 1830−211. In blue is the GPR predictions on 1-day binned data for the
paramaters with largest marginal likelihood (Middle panel]) ACF on 1-day
binned light curve of F5 period (Bottom panel) DPS on 1-day binned light
curve of F5 period
marginal likelihood. The optimal lag then corresponds to the max-
imum value of the likelihood metric. Given the probabilistic nature
of the method, the estimated lag is expected to exhibit a distribution
centered around the true lag value. The uncertainties in the derived
lag are quantified by analyzing the spread within this distribution.
2.3.4 Statistical significance
The significance of spurious peaks in ACF and DPS must be assessed
to determine whether the observed time delay is a result of chance
or represents an intrinsic time delay within the signal.
We simulated 105light curves using the techniques described
in Emmanoulopoulos et al. (2013) to generate artificial light curves
with similar flux distribution and temporal variability as the observed
light curve. This method allows for generating light curves with non-
Gaussian distributions, overcoming a limitation of Timmer & König
(1995). High-energy 𝛾-ray light curves of blazars typically follow a
Flare state 𝛼±𝛿 𝛼 𝛽 ±𝛿𝛽
F1 2.39 ±0.03 0.16 ±0.02
F2 2.29 ±0.03 0.10 ±0.02
F3 2.38 ±0.01 0.15 ±0.01
F4 2.55 ±0.04 0.13 ±0.04
F5 2.41 ±0.02 0.14 ±0.02
Quiet state 2.47 ±0.02 0.08 ±0.01
Table 2. Fermi-LAT Spectral parameters for the chosen flaring states and the
quiet state. The parameters are derived from the fitted log-parabola model.
log-normal flux distribution (Romoli et al. 2018;Bhatta 2021). As a
result, the simulated light curves have identical statistical properties
to the observed light curve.
To make the simulated light curves as realistic as possible, we
included data gaps identical to the observed periods and interpolated
them with zeros to ensure similar effects on the ACF and DPS of
the simulated signal. We constructed cumulative probability distri-
butions of the derived powers at each time delay. These distributions
were then analyzed for 1𝜎,2𝜎,3𝜎,4𝜎chances of detection. Any sig-
nificant (>3𝜎) powers corresponding to a time delay are considered
as intrinsic time delays in the signal.
3 RESULTS
The HE light curve of PKS 1830-211 appears quite complex, with
multiple flaring periods over the quiet states. Fig. 1shows significant
variability in the 10-day binned high-energy (200 MeV - 300 GeV)
flux over time. This variability is evident from the fluctuating flux
levels, with some periods displaying higher fractional variability than
others (see Table 1). The flaring periods exhibit dominant pink noise
behavior with a PSD power-law index of approximately 1. Notably, a
transition from pink to red noise behavior is observed for the brightest
flux state of the source, identified as F3 in this work (Table 1). We
use these flaring states to identify dominant emission zones, which
should appear as twin pairs of flares separated by a specific time
interval for a lensed blazar. The flaring epochs, which are above the
mean flux levels, are identified using BBs, indicated by grey patches
in Fig. 1. Multiple blocks spaced less than 70 days apart are merged
together, resulting in flaring states labeled F1, F2, F3, F4, and F5.
The flaring periods, except for F4, exhibit similar 𝛼parameters
in HE Fermi-LAT spectrum, indicating a consistent physical process
within a 3𝜎range. Additionally, the 𝛽values for all periods are
consistent, suggesting a similar influence of external seed photons
in the production of high-energy 𝛾-rays. The spectral parameters for
the flaring periods F1-F5 are summarized in Table 2.
The time lag between these counterpart flares from the lensed im-
ages is estimated using the three methods described in Section 2.3.
The maximum time delay between the lensed images is given by
∼6𝑧𝑔
0.1(2ℎ)−1days, where 𝑧𝑔is the redshift of the lensing galaxy
and ℎis the Hubble constant in units of 100 km s−1Mpc−1(Zhang
et al. 2008). Using a redshift of 𝑧𝑔=0.89 and ℎ=75,km s−1Mpc−1,
the maximum time lag is approximately 71 days. This represents the
maximum time delay between the mirage images when the source is
near the Einstein ring. For large delays, the magnification ratio is ex-
pected to be significant. Thus, detecting a large delay is unlikely since
the trailing component would be demagnified beyond the sensitivity
of Fermi-LAT. Additionally, to explore the full range of potential
MNRAS 000,1–11 (2025)

8S. Agarwal et. al
Figure 8. The likelihood metric for lags is derived using GPR. The bar represents the likelihood value for each lag, with the highest value indicating the most
probable lag. The red Gaussian fit over the likelihood values represents the mean lag value estimated and its corresponding error bar.
time delays, the length of the lightcurve must be at least twice the
longest time delay.
Detecting the trailing counterpart during a quiescent state of a
blazar is challenging due to its expected demagnification, which
impedes detection. However, the sensitivity of Fermi-LAT allows
for the detection of multiple flaring states (F1 to F5). Dominant
pink noise during flaring epochs creates spurious peaks, increasing
the chances of detecting both trailing and leading components. The
source shows the largest variability for F3 and F1, with respective
fractional variability values (F𝑣𝑎𝑟 ;Vaughan et al. 2003) of 0.88±0.01
and 0.66±0.02 (Table 1). Our objective is to identify time delays from
flares with sufficient magnification to detect both leading and trailing
components. Whenever possible, pairs of leading and trailing flares,
spaced within the identified time lag, are selected to study the spectral
properties of source and echo flare and draw the relationship between
lag and magnification. To analyze the flare variability properties, we
fitted exponential flares to sharp, distinct features in the light curve
using the form:
𝐹(𝑡)=𝐹0×exp 𝑡𝑜−𝑡
𝜏rise +exp 𝑡−𝑡𝑜
𝜏decay  (6)
Here, 𝜏rise and 𝜏decay represent the rise and decay timescales,
respectively; 𝑡𝑜is the peak time, and 𝐹0represents half of the peak
flux of the flare at time 𝑡𝑜.
3.1 Flare 1 - MJD 55453 - 55583
The 1-day and 12-hour binned light curve of flare F1 is shown in Fig.
3(a). The flaring period spans 150 days, with 18% of the flux points
not resulting in significant detection. The flare exhibits a sharp peak
from MJD 55483 to MJD 55488 and another distinct feature from
MJD 55553 to MJD 55573 in the 𝛾-ray light curve.
Flare LagACF LagDPS LagGPR
F1 - 17±1.5 (>3𝜎) 19.0±1.5
F2 20.3±2.3 20.0±0.5 (∼2𝜎) 22.1±2.6
F3 20.5±1.0 21.0 ±0.5 (∼3𝜎) 21.1±1.2
F4 - 14.0±0.5 (<2𝜎) 22.4±2.2
F5 - 17.0±0.5 (>2𝜎) 19.4±2.7
Table 3. Estimated lags for Flares F1, F2, F3, F4, F5 using the three methods
(a) Autocorrelation Function (b) Double Power Spectrum (c) Gaussian Pro-
cess regression
The light curve exhibits prominent pink noise behavior with a
power-law index of 𝑘=0.91 ±0.39. Monte Carlo simulations were
conducted to quantify the significance of the time delay by gener-
ating light curves with similar power spectral density indices. The
Autocorrelation Function did not result in a significant detection, but
a feature emerged at 55 ±2days (∼2𝜎), likely an artifact of the
Fermi light curve due to the spacecraft’s precession period of 53.4
days (Fig. 3(b)).
The DPS method is more prone to detecting spurious time delays.
The DPS on a 1-day binned light curve peaked at 17 ±1.5days with
above 3𝜎significance as shown in Fig. 3(c). Similarly, GPR analysis
on the 1-day binned light curve identified the maximum marginal
likelihood metric at 19.0±1.5days. The corresponding best-fit GPR
light curve is shown in Fig. 8(a). The marginal likelihood peaking at
9.8±2.9days could represent a lower harmonic of the 19.0±1.5day
delay, which aligns with the periodic nature of the selected kernel.
Consistent results have been reported in previous studies, such as
MNRAS 000,1–11 (2025)

Lensing delay in high states of PKS 1830−211 9
a lag of 19 ±1days by Abdo et al. (2015) and 17.9±7.1days by
Barnacka et al. (2015) using ACF.
The maximum peak in the light curve at MJD 55483 to MJD 55488
and the resulting echo flare (F11 in Fig. 9(a)) after 20.0±1.1 days can
be a proxy for evaluating the magnification ratio, resulting in a ratio
of ∼2.8. Another distinct feature at MJD 55553 to MJD 55573 (F12),
considering a similar time delay, shows a magnification ratio of ∼1.6
as in Fig. 9(a).
3.2 Flare 2 - MJD 56063 - 56173
Fig. 4(a) shows the 1-day and 12-hour binned light curves. This
epoch is above the mean flux level, with periods before and after
significantly below the mean. The 110-day-long light curve follows
a pink noise power-law time series with an index of 𝑘=0.72 ±0.33.
Simulated light curves with similar indices were generatedto measure
the significance of the lag in the signal.
The autocorrelation on F2 in Fig. 4(b) shows two features with
significance close to 2𝜎:11.0±2.3days and 20.3±0.5days, con-
sistent with Barnacka et al. (2015). An additional lag close to 2𝜎
at 55.7±2.2days appears in both the 1-day and 12-hour binned
light curves, possibly an artifact of the Fermi telescope’s processing
period, similar to Flare F1.
The DPS method on the 1-day binned light curve detected a time
delay of 20 ±0.5days with more than 2𝜎significance (Fig. 4(c)).
Consistent lags are found using GPR, with increased marginal like-
lihood metrics at 13.3±4.3and 22.1±2.6days, aligning with the
ACF results. The likelihood distribution of GPR on Flare F2 is as in
Fig. 8(b). The 13.3±4.3day lag in GPR could be a lower harmonic
of the 22.1day delay in the light curve.
Disentangling the light curve to identify flares and their echo flare
image is challenging for Flare F2. The double-peak structure from
MJD 56081 to MJD 56098 has a demagnified delayed counterpart
appearing from MJD 56101 to MJD 56118, with a demagnification
of approximately 1.9 as shown in Fig. 9(b)). No echo counterpart to
the broad feature from MJD 56142 to MJD 56156 is visible within the
20-day period. This absence suggests a much larger demagnification
corresponding to longer time delays.
3.3 Flare 3 - MJD 58363 - 58963
The 1-day and 12-hour binned light curve for Flare 3 is shown in
Fig. 5(a). This flare represents the brightest flux state of the source,
with the flux reaching 14 times its average level. Notably, it is also
the longest-lasting flare analyzed in this work, spanning a total of
600 days. The flare exhibits the highest power spectral index, with
characteristic noise behavior between pink and red noise types. Mul-
tiple flares appear to be superimposed on an underlying envelope, as
illustrated in Fig. 5(a).
The ACF, as displayed in Fig. 5(b)) reveals two features with
significance greater than 3𝜎: one at 12±1.8days and another at 21.1±
1.2days. For comparison, the DPS method calculates a prominent
lag at 21.0±0.5days (≳3.5𝜎) and 19.0±0.5days (=3𝜎). Less
significant detections, but still above ≳2.5𝜎, are found at 14 ±0.5
days and 25 ±0.5days. This suggests multiple lag values imprinted
on the flares. The estimated lag values could also be a reflection of
the time difference between subsets of flares.
A consistent lag detected by GPR at 14.0±2.2and 21.1±1.2days
is simultaneously observed by ACF and DPS (Figure 8(c)).
Due to the dense overlap of flaring periods, identifying associated
leading and trailing counterparts is extremely challenging during
such a flaring period. Therefore, estimating magnification by flare
identification is not performed for Flare 3.
3.4 Flare 4 - MJD 59063 - 59153
Fig. 6shows the 1-day and 12-hour binned light curve for Flare
4, which spans 90 days and is the least bright of the five states
analyzed. The power spectral density for the 1-day binned data yields
𝑘=0.88 ±0.47. Ideally, the sample length should be twice the
maximum expected time delay of 70 days, so the 90-day light curves
reduce the likelihood of detecting time delays. Additionally, F4 has
the least variability in lightcurve with 𝐹𝑣𝑎𝑟 , 𝐹4=0.19 ±0.05 days.
Such small fractional variability highlights the absence of significant
variable points in the light curve.
No significant time delay is detected by ACF and DPS in the 1-day
binned data. GPR estimates a lag of 22.4±2.2days as shown in Fig.
8(d). The absence of significant echo flares for the flux rise at MJD
59113 - 59123 (Fig. 6(a)) makes it difficult to conclude if a lensed
image is detectable within the telescope’s sensitivity.
3.5 Flare 5 - MJD 59683 - 59943
Fig. 7(a) shows the 1-day and 12-hour binned light curve for Flare 5,
with multiple peaks visible over the 260-day period. At least 18.1%
of the data contains gaps or periods with significantly low detection,
which we interpolated with zeros. No significant time lags were found
using ACF. However, DPS estimates a 17 ±0.5day time lag with
more than 2𝜎significance, consistent with the predicted time delay
from GPR at 19.4±2.7days (See Fig. 8(e)).
The bright peaks at MJD 59890 to MJD 59928 (F51) were fitted
with exponential function as shown in Fig. 9(c). However, due to
the emergence of multiple overlapping flares, associating flares to
identify magnification is challenging.
4 DISCUSSION
The time delay between two lensed images of a quasar, such as
PKS 1830−211, is typically evident from systematic changes in flux
density in the radio band for the leading and trailing components.
However, in the 𝛾-ray band, the resolution of high-energy telescopes
constrains the detection of resolved images. As a result, the combined
flux evolution from the two lensed images appears merged into a
single point source.
The time delay observed from the merged images reflects the distri-
bution of emission sites on the lens plane around the mass-weighted
center of the lens (Barnacka et al. 2014). Time delays, along with
magnifications, are fundamental for localizing the emitting region
relative to the black hole. Typically, a smaller time delay is associ-
ated with areas close to the base of the jet. Emissions originating
near the base of the jets are typically reflected as the fast variability
in high-energy 𝛾-ray light curves.
In this work, we employed three methods to investigate the ob-
served time lag in the stochastic time series. The resulting lags are
listed in Table 3. We devised a novel technique, Gaussian Process
Regression, to extract the lag in the signal. From our analysis, we
found that the autocorrelation function is unable to efficiently extract
the intrinsic time delay present in most signals, partially due to its
noise dependence. However, among the five flares studied, autocor-
relation successfully detected the intrinsic lag in the time series for
Flare 3 with a significance of more than 3𝜎. Flare 3 exhibited noise
behavior between pink and red, and the emergence of multiple flares
MNRAS 000,1–11 (2025)

10 S. Agarwal et. al
0 20 40 60 80 100 120 140
Time (MJD 55450.5 + )
0
10
20
30
40
50
Flux × 10 7
(ph cm 2 sec 1)
F11 F12
Mean Flux 1 day binned UL (TS<9)
0 20 40 60 80 100
Time (MJD 56063.5 + )
0
5
10
15
20
25
Flux × 10 7
(ph cm 2 sec 1)
F21
80 90 100 110 120 130 140
Time (MJD 59800.5 + )
0
10
20
30
40
Flux × 10 7
(ph cm 2 sec 1)
F51
Figure 9. High flux states of Flare F1, F2 and zoomed section of F5 (MJD
59880 - 59940) and fitted exponential flare using equation 6. The vertical
lines represent the source and echo pair for the lensed flares. The exponential
fits with similar colors are considered possible pairs of source and echo flares.
SOURCE ECHO
2.2
2.3
2.4
2.5
2.6
SOURCE ECHO
0.00
0.05
0.10
0.15
0.20
0.25
0.30
F11 F12 F21 F51
Figure 10. High energy spectral parameter of the flare and its associated echo
flare. (left) 𝛼and (right) 𝛽for the fitted log-parabola model.
resulted in significant detection. Similar results were obtained using
the Double Power Spectrum and Gaussian Process Regression.
Our results suggest the presence of a consistent time delay of
approximately 20 days during the flaring state of the source, as de-
termined by the three methods used in this work. This indicates
a similar orientation of the emitting site around the mass distri-
bution of the lens. Consequently, 𝛾-ray emission consistently oc-
curs in similar regions of the jet across all flaring states. The in-
ferred time delay aligns with the estimated lag reported in (Bar-
nacka et al. 2015) during the high states, indicating that the origin
of the 𝛾-rays is likely within the core. The detection of rapid vari-
ability, with 𝑡𝑣𝑎𝑟 ∼0.38 ±0.22 days, implies an emission region
size of 𝑟𝑒𝑚𝑚 =𝑐𝛿𝑡𝑣 𝑎𝑟 /(1+𝑧)=2.8×1015 cm at a distance of
𝑅𝑑𝑖𝑠 𝑠 =2𝑐Γ2𝑡𝑣𝑎 𝑟 =0.064 pc, thereby confirming that the high-
energy emission is localized within the core on sub-parsec scales.
Lovell et al. (1996) reported a lag of 26+4
−5days using the Aus-
tralian Telescope Compact Array at 8.6 GHz. Similar time delays
were estimated in the 𝛾-ray band during the quiet state of the source
(Barnacka et al. 2011). The lag observed during high states in this
study suggests a different origin for flaring 𝛾-ray emission. The in-
consistency between 𝛾-ray and radio lags indicates different dissi-
pation sites for these emissions, especially during the source’s high
state. Radio emissions are typically expected from the outer regions
of the parsec-scale jet. A small, compact jet leads to synchrotron
self-absorption, making radio emission unlikely in the inner parsec
scales. Shorter time delays during active states imply that 𝛾-ray dissi-
pation occurs closer to the central engine, whereas radio dissipation
occurs farther out in the jet. This has been observed as the absorption
of high-energy photons with energies greater than 10 GeV during
high states under the influence of BLR photons at sub-parsec scale
jet (Agarwal et al. 2024).
The high-energy spectral properties of the source and the echo
flare are consistent within 3𝜎(See Fig. 10). A change in the spectral
properties would imply a difference in the influence of soft seed pho-
tons on 𝛾−ray photon through 𝛾−𝛾absorption on passing through
a more luminous region of the lensing galaxy. A ∼2.8𝜎deviation
was observed in the spectral index for flare F21. The identical beta
parameters for the four possible lensed flares suggest a similar in-
fluence of external seed photons from the local jet environment, the
EBL, and the intervening galaxy on the high-energy spectrum. Since
absorption affects all of the flares in the same way, they originated
from similar regions of the jet. Our analysis focused on flares with a
clear source and a demagnified lensed echo flare at an average flux
level, leading us to select flares F1, F2, and F5. Due to the presence
of multiple overlapping flares, identifying individual flares and their
echoes for flare F3 was inefficient, likely due to the merging of mul-
tiple flares. Exponential fitting on individual flares reveals a linear
relationship between lag and magnification. However, further studies
are needed to identify more clean flares. This suggests that a smaller
emission region is confined close to the base of the jet, while a larger
magnification implies a larger emission region located farther out in
the jet.
5 SUMMARY
Strong gravitational lensing in 𝛾-ray bright blazars can identify the
locations of 𝛾-ray dissipation during both quiescent and active states.
The variation in time delays observed during periods of active 𝛾-ray
flux suggests different emission regions within the jet compared to
those during low flux states (Barnacka et al. 2011). The consistent lag
across five flaring states indicates a similar origin for the high-energy
𝛾-ray activity within the radio core. This contrasts with the larger lag
observed during quieter 𝛾-ray periods and the consistent time delays
from radio observations, which suggest that such emission occurs
farther from the central engine than that during flaring periods. Such
time delays caused by gravitational lensing of a background source
by a foreground object could help constrain the Hubble parameter
(Refsdal 1964).
MNRAS 000,1–11 (2025)

Lensing delay in high states of PKS 1830−211 11
We introduce a novel technique for estimating time delays in long,
continuous light curves from Fermi-LAT. Detecting these time delays
could be crucial in identifying hidden lensed blazars during flaring
periods that are not recognized as lensed sources in radio wave-
lengths. The signatures of such time delays could provide insights
into distant blazars and previously unidentified 𝛾-ray sources. Future
surveys, including those conducted by the SKA, will likely discover
many such lensed quasars. Extensive multi-wavelength searches of
these systems could offer valuable insights into the origins of radia-
tion and provide a magnified view limited by current telescopes.
6 ACKNOWLEDGEMENTS
We thank the refree, Dr. Nachiketa Chakraborty, for constructive
feedback, which has helped improve the manuscript. SA and AS ac-
knowledges Dr. Bhargav Vaidya for useful discussions on the work
which helped improve the manuscript. AS acknowledge support for
computational facility from DST-SERB grant CRG/2022/009332.
This research work has made use of archival data, software and
web tools obtained from NASA’s High Energy Astrophysics Science
Archive Research Center (HEASARC) and Fermi gamma-ray tele-
scope Support centre, a service of the Goddard Space Flight Center
and the Smithsonian Astrophysical Observatory.
DATA AVAILABILITY
The data will be made available upon reasonable request.
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