Active elastic metamaterials with equidistant solely-resonant bandgaps

Hasan B. Al Ba’ba’a‡

|  Department of Mechanical Engineering, Union College, Schenectady, NY 12308,
USA
---|---
|<EMAIL_ADDRESS>

Elastic metamaterials are man-made structures with properties that transcend
naturally occurring materials. One of the most predominant features of elastic
metamaterials is locally-resonant bandgaps, i.e., frequency ranges at which
wave propagation is blocked. Locally-resonant bandgaps appear at relatively
low frequency and arise from the existence of periodically placed mechanical
local resonators. Typically, elastic metamaterials exhibit both locally-
resonant and Bragg-scattering bandgaps, which can generally be different in
width and frequency ranges. This paper proposes two designs of active
metamaterials that only exhibit locally-resonant bandgaps, which are infinite
in number, evenly spaced in the frequency spectrum, and identical in width.
The mathematical model is established using the transfer matrix method (TMM)
and synthesis of locally-resonant bandgaps is achieved via an active elastic
element with carefully designed frequency-dependent stiffness. A single unit
cell of the active metamaterial is thoroughly studied, and its dispersion
relation is derived analytically, along with the bandgap limits and width.
Following the dispersion analysis and bandgap parametric studies, finite
arrays of the proposed active metamaterials are considered, and their
frequency response is investigated and used to verify the analytical
predictions from dispersion analyses. The proposed mathematical framework
establishes a design methodology that could be further used to synthesize
interesting dispersion behaviors in the future.

Keywords

Locally-resonant bandgaps; active elastic metamaterials; equidistant bandgaps;
frequency-dependent stiffness.

##  1 Introduction

Since their inception in early 2000s [1], locally-resonant materials have
witnessed a surge in research interest, aimed for realizing exotic phenomena
in elastic and acoustic wave propagation using locally-resonant bandgaps [2,
3, 4]. By definition, bandgaps are ranges of frequencies where wave
propagation is effectively blocked, without the need for any traditional
dissipative mechanisms. Mitigation of waves within locally-resonant bandgaps
is purely a consequence of local-resonance effects [5] (or negative effective
mass behavior [6]), unlike the Bragg-scattering phenomenon that is caused by
destructive interference of the incident and reflected waves [2]. Locally-
resonant bandgaps are superior to those Bragg-scattering bandgaps in terms of
their independence on the unit-cell size and considerably low operational
frequencies at the sub-wavelength regime [7, 8, 9].

A common configuration of elastic metamaterials (EMMs) is built via a periodic
placement of mechanical local resonators with relatively low stand-alone
natural frequencies to enable sub-wavelength bandgaps. Furthermore, EMMs may
be equipped with different resonator properties along the base structure body
to give the advantage of attenuating waves with multi-frequency harmonics [10]
as well as for bandgap widening [11, 12]. EMMs have been also crucial to
amplify damping due to metadamping effects [13, 14, 15, 16, 17] and to enable
enhanced performance of energy harvesting [18]. While EMMs are mainly designed
to exhibit single or multiple locally-resonant bandgaps, Bragg-scattering
bandgaps can still arise, and the conditions leading to the transition of one
to the other have been deeply studied in literature [19]. As a result, various
studies have focused on combining Bragg and locally-resonant bandgaps and
derive the conditions leading to such phenomenon and their relationship to
natural frequencies of structures as well as Bragg-conditions [20, 21, 22, 23,
24]. Another recent endeavour in this domain has been focused on identifying
patterns in bandgap formation (periodic versus non-periodic bandgap
widths/location) and conditions leading to bandgap closure for phononic
crystals undergoing axial vibrations [25]. Yet, a framework that enables
infinite number of locally-resonant-only bandgaps (i.e., no Bragg-type
bandgaps) that are equally spaced and having identical patterns of bandgap and
propagation-zone widths remains lacking.

In this paper, two designs of active metamaterials are introduced, with the
capability of exhibiting infinite number of locally-resonant bandgaps that are
identical in width, equidistant, and having repeating pattern of local-
resonance frequencies. The unique characteristics of the proposed designs are
achieved based on an elastic continuum rod with a periodically placed grounded
elastic springs of frequency-dependent stiffness, designed specifically to
realized the desired functionality. To engineer the bandgap patterns and
dispersion behaviors, the dispersion relations are derived analytically using
the transfer matrix method (TMM). A single unit cell of the active
metamaterial is thoroughly studied, and the resulting bandgap limits and width
derived analytically are verified using the frequency response of the proposed
active EMMs with a finite number of unit cells.

##  2 Design of Active Elastic Metamaterials

###  2.1 Background: discrete elastic metamaterials and diatomic lattices

Figure 1: Schematics of a typical discrete diatomic lattice (left) and EMM (right), showing the inertial and stiffness properties for the self-repeating unit cell. Typical dispersion relations of both cases are shown in the figure for reference. Bragg and locally-resonant (LR) bandgaps are observed in the diatomic lattice and EMM, respectively. Table 1: Expressions for the variable $\alpha$ in Equation (1) and functions $\Phi_{n}$ and $\Phi_{d}$ of the driven-wave formulation in Equation (2) for both the diatomic lattice and EMM models. For an interpretation of the mass and stiffness parameters of both systems, see Figure 1. Parameter | Diatomic Lattice | Elastic Metamaterial
---|---|---
$\alpha$ | $(m_{1}+m_{2})(k_{1}+k_{2})$ | $(m_{1}+m_{2})k_{2}+2m_{2}k_{1}(1-\cos(q))$
$\Phi_{n}$ | $m_{1}m_{2}\omega^{4}-(m_{1}+m_{2})(k_{1}+k_{2})\omega^{2}+2k_{1}k_{2}$ | $m_{1}m_{2}\omega^{4}-\left((m_{1}+m_{2})k_{2}+2m_{2}k_{1}\right)\omega^{2}+2k_{1}k_{2}$
$\Phi_{d}$ | $2k_{1}k_{2}$ | $2k_{1}(k_{2}-m_{2}\omega^{2})$

To demonstrate the design methodology of the proposed active EMMs, it is
essential to first revisit the dispersion relations of the discrete model of
an EMM and a diatomic lattice (Figure 1 details the inertial/stiffness
properties for both models), which generally have the following dispersion
relation [5, 26]:

$m_{1}m_{2}\omega^{4}-\alpha\omega^{2}+2k_{1}k_{2}(1-\cos(q))=0$ (1)

where $q$ is the non-dimensional wavenumber, $\omega$ is the excitation
frequency, and the parameter $\alpha$ is defined in Table 1 for both models.
Re-arranging the dispersion relations into the driven-wave formulation, i.e.,
the input being the frequency $\omega$ and the output being the wavenumber
$q$, one can show that:

$q=\cos^{-1}\left(\frac{\Phi_{n}}{\Phi_{d}}\right)$ (2)

and the functions $\Phi_{n}$ and $\Phi_{d}$ for the diatomic lattice and the
discrete EMM are listed in Table 1. Interestingly, the $\Phi_{n}$ function
becomes identical for both models if the constitutive masses are identical,
i.e., $m_{1}=m_{2}$, with the difference being in the denominator function
$\Phi_{d}$ for EMM, which has a zero at the stand-alone natural frequency of
the local resonator. In extension, having identical masses ($m_{1}=m_{2}$) and
stiffnesses ($k_{1}=k_{2}$) also yield identical $\Phi_{n}$ function for both
EMM and diatomic lattice, although the diatomic-lattice’s dispersion becomes
that of the monatomic lattice with no Bragg-bandgap. However, EMM exhibits a
locally-resonant bandgap even if the inertial and stiffness properties are
equal, owing to the existence of a zero of the function $\Phi_{d}$ at the
local-resonator’s natural frequency, inducing a locally-resonant bandgap as a
result. The conclusion here is that it is possible to artificially introduce a
function $\Phi_{d}$ with zeros at desired frequency(ies) to induce locally-
resonant bandgaps around them with a function $\Phi_{n}$ for a monatomic
lattice (as in the discrete EMM model) or in a uniform elastic rod as will be
elaborated next.

Figure 2: (top) Schematic of an elastic rod with a unit-cell length of $\ell$
and an equally-spaced elastic foundation elements $k_{\text{d}}$. The
continuous elastic rod has a Young’s modulus and density of $E$ and $\rho$,
respectively. (bottom) Frequency response of the frequency-dependent stiffness
$k_{\text{d}}$ relative to the effective stiffness of the continuum rod
segment, $k$, at different values of the tuning parameter $\gamma$.

###  2.2 First design: identical, equally-spaced and symmetric bandgaps

Consider a continuum elastic rod, with a unit-cell length of $\ell$, a cross-
sectional area of $A$ and a sonic speed of $c=\sqrt{E/\rho}$, where $E$ and
$\rho$ are the Young’s modulus and density of the rod, respectively.
Considering the longitudinal waves, the dispersion relation is linear, and it
can be written in the following non-dimensional form:

$\Omega=q$ (3)

where $\Omega=\omega\ell/c$ is a normalized version of excitation frequency
$\omega$. Equation (3) can be re-cast into a similar formula to that in
Equation (2) and one can show that $\Phi_{d}=1$ and $\Phi_{n}=\cos(\Omega)$.
To enable periodic, identical, and equally spaced locally-resonant bandgaps, I
propose the formula $\Phi_{d}=\sin(\Omega)/\gamma$, and based on Equation (2),
gives the following driven-wave dispersion relation:

$q=\cos^{-1}\left(\frac{\gamma\cos(\Omega)}{\sin(\Omega)}\right)$ (4)

where $\gamma$ is a tuning parameter for the bandgap’s limits. Having the term
$\sin(\Omega)$ in the denominator $\Phi_{d}$ means that evenly spaced (and
infinite in number) local-resonance frequencies emerge at $\Omega=n\pi$, where
$n\in\mathrm{N}_{0}$ and $\mathrm{N}_{0}$ denotes all natural numbers
including zero. Equivalently, the free-wave dispersion branches can be derived
from Equation (4) and are given by:

$\Omega=n\pi+\tan^{-1}\left(\frac{\gamma}{\cos(q)}\right)$ (5)

From the solution of Equation (5) at $q=0,\pi$, all lower and upper bandgap
limits, denoted here as $\Omega_{l}$ and $\Omega_{u}$, respectively, can be
found using the following equations for all $n\geq 1$:

$\Omega_{l}=n\pi-\tan^{-1}(\gamma)$ (6a) $\Omega_{u}=n\pi+\tan^{-1}(\gamma)$
(6b)

As inferred from the structure of the bandgap limits formulae, they are of
equal distance of $\tan^{-1}(\gamma)$ from the local resonance frequencies
$n\pi$. As a result, the bandgap limits are perfectly periodic and infinite in
number, and their corresponding widths are identical and equal to:

$\Delta\Omega=2\tan^{-1}(\gamma)$ (7)

Note that the lower limit of the very first locally-resonant bandgap is equal
to $\Omega=0$ (referred to here as zero-frequency bandgap) and its upper limit
being $\Omega_{0}=\tan^{-1}(\gamma)$, signifying a lower cutoff frequency of
this first design of active EMM. Such zero-frequency bandgap has half the
width of the rest of bandgaps.

To achieve the dispersion relation in Equation (4), a grounded elastic element
of stiffness $k_{\text{d}}$ is attached to the rod at the middle of the unit
cell, as shown in Figure 2. The elastic element is engineered such that it
exhibits a frequency-dependent stiffness, enabling repeating local resonances
induced by the zeros of the function $\Phi_{d}$. Here, the transfer matrix
method (TMM) is utilized to find the function of the frequency-dependent
stiffness $k_{\text{d}}$. The purpose of a transfer matrix $\mathbf{T}$ is to
relate the state vector (displacement and force) from one end of a unit cell
to its peer in the neighboring cell:

$\mathbf{z}_{i+1}=\mathbf{T}\mathbf{z}_{i}$ (8)

where $\mathbf{z}_{i}$ is the unit-cell’s state vector at one end of the $i$th
unit cell. To establish the transfer matrix for the unit cell in Figure 2, the
transfer matrices pertaining to the uniform elastic rod segments and ground
springs are needed. It can be shown that the transfer matrix of a uniform rod
for half of the unit cell length is found using the following transfer matrix:

$\mathbf{Y}=\begin{bmatrix}\cos(\Omega/2)&\frac{1}{k\Omega}\sin(\Omega/2)\\\
-k\Omega\sin(\Omega/2)&\cos(\Omega/2)\end{bmatrix}$ (9)

which is presented here in terms of the non-dimensional frequency $\Omega$
with $k=EA/\ell$ being the effective stiffness of the rod segment. For a
grounded spring, on the other hand, the following transfer matrix is applied
to account for its effect, which, in this case, is identical in form to that
of a transfer matrix of a typical local resonator [22]:

$\mathbf{P}=\begin{bmatrix}1&0\\\ k_{\text{d}}(\Omega)&1\end{bmatrix}$ (10)

Thus, the transfer matrix of the entire unit cell is given by the following
multiplication of the individual transfer matrices defined earlier:

$\mathbf{T}=\mathbf{Y}\mathbf{P}\mathbf{Y}=\begin{bmatrix}\cos(\Omega)+\frac{k_{\text{d}}}{2k\Omega}\sin(\Omega)&\frac{1}{2\Omega^{2}k^{2}}\left(k_{\text{d}}\left(1-\cos(\Omega)\right)+2\Omega
k\sin(\Omega)\right)\\\
\frac{k_{\text{d}}}{2}\left(1+\cos(\Omega)\right)-k\Omega\sin(\Omega)&\cos(\Omega)+\frac{k_{\text{d}}}{2k\Omega}\sin(\Omega)\end{bmatrix}$
(11)

Knowing that the trace of the transfer matrix $\mathbf{T}$ is:

$\text{tr}(\mathbf{T})=2\cos(\Omega)+\frac{k_{\text{d}}}{k\Omega}\sin(\Omega)$
(12)

the relation $\cos(q)=\text{tr}(\mathbf{T})/2$ is utilized to derive the
function $k_{\text{d}}(\Omega)$, which, after making use of Equation (5),
yields:

$k_{\text{d}}(\Omega)=-\frac{2k\Omega}{\tan(\Omega)}\left(1-\frac{\gamma}{\sin(\Omega)}\right)$
(13)

###  2.3 Second design: no zero-frequency bandgap

Figure 3: Band structure of the two designs of active EMM with two values of
the tuning parameter, namely $\gamma=1$ and $\gamma=2$. All cases exhibit
identical locally-resonant bandgaps and dispersion branches, and the bandgap
width gets larger with an increase in $\gamma$. For the first design, and as
$\gamma$ increases, the bandgap expands symmetrically about the center of the
bandgap (i.e., the marked local-resonance frequencies on the right vertical
axis) for all bandgaps, regardless of the value of the parameter $\gamma$,
except for the zero-frequency bandgap that always starts at $\Omega=0$. On the
other hand, no zero-frequency bandgap emerge in the second design given that
the first locally-resonant bandgap is centered around the local-resonance
frequency $\Omega=3\pi/2$, and the behavior recurs every $2\pi$ in the
frequency spectrum. As in the first case, the parameter $\gamma$ controls the
bandgap width, but it is generally not symmetric about the local-resonance
frequencies. Note that the initial group velocity (slope of dispersion branch)
at $\Omega=0$ in the second design is zero, which is not typical for
longitudinal waves with no zero-frequency bandgaps. As a final note, the
occurrence of local-resonance frequencies is evident from the drastic increase
of the imaginary component of the wavenumber around them and their location on
the frequency spectrum is independent of the value of $\gamma$.

For the dynamic stiffness $k_{\text{d}}$ function in the first active EMM
design in Equation (13), there exists a bandgap that opens starting at zero
frequency as mentioned earlier. If such a bandgap is not desired, then a small
modification to the dispersion relation in Equation (4) eliminates such gap
and the following engineered dispersion relation is proposed:

$q=\cos^{-1}\left(1-\gamma+\frac{\gamma\cos(\Omega)}{1+\sin(\Omega)}\right)$
(14)

The function in the argument of cosine inverse in Equation (14) is chosen such
that a zero wavenumber ($q=0$) is returned when $\Omega=0$, which mandates
that the zero-frequency bandgap is closed. Note that the local-resonance
frequencies in this case occur at:

$\Omega=(4n-1)\frac{\pi}{2}$ (15)

for integer values of $n\geq 1$. This second design of EMM results in solely-
resonant bandgaps having lower and upper limits of (for $n\in\mathrm{N}_{0}$):

$\Omega_{l}=2\left(\pi n+\tan^{-1}\left(\frac{1}{\gamma-1}\right)\right)$
(16a) $\Omega_{u}=2(n+1)\pi$ (16b)

with an identical bandgap width of:

$\Delta\Omega=2\pi-2\tan^{-1}\left(\frac{1}{\gamma-1}\right)$ (17)

Following an identical procedure to that presented earlier for Equation (13),
the function for $k_{\text{d}}(\Omega)$ in this second design can be derived
and it is given by:

$k_{\text{d}}(\Omega)=\frac{2k\Omega}{\sin(\Omega)}\left(1-\gamma-\cos(\Omega)\left(1-\frac{\gamma}{1+\sin(\Omega)}\right)\right)$
(18)

For reference, the frequency response of the dynamic stiffnesses
$k_{\text{d}}$ for both designs, i.e., Equations (13) and (18) are depicted in
Figure 2 at different values of $\gamma$, demonstrating its frequency
dependence and tunability with $\gamma$.

Figure 4: (a) Bandgap limits and (b) corresponding bandgap width for EMM first
and second designs as a function of $\gamma$. Shaded regions in sub-figure (a)
represent the width of the bandgap, which is quantitatively depicted in sub-
figure (b).

###  2.4 Results and discussion

The band structures for both active EMM designs are depicted in Figure 3 for
two values of $\gamma$, namely $\gamma=1$ and $\gamma=2$. As predicted earlier
for the first EMM design, the first locally-resonant bandgap starts at a zero
frequency and all bandgaps can be manipulated by changing the parameter
$\gamma$. It is interesting to observe that all bandgaps grow symmetrically
about the center of the bandgap as $\gamma$ increases, and the opposite is
true for a decreasing value of $\gamma$ (except for the zero-frequency
bandgap). The same can be said for the second design (i.e., the dispersion in
Equation (14)), yet the bandgap limits are not generally symmetric about the
local-resonance frequency and the bandgap grow/shrink only from changing the
lower limit. Peculiarly, the dispersion diagram of the second design exhibits
a zero group velocity at the long-wave limit (i.e., at $\Omega=0$), which is
not typical for longitudinal waves dispersion relations with no zero-frequency
bandgaps. A final difference between the band structures of the two designs is
that the second one exhibit local-resonance frequencies separated by $2\pi$,
starting at $\Omega=3\pi/2$, unlike the first design, which has resonances
separated by a range of $\pi$, starting from $\Omega=0$. This is clearly seen
from the imaginary component of the wavenumber, where the local-resonance
frequency corresponds to an imaginary component of the wavenumber going to
infinity.

The observations regarding bandgap limits made earlier can be further
confirmed by plotting them and their corresponding bandgap width as a function
of $\gamma$ (Figure 4). For the first design, it is observed that the bandgap
limits grow apart with increasing $\gamma$, occurring symmetrically around the
resonant frequencies $\Omega=n\pi$ for $n\geq 1$, as seen in Figure 4(a). The
exception here is the zero-frequency bandgap, which has a limit at $\Omega=0$,
confirming our observations from Figure 3. Notice that these bandgap limits
converge to odd multiples of $\pi/2$ (to multiples of $\pi$) as
$\gamma\rightarrow\infty$ ($\gamma\rightarrow 0$). As in the first design, the
bandgap widths in the second design increase with increasing $\gamma$, yet by
pushing the lower limit down as $\gamma$ increases with the upper limit
remaining constant at multiples of $2\pi$ (Figure 4(a)). Calculating the
bandgaps’ width as a function of $\gamma$, as seen in Figure 4(b), reveals
that such bandgaps’ width in the first (second) EMM design converge to $\pi$
($2\pi$) as $\gamma\rightarrow\infty$ and it diminishes (goes to $\pi/2$) if
$\gamma\rightarrow 0$, with the exception of the zero-frequency bandgap in the
first design that varies between 0 to $\pi/2$ for values of $\gamma\rightarrow
0$ and $\gamma\rightarrow\infty$. Finally, the bandgaps in the second design
remain open regardless the value of $\gamma$, which is not the case for the
first design as the bandgaps vanish at the limit of $\gamma\rightarrow 0$.

The predictions of the dispersion relations are further verified using the
frequency response of a finite structure of twenty unit cells ($\gamma=1,2$
are used as examples) for both EMM designs. The structures are base-excited at
one end, and the amplitude of the displacement near the opposite fixed end is
measured. The numerical computation is carried our using MATLAB and the
frequency response function is computed using the finite element method with
two-node linear rod elements [27]. The bandgaps are characterized in the
frequency response function as regions of drastic decrease in displacement
amplitude. As can be seen in Figure 5, it is evident that the first EMM design
has symmetric bandgaps around the resonant frequencies $\Omega=n\pi$ (for
integer $n\geq 1$), and that the zero-frequency bandgap is half the width of
the rest of them with the occurrence of a zero-frequency anti-resonance.
Analogously, the frequency response function for the second EMM design evinces
that anti-resonances occur according to Equation (15), yielding identical
bandgaps, yet generally unsymmetrical about the anti-resonance frequencies.
Finally, the lack of zero-frequency bandgap is also evident in the second
design.

##  3 Concluding remarks

Two active elastic metamaterial (EMM) designs are proposed, which exhibit
infinite number of locally-resonant bandgaps that are evenly spaced and
identical in width (with the exception of zero-frequency bandgaps in the first
design). Both designs are achieved via a grounded elastic element with a
frequency-dependent stiffness. The function describing the stiffness element
is carefully designed based on an engineered dispersion relation with infinite
number of local-resonance frequencies, with the aid of the transfer matrix
method (TMM). The dispersion analyses and bandgap predictions are verified
using a finite-element-based frequency response function of a finite array of
the active EMMs, showing excellent agreement. The proposed design methodology
is envision to be further used to synthesize interesting dispersion behaviors
incorporating locally-resonant bandgaps in the future.

Figure 5: Frequency response functions for both designs of active EMMs with 20
unit cells and $\gamma=1,2$. The rod is base-excited at one end and the
displacement amplitude is measured near the opposite fixed end. For the first
design, the bandgaps are symmetric around the resonant frequencies
$\Omega=n\pi$ for an integer $n\geq 1$, which are evident from the drastic
drop in displacement amplitude. The zero frequency bandgap is also observed,
which is half the width of the rest of them. On the other hand, no zero-
frequency bandgap is observed in the second design and the locally-resonant
bandgaps occur around the resonant frequencies according to Equation (15),
which, once again, are evident from the drastic drop in displacement
amplitude.

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# Dwarf AGNs from Variability for the Origins of Seeds (DAVOS): Optical
Variability of Broad-line Dwarf AGNs from the Zwicky Transient Facility

Z. Franklin Wang,1,2,3 Colin J. Burke,1 Xin Liu,1,4 Yue Shen1,4
1Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
West Green Street, Urbana, IL 61801, USA
2Department of Physics and Astronomy, Texas A&M University, College Station,
TX 77843-4242, USA
3George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, College Station, TX 77843-4242, USA
4National Center for Supercomputing Applications, 1205 West Clark Street,
Urbana, IL 61801, USA
E-mail<EMAIL_ADDRESS>(ZFW)E-mail<EMAIL_ADDRESS>(CJB)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We study the optical variability of a sample of candidate low-mass (dwarf ang
Seyfert) active galactic nuclei (AGNs) using Zwicky Transient Facility _g_
-band light curves. Our sample is compiled from broad-line AGNs in dwarf
galaxies reported in the literature with single-epoch virial black hole (BH)
masses in the range $M_{\rm{BH}}\sim 10^{4}$–$10^{8}\ M_{\odot}$. We measure
the characteristic “damping” timescale of the optical variability
$\tau_{\rm{DRW}}$, beyond which the power spectral density flattens, of a
final sample of 79 candidate low-mass AGNs with high-quality light curves. Our
results provide further confirmation of the $M_{\rm{BH}}-\tau_{\rm{DRW}}$
relation from Burke et al. (2022) within $1\sigma$ agreement, adding 78 new
low-mass AGNs to the relation. The agreement suggests that the virial BH mass
estimates for these AGNs are generally reasonable. We expect that the optical
light curve of an accreting intermediate-mass black hole (IMBH) to vary with a
rest-frame damping timescale of $\sim$ tens of hours, which could enable
detection and direct mass estimation of accreting IMBHs in wide-field time-
domain imaging surveys with sufficient cadence like with the Vera C. Rubin
Observatory.

###### keywords:

black hole physics, accretion, accretion discs, galaxies: dwarf, galaxies:
active

††pubyear: 2021††pagerange: Dwarf AGNs from Variability for the Origins of
Seeds (DAVOS): Optical Variability of Broad-line Dwarf AGNs from the Zwicky
Transient Facility–A

##  1 Introduction

Figure 1: Distributions of the virial BH mass estimates (_top left_), ZTF _g_
-band PSF magnitude (_top right_), spectroscopic redshift (_bottom left_), and
_g_ -band Petrosian radius (_bottom right_) for our parent sample of low-mass
AGNs selected from Greene & Ho (2007) ("GH07"), Reines & Volonteri (2015)
("RV15"), Chilingarian et al. (2018) ("Chinlingarian+18"), and Liu et al.
(2018) ("Liu+18").

Intermediate-mass black holes (IMBHs; $M_{\rm{BH}}=10^{2}-10^{5}\ M_{\odot}$)
are thought to be the missing link between stellar-mass and supermassive black
holes (SMBHs). The demographics of active galactic nuclei (AGN) in dwarf
galaxies hosting IMBHs traces the formation mechanisms of SMBH seeds which are
believed to have formed in nascent galaxies at high redshift (e.g., Volonteri,
2010; Reines & Comastri, 2016; Greene et al., 2020; Inayoshi et al., 2020).
Perhaps the best observational evidence for IMBHs is the gravitational wave
event GW190521 from the merger of two black holes (with a remnant mass of
$142^{+28}_{-16}$ $M_{\odot}$; LIGO Scientific Collaboration & Virgo
Collaboration 2020). Accreting IMBHs remain difficult to identify, with only a
few exceptional candidates such as the X-ray tidal disruption event 3XMM
J215022.4-055108 (Lin et al., 2018); and the $M_{\rm{BH}}\sim 10^{4}\
M_{\odot}$ hyper-luminous X-ray source ESO 243-49 HLX-1 (Farrell et al.,
2009).

Recent works demonstrated success in identifying dwarf AGNs with BH masses on
the boundary between IMBHs and SMBHs with $M_{\rm{BH}}\sim 10^{4}-10^{6}\
M_{\odot}$ using optical spectroscopic signatures (e.g., Greene & Ho 2005;
Reines et al. 2013; Baldassare et al. 2015; Baldassare et al. 2016; Cann et
al. 2020) or variability (e.g., Baldassare et al. 2018, 2020; Martínez-
Palomera et al. 2020; Guo et al. 2020; Ward et al. 2022; Burke et al. 2022).
We have compiled a parent sample of dwarf AGN candidates in the literature
with optical signatures of accretion activity (variability, narrow emission
line diagnostics, and/or broad emission lines). Then, we restricted our sample
to broad-line AGNs and those with virial BH mass estimates. These virial BH
masses are in the range $M_{\rm{BH}}\sim 10^{4}-10^{6}\ M_{\odot}$ (Greene &
Ho, 2007; Chilingarian et al., 2018; Liu et al., 2018). We add the sample of
low-mass (dwarf and Seyfert) AGNs with $M_{\rm{BH}}\sim 10^{5}-10^{8}\
M_{\odot}$ from Reines & Volonteri 2015 used to calibrate their local BH -
host galaxy stellar mass relation, because we will attempt to benchmark our
variability timescale -derived BH masses against the Reines & Volonteri 2015
relation. These low-mass AGNs offer the best cases currently to test our
understanding of accretion physics close to the IMBH regime.

Optical variability in AGN light curves can be used to probe accretion disk
structure and the dominant physical processes at the radius of optical
emission. The power spectral density (PSD) of optical AGN light curves are
well-described by a damped random walk (DRW) model from hours to years
timescales, where the PSD transitions from white noise to red noise ($f^{-2}$)
at a characteristic frequency ($f_{\rm DRW}=(2\pi\tau_{\rm DRW})^{-1}$) (e.g.,
Kelly et al., 2009; MacLeod et al., 2010). The corresponding rest-frame
“damping” timescale $\tau_{\rm{DRW}}$ correlates strongly with $M_{\rm{BH}}$
and is likely associated with the thermal timescale in the accretion disk at
the UV-emitting radius (Burke et al., 2021). Ward et al. (2022) found evidence
that their sample of variability-selected dwarf AGNs have more rapid
variability compared to a control sample of massive AGNs using ZTF light
curves.

The $M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation of Burke et al. (2021) predicts a
$\tau_{\rm{DRW}}$ of tens of days for a $10^{6}\ M_{\odot}$ BH as opposed to
several hundred days for a $10^{9}\ M_{\odot}$ SMBH. Therefore, the
variability timescales for dwarf AGNs are well-sampled by Zwicky Transient
Facility (ZTF; Masci et al. 2019; Bellm et al. 2019) light curves with a
typical baseline and cadence of $\sim 1000$ days and $\sim 3$ days
respectively. We will test the $M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation in the
dwarf AGN regime, motivated by the possibility of extrapolating the relation
to even lower masses to enable not only IMBH discovery, but also host-
independent BH mass estimation, which may not be feasible with traditional
spectroscopy-based methods due to the lack of broad emission lines
(Chakravorty et al., 2014).

In this work, we report our time series analysis of ZTF light curves of 79
low-mass AGNs with virial BH mass estimates of $10^{5.69}\leq
M_{\rm{BH}}(M_{\odot})\leq 10^{7.80}$. We focus on spectroscopically-confirmed
broad-line dwarf AGNs. This includes 78 new objects which we place on the
$M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation compared to the previous Burke et al.
(2021) sample with $\sim 4$ years long light curves, sufficient to constrain
the damping timescale for dwarf AGNs. These objects were not included in the
Burke et al. (2021) sample, which only considered objects from Reines et al.
(2013) and used a previous ZTF data release with shorter light curve lengths
of $\sim 2$ years. We describe the AGN parent sample and data in §2, present
our results in §3, discuss the possible implications in §4, and conclude in
§5. All uncertainties and plotted error bars are $1\sigma$ unless otherwise
specified.

##  2 Observations and Data Analysis

###  2.1 Dwarf AGN Parent Sample

We searched the literature for dwarf AGN candidates with broad-line
measurements that enabled single-epoch virial BH mass estimates (Shen, 2013).
Our parent sample of low-mass AGNs come from the following sources: Greene &
Ho (2007) ("GH07"), Reines & Volonteri (2015) ("RV15"), Chilingarian et al.
(2018) ("Chinlingarian+18"), and Liu et al. (2018) ("Liu+18"). These AGNs were
identified using one of two methods. First, dwarf galaxies were identified
with narrow emission line ratios consistent with an AGN based on the classical
BPT diagram (Baldwin et al., 1981; Veilleux & Osterbrock, 1987; Kewley et al.,
2006) with broad emission lines detected afterwards. Second, AGNs were
selected via detection of the broad emission line itself from Sloan Digital
Sky Survey (SDSS) spectra (York et al., 2000; Abazajian et al., 2009) even if
the line ratios are consistent with star-formation. In the later case, the
broad emission lines can be due to a supernova or stellar transient, requiring
follow-up observations years later (Baldassare et al., 2016). However, long-
duration variability can also be used to test the AGN scenario. In this work,
we only included sources with published BH masses from broad emission line
detections with persistent, AGN-like variability. We show the virial BH mass
estimate, $g$-band PSF magnitude, spectroscopic redshift, and host-galaxy _g_
-band Petrosian radii (Petrosian, 1976; Graham et al., 2005) distributions for
the parent sample sources in Figure 1. We checked for duplicate sources
between the samples, and used the $M_{\rm{BH}}$ value from the more recent
publication in those cases.

Figure 2: Example DRW modeling of SDSS J090320.97+045738.0 with
$M_{\rm{BH}}=10^{6.1\,\pm\,0.3}M_{\odot}$ and $z=0.0567$ (Greene & Ho, 2007).
We show the $g$-band difference imaging light curve and the best-fit DRW model
with 1$\sigma$ uncertainty (orange shaded area) (_top panel_). We show the
posterior probability distributions for the fitted DRW parameters and their
covariance (Equation 1). In the covariance panels, the contours trace the 1,
2, 3$\sigma$ levels overplotted on the sample density map (black being higher
density) with individual samples in the lowest-density regions shown as black
points (_bottom left panel_). Finally, we show the normalized Lomb-Scargle PSD
in gray and the 1$\sigma$ range of the DRW PSD from the posterior prediction
as the orange shaded area (_bottom right panel_). Note the Lomb-Scargle PSD
includes a white noise floor at high frequency that is not included in the DRW
model. The red shaded regions correspond to periods greater than 20% the light
curve length and less than the mean cadence, where the PSD is not well
sampled.

###  2.2 Light Curve Data and Cleaning

ZTF is a time-domain optical (primarily _g_ and _r_ with some _i_ -band)
imaging survey using the Samuel Oschin 48-inch Schmidt telescope at Palomar
Observatory. ZTF light curves began in 2018 with a nominal cadence of 2 or 3
days (with seasonal gaps), covering declinations above $-30$ degrees with a
single-epoch imaging depth of 20.5 mag ($5\sigma$, _r_ band). The ZTF data
release light curves are PSF-based, and will introduce additional measurement
error for resolved sources given the median $g$-band PSF full width at half
maximum (FWHM) of $2\aas@@fstack{\prime\prime}1$ (Bellm et al., 2019). For
this reason, we use difference imaging-based photometry as described below.

We downloaded difference images from the ZTF data release 13 (March 2018 $-$
July 2022) in the _g_ band from the NASA/IPAC Infrared Science
Archive111https://irsa.ipac.caltech.edu/Missions/ztf.html which had coverage
of any one of our sample targets. After collecting the images, we extracted
photometry using a $2\aas@@fstack{\prime\prime}5$ diameter aperture, which is
well-matched to the ZTF seeing, using the Photutils code (Bradley et al.,
2022). After constructing the light curves, we performed outlier rejection at
the $3\sigma$ level using a moving window size of 1 year. Given AGNs tend to
be more variable in at bluer wavelengths, we restrict our analysis to the _g_
band. This effect may be even stronger for dwarf AGNs, as the disk temperature
of an IMBH shifts the SED into the UV (e.g., Cann et al. 2018).

###  2.3 Light Curve Modeling

It has become increasingly popular to model AGN light curves in the time
domain (as opposed to the frequency domain) with a Continuous Auto-Regressive
Moving Average (CARMA) model (e.g., Kelly et al., 2009, 2014; Simm et al.,
2016; Suberlak et al., 2021). The CARMA models are a set of flexible Gaussian
process models (Aigrain & Foreman-Mackey, 2022) to describe stochastic time
series, with two parameters, $p$ and $q$, describing the orders of the auto-
regression (AR) part and the moving average (MA) part. The DRW model is the
lowest order CARMA model with $p=1$ and $q=0$, or a CAR(1) process. In this
work we use the DRW model to constrain the characteristic damping timescale of
an AGN light curve. Deviations from the simple DRW model for SMBHs have been
reported on short timescales with space- or ground-based photometry (e.g.,
Mushotzky et al., 2011; Kasliwal et al., 2015; Stone et al., 2022). To check
that the DRW model is sufficient for our data, we visually confirmed that the
ACF of the $\chi^{2}$ (model$-$data) residuals are consistent with Gaussian
white noise using Bartlett’s formula. Given that the best-fitting DRW models
have sufficiently small residuals, higher-order CARMA models are not justified
given the data quality. Burke et al. (2021) provided a more detailed
discussion on the advantages and caveats of the DRW model compared with the
more complicated higher-order CARMA models.

The major advantage of the Gaussian process models is that they can provide an
accurate description of irregularly-sampled stochastic light curves and the
underlying PSD while taking into account measurement uncertainties. Because
the modeling is performed in the time domain rather than in the frequency
domain, the inferred PSD is more robust against windowing effects (such as
aliasing and red noise leakage) than traditional power spectral analysis.
Fitting a broken power law to the PSD of light curves can produce consistent
results with time domain DRW fitting, albeit with larger intrinsic scatter and
greater demands on the light curve sampling and quality (Burke et al., 2021).
We compute the PSD using the Lomb$-$Scargle periodogram (Lomb, 1976; Scargle,
1982) as a consistency check. A typical example of our time series analysis
and DRW fitting is shown in Figure 2. The light curves and their best-fit DRW
models of our final sample are shown in Appendix A.

The covariance function for a DRW plus extra white noise model is,

$k(t_{nm})=2\sigma_{\rm{DRW}}^{2}\exp{(-t_{nm}/\tau_{\rm{DRW}})}+\sigma_{n}^{2}\delta_{nm}\
$ (1)

where $t_{nm}=|t_{n}-t_{m}|$ is the time lag between measurements $m$ and $n$,
$\sigma_{\rm{DRW}}$ is the amplitude term, $\tau_{\rm{DRW}}$ is the damping
timescale, $\sigma_{\rm{n}}$ is the white noise amplitude, and $\delta_{nm}$
is the Kronecker $\delta$ function. We include the white noise term to account
for excess measurement noise in the ZTF light curves due to e.g., seeing
variations or under-estimated uncertainties from our photometric extraction
procedure, which does not include systematic sources of uncertainty in
addition to the statistical uncertainties. This model has an exact
correspondence to the structure function (SF) definition,

$\displaystyle\begin{split}{\rm{SF}}^{2}&=\rm{SF}_{\infty}^{2}(1-{\rm{ACF}}(t_{nm}))\\\
&=2\sigma_{\rm{DRW}}^{2}(1-\exp{(-t_{nm}/\tau_{\rm{DRW}}}))\ \end{split}$ (2)

where the asymptotic variability amplitude
$\rm{SF}_{\infty}=\sqrt{2}\sigma_{\rm{DRW}}$ and the autocorrelation function
${\rm{ACF}}(t_{nm})=\exp{(-t_{nm}/\tau_{\rm{DRW}}})$. Therefore, we have three
free parameters to fit in the model: $\sigma_{\rm{DRW}}$, $\tau_{\rm{DRW}}$,
and $\sigma_{\rm{n}}$. We use the celerite package (Foreman-Mackey et al.,
2017) to perform the model fitting. We use Markov chain Monte Carlo (MCMC)
implemented in the emcee package (Foreman-Mackey et al., 2013) to sample the
joint posterior probability density, with uniform priors for all parameters.
We take the 16th and 84th percentiles of the marginalized posterior
distributions for each parameter to estimate the $1\sigma$ uncertainties.

###  2.4 Recoverability of Input Damping Timescale

Figure 3: Input versus recovered damping timescale for simulated realistic DRW
light curves. _Left panel_ : Recoverability using the input $\tau_{\rm{DRW}}$
from our parent sample with no quality constraints (grey points), with
constrains (ii) ($\tau_{\rm{DRW}}>{\rm{cadence}}$) and (iv)
($\sigma_{\rm{LB}}>3$) (red points), and with constraints (i) ($\rm{SNR}>1$),
(ii), and (iii) (blue circles). _Center panel_ : Recoverability using input
$\tau_{\rm{DRW}}$ of $10$ times the result from our parent sample after
applying constraints (i), (ii), and (iii) on the original light curves. The
recovered $\tau_{\rm{DRW}}$ are biased and saturate at values near $20-30\%$
the light curve length when the input $\tau_{\rm{DRW}}$ is greater than about
$10\%$ the light curve length. _Right panel_ : Recoverability using input
$\tau_{\rm{DRW}}$ of $0.1$ times the result from our parent sample after
applying constraints (i), (ii), and (iii) on the original light curves. Light
curves with input $\tau_{\rm{DRW}}$ smaller than the mean cadence typically
have unreliable recovered $\tau_{\rm{DRW}}$. No duration constraints were
applied in our simulations to demonstrate the bias for light curves of
insufficient duration (Kozłowski, 2017). The pair of (vertical or horizontal)
gray lines corresponds to the cadence and $0.1\times$ the light curve length
of a typical ZTF light curve in our sample. The dashed gray line shows the
$y=x$ line.

Following Burke et al. (2021), we impose the following minimal quality
constraints on our light curves for our final sample:

  1. 1.

Length: $\tau_{\rm{DRW}}<0.1\times{\rm{baseline}}$.

  2. 2.

Sampling: $\tau_{\rm{DRW}}>6\ {\rm{days}}$.

  3. 3.

Signal-to-noise ratio (SNR): ${\rm{SNR}}>1$, where
${\rm{SNR}}=\sigma_{\rm{DRW}}/\sqrt{\sigma_{\rm{n}}^{2}+\overline{dy}^{2}}$
where $\overline{dy}$ is the mean uncertainty of the light curve measurements.

  4. 4.

AGN-like variability: $\sigma_{\rm{LB}}>3$, where $\sigma_{\rm{LB}}$ is the
Ljung$-$Box test (Ljung & Box, 1978) significance under the null hypothesis
that the light curve is white noise.

Constraint (i) is required to avoid a bias in the value of $\tau_{\rm{DRW}}$
if the light curve is of insufficient length. If the light curve is too short,
the most likely value of the recovered $\tau_{\rm{DRW}}$ is $20-30\%$ the
light curve length with a large scatter (Kozłowski, 2017). Constraint (ii) is
required so that the measured damping timescale does not artificially converge
near the cadence. Given the uncertainty of a factor of $0.3$ dex, we require
$\tau_{\rm{DRW}}$ to be a factor of $\sim 2$ greater than the typical ZTF
cadence of 3 days (i.e., $\tau_{\rm{DRW}}\lesssim 6$ days could be consistent
with fitting noise). Constraint (iii) ensures the fitted model amplitude is
reasonable compared to the light curve measurement errors and the DRW SNR is
sufficient for a reliable fit. Constraints (iii) and (iv) are both required,
because celerite will attempt to fit a DRW to a light curve even if it is
dominated by measurement noise, resulting in ${\rm{SNR}}>1$. Both these
constraints assure the DRW is properly fitted to a light curve with AGN-like
(correlated) variability.

To justify each of the constraints above, we perform DRW simulations to test
the recoverability of input $\tau_{\rm{DRW}}$. For each of the light curves in
our _g_ -band parent sample, we create a simulated DRW light with the same
sampling/cadence, length, and noise as the real data. For the first test, we
use the same amplitude $\sigma_{\rm{DRW}}$ and damping timescale
$\tau_{\rm{DRW}}$ as the real light curve. Then, we performed a set of
simulations in the same fashion with input $\tau_{\rm{DRW}}$ $10$ times and
$0.1$ times the values from the real light curves. Figure 3 shows the results
of these simulations. When all quality and duration constraints are applied,
the recovered $\tau_{\rm{DRW}}$ are shown to be reliable (blue circles in the
central box in the left panel of Figure 3).

In our final sample of 79 low-mass AGN candidates that pass both our initial
criteria (compactness and variability) as well as the DRW model fitting
quality checks above (criteria i-iii), we found one object in common with the
final Burke et al. (2021) sample. The low-mass end of the Burke et al. (2021)
relation is dominated by a few dwarf AGNs from Reines et al. (2013) with
shorter light curve durations from a previous ZTF data release and a light
curve of NGC 4395 from the Transiting Exoplanet Survey Satellite (Burke et
al., 2020). Of the five ZTF sources in the final sample from Burke et al.
(2021), we excluded four from this work for the following reasons: Mrk 1044
(SNR < 1.0), NGC 4395 (insufficient cadence/sampling), RGG 127 (SNR < 1.0),
and RGG 123 (SNR < 1.0). This does not indicate the Burke et al. (2021)
measurements for these sources were incorrect, because in this work we
enforced stricter quality controls starting from a larger sample of ZTF light
curves. Compared to the previous work, more of our ZTF light curves would pass
the length requirement given the longer ZTF light curves from the more recent
data release. However, we found that many of those light curves which were too
short for inclusion in the final Burke et al. (2021) sample are noisy from
e.g., variations in weather across the light curve. Given the much worse
average quality of parent AGN sample of ZTF data release light curves, we
required stricter quality cuts in this work at a cost of sample completeness.
Indeed, visual inspection of the model fitting results for these four sources
indicates that the fits may be reasonable and their damping timescale are
within $1\sigma$ agreement. A summary table of our DRW fitting results for the
final sample is given in Table 1.

##  3 Results

Figure 4: Rest-frame damping timescale versus black hole mass for our final
sample (red symbols) with $1\sigma$ uncertainty measured using _g_ -band ZTF
light curves. We also show the data (blue square symbols) and best-fit
relation (blue line) with $1\sigma$ uncertainty (light blue band) from Burke
et al. (2021). Our best-fit relation is consistent with the relation of Burke
et al. (2021) within $1\sigma$.

We perform linear regression between $\tau_{\rm DRW}$ (rest-frame) and
$M_{\rm{BH}}$ incorporating measurement uncertainties of both quantities. We
do not correct for differences between observed wavelength at different
redshifts, because the observed scaling of $\tau_{\rm
DRW}\propto\lambda^{0.17}$ (MacLeod et al., 2010; Suberlak et al., 2021; Stone
et al., 2022) is generally smaller than the measurement uncertainties given
the moderate redshift range of our AGNs. These measurement uncertainties are
slightly asymmetric in general, therefore we symmetrize them by taking the
mean. We use a hierarchical Bayesian model for fitting a line to data with
measurement uncertainties (Kelly, 2007). The best-fit relation and $1\sigma$
uncertainties for our parent sample of 79 low-mass AGNs follow the simple
linear relation,

$\tau_{\rm{DRW}}=\alpha\ {\rm{days}}\ \left(\frac{M_{\rm{BH}}}{10^{6}\
M_{\odot}}\right)^{\beta}\ $ (3)

where $\alpha=22.19^{+2.66}_{-1.68}$, $\beta=0.31^{+0.03}_{-0.03}$ with a
$1\sigma$ intrinsic scatter of $0.20^{+0.03}_{-0.03}$ dex in
$\tau_{\rm{DRW}}$. The data and best-fit relation are shown in Figure 4. Our
best-fitting linear model is consistent with the Burke et al. (2021) results
within $1\sigma$. However, the dynamic range in BH mass for the sample of
dwarf AGNs is small, which results in a much shallower slope with larger
uncertainties than the best-fitting relation in Burke et al. (2021). Given the
small dynamic range in mass of dwarf AGNs, it is not surprising that a
correlation is not statistically significant using our sample alone.

###  3.1 Extension to Stellar Mass Accretion Disks

Figure 5: Rest-frame damping timescale versus black hole mass for our final
_g_ -band ZTF sample combined with the final sample from Burke et al. (2021)
(black symbols). We show the best-fit relation (black line) with $1\sigma$
uncertainty (gray band) measured using the combined sample data with duplicate
sources removed (see text). We also show the data and best-fit relation (blue
line) with $1\sigma$ uncertainty (light blue band) from Burke et al. (2021).
For comparison, we show the values for accreting white dwarfs (Scaringi et
al., 2015), scaled to the typical AGN Eddington luminosity ratio of $0.15$, in
magenta. The typical uncertainties on $M_{\rm WD}$ and the white dwarf damping
timescale are 0.2 dex and 0.01 days respectively (Scaringi et al., 2015).

Next, we re-fit the combined final sample of _g_ -band ZTF dwarf AGNs with the
final sample from Burke et al. (2021). The results are shown in Figure 5. We
removed the one _g_ -band duplicate source from the Burke et al. (2021) sample
which was included in our sample with longer ZTF light curves and updated BH
masses from the relation of Reines & Volonteri (2015). For comparison to
stellar mass accretion disks, we also show accreting white dwarfs of the nova-
like class from Scaringi et al. (2015). However, we do not include the white
dwarfs in the fitting. Although our combined sample results appear consistent
with previous work, we caution that the ZTF light curves are generally of
poorer quality and the virial BH masses of dwarf AGNs may have greater
uncertainties, results in a larger scatter for these dwarf AGNs. Nevertheless,
this provides an important confirmation to the Burke et al. (2021) results
after better populating the AGN mass range of $M_{\rm{BH}}\sim 10^{6}-10^{8}\
M_{\odot}$.

Table 1: Final sample parameters and variability timing results. We list the J2000 right ascension (RA), declination (DEC), spectroscopic SDSS redshift ($z$), black hole mass, fitted damping timescale and amplitude, and light curve baseline, Ljung–Box test significance ($\sigma_{\rm{LB}}$), and signal to noise ratio (SNR) as defined in §2.4 in the text. $\sigma_{\rm{LB}}$ values of “inf” indicates a high significance beyond the numerical precision limits of our code. The reference refers to the source of the broad-line BH mass estimate we adopted. Logarithms are 10 based. Used Name | RA | DEC | z | $\log(M_{\rm{BH}}/M_{\odot})$ | $\log(\tau_{\rm{DRW}}/{\rm{days}})$ | $\log(\sigma_{\rm{DRW}}/{\rm{mag}})$ | baseline | $\sigma_{\rm{LB}}$ | SNR | reference
---|---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) |  | (dex) | (dex) | (dex) | (days) |  |  |
Liu+18 5 | 122.29468 | 11.10531 | 0.0526 | 5.95 ± 0.3 | 1.2 ± 0.1 | -1.5 ± 0.4 | 1505 | 17.0 | 1.4 | Liu et al. (2018)
Liu+18 33 | 143.53584 | 17.94556 | 0.0271 | 5.69 ± 0.3 | 1.6 ± 0.2 | -1.4 ± 0.4 | 1512 | 26.3 | 1.5 | Liu et al. (2018)
Liu+18 34 | 143.77839 | 14.97909 | 0.0831 | 6.04 ± 0.3 | 1.9 ± 0.3 | -1.6 ± 0.5 | 1512 | 20.1 | 1.4 | Liu et al. (2018)
Liu+18 100 | 188.65214 | 16.53123 | 0.0736 | 6.14 ± 0.3 | 1.8 ± 0.2 | -1.2 ± 0.4 | 1553 | inf | 2.9 | Liu et al. (2018)
Liu+18 116 | 196.20814 | 20.64553 | 0.0644 | 6.17 ± 0.3 | 1.8 ± 0.3 | -1.5 ± 0.5 | 1561 | 33.6 | 1.6 | Liu et al. (2018)
Liu+18 117 | 196.23734 | 39.92492 | 0.0274 | 5.90 ± 0.3 | 1.3 ± 0.1 | -1.7 ± 0.4 | 1563 | inf | 1.1 | Liu et al. (2018)
Liu+18 125 | 200.32909 | 30.07328 | 0.0916 | 6.28 ± 0.3 | 1.8 ± 0.2 | -1.4 ± 0.4 | 1564 | inf | 1.5 | Liu et al. (2018)
Liu+18 126 | 201.16139 | 29.17003 | 0.0727 | 6.27 ± 0.3 | 1.7 ± 0.2 | -1.4 ± 0.5 | 1563 | inf | 1.6 | Liu et al. (2018)
Liu+18 157 | 218.54809 | 16.84901 | 0.0936 | 6.20 ± 0.3 | 1.8 ± 0.3 | -1.5 ± 0.5 | 1564 | inf | 1.3 | Liu et al. (2018)
Liu+18 173 | 226.26005 | 13.03378 | 0.2045 | 6.19 ± 0.3 | 2.1 ± 0.4 | -1.7 ± 0.5 | 1562 | 31.5 | 1.1 | Liu et al. (2018)
Liu+18 186 | 238.10368 | 4.29062 | 0.0459 | 6.27 ± 0.3 | 2.0 ± 0.3 | -1.5 ± 0.5 | 1563 | 31.8 | 1.2 | Liu et al. (2018)
Liu+18 199 | 248.75609 | 30.90337 | 0.0543 | 6.27 ± 0.3 | 1.7 ± 0.2 | -1.5 ± 0.4 | 1562 | inf | 2.5 | Liu et al. (2018)
GH07 60 | 132.96933 | 52.47582 | 0.0644 | 6.20 ± 0.3 | 1.6 ± 0.2 | -1.8 ± 0.4 | 1515 | inf | 1.1 | Greene & Ho (2007)
GH07 63 | 135.83736 | 4.96059 | 0.0567 | 6.10 ± 0.3 | 1.7 ± 0.2 | -1.4 ± 0.5 | 1512 | 19.5 | 1.9 | Greene & Ho (2007)
GH07 105 | 165.74471 | 46.63652 | 0.1490 | 6.10 ± 0.3 | 1.9 ± 0.2 | -1.7 ± 0.5 | 1538 | 37.7 | 1.3 | Greene & Ho (2007)
GH07 124 | 174.15992 | 42.76473 | 0.0710 | 6.00 ± 0.3 | 1.2 ± 0.1 | -1.7 ± 0.4 | 1540 | inf | 1.1 | Greene & Ho (2007)
GH07 160 | 199.86049 | 10.93638 | 0.0644 | 5.90 ± 0.3 | 1.8 ± 0.2 | -1.6 ± 0.5 | 1563 | 29.3 | 1.4 | Greene & Ho (2007)
GH07 171 | 209.35217 | 65.41832 | 0.1060 | 6.20 ± 0.3 | 1.8 ± 0.2 | -1.4 ± 0.5 | 1563 | inf | 2.2 | Greene & Ho (2007)
GH07 177 | 215.32969 | 63.21673 | 0.1340 | 6.20 ± 0.3 | 1.7 ± 0.2 | -1.6 ± 0.5 | 1563 | inf | 1.5 | Greene & Ho (2007)
GH07 203 | 239.79010 | 35.02988 | 0.0310 | 6.20 ± 0.3 | 1.6 ± 0.1 | -1.1 ± 0.4 | 1564 | inf | 3.2 | Greene & Ho (2007)
GH07 204 | 240.18746 | 50.87044 | 0.1000 | 6.10 ± 0.3 | 1.6 ± 0.1 | -1.8 ± 0.4 | 1562 | inf | 1.1 | Greene & Ho (2007)
GH07 215 | 254.15411 | 37.24434 | 0.0626 | 6.30 ± 0.3 | 1.1 ± 0.1 | -1.6 ± 0.4 | 1562 | inf | 2.3 | Greene & Ho (2007)
GH07 219 | 314.59226 | -6.83455 | 0.0737 | 6.20 ± 0.3 | 2.1 ± 0.6 | -1.4 ± 0.5 | 1506 | 23.8 | 1.3 | Greene & Ho (2007)
RV15 4 | 32.54789 | -9.05987 | 0.0419 | 8.11 ± 0.5 | 1.8 ± 0.3 | -1.9 ± 0.5 | 1303 | 22.8 | 1.1 | Reines & Volonteri (2015)
RV15 6 | 46.07405 | 0.47423 | 0.0447 | 6.50 ± 0.5 | 1.8 ± 0.3 | -1.4 ± 0.5 | 1315 | inf | 1.4 | Reines & Volonteri (2015)
RV15 7 | 50.02634 | 40.36628 | 0.0479 | 6.72 ± 0.5 | 1.4 ± 0.1 | -1.7 ± 0.4 | 1553 | 34.8 | 1.1 | Reines & Volonteri (2015)
RV15 12 | 118.85534 | 39.18606 | 0.0333 | 6.74 ± 0.5 | 0.9 ± 0.1 | -1.3 ± 0.4 | 1508 | inf | 2.4 | Reines & Volonteri (2015)
RV15 14 | 119.70951 | 9.58909 | 0.0455 | 7.25 ± 0.5 | 1.7 ± 0.2 | -1.4 ± 0.5 | 1505 | inf | 1.3 | Reines & Volonteri (2015)
RV15 17 | 120.68084 | 31.06759 | 0.0410 | 7.80 ± 0.5 | 2.1 ± 0.3 | -1.1 ± 0.5 | 1506 | inf | 1.9 | Reines & Volonteri (2015)
RV15 21 | 122.00730 | 38.32648 | 0.0413 | 6.46 ± 0.5 | 1.1 ± 0.1 | -1.8 ± 0.4 | 1508 | 31.2 | 1.0 | Reines & Volonteri (2015)
RV15 23 | 123.33051 | 46.14712 | 0.0539 | 7.08 ± 0.5 | 1.7 ± 0.2 | -1.2 ± 0.4 | 1496 | inf | 3.6 | Reines & Volonteri (2015)
RV15 26 | 124.36005 | 10.20281 | 0.0455 | 7.40 ± 0.5 | 2.2 ± 0.4 | -0.9 ± 0.5 | 1505 | inf | 1.4 | Reines & Volonteri (2015)
RV15 37 | 131.72539 | 25.37009 | 0.0506 | 7.73 ± 0.5 | 1.7 ± 0.2 | -1.0 ± 0.5 | 1507 | 30.5 | 1.8 | Reines & Volonteri (2015)
RV15 45 | 139.14301 | 17.91898 | 0.0268 | 7.40 ± 0.5 | 1.2 ± 0.1 | -1.7 ± 0.4 | 1512 | 21.6 | 1.8 | Reines & Volonteri (2015)
RV15 47 | 140.31484 | 10.29473 | 0.0390 | 7.36 ± 0.5 | 1.7 ± 0.3 | -1.6 ± 0.5 | 1512 | 4.9 | 2.8 | Reines & Volonteri (2015)
RV15 56 | 145.51997 | 23.68526 | 0.0211 | 7.10 ± 0.5 | 2.1 ± 0.4 | -1.3 ± 0.5 | 1512 | inf | 1.5 | Reines & Volonteri (2015)
RV15 58 | 146.37234 | 9.60290 | 0.0137 | 6.35 ± 0.5 | 1.3 ± 0.2 | -1.6 ± 0.4 | 1512 | 22.5 | 2.3 | Reines & Volonteri (2015)
RV15 68 | 150.52930 | 3.05767 | 0.0228 | 7.22 ± 0.5 | 1.6 ± 0.2 | -1.4 ± 0.4 | 1513 | 31.3 | 4.5 | Reines & Volonteri (2015)
RV15 69 | 150.57826 | 26.80159 | 0.0518 | 7.52 ± 0.5 | 1.4 ± 0.2 | -1.6 ± 0.4 | 1515 | inf | 2.1 | Reines & Volonteri (2015)
RV15 73 | 153.01080 | 30.21751 | 0.0497 | 7.29 ± 0.5 | 1.8 ± 0.2 | -1.3 ± 0.5 | 1523 | inf | 1.3 | Reines & Volonteri (2015)
RV15 83 | 159.46589 | 33.81392 | 0.0510 | 6.43 ± 0.5 | 1.5 ± 0.2 | -1.7 ± 0.4 | 1523 | 28.0 | 1.1 | Reines & Volonteri (2015)
RV15 85 | 160.72055 | 4.24476 | 0.0519 | 7.29 ± 0.5 | 1.8 ± 0.3 | -1.4 ± 0.5 | 1529 | 26.9 | 1.8 | Reines & Volonteri (2015)
RV15 92 | 165.06680 | 46.27092 | 0.0315 | 6.44 ± 0.5 | 1.5 ± 0.1 | -1.5 ± 0.4 | 1538 | inf | 2.4 | Reines & Volonteri (2015)
RV15 98 | 167.69151 | 11.61153 | 0.0417 | 7.18 ± 0.5 | 1.4 ± 0.2 | -1.5 ± 0.4 | 1527 | 15.7 | 1.1 | Reines & Volonteri (2015)
RV15 102 | 169.74039 | 58.05665 | 0.0282 | 7.10 ± 0.5 | 1.4 ± 0.2 | -1.5 ± 0.4 | 1540 | inf | 1.8 | Reines & Volonteri (2015)
RV15 106 | 171.61964 | 47.41495 | 0.0331 | 6.35 ± 0.5 | 2.1 ± 0.3 | -1.6 ± 0.5 | 1540 | inf | 1.3 | Reines & Volonteri (2015)
RV15 113 | 175.05809 | 24.69706 | 0.0127 | 7.00 ± 0.5 | 2.1 ± 0.5 | -1.9 ± 0.5 | 1538 | 33.1 | 1.6 | Reines & Volonteri (2015)
RV15 119 | 176.43826 | 55.79998 | 0.0540 | 7.16 ± 0.5 | 1.2 ± 0.1 | -2.0 ± 0.4 | 1547 | inf | 1.3 | Reines & Volonteri (2015)
RV15 122 | 179.29451 | 22.29617 | 0.0520 | 7.30 ± 0.5 | 1.9 ± 0.3 | -1.5 ± 0.5 | 1547 | 27.4 | 2.3 | Reines & Volonteri (2015)
RV15 123 | 180.30980 | -3.67809 | 0.0193 | 6.96 ± 0.5 | 1.7 ± 0.3 | -1.4 ± 0.5 | 1536 | 10.8 | 1.8 | Reines & Volonteri (2015)
RV15 129 | 181.60947 | 42.74056 | 0.0522 | 7.33 ± 0.5 | 1.8 ± 0.2 | -1.1 ± 0.5 | 1549 | inf | 3.7 | Reines & Volonteri (2015)
RV15 130 | 181.86655 | 25.60303 | 0.0481 | 6.64 ± 0.5 | 1.5 ± 0.1 | -1.6 ± 0.4 | 1545 | inf | 1.5 | Reines & Volonteri (2015)
RV15 133 | 184.02951 | 50.82512 | 0.0313 | 8.06 ± 0.5 | 2.0 ± 0.3 | -0.9 ± 0.5 | 1553 | inf | 2.5 | Reines & Volonteri (2015)
RV15 134 | 184.61134 | -0.13059 | 0.0499 | 6.56 ± 0.5 | 1.5 ± 0.2 | -1.7 ± 0.4 | 1545 | 12.9 | 1.7 | Reines & Volonteri (2015)
RV15 137 | 185.56405 | 28.35659 | 0.0277 | 6.84 ± 0.5 | 1.4 ± 0.1 | -1.4 ± 0.4 | 1551 | inf | 1.6 | Reines & Volonteri (2015)
RV15 140 | 188.15618 | 66.41459 | 0.0473 | 7.40 ± 0.5 | 1.3 ± 0.1 | -1.1 ± 0.4 | 1563 | inf | 2.8 | Reines & Volonteri (2015)
RV15 152 | 195.08309 | 61.65512 | 0.0526 | 7.60 ± 0.5 | 1.3 ± 0.1 | -1.2 ± 0.4 | 1561 | inf | 3.3 | Reines & Volonteri (2015)
RV15 157 | 198.27422 | 1.46553 | 0.0288 | 6.44 ± 0.5 | 2.0 ± 0.3 | -1.5 ± 0.5 | 1563 | 25.0 | 2.1 | Reines & Volonteri (2015)
RV15 164 | 204.41614 | 39.15456 | 0.0202 | 6.70 ± 0.5 | 1.6 ± 0.2 | -1.8 ± 0.4 | 1563 | inf | 1.4 | Reines & Volonteri (2015)
RV15 167 | 206.11005 | 44.27223 | 0.0546 | 6.82 ± 0.5 | 1.5 ± 0.1 | -1.1 ± 0.4 | 1564 | inf | 2.0 | Reines & Volonteri (2015)
RV15 174 | 211.31193 | -2.98367 | 0.0535 | 7.21 ± 0.5 | 1.5 ± 0.2 | -1.6 ± 0.4 | 1559 | 16.8 | 1.4 | Reines & Volonteri (2015)
RV15 176 | 214.12843 | 1.61890 | 0.0536 | 7.22 ± 0.5 | 1.6 ± 0.2 | -1.4 ± 0.4 | 1564 | 32.9 | 2.7 | Reines & Volonteri (2015)
RV15 177 | 214.49809 | 25.13687 | 0.0167 | 7.94 ± 0.5 | 2.1 ± 0.3 | -0.3 ± 0.5 | 1564 | inf | 1.1 | Reines & Volonteri (2015)
RV15 180 | 216.03493 | 21.08848 | 0.0471 | 7.03 ± 0.5 | 1.7 ± 0.2 | -1.4 ± 0.5 | 1564 | inf | 1.9 | Reines & Volonteri (2015)
RV15 186 | 222.73097 | 27.57853 | 0.0303 | 6.79 ± 0.5 | 1.6 ± 0.2 | -1.5 ± 0.4 | 1563 | 30.2 | 1.7 | Reines & Volonteri (2015)
RV15 191 | 225.41164 | 10.42040 | 0.0302 | 7.07 ± 0.5 | 1.7 ± 0.2 | -1.2 ± 0.4 | 1564 | inf | 1.5 | Reines & Volonteri (2015)
RV15 200 | 230.85180 | 55.31542 | 0.0394 | 6.72 ± 0.5 | 1.9 ± 0.2 | -1.9 ± 0.5 | 1564 | inf | 1.0 | Reines & Volonteri (2015)
RV15 202 | 232.41909 | 30.48592 | 0.0359 | 6.97 ± 0.5 | 1.8 ± 0.2 | -1.5 ± 0.5 | 1564 | inf | 1.2 | Reines & Volonteri (2015)
RV15 211 | 238.57264 | 32.64387 | 0.0486 | 7.73 ± 0.5 | 1.5 ± 0.1 | -1.3 ± 0.4 | 1564 | inf | 2.4 | Reines & Volonteri (2015)
RV15 212 | 239.17043 | 12.28831 | 0.0355 | 6.66 ± 0.5 | 1.5 ± 0.2 | -1.6 ± 0.4 | 1562 | 31.9 | 1.6 | Reines & Volonteri (2015)
RV15 215 | 241.26026 | 33.09584 | 0.0532 | 7.80 ± 0.5 | 1.6 ± 0.1 | -1.3 ± 0.4 | 1563 | inf | 3.9 | Reines & Volonteri (2015)
RV15 220 | 242.98464 | 52.18809 | 0.0416 | 7.26 ± 0.5 | 1.5 ± 0.1 | -1.4 ± 0.4 | 1562 | inf | 2.1 | Reines & Volonteri (2015)
RV15 223 | 244.68689 | 25.65212 | 0.0483 | 7.20 ± 0.5 | 1.5 ± 0.1 | -1.3 ± 0.4 | 1563 | inf | 2.0 | Reines & Volonteri (2015)
RV15 224 | 244.96380 | 40.97981 | 0.0383 | 6.78 ± 0.5 | 1.8 ± 0.2 | -1.0 ± 0.5 | 1563 | inf | 4.2 | Reines & Volonteri (2015)
RV15 228 | 247.53080 | -0.19345 | 0.0467 | 7.14 ± 0.5 | 1.3 ± 0.1 | -1.7 ± 0.4 | 1555 | 8.3 | 1.3 | Reines & Volonteri (2015)
RV15 233 | 258.11851 | 35.88406 | 0.0269 | 6.97 ± 0.5 | 2.0 ± 0.3 | -1.5 ± 0.5 | 1562 | inf | 2.2 | Reines & Volonteri (2015)
RV15 236 | 316.38934 | 0.47478 | 0.0549 | 6.49 ± 0.5 | 1.6 ± 0.2 | -1.6 ± 0.4 | 1532 | 25.2 | 1.4 | Reines & Volonteri (2015)
RV15 238 | 323.43439 | -8.27101 | 0.0533 | 7.57 ± 0.5 | 1.5 ± 0.2 | -1.8 ± 0.4 | 1503 | 7.9 | 1.1 | Reines & Volonteri (2015)
RV15 242 | 342.10264 | 0.15573 | 0.0544 | 6.62 ± 0.5 | 1.3 ± 0.1 | -1.6 ± 0.4 | 1496 | 20.8 | 1.2 | Reines & Volonteri (2015)

##  4 Discussion

Figure 6: Comparison between $M_{\rm{BH}}-M_{\ast}$ and
$M_{\rm{BH}}-\tau_{\rm{DRW}}$ relations. We plot the virial BH mass
$M_{\rm{BH,virial}}$ versus $M_{\ast}$ against the relations of Reines &
Volonteri (2015) and Kormendy & Ho (2013) (_left_). We also plot the BH mass
predicted from the $M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation of Burke et al.
(2021) using our ZTF light curves versus the stellar masses from Reines &
Volonteri (2015) in the _g_ band (_right_). The gray (blue) line and shaded
band is the best-fit local AGN scaling relation and $1\sigma$ uncertainty from
Reines & Volonteri (2015) (Kormendy & Ho (2013)) relations. The fitted
intrinsic scatter of our data in each panel is, from left to right:
$0.14^{+0.09}_{-0.06}$ and $0.46^{+0.09}_{-0.10}$ dex.

To test the usefulness of the $M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation, we
obtained host galaxy stellar masses $M_{\ast}$ for objects in our final sample
when available from Reines & Volonteri (2015). These stellar masses were
estimated from Zibetti et al. (2009) mass-to-light ratios as a function of
their $g-i$ colors after subtracting the contribution from AGN emission using
mock AGN spectra (Reines & Volonteri, 2015). We then compared the predicted BH
masses from the $M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation of Burke et al. (2021),
$M_{\rm{BH},\tau_{\rm{DRW}}}$, to the expectation from the
$M_{\rm{BH}}-M_{\ast}$ relation. The results are shown in Figure 6.

We find that the BH masses predicted from $M_{\rm{BH}}-\tau_{\rm{DRW}}$
relation are less precise (intrinsic scatter of $0.46^{+0.09}_{-0.10}$ dex in
predicted BH mass but total rms scatter of $0.77^{+0.02}_{-0.01}$ dex) than
the $M_{\rm{BH}}-M_{\ast}$ relation (intrinsic scatter of
$0.14^{+0.09}_{-0.06}$ dex with total rms scatter of $0.51^{+0.01}_{-0.01}$
dex). This intrinsic scatter in predicted BH mass from the
$M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation is broadly consistent with the
intrinsic scatter of $0.33^{+}{0.11}_{-0.11}$ dex found by Burke et al.
(2021), and similar to the scatter in single-epoch virial BH masses (Shen,
2013). The larger total rms scatter could be due to the larger measurement
uncertainties in our ZTF damping timescale measurements from poorer light
curve quality. The BH mass prediction is proportional to $\tau_{\rm
DRW}^{2.54}$, hence a 0.2 dex measurement uncertainty in $\tau_{\rm DRW}$
translates to $\sim 0.5$ dex uncertainty in BH mass, leading to the large rms
scatter seen in Figure 6. Given such large measurement uncertainties in
predicted BH mass, the nominal intrinsic scatter constrained from the
regression fit in this work is more difficult to constrain. Therefore, with
higher quality light curves of dwarf AGNs, we expect the predicted BH masses
to improve. Alternatively, the large scatter could be due to larger variations
of other parameters in dwarf AGNs which were unaccounted for, such as black
hole spin and accretion rate. Notably, the Greene & Ho (2007) sample, which
may have systematically longer damping timescales in Figure 6, have relatively
high Eddington ratios of $\log(L_{\rm{bol}}/L_{\rm{Edd}})\sim-0.5$ (Greene &
Ho, 2007). Future studies with a sample spanning a large range of Eddington
luminosity ratios could help inform this work.

We caution that the single-epoch virial BH masses in this work have a typical
systematic uncertainty of $\sim 0.4$ dex (Shen, 2013). For a dwarf AGN, the
uncertainty may be even larger, because of enhanced complications from
outflows or stellar continuum absorption features to measure the weak broad-
line emission from the AGN. In addition, the ZTF data release light curves are
generally noisier than composite Stripe 82 light curves adopted in Burke et
al. (2021) or light curves from dedicated AGN monitoring programs despite the
quality constraints we applied. These sources of systematic uncertainty will
contribute to the intrinsic scatter in our relations.

##  5 Conclusion

We have analyzed the characteristic “damping” timescale of a final sample of
79 low-mass (dwarf and Seyfert) AGNs with virial BH mass estimates of
$M_{\rm{BH}}\sim 10^{6}-10^{8}\ M_{\odot}$ at a redshift range of $z\sim
0-0.3$ using ZTF data release 13 with $\sim 4$ year long light curves with
sufficient quality to enable reliable DRW model fitting. Our best-fit linear
model is consistent with the Burke et al. (2021) results within $1\sigma$
where the IMBH range is better populated by the new ZTF sample. This work
demonstrates consistency of dwarf AGN variability with the
$M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation, which indicates their single-epoch
virial masses are reasonable on average. Given their short damping timescales,
we conclude that the dwarf AGN candidates in our sample may indeed be dwarf
AGNs, rather than long-lived transients (Burke et al., 2020) or more massive
AGNs. This provides a useful and independent line of evidence for the AGN
nature of these sources, even for those with narrow emission line ratios
consistent with star formation.

High-quality, long-duration light curves will improve the uncertainties on
individual $\tau_{\rm{DRW}}$ measurements and improve the robustness of the
$M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation, particularly at the high-mass end
($M_{\rm{BH}}\gtrsim 10^{9}\ M_{\odot}$) where exclusion of AGNs with
$\tau_{\rm{DRW}}$ values greater than several hundred days with insufficient
light curve baselines may result in a slope slightly shallower than the
predicted slope of $0.5$ from the standard model of accretion disks (Burke et
al., 2021).

High-cadence monitoring of the lowest-mass dwarf AGNs ($M_{\rm{BH}}\lesssim
10^{5}\ M_{\odot}$) can be used to estimate the BH mass without the need for
spectroscopy. The $M_{\rm{BH}}-\tau_{\rm{DRW}}$ of Burke et al. (2021)
relation predicts a damping timescale of about 13 hours (3 days) for a
$10^{2}\ M_{\rm{\odot}}$ ($10^{4}\ M_{\rm{\odot}}$) IMBH. In this case, the
$M_{\rm{BH}}-\tau_{\rm{DRW}}$ relation can serve as a discovery machine for
IMBHs which fail to produce a broad line region (Chakravorty et al., 2014) or
where the broad emission line is too weak to detect, as long as the optical
variability is dominated by the accretion disk emission rather than from a
companion star or reprocessed light from X-rays. This motivates optical
surveys with intra-day cadence to enable discovery and mass estimation of
IMBHs using variability (Bellm et al., 2022). The variable sky at these
timescales is still relatively unexplored at sufficient volumes and depths to
enable a systematic search for rapidly-accreting IMBHs.

## Acknowledgements

We thank Gautham Narayan for useful discussions. C.J.B. acknowledges the
Illinois Graduate Survey Science Fellowship for partial support. Z.F.W. and
X.L. acknowledge support from the University of Illinois Campus Research Board
and NSF grant AST-2206499. Y.S. acknowledges support from NSF grant
AST-2009947. This research was supported in part by the National Science
Foundation under PHY-1748958. This research made use of Photutils, an Astropy
package for detection and photometry of astronomical sources (Bradley et al.,
2022).

Based on observations obtained with the Samuel Oschin Telescope 48-inch and
the 60-inch Telescope at the Palomar Observatory as part of the Zwicky
Transient Facility project. ZTF is supported by the National Science
Foundation under Grant No. AST-2034437 and a collaboration including Caltech,
IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm
University, the University of Maryland, Deutsches Elektronen-Synchrotron and
Humboldt University, the TANGO Consortium of Taiwan, the University of
Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National
Laboratories, and IN2P3, France. Operations are conducted by COO, IPAC, and
UW.

This work makes use of SDSS-I/II and SDSS-III/IV data (http://www.sdss.org/
and http://www.sdss3.org/).

Facilities: ZTF, Sloan

## Data Availability

All ZTF light curves can be downloaded at
https://irsa.ipac.caltech.edu/Missions/ztf.html. Our timing analysis code is
available at https://github.com/burke86/taufit.

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##  Appendix A Final Sample Light Curves and Best-fit DRW Models

Figure 7: ZTF _g_ -band difference imaging light curves and DRW model
predictions for dwarf AGNs in our final sample (continued below).

Figure 8: Continuation of Figure 7.

Figure 9: Continuation of Figure 7.

Figure 10: Continuation of Figure 7.

Figure 11: Continuation of Figure 7.

Figure 12: Continuation of Figure 7.

We show the difference imaging light curves and DRW modeling for all sources
in our final, high-quality sample in Figure 7.