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CERN-TH-2022-061

# Dynamically Induced Topological Inflation

Gongjun Choi<EMAIL_ADDRESS>Theoretical Physics Department, CERN,
CH-1211 Genève 23, Switzerland  Weikang Lin<EMAIL_ADDRESS>Tsung-Dao
Lee Institute (TDLI) & School of Physics and Astronomy, Shanghai Jiao Tong
University,
Shengrong Road 520, 201210 Shanghai, P. R. China  Tsutomu T. Yanagida
<EMAIL_ADDRESS>Tsung-Dao Lee Institute (TDLI) & School of
Physics and Astronomy, Shanghai Jiao Tong University,
Shengrong Road 520, 201210 Shanghai, P. R. China  Kavli IPMU (WPI), The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

We propose an inflation model in which the inflationary era is driven by the
strong dynamics of $\displaystyle Sp(2)$ gauge theory. The quark condensation
in the confined phase of $\displaystyle Sp(2)$ gauge theory generates the
inflaton potential comparable to the energy of the thermal bath at the time of
phase transition. Afterwards, with super-Planckian global minimum, the
inflation commences at a false vacuum region lying between true vacuum regions
and hence the name “topological inflation”. Featured by the huge separation
between the scale of the false vacuum ($\displaystyle V(0)^{1/4}\sim
10^{15}{\rm GeV}$) and the global minimum
($\displaystyle\langle\phi\rangle\sim M_{P}$), the model can be consistent
with CMB observables without suffering from the initial condition problem.
Crucially, this is achieved without any fine-tuning of parameters in
$\displaystyle V(\phi)$. In addition to $\displaystyle Sp(2)$, this model is
based on an anomaly free $\displaystyle Z_{6R}$ discrete $\displaystyle R$
symmetry. Remarkably, while all parameters are fixed by CMB observations, the
model predicts a hierarchy of energy scales including the inflation scale,
SUSY-breaking scale, R-symmetry breaking scale, Higgsino mass and the right-
handed neutrino mass given in terms of the dynamical scale of $\displaystyle
Sp(2)$.

Introduction — The cosmic inflation has been the firmly established solution
to the main cosmological issues including the flatness problem and the horizon
problem [1, 2, 3, 4]. The requirement for the de Sitter period and its end
leads on to the slow-roll inflation models [3, 4]. In these models, starting
from the origin in the field space, the inflaton field ($\displaystyle\phi$)
goes through the slow-rolling and eventually arrives at the global minimum,
which results in the end of inflation. The requisite flatness near the origin,
however, may require the unnatural fine-tuning of parameters of an inflaton
potential. Moreover, more severe is how to justify the initial location of
$\displaystyle\phi$ near the origin from the outset.

These problems bothering the new inflation type models, nevertheless, could be
overcome provided the inflaton potential ($\displaystyle V(\phi)$) is featured
by a noticeable hierarchy between the scale of the false vacuum
($\displaystyle V(0)^{1/4}$) and the global minimum ($\displaystyle\phi_{\rm
min}$), and $\displaystyle\mathcal{O}(1)$ parameters in $\displaystyle
V(\phi)$. Particularly the hierarchy concerns the initial location of the
inflaton ($\displaystyle\phi_{\rm ini}$). In the case where $\displaystyle
V(0)^{1/4}$ and $\displaystyle\langle\phi\rangle$ coincide with a scale of the
phase transition (PT) in the pre-inflationary era, $\displaystyle\phi_{\rm
ini}$ on the hill top is not guaranteed because the field fluctuation
$\displaystyle\delta\phi\sim H\sim T^{2}$111In this paper, we invoke the
Planck unit in which the reduced Planck scale is set to the unity, i.e.
$\displaystyle M_{P}=(8\pi G)^{-1/2}=1$. becomes comparable to
$\displaystyle\phi_{\rm min}$. For that reason, it becomes questionable
whether the inflationary era can start in the small field inflation models
after PT.222By the small field inflation models, we mean the models where
$\displaystyle\phi_{\rm ini}$ starts the slow-roll near the false vacuum of
$\displaystyle V(\phi)$ which is taken to be located at $\displaystyle\phi=0$.
Usually a PT from a symmetry breaking precedes the inflationary era,
generating $\displaystyle V(\phi)$ for inflation. The “new inflation” [3, 4]
and the “natural inflation” [5] are the well-known examples of the small field
inflation scenario. Aside from that, it is challenging in most cases to
establish the long enough flatness for the successful inflation consistent
with CMB observables without relying on the artificial tuning of parameters in
$\displaystyle V(\phi)$.

In this work, given the aforementioned problems of the small field inflation,
we propose an inflation model which establishes the separation of scales of
$\displaystyle V(0)^{1/4}$ and $\displaystyle\langle\phi\rangle$ without any
fine-tuning of parameters appearing in $\displaystyle V(\phi)$. To this end,
we consider the supersymmetric $\displaystyle Sp(2)$ gauge theory with
$\displaystyle N_{F}=6$ flavors of quarks $\displaystyle Q_{i}$ transforming
as the fundamental representation. With an integer $\displaystyle R$-charge of
$\displaystyle Q_{i}$, $\displaystyle Z_{6R}$ can be easily shown to be free
of the mixed anomaly $\displaystyle Z_{6R}-[Sp(2)]^{2}$ and thus we choose
$\displaystyle Z_{6R}$ as the R-symmetry of the theory. On top of this,
because the discrete $\displaystyle Z_{6}$ is also free of the mixed anomaly
$\displaystyle Z_{6}-[Sp(2)]^{2}$ within $\displaystyle Sp(2)$ gauge theory,
we introduce $\displaystyle Z_{6}$ as a gauge symmetry of the theory. Once the
$\displaystyle Sp(2)$ gauge theory enters the confinement phase at the
dynamical scale $\displaystyle\Lambda_{*}$, the quark condensate
$\displaystyle\langle QQ\rangle$ forms to produce $\displaystyle V(\phi)$ [6].
As such, the scale of $\displaystyle\langle QQ\rangle\sim\Lambda_{*}^{2}$
determines the inflation scale. We shall show that how the presence of
$\displaystyle Z_{6}$ resolves the fine-tuning of parameters in $\displaystyle
V(\phi)$ which the similar set-up without $\displaystyle Z_{6}$ suffers from
[7, 8]. In this way, we trade the parameter fine-tuning problem with the
additional symmetry, which is nonetheless non-trivial.

Quark fields $\displaystyle Q_{i}$ being charged under $\displaystyle Z_{6R}$,
its condensation in the non-perturbative regime of $\displaystyle Sp(2)$
drives the spontaneous breaking of $\displaystyle Z_{6R}$ to $\displaystyle
Z_{2R}$. However, in our model, there is no domain wall problem associated
with the discrete $\displaystyle R$-symmetry since the domain walls are
diluted away by the inflation. The model gains more attractiveness when we
specify the role of quark condensate more than generating the inflaton
potential: as a spurion field arising from the spontaneous breaking of
$\displaystyle Z_{6R}$, the quark condensate explains the dimensionful
parameters in the MSSM. This interesting attribute of the model unifies the
origin of various energy scales including inflation scale ($\displaystyle
H_{\rm inf}$), SUSY-breaking scale ($\displaystyle\sqrt{F_{Z}}$), R-symmetry
breaking scale, Higgsino mass ($\displaystyle\mu_{H}$) and the right-handed
(RH) neutrino mass ($\displaystyle m_{N}$) and explains those based on a
single energy scale $\displaystyle\Lambda_{*}$ inferred from the CMB
observable.

Model — On top of the MSSM gauge group, we introduce

$G=Sp(2)\otimes Z_{6R}\otimes Z_{6}\,.$ (1)

as the additional symmetry group. The matter contents we assume are shown in
Table. Dynamically Induced Topological Inflation. In addition to the MSSM
particle contents333$\displaystyle H_{u}$ and $\displaystyle H_{d}$ are the
MSSM up-type and down-type Higgs $\displaystyle SU(2)_{L}$ doublets
respectively. We denote other matter contents in the MSSM
($\displaystyle\bm{5^{*}}$ and $\displaystyle\bm{10}$) by using the
representations of $\displaystyle SU(5)_{\rm GUT}$. and the right-handed
neutrino ($\displaystyle N$), $\displaystyle Sp(2)$ quark chiral multiplet
$\displaystyle Q_{i}$, the singlet anti-symmetric field $\displaystyle
S_{ij}=-S_{ji}$, the inflaton chiral multiplet $\displaystyle\Phi$ and the
SUSY-breaking field $\displaystyle Z$ are newly introduced. The indices of
$\displaystyle Q_{i}$ and $\displaystyle S_{ij}$ run from $\displaystyle 1$ to
$\displaystyle N_{F}=6$, which makes the mixed anomaly of $\displaystyle
Z_{6R}-[Sp(2)]^{2}$ vanish with account taken of the contribution from the
$\displaystyle Sp(2)$ gaugino [9, 10, 11]. Thanks to this, the symmetry group
$\displaystyle Sp(2)\otimes Z_{6R}$ is the gauged one.444One can also check
that the charge assignment of the MSSM particle contents under $\displaystyle
Z_{6R}$ makes $\displaystyle Z_{6R}$ anomaly free with respect to the MSSM
gauge group [12]. With $\displaystyle Q_{i}$ assigned the same $\displaystyle
Z_{6}$ charge as the $\displaystyle Z_{6R}$ charge, it is clear that
$\displaystyle Z_{6}$ is free of anomaly with respect to $\displaystyle Sp(2)$
within $\displaystyle Sp(2)$ sector. But since $\displaystyle Z_{6}$ is
anomalous with respect to $\displaystyle SU(2)_{L}$, for arguing the gauged
$\displaystyle Z_{6}$, we need additional fields contributing to
$\displaystyle Z_{6R}-[SU(2)_{L}]^{2}$. We will get back to this point later.

| $\displaystyle Q_{i}$ | $\displaystyle S_{ij}$ | $\displaystyle\Phi$ | $\displaystyle\bm{5^{*}}$ | $\displaystyle\bm{10}$ | $\displaystyle H_{u}$ | $\displaystyle H_{d}$ | $\displaystyle N$ | $\displaystyle Z$
---|---|---|---|---|---|---|---|---|---
$\displaystyle Sp(2)$ | $\displaystyle\ytableausetup{textmode,centertableaux,boxsize=0.6em}$ |  |  |  |  |  |  |  |
| - | - | - | - | - | - | - | - |
Z6R | 1 | 0 | 3 | 0 | 0 | 2 | 2 | 0 | 4
Z6 | 1 | 4 | 3 | 4 | 0 | 0 | 2 | 2 | 2

Table 1: Quantum numbers of the matter contents of the model under the
additional symmetry group in Eq. (1). All the MSSM non-gauge interactions are
consistent with the charge assignment.

For a high enough energy scale where $\displaystyle Sp(2)$ is in its
perturbative regime, the symmetry group in Eq. (1) allows for the
superpotential

$\displaystyle\displaystyle W$ $\displaystyle\displaystyle\supset$
$\displaystyle\displaystyle-\lambda_{ij}S_{ij}Q_{i}Q_{j}+\lambda_{ij}gS_{ij}Q_{i}Q_{j}\Phi^{2}$
(2) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\lambda_{H,ijk\ell}Q_{i}Q_{j}Q_{k}Q_{\ell}H_{u}H_{d}$
(4) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\lambda_{N,ij}Q_{i}Q_{j}NN$ (6)
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\lambda_{Z,ijk\ell}Q_{i}Q_{j}Q_{k}Q_{\ell}Z\,.$ (8)

where all $\displaystyle\lambda$’s and $\displaystyle g$ are
$\displaystyle\mathcal{O}(1)$ dimensionless coupling constants.555Although in
principle the higher dimension operators
$\displaystyle(c_{2n}/(2n)!)QQS(\Phi)^{2n}$ are allowed, for perturbative
$\displaystyle c_{2n}$s they are negligible. So it suffices for us to consider
the second term in LHS of Eq. (8). The superpotential in the first line of Eq.
(8) is responsible for the inflationary dynamics, the second line for the
Higgsino mass term (a.k.a $\displaystyle\mu$-term), the third line for the
right-handed neutrino mass and the final line for the SUSY-breaking. As we
shall see shortly, in the non-perturbative regime of $\displaystyle Sp(2)$,
the quark condensate $\displaystyle\langle QQ\rangle\sim\Lambda_{*}^{2}$ and
its powers generate dimensionful parameters, explaining various energy scales
in the theory at the tree-level.

As the $\displaystyle Sp(2)$ gauge theory becomes strongly coupled for the
energy scale below $\displaystyle\Lambda_{*}$, it is described by 15 composite
meson fields $\displaystyle\mathcal{M}_{ij}\equiv(4\pi)\langle
Q_{i}Q_{j}\rangle/\Lambda_{*}$ with the deformed moduli constraint
$\displaystyle{\rm Pf}(\mathcal{M}_{ij})=\Lambda_{*}^{3}$ [6]. For simplicity,
we can make a choice of vacuum expectation values (VEV) of the quark fields
($\displaystyle Q_{i}$) such that the only nonvanishing meson fields are

$\langle Q_{i}Q_{i+1}\rangle=v^{2}=\frac{\Lambda_{*}^{2}}{4\pi}\quad{\rm
for}\quad i=1,3,5\,,$ (9)

where $\displaystyle\Lambda_{*}$ is the dynamical scale of $\displaystyle
Sp(2)$. The quark condensation induces the spontaneous breaking of
$\displaystyle Z_{6R}$ to $\displaystyle Z_{2R}$. This physics will be taken
as the fundamental origin of the various energy scales in our scenario.

Given Eq. (9), in the non-perturbative regime of $\displaystyle Sp(2)$, the
superpotential in Eq. (8) transforms to

$\displaystyle\displaystyle W_{\rm eff}$ $\displaystyle\displaystyle\supset$
$\displaystyle\displaystyle-\lambda\Lambda_{*}^{2}S(1+g\Phi^{2})$ (10)
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\lambda_{H}\Lambda_{*}^{4}H_{u}H_{d}+\lambda_{N}\Lambda_{*}^{2}NN+\lambda_{Z}\Lambda_{*}^{4}Z$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle W_{\rm
inf}+W_{H}+W_{N}+W_{\cancel{SUSY}}\,,$

where all $\displaystyle\lambda$s without the flavor indices are
$\displaystyle\mathcal{O}(1)$ dimensionless coupling constants obtained after
re-scaling $\displaystyle\lambda$s in Eq. (8). Here $\displaystyle S$ is a
linear combination of $\displaystyle S_{ii+1}$ with $\displaystyle i=1,3,5$.

Inflation ($\displaystyle W_{\rm inf}$) — As mentioned above, the inflation of
the universe in our scenario is attributed to the first term of Eq.
(LABEL:eq:superpotential2). Along with the Kähler potential of the form

$K(\Phi,S)\supset|S|^{2}+|\Phi|^{2}+c|S|^{2}|\Phi|^{2}+...\,,$ (15)

the $\displaystyle F$-term contribution of $\displaystyle S$ to the scalar
potential of the model yields the inflaton potential below [8]

$V(\phi)\simeq\Lambda_{*}^{4}e^{\frac{\phi^{2}}{2}}\left(1-g\frac{\phi^{2}}{2}\right)^{2}\left(1+c\frac{\phi^{2}}{2}\right)^{-1}\,,$
(16)

where $\displaystyle\phi$ is the real part of the scalar component of
$\displaystyle\Phi$.

In order for the slow-roll of $\displaystyle\phi$ from somewhere near
$\displaystyle\phi=0$ to the global minimum to be responsible for the early
inflationary era of the universe, the power spectrum of the curvature
perturbation ($\displaystyle P_{\zeta}(k)=A_{s}(k/k_{\star})^{n_{s}-1}$) needs
to satisfy $\displaystyle A_{s}=2.1\times 10^{-9}$, $\displaystyle
n_{s}=0.9649\pm 0.0042$ ($\displaystyle 68\%$ C.L., Planck
TT,TE,EE+lowE+lensing) [13] and $\displaystyle r<0.036$ ($\displaystyle 95\%$
C.L., BICEP/Keck) [14] should hold at the CMB pivot scale $\displaystyle
k_{\star}=0.05{\rm Mpc}^{-1}$. According to the scanning of the parameter
space ($\displaystyle\Lambda_{*},c,g$) performed in [8] with $\displaystyle
0<\phi_{\star}<1$, $\displaystyle\Lambda_{*}\sim 10^{-3}$, $\displaystyle
c\gtrsim 0.4$, $\displaystyle g\sim 0.3$ with $\displaystyle\phi_{\star}\sim
0.6$ were found to provide a good fit to the CMB observables. The model
predicts a rather low tensor-to-scalar ratio $\displaystyle
r=\mathcal{O}(10^{-4})$. We note that it is remarkable for the inflation model
in Eq. (16) to accomplish the successful fit to CMB observables with
$\displaystyle\mathcal{O}(1)$ dimensionless parameters, i.e.
$\displaystyle\lambda$, $\displaystyle g$ and $\displaystyle c$.

With the concrete inflaton potential specified above, now we discuss appealing
points of our inflation model. As is clear from Eq. (16), the potential is of
the typical new inflation type. Albeit similar in the shape, there is a
crucial distinction of Eq. (16) as compared to the original new inflation [3,
4]: our inflation model is featured by the global minimum at
$\displaystyle\phi_{\rm min}\sim 2.5$ and the inflation scale $\displaystyle
H_{\rm inf}\simeq\Lambda_{*}^{2}\sim 10^{-6}<\\!\\!<1$. Thanks to the (super-
Planckian) large enough global minimum, the inflationary expansion in a false
vacuum residing in a wall (the spatial region in-between domains with
$\displaystyle\langle\phi\rangle=\pm\phi_{\rm min}$) is guaranteed when
$\displaystyle\phi_{\rm min}>1$ holds [15, 16].666When the wall thickness
$\displaystyle\delta\sim\phi_{\rm min}V(0)^{-1/2}$ is greater than
$\displaystyle H^{-1}\sim V(0)^{-1/2}$, the false vacuum region located in the
wall experiences the inflationary expansion [15, 16]. The condition
$\displaystyle\delta>H^{-1}$ is converted to $\displaystyle\phi_{\rm min}>1$.
Hence, our model does not suffer from the initial condition problem. What’s
remarkable is that the separation of the scales of $\displaystyle V^{1/4}$ and
$\displaystyle\phi_{\rm min}$, which is required by consistency with CMB
observables and the use of logics in the topological inflation, is achieved
without any fine-tuning of parameters in $\displaystyle V(\phi)$.

Pre-Inflation Era — At the Planck time $\displaystyle t\sim 1/M_{p}^{-1}$,
there might be particle creation in an expanding background [17, 18, 19].
Afterwards, when $\displaystyle Sp(2)$ gauge theory becomes strong enough, the
thermal bath made of multiplets of $\displaystyle Q_{i}$, $\displaystyle S$
and $\displaystyle Sp(2)$ gluons is expected to form. With a certain
distribution of $\displaystyle T$ of thermal baths in mind, we may consider
for simplicity two classes of horizons with $\displaystyle T<\Lambda_{*}$ and
$\displaystyle T>\Lambda_{*}$. For the former, $\displaystyle V(\phi)$ in Eq.
(16) applies since the horizon is already in the confined phase of
$\displaystyle Sp(2)$. In contrast, for the later case, starting from zero
potential, $\displaystyle V(\phi)$ eventually develops to the form in Eq. (16)
at $\displaystyle t\sim(M_{P}/\Lambda_{*})\Lambda_{*}^{-1}$.

Once the inflaton potential is described by Eq. (16) everywhere, we can expect
that there arise a pair of neighboring Hubble patches with
$\displaystyle\langle\phi\rangle\sim\pm\phi_{\rm min}$ after a bit of time for
homogenizing $\displaystyle\phi$ within horizons. Then inflation is initiated
at $\displaystyle\phi=0$ residing in the wall between the Hubble patches (the
space with $\displaystyle\phi$ field variation by $\displaystyle\sim
2\phi_{\rm min}$) [15, 16]. Hence our model serves as a UV model for
topological inflation scenario, providing the concrete picture for development
of $\displaystyle V(\phi)$ at the pre-inflation era in accordance with the
$\displaystyle Sp(2)$ strong dynamics.

SUSY Breaking ($\displaystyle W_{\cancel{SUSY}}$) — The last term in Eq.
(LABEL:eq:superpotential2) explains the (F-term) SUSY breaking. Now that the
Polonyi field $\displaystyle Z$ is charged under both $\displaystyle Z_{6R}$
and $\displaystyle Z_{6}$, there cannot be a marginal or relevant operator
purely composed of $\displaystyle Z$. This fact makes the last operator in Eq.
(8), i.e. $\displaystyle\mathcal{O}\sim QQQQZ$, most relevant in the
$\displaystyle F$-term contribution of $\displaystyle Z$ to the scalar
potential777We introduce a pair of massive chiral multiplets $\displaystyle X$
and $\displaystyle{\bar{X}}$ which have a superpotential $\displaystyle
W=ZX^{2}+MX{\bar{X}}$. One-loop diagrams of the $\displaystyle X$ and
$\displaystyle{\bar{X}}$ give a large positive soft mass squared for
$\displaystyle Z$ to stabilize the potential of $\displaystyle Z$ at the
origin [20]. and thus SUSY-breaking is indeed safely guaranteed once
$\displaystyle Sp(2)$ enters the confined phase.888When either of marginal or
relevant operator purely made up of $\displaystyle Z$ is allowed in the
superpotential, the scalar potential $\displaystyle|F_{Z}|^{2}$ can possibly
have a SUSY-preserving global minimum. Then SUSY remains unbroken. Were it not
for $\displaystyle Z_{6}$, for instance, the operator $\displaystyle
m_{Z}Z^{2}$ with a dimensionful parameter $\displaystyle m_{Z}$ is allowed.
Therefore, having other discrete symmetry than the discrete $\displaystyle
R$-symmetry is the crucial point in constructing the SUSY-breaking sector in
the model. One may be concerned about the unwanted operator $\displaystyle\sim
ZH_{u}H_{d}N$. This one is not a problem since there is the parity as the
remnant of $\displaystyle U(1)_{B-L}$ to suppress $\displaystyle\sim
ZH_{u}H_{d}N$ to stabilize the SUSY vacuum. Under the parity, only
$\displaystyle\bf 5^{*},10$ and $\displaystyle N$ transform as odd. Even if
the Polonyi field has no anti-symmetric flavor indices of $\displaystyle
Sp(2)$ quarks from the beginning, we see that this way of SUSY-breaking is
similar to IYIT dynamical SUSY-breaking [21, 22] in that $\displaystyle
F_{Z}\neq 0$ is attributable to quark condensation with the deformed moduli
constraint. Given $\displaystyle F_{Z}\sim\Lambda_{*}^{4}$, the model’s
prediction for the gravitino mass becomes $\displaystyle
m_{3/2}=F_{Z}/(\sqrt{3}M_{P})\sim\Lambda_{*}^{4}=\mathcal{O}(10^{-12})$.

Higgsino ($\displaystyle W_{H}$) and RH Neutrino Mass ($\displaystyle W_{N}$)
— With $\displaystyle\lambda_{H},\lambda_{N}=\mathcal{O}(1)$, the second and
third term in Eq. (LABEL:eq:superpotential2) can account for the Higgsino mass
($\displaystyle\mu_{H}$) term in the MSSM and the right-handed (RH) neutrino
masses. Given $\displaystyle\Lambda_{*}\sim 10^{-3}$ inferred from CMB
observables, the model predicts $\displaystyle\mu_{H}=\mathcal{O}(10^{-12})$
and $\displaystyle m_{N}=\mathcal{O}(10^{-6})$. Below we discuss implications
of the prediction.

Now that the soft masses are of the order $\displaystyle
m_{3/2}=F_{Z}/(\sqrt{3}M_{P})$ in SUGRA [23], we expect all of $\displaystyle
m_{H_{u}}$, $\displaystyle m_{H_{d}}$ and $\displaystyle B$ to be
$\displaystyle\mathcal{O}(m_{3/2})$. Hence, the prediction
$\displaystyle\mu_{H}=\mathcal{O}(10^{-12})$ renders the electroweak symmetry
breaking (EWSB) condition
$\displaystyle(|\mu_{H}|^{2}+m_{H_{u}}^{2})(|\mu_{H}|^{2}+m_{H_{d}}^{2})\simeq(B\mu_{H})^{2}$
nicely satisfied.

On the other hand, the heavy Majorana mass term $\displaystyle m_{N}NN$ and
the symmetry allowed Yukawa interaction $\displaystyle\bm{5^{*}}H_{u}N$ in the
superpotential can explain the tiny mass for the active neutrinos in the MSSM
via the seesaw mechanism [24, *Yanagida:1979gs, 26, 27]. Also we note that
$\displaystyle m_{N}=\mathcal{O}(10^{-6})$ is sufficiently heavy enough not to
exceed the maximal baryon asymmetry in the leptogenesis [28, 29].

Constant Term in $\displaystyle W$ — The constant term in the superpotential
($\displaystyle W_{0}$) needs to be generated so as to properly cancel the
SUSY-breaking $\displaystyle F$-term contributions to the scalar potential of
the model. To this end, we may consider $\displaystyle SU(3)$ pure gauge
theory, i.e. without any matter fields, with which $\displaystyle Z_{6R}$
still remains gauge anomaly free. Then, once $\displaystyle SU(3)$ pure gauge
theory enters its confinement at the scale $\displaystyle\Lambda^{\prime}$,
there arises the gaugino condensation [30, 31], i.e.
$\displaystyle\langle\lambda^{a}\lambda^{a}\rangle=32\pi^{2}\Lambda^{\prime
3}$. This results in the effective superpotential which take responsible for
the constant term in the superpotential, i.e.

$W_{0}=3\Lambda^{\prime
3}=m_{3/2}\quad\Rightarrow\quad\Lambda^{\prime}=\mathcal{O}(10^{-4})\,.$ (17)

Given that $\displaystyle\Lambda^{\prime}$ is one order of magnitude smaller
than $\displaystyle\Lambda_{*}$, we expect that $\displaystyle W_{0}$ is
generated after PT of $\displaystyle Sp(2)$ and during inflation.
$\displaystyle\langle\lambda^{a}\lambda^{a}\rangle$ respecting $\displaystyle
Z_{2R}$ but not $\displaystyle Z_{6R}$, there could be formation of the domain
wall due to the gaugino condensation. This wall, however, will be diluted away
during inflation

Cosmological Constant — Because the model is embedded in the SUGRA framework,
the vanishingly small vacuum energy (cosmological constant) remains to be
discussed given the scales for the SUSY-breaking $\displaystyle F_{Z}$ and the
R-symmetry breaking $\displaystyle m_{3/2}$. In a SUGRA model, the difference
between $\displaystyle|F_{Z}|^{2}$ and $\displaystyle m_{3/2}^{2}$ determines
the scalar potential at the leading order. In our model, since both of
$\displaystyle F_{Z}$ and $\displaystyle m_{3/2}$ are of the order
$\displaystyle\mathcal{O}(\Lambda_{*}^{4})$, the required cancellation among
the two can be indeed well achieved. Behind this cancellation is the
$\displaystyle Sp(2)$ strong dynamics as the common origin of operators
$\displaystyle QQQQH_{u}H_{d}$ and $\displaystyle QQQQZ$. Because
$\displaystyle\mu_{H}$-parameter along with the EWSB was decisive for
determining the scale of $\displaystyle m_{3/2}$, the quantum number of
$\displaystyle H_{u}H_{d}$ was referred in determining that of $\displaystyle
Z$.

Discussion and Outlook — In this letter, we proposed a new inflation model
with the huge separation between the scale of the false vacuum ($\displaystyle
V(\phi)^{1/4}\sim 10^{15}{\rm GeV}$) and the global minimum
($\displaystyle\langle\phi\rangle\simeq M_{P}$). We discussed how the physics
in the pre-inflationary era in our model can justify occurrence of the
inflation in the similar way to topological inflation. Thereby the model was
shown free from the initial condition problem. What’s new as compared to the
arguments made in the topological inflation scenarios [15, 16] lies in
$\displaystyle Sp(2)$ strong dynamics-driven inflaton potential generation
which importantly does not have any fine-tuned parameter.

The interesting observation that $\displaystyle Z_{6R}$ and $\displaystyle
Z_{6}$ are gauge anomaly free with respect to $\displaystyle Sp(2)$ within the
$\displaystyle Sp(2)$ sector was invoked for achieving $\displaystyle V(\phi)$
without any tuning. The inflation scale was determined by the R-charged quark
condensate and thus $\displaystyle R$-symmetry breaking scale,
$\displaystyle\Lambda_{*}\sim 10^{-3}$, could be inferred from CMB
observables. As the spurion field of $\displaystyle Z_{6R}$ breaking, the
quark condensate $\displaystyle\langle QQ\rangle\sim\Lambda_{*}^{2}$ also
determines the Higgsino mass and the RH neutrino mass and thereby unifies the
origin of various energy scales (the inflation scale, SUSY-breaking scale,
R-symmetry breaking scale, Higgsino mass and RH neutrino mass).

We also emphasize the key idea that the model underlies. The specialty of
$\displaystyle R$-symmetry to apply to every operator in a superpotential
$\displaystyle W$ was used for explaining various energy scales and
dimensionful parameters appearing in $\displaystyle W$. When every physics and
operator yielding certain energy scales of interest is associated with a
common symmetry, if experimental consistency allows, one may dream of
explaining those energy scales as proper powers of spurion field of the
symmetry. For our work, the symmetry was $\displaystyle Z_{6R}$.

With all phenomenologies handled in this work described by a single energy
scale $\displaystyle\Lambda_{*}$, the model proposes the strong dynamics of
$\displaystyle Sp(2)$ as the fundamental underlying physics governing the
universe. Albeit ambitious, there are still open questions to be answered.
$\displaystyle Z_{6}$ in Eq. (1) still remains anomalous for $\displaystyle
SU(2)_{L}$ and thus poses the question about extension of $\displaystyle
SU(2)_{L}$-charged particle contents999One way to eliminate such an anomaly is
to introduce $\displaystyle H^{\prime}_{d}$ and $\displaystyle H^{\prime}_{u}$
whose product $\displaystyle H^{\prime}_{u}H^{\prime}_{d}$ has charges
$\displaystyle(2,4)$ for $\displaystyle Z_{6R}$ and $\displaystyle Z_{6}$
respectively.. On the other hand, as the SUGRA inflation model, the model is
still subject to the long-standing $\displaystyle\eta$-problem, i.e. how to
justify suppression of all the unwanted higher dimension operators in Eq. (15)
contributing to $\displaystyle V(\phi)$. We leave these structural problems of
the model as the future work.

###### Acknowledgements.

T.T.Y. appreciates Li Fu for encouraging him to consider the origin of the
universe. T. T. Y. is supported in part by the China Grant for Talent
Scientific Start-Up Project and by Natural Science Foundation of China (NSFC)
under grant No. 12175134 as well as by World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan.

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# Image Flux Ratios of Gravitationally Lensed HS 0810+2554 with High
Resolution Infrared Imaging

Terry Jay Jones, Liliya L. R. Williams  Minnesota Institute for Astrophysics,
School of Physics and Astronomy
116 Church St SE, University of Minnesota, Minneapolis, MN 55455, USA  Steve
Ertel  Steward Observatory, Department of Astronomy, University of Arizona,
993 N. Cherry Ave, Tucson, AZ 85721, USA  Large Binocular Telescope
Observatory, 933 Cherry Avenue, Tucson, AZ 85721, USA  Philip M. Hinz  Steward
Observatory, Department of Astronomy, University of Arizona, 993 N. Cherry
Ave, Tucson, AZ 85721, USA  Center for Adaptive Optics, UC Santa Cruz, 1156
High Street, Santa Cruz, CA 95064  Amali Vaz  Steward Observatory, Department
of Astronomy, University of Arizona, 993 N. Cherry Ave, Tucson, AZ 85721, USA
Shane Walsh  Large Binocular Telescope Observatory, 933 Cherry Avenue, Tucson,
AZ 85721, USA  Ryan Webster  Steward Observatory, Department of Astronomy,
University of Arizona, 993 N. Cherry Ave, Tucson, AZ 85721, USA

###### Abstract

We report near simultaneous imaging using LMIRCam on the LBTI of the quadruply
imaged lensed quasar HS 0810+2554 at wavelengths of 2.16, 3.7 and
$4.78~{}\micron$ with a Full Width Half Max (FWHM) spatial resolution of
$0\farcs 13$, $0\farcs 12$ and $0\farcs 15$ respectively, comparable to HST
optical imaging. In the $\rm{z}=1.5$ rest frame of the quasar, the observed
wavelengths correspond to 0.86, 1.48, and $1.91~{}\micron$ respectively. The
two brightest images in the quad, A and B, are clearly resolved from each
other with a separation of $0.187\arcsec$. The flux ratio of these two images
(A/B) trends from 1.79 to 1.23 from 2.16 to $4.78~{}\micron$. The trend in
flux ratio is consistent with the $2.16~{}\micron$ flux originating from a
small sized accretion disk in the quasar that experiences only microlensing.
The excess flux above the contribution from the accretion disk at the two
longer wavelengths originates from a larger sized region that experiences no
microlensing. A simple model employing multiplicative factors for image B due
to stellar microlensing $(m)$ and sub-structure millilensing $(M)$ is
presented. The result is tightly constrained to the product $m\times M=1.79$.
Given the observational errors, the 60% probability contour for this product
stretches from $m=2.6$, $M=0.69$ to $m=1.79$, $M=1.0$, where the later is
consistent with microlensing only.

gravitational lensing: strong, cosmology: dark matter

##  1 Introduction

High resolution N-body simulations of the cold dark matter (CDM) structure
formation of the universe predicts that there should be hundreds of dark
matter sub-halos with mass $\sim 10^{4}-10^{9}{\rm{M}}_{\odot}$ within a
massive halo $(\sim 10^{12}{\rm{M}}_{\odot})$. The fact that only a few dozen
dwarf galaxies have been observed within our own Milky Way (McConnachie, 2012)
has led to the so called ”missing satellite” problem (Klypin et al., 1999;
Moore et al., 1999; Sawala et al., 2014). The inclusion of baryonic physics
(Garrison-Kimmel et al., 2019; Bullock & Boylan-Kolchin, 2017; Brooks &
Zolotov, 2014; Buck et al., 2018; Dutton et al., 2016) in models of the
formation of galaxies has made progress on resolving this issue, but many
aspects of dark matter substructure are still uncertain (Despali & Vegetti,
2017; Gomer & Williams, 2018; Despali et al., 2019; Hsueh et al., 2019).

While these sub-halos would remain undetected due to their lack of visible
matter, they should still be detectable through strong gravitational lensing
(e.g. Kochanek & Dalal, 2004). Gravitational lensing provides a unique method
for detecting the presence or absence of substructure in a galaxy outside of
the Local Group. Any perturbation in the lens potential, be it stars or dark
matter substructure, will create perturbations in the observed flux from some
of the lensed images, if it is sufficiently compact and close to the images.
It has been shown by Mao & Schneider (1998), Metcalf & Madau (2001), and Chiba
(2002) that the presence of substructure in the lens mass model can reproduce
the observed flux ratios in multiply imaged lensed quasars, something that
smooth lens models can not do.

Hsueh et al. (2019) use a sample of seven quadruply imaged lensing systems and
assuming a CDM cosmology and contribution from low mass halos along the line-
of-sight, they infer an average total mass fraction in substructure that is in
rough agreement with the predictions from CDM hydrodynamical simulations.
Their result is significantly different when compared to previous studies that
did not include line-of-sight haloes (Xu et al., 2015). Their data set is
dominated by images with good flux ratios at radio wavelengths, but with less
contribution to the analysis from Mid-Infrared (MIR) observations which probe
smaller scales. Several lensed systems, including HS 0810+2554, have images
that are probably resolved at radio wavelengths and were dropped from their
initial sample of 14. Nierenberg et al. (2019) report observations of 8 quads,
separating out emission from the quasar narrow line emission region (which is
not subject to microlensing) and conclude that simple, smooth halo mass
distributions are inadequate, indicating the need for more complex mass
distributions. Although HS 0810 was in their sample, the small angular
distance between images A and B made the analysis by Nierenberg et al. (2019)
difficult, and narrow line flux ratios were not reported.

Dobler & Keeton (2006) point out that the angular size of the source and the
size of the perturber must be on the same order of magnitude for the source to
be appreciably magnified. MIR observations have the advantage that in the rest
frame of the lensed quasar, they span a range in wavelengths that are
sensitive to the emission of both the very small sized accretion disk in the
quasar which is subject to the effects of both stellar microlensing and
substructure millilensing, and the much larger dusty torus which is subject
only to millilensing (Sluse et al., 2013). Therefore, if we compare the flux
ratios in a lensed quasar at different wavelengths, corresponding to different
source sizes, we can quantify the size and location of substructure in the
lens galaxy. This approach was taken by Fadely & Keeton (2011, 2012) using K
and L’ bands for six lenses with image separations $>1\arcsec$ and by Chiba et
al. (2005) at $11\micron$ for two lenses with image separations $\sim 1.0$ and
$0.5\arcsec$. The closer the two brightest images in a quad are together, the
less influence larger scale variations in the halo potential (e.g.
ellipticity) and the effects of time delay will have (Lemon et al., 2017).
Thus, quad images with very close separation are preferred in the search for
evidence of halo substructure.

In this paper we present observations of the quad lensed system HS 0810+2554
(Reimers et al., 2002; Chartas et al., 2014; Jackson et al., 2015; Nierenberg
et al., 2019) at wavelengths from $2-5\micron$. The background quasar is at a
redshift of ${\rm{z}}=1.5$ but the lensing galaxy is at an unknown redshift.
The two brightest images are separated by only $0.19\arcsec$ and at a redshift
of z = 1.5, the observations span a range in rest frame wavelength of
$0.8-2.0~{}\micron$. Over this wavelength range the flux arises purely from
the accretion disk at the shorter wavelength, and from a combination of
accretion disk and dusty torus at the two longer wavelengths (e.g. Kobayashi
et al., 1993; Suganuma et al., 2006).

##  2 OBSERVATIONS

We observed HS 0810 on UT 2019 February 24 with LMIRCam (Skrutskie et al.,
2010) on the LBTI (Hinz et al., 2016) using a single 8.4 m primary mirror of
the Large Binocular Telescope (LBT). The focal plane scale was $0.0108\arcsec$
per pixel. Images were made in the Ks $(2.16~{}\micron)$, L’ $(3.7~{}\micron)$
and M $(4.78~{}\micron)$ filters and the natural seeing was $0.8\arcsec$. We
also made a test observation along with a standard star on UT 2017 February 17
in the L’ filter (only) with natural seeing of $1.2\arcsec$. Filters and
integration times are listed in Table 1. The AO secondary was operating in
natural guide star mode for both epochs and had to lock onto the optical
images of the two adjacent bright images in the quad together. This probably
explains the slightly elongated images we obtained for the quad compared to
the standard star.

Table 1: LMIRCam Observations Date | Filter | frame time (ms) | Int Time (sec) | Total Int Time (min)
---|---|---|---|---
2019 | Ks $(2.16~{}\micron)$ | 1475 | 30.1 | 30.1
2019 | L’ $(3.7~{}\micron)$ | 13.7 | 1.0 | 26.7
2019 | M $(4.78~{}\micron)$ | 13.7 | 0.412 | 32.9
2017 | L’ $(3.7~{}\micron)$ | 290 | 0.874 | 9.6

For the 2017 observations in the L’ filter, several dither positions on the
array were used, but most often with a separation of $4.5\arcsec$. In 2019 we
imaged HS 0810 at two dither positions on the array, separated by $8\arcsec$.
Images at each dither position in the 2019 observations were taken in sets of
10, 100 and 200 integrations at Ks, L’, and M respectively. Slightly elongated
images, probably due to the AO locking onto the blended image, were evident.
The axis of elongation rotates between wavelengths due to the rotation of the
sky on the focal plane over the length of time the observations were made. The
images occupied a small portion of the array, minimizing errors in flat
fielding. Simple aperture photometry was used to determine the brightness of
the well isolated sources C and D in the image, whereas an elliptical point
spread function (PSF) was used to separate the slightly blended images of A
and B, as explained in Section 3. Formal statistical errors were computed
using the measured rms fluctuations in the background, which was determined to
be flat within the vicinity of sources C and D. For blended sources A and B, a
small systematic error of 0.01 mag. was added to account for any potential
residual uncertainties in the background level (see Section 3) due to the more
complicated PSF fitting.

Due to time and weather constraints in 2019, we were unable to observe
standard stars on the same night as HS 0810 to fix the fluxes at each
wavelength on a photometric scale. We have examined standard star observations
on previous nights at Ks, L’ and M and found that the relative response of the
detector between filters, adjusted for airmass and stellar colors, is
consistent to within a few percent. We used this information to build the
relative fluxes between filters for our observations of HS 0810. In this way
we can place the fluxes of the images in HS 0810 on a spectral energy
distribution (with a flux normalization) with confidence. This does not affect
the relative brightness between images at each wavelength which are, within
the photometric and systematic errors, very well determined.

##  3 Analysis

Gray scale images of a star (HD 74721) and HS 0810 from the 2017 test
observations are shown in Figure 1. The stellar image has had dithers
subtracted, but has not been processed for bad pixel and pattern removal nor
has the image been flat fielded. The image of HS 0810 has been fully
processed. The individual lensed images are labeled according to Reimers et
al. (2002). A radial profile of the stellar image is shown in Figure 2, where
the measured FWHM is $0.10\arcsec$.

Gray scale images of the quad from the 2019 observations are shown in Figure
3. The lensing galaxy itself is barely visible in the Ks band, and finding its
centroid is impossible using our data. Consequently we can not measure the
positions of the lensed images with respect to the lensing galaxy. The
positions relative to image A are, within an error of $\pm 0.003\arcsec$,
identical to the values listed in the CfA-Arizona Space Telescope LEns Survey
(CASTLES) website (Muñoz et al., 1998). The brightness for images C and D were
computed using standard aperture photometry on the images. For images A and B,
their individual point spread functions (PSF) overlap. For these two images,
the PSF was modeled with a simple two dimensional Voigt profile with a slight
ellipticity. The Gaussian width, dampening wings parameter, and orientation
(long axis) were derived from image C at each wavelength. The measured values
for the Gaussian width ($1\sigma$) were $0.055\arcsec$, $0.051\arcsec$, and
$0.064\arcsec$ at Ks, L’, and M respectively. The dampening wings made a
small, but detectable contribution to the overall PSF. The ellipticity was 15%
at Ks and L’, and 5% at M.

A synthetic image of the blend was then formed using two model Voigt profiles
with adjustable spacing and peak intensity. This model was subtracted from the
image and the spacing and peak intensity of each profile adjusted until the
residual intensity was less than 1% of the total intensity of the A+B
combination. A cut through the centers of images A and B in the L’ filter for
the 2019 observations is shown in Figure 4 along with the model fit. A slight
tilt to the baseline background can be seen in the image cut that is not
modeled. Since our PSF modeling is more complicated that the simple aperature
photometry used for images C and D, a systematic error of 0.01 mag. was added
to the error budget to account for uncertainties in method. The rest of the
error budget was determined by comparing separate sets of observations that
were co-added to make the final images and computing the statistical variation
between these subsets. The results are given in Table 2, with the ratios of
images B, C and D to A expressed as magnitude differences, e.g., for images B
and A, $\Delta m=-2.5\log({\rm{B/A}})$.

Table 2: Observed and Model Flux Ratios Expressed As Magnitude Differences Date | Filter |  $\Delta$m(B/A) | Model $\Delta$m (sec. 4) |  $\Delta$m(C/A) |  $\Delta$m(D/A)
---|---|---|---|---|---
2019 | $2.16~{}\micron$ | $0.635\pm 0.02$ | 0.638 | $1.60\pm 0.03$ | $2.75\pm 0.05$
2019 | $3.7~{}\micron$ | $0.339\pm 0.02$ | 0.358 | $1.56\pm 0.03$ | $2.45\pm 0.05$
2019 | $4.78~{}\micron$ | $0.224\pm 0.03$ | 0.223 | $1.48\pm 0.12$ | $2.39\pm 0.22$
2017 | $3.7~{}\micron$ | $0.40\pm 0.03$ | – | $1.42\pm 0.04$ | $2.22\pm 0.06$
Table 3: Observed Flux Ratio Comparison Band | Year |  $\Delta$m(B/A) |  $\Delta$m(C/A) |  $\Delta$m(D/A)
---|---|---|---|---
0.2–10 keV | 2002 | 0.20 | 0.73 | 2.3
0.2–10 keV | 2013 | 0.35 | 0.26 | 1.8
V | – | 0.78 | 1.16 | 2.07
$0.72~{}\micron$ | 2001 | 0.7 | 1.4 | 2.8
I | – | 0.85 | 1.66 | 2.98
H | – | 0.61 | 1.44 | 2.76
Ks | 2019 | 0.635 | 1.60 | 2.75
L’ | 2019 | 0.339 | 1.56 | 2.45
L’ | 2017 | 0.40 | 1.42 | 2.22
M | 2019 | 0.224 | 1.48 | 2.39
20 cm | 2014 | 0.02 | 0.37 | 0.59

Figure 5 shows images A and B with scaled contours overlaying the gray scale
image at each wavelength. In all 3 filters image A, which is the minimum in
the lensing arrival time surface, is brighter than image B, which is the
saddle point. This is consistent with these images being affected by
substructure–stars or subhalos, where the saddle is expected to be demagnified
with respect to the neighboring minimum (Schechter, & Wambsganss, 2002; Saha &
Williams, 2011). Clearly visible is the trend towards more equal brightness
with increasing wavelength. Our goal is to determine if the ratio of B/A as a
function of wavelength can be used to determine if there is substructure in
the dark matter halo of the lensing galaxy or if all of the B/A ratios are
consistent with microlensing only. Because the two images are very close on
the sky in comparison to the scale of the quad system, their light paths are
traveling through nearly identical regions in the gravitational potential of
the lensing galaxy. If there were no microlensing and the dark matter halo
associated with the lensing galaxy was smooth azimuthally in the region of
images A and B, images A and B should have identical fluxes.

We estimated the time delay between images A and B using a free-form lens
modeling software PixeLens (Williams & Saha, 2004), which is dependent on the
(unknown) redshift of the lensing galaxy. Mosquera, & Kochanek (2011) made an
estimate of $z=0.89$ for the lensing system based on basic photometric
properties of the galaxy. For $z=0.3$ ($z=0.6$, $z=0.89$), the time delay is
$\sim 0.05$ days ($\sim 0.15$ days, $\sim 0.33$ days) respectively. These
delays are much shorter than typical intrinsic quasar variability (MacLeod et
al., 2012), and comparable to the $\sim$3 hours of telescope time we spent
making the observations. For all practical purposes, our observations of the A
and B quasar images can be considered simultaneous in the rest frame.

Our flux ratios, expressed as magnitude differences, are also listed in Table
3, which includes ratios at other wavelengths from Chartas et al. (2016),
Jackson et al. (2015) and Reimers et al. (2002), along the the epoch of
observation. We have been unable to find information on when the CASTLES
observations cited in Chartas et al. (2016) were made, so no epoch is given in
the Table. Note that there is clear variation in the short wavelength (X-Ray –
Ks band) B/A ratio over time, ranging from $\Delta{\rm{m}}=0.2$ in the X-Rays
in 2002, to 0.635 at Ks in 2019. According to Mosquera, & Kochanek (2011), a
typical timescale for a microlensing event in the HS 0810 system due to
crossing the Einstein radius of a single star is $\sim 24$ yrs, a timescale
comparable to the elapsed time between the earliest and most recent
observations. For a typical projected density of stars in the lensing galaxy,
microlensing will likely result from the combined effect of many stars along
the line-of-sight, rather than a single star. Modeling by Wambsganss (1990)
shows that microlensing is a non-linear process, and both a longer term trend
in magnification in addition to variations on time scales shorter than the
crossing time, combine to create the light curve of an image. This is similar
to what we observe for the B/A ratio, which is always in the same sense
$({\rm{A}}>{\rm{B}})$, but with significant variations over the last 20 years.

To establish the range in mass and size of halo sub-structure our observations
are sensitive to, we list the relevant $source$ plane sizes in Table 4. The
values for the stellar Einstein radius RE, the accretion disk radius RAD, and
the radius of the broad line region RBLR, were taken from Mosquera, & Kochanek
(2011). We estimated the size of the outer radius of the dusty torus from
Table 1 in Sluse et al. (2013) assuming HS 0810 has a luminosity of $L\sim
5\times 10^{45}-{\rm{ergs}}-{{\rm{s}}^{-1}}$ and that the torus is represented
by the ’extended’ model. The luminosity estimate was made using the black hole
mass given in Mosquera, & Kochanek (2011) and the mass – luminosity relation
in Woo, & Urry (2002). We chose the extended torus model since the infrared
flux from HS 0810 is greater than for the typical quasar (Figure 6),
suggesting a more extensive emitting region.

Table 4: Source Plane Sizes Source | Size (pc)
---|---
RE (stellar) | 0.006
RAD | 0.0003
RBLR | 0.03
RIR (outer) | 1.0

With these size estimates we can roughly estimate the mass range for any halo
sub-structure that can significantly influence the flux ratios we measure by
using a Sersic radial profile to compute an Einstein radius. Using a suite of
numerical simulations and fitting radial profiles with an Einasto function,
Navarro et al. (2004) derive a shape parameter of $\sim 0.17$ that fits the
simulated haloes well. Using Figure 8 in Dhar, & Williams (2010), this
approximately corresponds to a Sersic profile index of $m_{S}\sim 6$. We use a
Sersic profile because the Einstein radius can be analytically calculated for
a given index and half-mass radius ${{\rm{R}}_{1/2}}$. For a mass of
$10^{4}$M⊙ and assuming the lensing galaxy is at z = 0.89, the Einstein radius
ranges from 1.1 pc for ${{\rm{R}}_{1/2}}=0$ (point mass) to 0.5 pc for
${{\rm{R}}_{1/2}}=2$ pc projected onto the source plane. These sizes are
comparable to the size of the infrared torus and represents a rough lower
limit to the mass we can detect. The distance between images A and B in the
lensing plane is $\sim 1.45$ kpc, again assuming the lensing galaxy is at z =
0.89. This corresponds to an Einstein radius for a point mass of $2.1\times
10^{10}$M⊙, or $1\times 10^{10}$M⊙ for ${{\rm{R}}_{1/2}}=2$ kpc. This
represents a rough upper limit to the mass that can influence our
observations.

The mean spectral energy distribution (SED) for quasars has been computed by
Padovani et al. (2017) and Krawczyk et al. (2015). They find that the
combination of a small, sub-parsec sized accretion disk with a simple power
law SED well matches the rest frame optical flux. At longer wavelengths,
starting at about $1.2~{}\micron$, flux from a much larger (few parsecs) dusty
torus begins to contribute and eventually dominates the SED. Krawczyk et al.
(2015) found a mean power law index for the accretion disk of
F${}_{\nu}\propto\nu^{-0.4}$. The dust in the torus is at a temperature of
$\sim 1500$K (e.g. Kobayashi et al., 1993; Suganuma et al., 2006), and the
flux from the dusty torus drops very rapidly shortward of $1.2~{}\micron$,
which is on the Wien side of the emission spectrum for the dust. Our 3.7 and
$2.16~{}\micron$ filters correspond to wavelengths of 1.48 and
$0.86~{}\micron$ in the rest frame of the lensed quasar. At these rest
wavelengths, the flux from a ${\rm{T}}=1500$K black body in
$\nu{\rm{F}}_{\nu}$ at the shorter wavelength is a factor of 12 lower than at
the longer wavelength. Given that the dust emission in our $3.7~{}\micron$
filter is only about 40% (see below) of the total, the contribution to our
$2.16~{}\micron$ filter from hot dust emission is only $\sim 3$%, and can be
ignored.

We plot the SED for the entire (within a single photometric aperture) HS 0810
quad in Figure 6, along with the mean SED for quasars taken from Krawczyk et
al. (2015). The flux values plotted for HS 0810 were taken from the WISE,
2MASS (Skrutskie et al., 2006), UKDISS (Hambly et al., 2008) catalogs, and a
quasar catalog by Souchay et al. (2015). As noted above, the K filter is
likely dominated by the small accretion disk, and consequently could show
considerable variation in flux over time due to stellar microlensing. For this
reason we plot the observed J and K band fluxes as a vertical bar, indicating
the range of fluxes between these data sets. For each of 2MASS and UKIDSS
surveys, the J, H and K observations were made at the same time, so the J–K
colors should represent the quasar accretion disk continuum at the epoch of
observation. The 2MASS and UKIDSS H filter observations are strongly
contaminated by H$\alpha$ and do not represent the continuum. The J band
observations suggest a somewhat steeper (bluer) power law index of -0.3, but
this filter could be contaminated by [OIII], although the effect should be
small.

We only have relative flux levels between our infrared filters and we have no
simultaneous red and optical photometry. This makes it difficult to
empirically determine the power law slope for the accretion disk in HS 0810 at
the epoch of our observations. Based on the J–K colors from 2MASS and UKIDSS,
we will fix the power law slope at -0.3 as shown in Figure 7. We have
vertically scaled our relative flux measurements for the A+B pair to match the
trend between the WISE W1 and W2 filters. Flux from images A and B by far
dominate our images and would be expected to dominate the overall SED, and we
will assume the $\bf{shape}$ of the overall SED represents the $\bf{shape}$ of
the SED for the combination of images A and B. With this normalization, our Ks
($2.16~{}\micron$) point fits within the lower range of previous K band
observations, although this is not a given.

##  4 Lensing Model

In Figure 7 we plot the SED for HS 0810 where the dashed blue line shows a
simple accretion disk power law with ${{\rm{F}}_{\nu}}\propto{\nu^{-0.3}}$
that passes through our Ks flux. Working with this figure, we developed a
simple model for the flux from images A and B by making the following
assumptions. 1) The Ks flux is due entirely to a small sized (fraction of a
parsec) accretion disk that can suffer microlensing in the lensing galaxy. 2)
The longer wavelength flux consists of a contribution from the accretion disk
(extrapolated from the Ks flux) and a contribution from a larger dusty torus
that does not experience any microlensing. Expressed mathematically, we have:

$\displaystyle{I_{\lambda}}$ $\displaystyle=$
$\displaystyle{I_{{{\rm{B}}_{\lambda}}}}+{I_{{{\rm{A}}_{\lambda}}}}$
$\displaystyle{I_{{{\rm{B}}_{\lambda}}}}$ $\displaystyle=$
$\displaystyle{I_{{{\rm{B}}_{\lambda{\rm{AD}}}}}}+{I_{{{\rm{B}}_{\lambda{\rm{IR}}}}}}$
(1) $\displaystyle{I_{{{\rm{A}}_{\lambda}}}}$ $\displaystyle=$
$\displaystyle{MmI_{{{\rm{B}}_{\lambda{\rm{AD}}}}}}+{MI_{{{\rm{B}}_{\lambda{\rm{IR}}}}}}$

$I_{\lambda}$ is the total flux from A and B, AD indicates flux from the
accretion disk alone, and IR represents the flux from the torus alone, as
indicated in Figure 7. The factor $m$ is the ratio of $I_{\rm{A}}/I_{\rm{B}}$
due to microlensing. If lensing due to substructure in the dark matter halo
was present in addition to microlensing, an additional magnification factor
$M$ must be added. This factor, which we will call millilensing, effects both
the emission from the accretion disk and the infrared emission from the torus.
Observations of HS 0810 at radio wavelengths (Jackson et al., 2015) and in
optical narrow emission lines (Nierenberg et al., 2019), neither of which
should be subject to microlensing, are consistent with no millilensing. For
this reason we initially set this value to $M=1.0$. Given the observed SED and
our assumptions listed above, we can solve for the factor $m$ that provides a
best fit to the data at all wavelengths and compare to the observations,
where:

$\displaystyle{R_{\lambda}}$ $\displaystyle=$
$\displaystyle{\left({\frac{{{I_{{\rm{IR}}}}}}{{{I_{{\rm{AD}}}}}}}\right)_{\lambda}}$
(2)
$\displaystyle\frac{{{I_{{{\rm{B}}_{\lambda}}}}}}{{{I_{{{\rm{A}}_{\lambda}}}}}}$
$\displaystyle=$
$\displaystyle\frac{{1+M+{R_{\lambda}}\left({1+mM}\right)}}{{mM(1+M)+{R_{\lambda}}(1+mM)}}$

The results of our model fit for the case with only microlensing, no
millilensing $(M=1)$, are shown in Figure 8, where we have expressed the flux
ratios as a magnitude difference for comparison with the data (section 3). The
data from Table 2 are plotted as points and the model fit corresponding to
$m=1.79$ is plotted as a solid line. The predicted magnitude differences from
the model are listed in Table 2 along with the observed differences.

Our simple model fits the data well, and are consistent with A/B ratios at
radio wavelengths (Jackson et al., 2015) of 1.02+/- 0.06 and model results
using narrow line emission (Nierenberg et al., 2019) that imply a flux ratio
of 1.0 +/- 0.07. If substructure were present at a level that could
significantly influence the observed flux ratios, then the measured ratios for
B/A at the two longer wavelengths would lie above or below the simple model
that is compatible with the flux ratio at Ks. In this case, the $\bf{product}$
$m\times M$ determines the flux ratio at Ks and the inclusion of the
millilensing factor changes the flux ratios at L’ and M from what is observed.
In Figure 9 we plot the probability contours for our model in the $m,M$ plane.
The red point is the case for microlensing only, no millilensing. The contours
take on the shape of a very narrow band and there is a significant degeneracy
along the line $m\times M=1.79$. The extent of this degeneracy can be reduced
only by the inclusion of longer wavelengths where the contribution of the
accretion disk becomes insignificant. The small errors on our flux ratios make
the contours very tight perpendicular to the $m\times M=1.79$ line. Changing
the power law slope we used for the accretion disk moves the contours along
the line $m\times M=1.79$ where steepening the power law index (bluer) will
move the contours up to the left, and vice verse.

The contribution for HS 0810 from the accretion disk in our simple model
continues to be a factor even at longer wavelengths (see Sluse et al., 2013).
Our model predicts a flux ratio of B/A = 0.9/1 will not occur until
wavelengths longer than $\sim 10\micron$. The FWHM of the LBT at this
wavelength is $0.3\arcsec$, but with a high signal-to-noise observation it is
still possible to extract an accurate flux ratio given the very stable
telescope PSF at this wavelength and the known separation of the two point
sources. The source brightness at $10~{}\micron$ is $\sim 30$ mJy, which is
sufficient given LBTI’s high sensitivity at infrared wavelengths. Although the
upcoming JWST will be able to image much deeper at thermal MIR wavelengths
than ground based telescopes such as the LBT, with a smaller 6 meter primary
it will be difficult to resolve adjacent images in quad systems such at HS
0810. Davidson et al. (2019) report detection of all four images at a
wavelength of 2.1mm in the continuum and in CO line emission using ALMA. At
this long wavelength the emitting region is likely considerably larger than
the hot, inner dusty torus we observe in the MIR, and the images may be
insensitive to millilensing on scales smaller than a few parsec, but capable
of detecting substructure on larger scales. MIR imaging on the next generation
of very large, ground based telescopes equipped with an infrared AO system
presents a future opportunity to further constrain the presence of
millilensing due to sub-structure in dark matter halos. Alternative to long
wavelength imaging, observations of emission from the quasar narrow line
region, also unaffected by microlensing, shows promise (e.g. Nierenberg et
al., 2019), but also requires high angular resolution.

##  5 Conclusions

We have imaged the lensed quasar system HS 0810+2554 at wavelengths spanning
$2-5\micron$ with LMIRCam on the LBTI using the adaptive optics system. Flux
ratios were computed for all four images in each filter. The two brightest
images are $0.187\arcsec$ apart, and have very high signal to noise at all
wavelengths. At the shorter wavelength of $2.16\micron$ (rest wavelength
$0.86\micron$) the flux is entirely consistent with flux from only the sub-
parsec accretion disk of the quasar. At the two longer wavelengths, the flux
is a combination of flux from the accretion disk and flux from a larger (few
pc) dusty torus. A simple model employing multiplicative factors for image B
due to stellar microlensing $(m)$ and sub-structure millilensing $(M)$ was
presented. The result is tightly constrained to the product $m\times M=1.79$.
The 60% probability contour for this product stretches from $m=2.6$, $M=0.69$
to $m=1.79$, $M=1.0$, where the later is consistent with microlensing only.
Note that we have not ruled out the presence of millilensing, but have shown
that the microlensing only case is within the 60% probability contour.

##  6 Acknowledgments

This publication makes use of data products from the Wide-field Infrared
Survey Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration.

This publication makes use of data products from the Two Micron All Sky
Survey, which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation.

We thank Gerd Weigelt and Charles (Chick) Woodward for providing standard star
observations using LMIRCam on different dates.

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Figure 1: Gray scale images of a standard star and the HS 0810 quad system in
the L’ band on 2017 February 17. The image of the star is only partially
processed. The image of HS 0810 is fully processed and all four images of the
source quasar are easily seen. Figure 2: Radial profile of the standard star
shown in Figure 1. The profile is consistent with a Gaussian with a FWHM of
$0.10\arcsec$. Figure 3: Gray scale images of the HS 0810 Quad system at the
three observed wavelengths. The lensing galaxy is just detectable at Ks left
of center in the image. Figure 4: Intensity (in ADUs) cut along an axis
through the centers of images A and B at L’ plotted as a black line. The model
fit using two Voight functions is shown in red. The separation of the two
peaks in the model is $0.187\arcsec\pm 0.003\arcsec$ Figure 5: Contours for
images A and B arbitrarily scaled to produce 11 contours from sky to peak
brightness. Note the clear trend towards more equal brightness with increasing
wavelength. Figure 6: Filled Circles: the spectral energy distribution of HS
0810 taken from the literature (see text). Blue Line: the mean quasar SED from
Krawczyk et al. (2015). Red open circles: combined flux in our filters scaled
vertically to match the trend between the WISE W1 and W2 filters. The vertical
bars indicate the range in fluxes between 2MASS and UKIDSS J and K band
observations. The H band $(1.65~{}\micron)$flux is not plotted due to strong
contamination by H$\alpha$. Figure 7: Similar to Figure 6 except except a
power law fit (dashed blue line) to the short wavelength portion is used to
extrapolate the flux contribution from the physically small accretion disk to
longer wavelengths. This power law fit is slightly steeper than the mean power
law index shown in Figure 6 (see text). This yields the ‘AD’ terms in Eq. 1.
The vertical dashed lines indicate the location of our three filter
bandpasses, and the difference between the accretion disk SED and the observed
flux gives the ‘IR’ terms in Eq. 1. Figure 8: Magnitude difference in
brightness between images A and B as a function of observing wavelength. The
red line is the model discussed in the text which has only microlensing, no
millilensing $({\rm{M}}=1)$.  Figure 9: Goodness of fit for the model as a
function of microlensing (m) and millilensing (M). The outer contour is the
99% confidence level with interior contours of 90, 80, and 70%. The red point
is the best fit model with microlensing only $(M=1)$. The Blue point
corresponds to the minimum in $\chi^{2}$ with $m=2.11$ and $M=0.85$.