OU-HEP-240301

Weak scale supersymmetry
emergent from the string landscape

Howard Baer1111Email<EMAIL_ADDRESS>, Vernon Barger2222Email:
<EMAIL_ADDRESS>Dakotah Martinez1444Email<EMAIL_ADDRESS>and Shadman Salam3444Email<EMAIL_ADDRESS>
1Homer L. Dodge Department of Physics and Astronomy,
University of Oklahoma, Norman, OK 73019, USA
2Department of Physics, University of Wisconsin, Madison, WI 53706 USA
3Department of Mathematics and Natural Sciences, Brac University, Dhaka 1212,
Bangladesh

Superstring flux compactifications can stabilize all moduli while leading to
an enormous number of vacua solutions, each leading to different $4-d$ laws of
physics. While the string landscape provides at present the only plausible
explanation for the size of the cosmological constant, it may also predict the
form of weak scale supersymmetry which is expected to emerge. Rather general
arguments suggest a power-law draw to large soft terms, but these are subject
to an anthropic selection of not-too-large a value for the weak scale. The
combined selection allows one to compute relative probabilities for the
emergence of supersymmetric models from the landscape. Models with weak scale
naturalness appear most likely to emerge since they have the largest parameter
space on the landscape. For finetuned models such as high scale SUSY or split
SUSY, the required weak scale finetuning shrinks their parameter space to tiny
volumes, making them much less likely to appear compared to natural models.
Probability distributions for sparticle and Higgs masses from natural models
show a preference for Higgs mass $m_{h}\sim 125$ GeV with sparticles typically
beyond present LHC limits, in accord with data. From these considerations, we
briefly describe how natural SUSY is expected to be revealed at future LHC
upgrades. This article is a contribution to the Special Edition of the journal
Entropy honoring Paul Frampton on his 80th birthday.

##  1 Introduction

Superstring theory provides the most promising avenue for unifying the
Standard Model with gravity, but at the cost of requiring 6 or 7 extra spatial
dimensions[1, 2, 3, 4, 5]. The low energy limit $E<m_{P}$ (where $m_{P}$ is
the reduced Planck mass) of string theory, once Kaluza-Klein modes are
integrated out, is expected to be $10-d$ supergravity (SUGRA). The $10-d$
SUGRA theory is then assumed to be compactified to a tiny $6-d$ space $K$
tensored with our usual $4-d$ (approximately) Minkowski spacetime $M_{4}$:
$M_{10}=M_{4}\times K$. Originally, $K$ was taken to be a $6-d$ compact Ricci-
flat Kähler manifold with special holonomy[6]; such a Calabi-Yau manifold
admits a conserved Killing spinor leading to a remnant $N=1$ supersymmetry
(SUSY) on $M_{4}$.

The cosmological constant (CC) problem remained a thorny issue until the early
2000s when it was realized that string flux compactifications could lead to an
enormous number of vacuum states each with different $4-d$ laws of physics,
and in particular, different $\Lambda_{CC}$ values[7]. Such large numbers of
vacuum states ($N_{vac}\sim 10^{500}$ is an oft-quoted number[8]) provided a
setting for Weinberg’s anthropic solution to the CC problem[9]. But if the
landscape[10] of string vacua provides a solution to the CC problem, might it
also enter into other naturalness problems, such as the $m_{weak}/m_{P}$ (or
related, $m_{SUSY}/m_{P}$) hierarchy problems (where $m_{P}\simeq 2.4\times
10^{18}$ GeV)?

In this contribution to the volume of the journal Entropy honoring Paul
Frampton on his 80th birthday, we address this question. Here, we will put
forward arguments in favor not only of weak scale SUSY as emergent from the
string landscape, but indeed of a special form of weak scale SUSY (WSS)– SUSY
with radiatively-driven naturalness[11, 12], or stringy-natural SUSY[13]. The
specific form of WSS predicts at present that a light SUSY Higgs boson with
mass $m_{h}\simeq 125$ GeV should emerge, whilst sparticles masses are at
present somewhat or well beyond reach of the CERN Large Hadron Collider
(LHC)[14]. It also allows us to predict a variety of SUSY signatures which may
allow for SUSY discovery at LHC luminosity upgrades over the upcoming years.
Perhaps most important of these are the soft isolated opposite-sign
dileptons+MET which arise from light higgsino pair production[15] and which
recoil against a hard initial state jet radiation[16, 17, 18, 19]. At present,
both ATLAS[20] and CMS[21] with 139 fb-1 seem to have $2\sigma$ excesses in
this channel, and an associated monojet signal may also be emerging[22].

##  2 Approximate supersymmetric vacua from string theory

The main motivation for SUSY is that it provides a ’tHooft technical
naturalness solution to the gauge hierarchy problem via the cancellation of
quadratic divergences associated with the Higgs sector. This is true for SUSY
breaking at any energy scale below $m_{P}$, since in the limit of
$m_{SUSY}\rightarrow 0$, the model becomes more (super)symmetric. Thus, SUSY
provides a technically natural solution to the so-called Big Hierarchy
problem.

Specific motivation for weak scale SUSY comes from the Little Hierarchy
problem and what we call practical naturalness[23, 24]: an observable ${\cal
O}$ is practically natural if all independent contributions to ${\cal O}$ are
comparable to or less than ${\cal O}$. For the case of WSS, we can relate the
weak scale $m_{weak}\sim m_{W,Z,h}\sim 100$ GeV to the weak scale soft SUSY
breaking terms and SUSY conserving $\mu$ term via the scalar potential
minimization conditions:

$\frac{m_{Z}^{2}}{2}=\frac{m_{H_{d}}^{2}+\Sigma_{d}^{d}-(m_{H_{u}}^{2}+\Sigma_{u}^{u})\tan^{2}\beta}{\tan^{2}\beta-1}-\mu^{2}\sim-
m_{H_{u}}^{2}-\Sigma_{u}^{u}(\tilde{t}_{1,2})-\mu^{2}$ (1)

where $m_{H_{u,d}}^{2}$ are the soft SUSY breaking Higgs masses and the
$\Sigma_{u,d}^{u,d}$ contain an assortment of loop corrections to the scalar
potential (explicit formulae are included in Ref’s [12]). A measure of
practical naturalness $\Delta_{EW}$ can be defined which compares the largest
(absolute) contribution to the right-hand-side of Eq. 1 to $m_{Z}^{2}/2$.
Requiring $\Delta_{EW}\lesssim 30$ fulfills the practical naturalness
condition. From Eq. 1, we see immediately that $m_{H_{u}}^{2}$ must be driven
to small negative values at the weak scale while the $\mu$ term must also be
$\mu\sim 100-350$ GeV. The latter condition means the higgsinos are usually
the lightest SUSY particles, and the only ones required to be $\sim m_{weak}$.
The other sparticles enter via the $\Sigma_{u,d}^{u,d}$ terms and hence are
suppressed by loop factors, and so can live in the TeV or beyond range. We
shall see shortly that practical naturalness is closely linked to selection of
SUSY models on the landscape.

On the theory side, we expect the $4-d$ vacua emergent from the landscape to
often contain some remnant SUSY.

  * •

Remnant spacetime SUSY: In Ref. [25], Acharya argues that all stable, Ricci-
flat manifolds in dimensions $<11$ have special holonomy, and consequently a
conserved Killing spinor. If so, then some remnant spacetime SUSY should exist
in the $4-d$ low-energy effective field theory (LE-EFT).

  * •

EW stability: A problem for the Standard Model to be the low-energy effective
field theory for $E<m_{P}$ is electroweak stability in that the Higgs quartic
term $\lambda$ may evolve to negative values at $E>m_{GUT}$ leading to a
runaway scalar potential. For $m_{t}\sim 173.2$ GeV, the SM is just on the
edge of metastability/runaway[26, 27]. The Minimal Supersymmetric Standard
Model (MSSM) has no such problem since for the MSSM the quartic couplings
involve the gauge couplings which are always positive.

  * •

Landscape vacua stability: In Ref’s [28, 29], Dine et al. ask what sort of
conditions can stabilize landscape deSitter vacua against decay to AdS vacua,
leading to a big crunch. The presence of SUSY leads to absolutely stable vacua
whilst the presence of approximate (broken) SUSY leads to (metastable) vacua
decay rates $\Gamma\sim m_{P}e^{-m_{P}^{2}/m_{3/2}^{2}}$ far beyond the age of
the universe.

  * •

Hierarchy of scales: While a hierarchy of scales is typically hard to come by
in many BSM models, SUSY models allow for dynamical SUSY breaking[30] where
the SUSY breaking scale $m_{hidden}$ is gained via dimensional transmutation
$m_{hidden}\sim m_{P}e^{-8\pi^{2}/bg^{2}}$ and where the soft terms are
developed as $m_{soft}\sim m_{hidden}^{2}/m_{P}$ under gravity-
mediation.111Here, we only consider gravity-mediation since gauge mediation
leads to tiny trilinear soft terms $A$ which then require unnatural top-squark
contributions $\Sigma_{u}^{u}$ to gain $m_{h}\sim 125$ GeV[31].

  * •

Harmony: Witten emphasizes that consistent QFTs exist for spin-0, 1/2, 1, 3/2
and 2. The graviton is the physical spin-2 particle and the spin-3/2 Rarita-
Schwinger gravitino field would exist as the superpartner of the graviton,
thus filling out all allowed spin states.

In addition, WSS is motivated experimentally by a variety of measurements.

  * •

The measured values of the gauge couplings unify under MSSM RG evolution while
not under most other BSM extensions, including the SM itself[32].

  * •

The measured top-quark mass is large enough to seed the required radiative
breakdown of EW symmetry[33].

  * •

The measured value of $m_{h}\simeq 125$ GeV falls squarely into the range
allowed by the MSSM: $m_{h}\lesssim 130$ GeV[34].

  * •

Precision EW corrections tend to prefer the (heavy spectra) MSSM over the
SM[35].

It is often complained, with good reason, that gravity mediation has its own
flavor and CP problems, the former arising from operators such as $\int
d^{4}\theta S^{\dagger}SQ_{i}^{\dagger}Q_{j}/m_{P}^{2}$ where $S$ is a hidden
sector superfield obtaining a SUSY breaking vev $F_{S}\sim 10^{11}$ GeV and
the $Q_{i}$ are visible sector chiral superfields with generation index
$i,j=1-3$. Since no symmetry forbids such flavor mixing, then FCNCs are
expected to be large in gravity-mediated SUSY breaking (historically, this
strongly motivated the search for flavor conserving models such as gauge-
mediation and sometimes anomaly-mediation.) It is pointed out in Ref. [36]
that the landscape provides its own decoupling/quasi-degeneracy solution to
the SUSY flavor and CP problems by pulling first/second generation matter
scalars to a flavor-independent upper bound in the 20-40 TeV range.

For these reasons, we will assume a so-called “fertile patch” or friendly
neighborhood[37] of the string landscape: those vacua which include the MSSM
as the LE-EFT and where only the CC and the magnitude of the weak scale scan
within the landscape. In this case, Yukawa couplings and gauge couplings are
instead fixed by string dynamics rather than environmental selection. This
leads to predictive landscape models[37]: if the CC is too large, then large
scale structure will not form which seems required for complexity to emerge
(the structure principle, leading to Weinberg’s successful prediction of
$\Lambda_{CC}$). Only the magnitude of the weak scale scans. If
$m_{weak}^{PU}\gtrsim 4m_{weak}^{OU}$, then the down-up quark mass difference
becomes so large that neutrons are no longer stable in nuclei and the only
atoms formed in the early universe are Hydrogen. If $m_{weak}^{PU}\lesssim
0.5m_{weak}^{OU}$, then we get a universe with only neutrons. This is the
atomic principle[38], since complex nuclei are also apparently needed for
complexity to emerge in any pocket universe (PU) within the greater multiverse
(and where OU refers to $m_{weak}$ in our universe).

##  3 Natural SUSY from the landscape

It is emphasized by Douglas that the CC scans independently of the SUSY
breaking scale in the landscape[39]. For the SUSY breaking scale, we expect
the vacua to be distributed as

$dN_{vac}\sim f_{SUSY}\cdot f_{EWSB}\cdot dm_{SUSY}^{2}$ (2)

where $m_{SUSY}$ is the overall hidden sector SUSY breaking scale expected to
be $\sim 10^{11}$ GeV such that the scale of soft terms is given by
$m_{soft}\sim m_{SUSY}^{2}/m_{P}$.

###  3.1 Distribution of soft breaking terms on the landscape

How is $f_{SUSY}$ distributed? Douglas[39] emphasizes that there is nothing in
string theory to favor any particular SUSY breaking vev over another, and
hence $m_{soft}$ would be distributed as a power-law

$f_{SUSY}\sim m_{soft}^{2n_{F}+n_{D}-1}$ (3)

where $n_{F}$ is the total number of (complex-valued) $F$-breaking fields and
$n_{D}$ is the total number of (real-valued) $D$-breaking fields contributing
to the overall scale of SUSY breaking
$m_{SUSY}^{4}=\sum_{i}|F_{i}|^{2}+\sum_{\alpha}D_{\alpha}^{2}$. The prefactor
of 2 in the exponent comes since the $F_{i}$ are distributed randomly as
complex numbers. For the textbook case of SUSY breaking via a single $F$ term,
then we expect $f_{SUSY}\sim m_{soft}^{1}$, i.e. a linear draw to large soft
terms. If more hidden fields contribute to the overall SUSY breaking scale,
then the draw to large soft terms will be a stronger power-law.

While the overall SUSY breaking scale is distributed as a power-law, the
different functional dependence[40, 41, 42] of the soft terms on the hidden
sector SUSY breaking fields means that gaugino masses, the trilinear soft
terms and the various scalar masses will effectively scan independently on the
landscape[43]. Now it is an advantage that different scalar mass-squared terms
scan independently (as expected in SUGRA) since the first/second generation
scalars get pulled to much higher values than 3rd generation scalars, while
the two Higgs soft masses are also non-universal and scan independently. This
situation is borne out in Nilles et al. mini-landscape where different fields
gain different soft masses due to their different geographical locations on
the compactification manifold[44]. In terms of gravity mediation, then the so-
called $n$-extra-parameter non-universal Higgs model (NUHMn) with
parameters[45, 46]

$m_{0}(i),\ m_{H_{u}},\ m_{H_{d}},\ m_{1/2},A_{0},\ \tan\beta\ \ \ \ \
(NUHM4)$ (4)

provides the proper template. Since the matter scalars fill out a complete
spinor rep of $SO(10)$, we assume each generation $i=1-3$ is unified to
$m_{0}(i)$. Also, for convenience one may ultimately trade $m_{H_{u}}$ and
$m_{H_{d}}$ for the more convenient weak scale parameters $m_{A}$ and $\mu$.
One may also build (and scan separately) the natural anomaly-mediated SUSY
breaking model[47, 48] (nAMSB) and the natural generalized mirage mediation
model[49] (nGMM).

###  3.2 The ABDS window

The anthropic selection on the landscape comes from $f_{EWSB}$. This involves
a rather unheralded prediction of the MSSM: the value of the weak scale in
terms of soft SUSY breaking parameters and $\mu$, as displayed in Eq. 1.
However, in the multiverse, here we rely on the existence of a friendly
neighborhood[37] wherein the LE-EFT contains the MSSM but where only
dimensionful quantities such as $\Lambda_{CC}$ and $v_{u}^{2}+v_{d}^{2}$ scan,
whilst dimensionless quantities like gauge and Yukawa couplings are determined
by dynamics. This assumption leads to predictivity as we shall see.

Under these assumptions, then we ask what conditions lead to complex nuclei,
atoms as we know them, and hence the ability to generate complex lifeforms in
a pocket universe? For different values of soft terms, frequently one is
pushed into a weak scale scalar potential with charge-or-color breaking minima
(CCB) where one or more charged or colored scalar mass squared is driven
tachyonic (i.e., $m^{2}<0$). Such CCB minima must be vetoed. Also, for too
large of values of $m_{H_{u}}^{2}$, then its value is not driven to negative
values and EW symmetry is not broken. These we label as “no EWSB” and veto
them as well. In practice, we must check boundedness of the scalar potential
from below in the vacuum stability conditions and that the origin of field
space has been destabilized at tree-level.

At this point, we are left with (MS)SM vacua where EW symmetry is properly
broken, but where $m_{weak}\sim m_{W,Z,h}$ is at a different value from what
we see in our universe. Here, we rely on the prescient analysis of Agrawal,
Barr, Donoghue and Seckel (ABDS)[50, 38]. If the derived value of the weak
scale is bigger than ours by a factor $(2-5)$, then the light quark mass
difference $m_{d}-m_{u}$ becomes so large that neutrons are no longer stable
in the nucleus and nuclei with $Z\gg N$ are not bound; such pocket universes
would have nuclei of single protons only, and would be chemically inert.
Likewise, if the PU value of the weak scale is a factor $\sim 0.5$ less than
our measured value, then one obtains a universe with only neutrons– also
chemically inert. The ABDS window of allowed values is that

$0.5m_{weak}^{OU}<m_{weak}^{PU}\lesssim 4m_{weak}^{OU}$ (5)

where we take the $(2-5)m_{weak}^{OU}$ to be $\sim 4m_{weak}^{OU}$ for
definiteness, which is probably a conservative value. It is very central to
our analysis and so is displayed in Fig. 1. Our anthropic condition $f_{EWSB}$
is then that the scalar potential lead to minima with not only appropriate
EWSB, but also that the derived value of the weak scale lie within the ABDS
window. Vacua not fulfilling these conditions must be vetoed.222Early papers
on this topic used instead a naturalness “penalty” of $f_{EWSB}\sim
m_{weak}^{2}/m_{SUSY}^{2}$; this condition would allow for many of the vacua
which are forbidden by our approach.

Figure 1: The ABDS-allowed window within the range of $m_{Z}^{PU}$ values.

###  3.3 EW natural SUSY emergent from the landscape

The next goal is to build a toy simulation of our friendly neighborhood of the
string landscape. We can generate the soft terms of Eq. 4 according to a
power-law selection, usually taken to be $n=2n_{F}+n_{D}-1=1$ (linear draw).
While Eq. 3 favors the largest possible soft terms, the anthropic veto
$f_{EWSB}$ places an upper bound on such terms because usually large soft
terms lead to too large a value of $m_{weak}^{PU}$ beyond the ABDS window. The
trick is to take the upper bound on scan limits beyond the upper bound posed
by $f_{EWSB}$. However, in some cases larger soft terms are more apt to lie
within the ABDS window. A case in point is $m_{H_{u}}^{2}$: the smaller its
value, the deeper negative it runs to unnatural values at the weak scale,
while as it gets larger, then it barely runs negative: EW symmetry is barely
broken. As its high scale value becomes even larger, it doesn’t run negative
by $m_{weak}$, and EW symmetry is not broken– such vacua are vetoed. Also, for
small $A_{0}$, the $\Sigma_{u}^{u}(\tilde{t}_{1,2})$ terms can be large. When
$A_{0}$ becomes large negative, then canellations occur in
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ such that these loop corrections then lie
within the ABDS window: large negative weak scale $A$ terms make
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ more natural while raising the light Higgs
mass $m_{h}\sim 125$ GeV.

A plot of the weak scale values of $m_{H_{u}}$ and $\mu$ is shown in Fig. 2
(taken from Ref. [51]) for the case where all radiative corrections– some
negative and some positive[52]– lie within the ABDS window. The ABDS window
lies between the red and green curves. Imagine playing darts with this target,
trying to land your dart within the ABDS window. There is a large region to
the lower-left where both $m_{H_{u}}$ and $\mu$ are $\lesssim 350$ GeV which
leads to PUs with complexity. Alternatively, if you want to land your dart at
a point with $\mu\sim 1000$ GeV, then the target space has pinched off to a
tiny volume: the target space is finetuned and your dart will almost never
land there. The EW natural SUSY models live in the lower-left ABDS window
while finetuned SUSY models with large $\Delta_{EW}$ lie within the extremely
small volume between the red and green curves in the upper-right plane. This
tightly-constrained region is labeled by split SUSY[53], high scale SUSY[54]
and minisplit[55].

It is often said that landscape selection offers an alternative to naturalness
and allows for finetuned SUSY models. After all, isn’t the CC finetuned?
However, from Fig. 2 we see that models with EW naturalness (low
$\Delta_{EW}$) have a far greater relative probability to emerge from
landscape selection than do finetuned SUSY models.

Figure 2: The $\mu^{PU}$ vs. $\sqrt{-m_{H_{u}}^{2}(weak)}$ parameter space in
a toy model ignoring radiative corrections to the Higgs potential. The region
between red and green curves leads to $m_{weak}^{PU}<4m_{weak}^{OU}$ so that
the atomic principle is satisfied.  Figure 3: The value of $m_{H_{u}}(weak)$
vs. $\mu^{PU}$ The green points denote vacua with appropriate EWSB and with
$m_{weak}^{PU}<4m_{weak}^{OU}$ so that the atomic principle is satisfied. Blue
points have $m_{weak}^{PU}>4m_{weak}^{OU}$.

In Fig. 3 (from Ref. [51]), we perform a numerical exercise to generate high
scale SUSY soft terms in accord with an $n=1$ draw in Eq. 3. The green dots
are viable vacua states with appropriate EWSB and $m_{weak}^{PU}$ within the
ABDS window. While some dots do land in the finetuned region, the bulk of
points lie within the EW natural SUSY parameter space.

An alternative view is gained from Fig. 4 from Ref. [56]. Here, we compute
contributions to the scalar potential within a variety of SUSY models
including RNS (radiatively-driven natural SUSY[11]), CMSSM[57], G2MSSM[58],
high scale SUSY[59], spread SUSY[60], minisplit[55], split SUSY[53] and the SM
with cutoff $\Lambda=10^{13}$ TeV, indicative of the neutrino see-saw
scale[61]. The $x$-axis is either the SM $\mu$ parameter or the SUSY $\mu$
parameter while the $y$-axis is the calculated value of $m_{Z}$ within the PU.
The ABDS window is the horizontal blue-shaded region. For $\mu$ distributed as
equally likely at all scales (integrates to a log), then the length of the
$x$-axis interval leading to $m_{Z}^{PU}$ within the ABDS window can be
regarded as a relative probability measure $P_{\mu}$ for the model to emerge
from the landscape. There is a substantial interval for the RNS model, but for
finetuned SUSY models, the interval is typically much more narrow than the
width of the printed curves. We can see that finetuned models have only a tiny
range of $\mu$ values which allow habitation within the ABDS window.

Figure 4: Values of $m_{Z}^{PU}$ vs. $\mu_{PU}$ or $\mu_{SM}$ for various
natural (RNS) and unnatural SUSY models and the SM. The ABDS window extends
here from $m_{Z}^{PU}\sim 50-500$ GeV.

Using the magic of algebra, the width of the $\mu$ intervals can be computed,
and the results are in Table 1. Here, $P_{\mu}$ is to be considered as a
relative probability. From the Table, we see that the SM is about $10^{-27}$
times likely to emerge as compared to RNS. Minisplit is $10^{-4}-10^{-8}$
times likely to emerge (depending on the version of minisplit). Even the once-
popular CMSSM model is $\sim 10^{-3}$ times likely than RNS to emerge from the
landscape.

model | $\tilde{m}(1,2)$ | $\tilde{m}(3)$ | gauginos | higgsinos | $m_{h}$ | $P_{\mu}$
---|---|---|---|---|---|---
SM | - | - | - | - | - | $7\cdot 10^{-27}$
CMSSM ($\Delta_{EW}=2641$) | $\sim 1$ | $\sim 1$ | $\sim 1$ | $\sim 1$ | $0.1-0.13$ | $5\cdot 10^{-3}$
PeV SUSY | $\sim 10^{3}$ | $\sim 10^{3}$ | $\sim 1$ | $1-10^{3}$ | $0.125-0.155$ | $5\cdot 10^{-6}$
Split SUSY | $\sim 10^{6}$ | $\sim 10^{6}$ | $\sim 1$ | $\sim 1$ | $0.13-0.155$ | $7\cdot 10^{-12}$
HS-SUSY | $\gtrsim 10^{2}$ | $\gtrsim 10^{2}$ | $\gtrsim 10^{2}$ | $\gtrsim 10^{2}$ | $0.125-0.16$ | $6\cdot 10^{-4}$
Spread ($\tilde{h}$LSP) | $10^{5}$ | $10^{5}$ | $10^{2}$ | $\sim 1$ | $0.125-0.15$ | $9\cdot 10^{-10}$
Spread ($\tilde{w}$LSP) | $10^{3}$ | $10^{3}$ | $\sim 1$ | $\sim 10^{2}$ | $0.125-0.14$ | $5\cdot 10^{-6}$
Mini-Split ($\tilde{h}$LSP) | $\sim 10^{4}$ | $\sim 10^{4}$ | $\sim 10^{2}$ | $\sim 1$ | $0.125-0.14$ | $8\cdot 10^{-8}$
Mini-Split ($\tilde{w}$LSP) | $\sim 10^{2}$ | $\sim 10^{2}$ | $\sim 1$ | $\sim 10^{2}$ | $0.11-0.13$ | $4\cdot 10^{-4}$
SUN-SUSY | $\sim 10^{2}$ | $\sim 10^{2}$ | $\sim 1$ | $\sim 10^{2}$ | $0.125$ | $4\cdot 10^{-4}$
G2MSSM | $30-100$ | $30-100$ | $\sim 1$ | $\sim 1$ | $0.11-0.13$ | $2\cdot 10^{-3}$
RNS/landscape | $5-40$ | $0.5-3$ | $\sim 1$ | $0.1-0.35$ | $0.123-0.126$ | $1.4$
Table 1: A survey of some unnatural and natural SUSY models along with general
expectations for sparticle and Higgs mass spectra in TeV units. We also show
relative probability measure $P_{\mu}$ for the model to emerge from the
landscape. For RNS, we take $\mu_{min}=10$ GeV.

##  4 Radiatively-driven natural SUSY

Along with probability distributions for models to emerge from the landscape,
one can compute probability distributions for sparticle and Higgs mass values
from particular models given an assumed value of $n$ in $f_{SUSY}$. Here, we
use a linear draw, $n=1$, to large soft terms with the NUHM4 model as the LE-
EFT. We capture non-finetuned models by requiring $\Delta_{EW}\lesssim 30$,
i.e., that the largest independent contribution to $m_{Z}$ lies within the
ABDS window. These models have radiatively-driven naturalness (RNS) where RG
running drives various soft terms to natural values at the weak scale.

The distribution for the light Higgs mass is shown in Fig. 5 (taken from Ref.
[64]). We see for $n=1$ that the blue distribution rises to a maximum at
$m_{h}\sim 125$ GeV. This is where $A_{t}$ is large enough to yield
cancellations in the $\Sigma_{u}^{u}(\tilde{t}_{1,2})$ terms, but also lifts
$m_{h}$ up to $\sim 125$ GeV via maximal stop mixing[11]. For comparison, we
also show the orange histogram for $n=-1$ where soft terms are equally favored
at any mass scale. Here, the distribution peaks at $m_{h}\sim 118$ GeV with
hardly any probability at $m_{h}\sim 125$ GeV.

Figure 5: Probability distributions for the light Higgs scalar mass $m_{h}$
from the $f_{SUSY}=m_{soft}^{\pm 1}$ distributions of soft terms in the string
landscape with $\mu=150$ GeV.

In Fig. 6, we show the corresponding probability distribution for the gluino
mass. Here, for $n=1$ the curve begins around $m_{\tilde{g}}\sim 1$ TeV and
reaches a broad maximum around 3-4 TeV, while petering out beyond
$m_{\tilde{g}}\sim 6$ TeV. The present LHC Run 2 limit from ATLAS/CMS[62, 63]
is $m_{\tilde{g}}\gtrsim 2.2$ TeV from searches within the simplified model
context. From the plot, we see that LHC is only beginning to probe the
expected range of $m_{\tilde{g}}$ values from the landscape.

Figure 6: Probability distribution for $m_{\tilde{g}}$ from the
$f_{SUSY}=m_{soft}^{\pm 1}$ distributions of soft terms in the string
landscape with $\mu=150$ GeV.

Other sparticle and Higgs mass distributions from the landscape are shown in
Ref’s [64, 14], and they are typically beyond or even well-beyond present LHC
limits. For instance, light top-squarks are expected around
$m_{\tilde{t}_{1}}\sim 1-2.5$ TeV whilst first/second generation squarks and
sleptons are expected near $m_{\tilde{q},\tilde{\ell}}\sim 10-30$ TeV. From
this point of view, LHC is at present seeing exactly what the string landscape
predicts.

##  5 Conclusions

Theoretical arguments suggest many models which include a remnant spacetime
SUSY to populate the string landscape of $4-d$ vacua. We assume a friendly
neighborhood of the landscape populated by the MSSM as the LE-EFT, but where
the CC and also the soft SUSY breaking gaugino masses, scalar mases and
$A$-terms scan via a power-law draw to large values. Landscape selection of
soft terms then allows for a derived value of the weak scale which must lie
within the ABDS window in order for the atomic principle to be obeyed, leading
to complex nuclei and hence atoms which are needed for complexity.

Under the landscape selection of soft SUSY breaking terms, one expects
radiative natural SUSY, or RNS, to be much more prevalent than finetuned SUSY
models such as CMSSM, G2MSSM, high scale SUSY, split SUSY or minisplit SUSY.
This is evident because in RNS, where all contributions to the weak scale lie
within the ABDS window, there is a much larger volume of scan space leading to
$m_{weak}\in ABDS$. Alternatively, if even one contribution to the weak scale
lies outside the ABDS window, then the remaining volume of parameter space
leading to $m_{weak}\in ABDS$ shrinks to tiny values, and is relatively less
likely. This is borne out by toy simulations of the string landscape and also
allows for a relative probability measure $P_{\mu}$ for different models to
emerge from the landscape. For instance, $P_{\mu}(RNS)\sim 1.4$ compared to
for instance $P_{\mu}(HS\text{-}SUSY)\sim 6\times 10^{-4}$. Finally, we show
probability distributions of the light Higgs mass and gluinos, showing that
the present LHC is seeing what one would expect from the string landscape. New
SUSY signals especially from higgsino pair production could arise within the
next few years at LHC. With all these beautiful results, we anticipate that
Paul will begin to work on landscape SUSY as well ;).

## Acknowledgments

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences Energy Frontier Research
Centers program under Award Number DE-SC-0009956 and DE-SC-0017647. VB
gratefully acknowledges support from the William F. Vilas estate.

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# Type Ia Supernova Models: Asymmetric Remnants and Supernova Remnant G1.9+0.3

Alice Griffeth Stone  Department of Physics, North Carolina State University,
Raleigh, NC 27695-8202  Heather T. Johnson  Department of Physics, North
Carolina State University, Raleigh, NC 27695-8202  Department of Physics,
University of Texas, 2515 Speedway, C1600, Austin, TX 78712-1192  John M.
Blondin  Department of Physics, North Carolina State University, Raleigh, NC
27695-8202  Richard A. Watson  Department of Physics, North Carolina State
University, Raleigh, NC 27695-8202  Kazimierz J. Borkowski Department of
Physics, North Carolina State University, Raleigh, NC 27695-8202  Carla
Fröhlich Department of Physics, North Carolina State University, Raleigh, NC
27695-8202  Ivo R. Seitenzahl  School of Science, The University of New South
Wales, Australian Defence Force Academy, Northcott Drive, Canberra, ACT 2600,
Australia  Stephen P. Reynolds Department of Physics, North Carolina State
University, Raleigh, NC 27695-8202

###### Abstract

The youngest Galactic supernova remnant G1.9+0.3, probably the result of a
Type Ia supernova, shows surprising anomalies in the distribution of its
ejecta in space and velocity. In particular, high-velocity shocked iron is
seen in several locations far from the remnant center, in some cases beyond
prominent silicon and sulfur emission. These asymmetries strongly suggest a
highly asymmetric explosion. We present high-resolution hydrodynamic
simulations in two and three dimensions of the evolution from ages of 100
seconds to hundreds of years of two asymmetric Type Ia models, expanding into
a uniform medium. At the age of G1.9+0.3 (about 100 years), our 2D model shows
almost no iron shocked to become visible in X-rays. Only in a much higher-
density environment could significant iron be shocked, at which time the
model’s expansion speed is completely inconsistent with the observations of
G1.9+0.3. Our 3D model, evolving the most asymmetric of a suite of Type Ia SN
models from Seitenzahl et al. (2013), shows some features resembling G1.9+0.3.
We characterize its evolution with images of composition in three classes: C
and O, intermediate-mass elements (IMEs), and iron-group elements (IGEs). From
ages of 13 to 1800 years, we follow the evolution of the highly asymmetric
initial remnant as the explosion asymmetries decrease in relative strength to
be replaced by asymmetries due to evolutionary hydrodynamic instabilities. At
an age of about 100 years, our 3D model has comparable shocked masses of C+O,
IMEs, and IGEs, with about 0.03 $M_{\odot}$ each. Evolutionary changes appear
to be rapid enough that continued monitoring with the Chandra X-ray
Observatory may show significant variations.

##  1 Introduction

Type Ia supernovae perform vital functions in the Universe: providing the
majority of iron, as well as significant quantities of other high-$Z$
elements; serving as standardizable candles for cosmography; accelerating
cosmic rays with very fast shocks in low-density media. Given that importance,
our poor understanding of the progenitor systems, the immediate surroundings,
and the fundamental mechanisms of the explosion is a serious embarrassment.
The general agreement that SNe Ia result from exploding white dwarfs is now
over 30 years old. However, the decades of work since then have failed to
resolve the question of the ignition mechanism or the nature of the nuclear
burning front that disrupts the star. Detonation, deflagration, delayed
detonation, pulsed delayed detonation, and gravitationally confined detonation
models are still discussed and considered as viable models. The ignition of
either a subsonic (deflagration) or a supersonic (detonation) thermonuclear
burning front is a very complex astrophysical problem, and consequently
initial conditions for both are often implemented in astrophysical simulations
in an ad-hoc manner. Even putting aside the fraught question of the nature of
the progenitor systems (degenerate or non-degenerate companion, that is,
single-degenerate [SD] or double-degenerate [DD] systems), tremendous
uncertainty surrounds the basic nature of the explosion. See Hillebrandt &
Niemeyer (2000), Hillebrandt et al. (2013), and Ruiter (2020) for recent
reviews.

While the basic idea of a Chandrasekhar-mass white dwarf exploding (somehow)
attractively explains the general homogeneity of SN Ia light curves and
spectra, at least compared to core-collapse events, evidence for SN Ia
diversity continues to increase (Benetti et al., 2005; Taubenberger, 2017)
from the seminal discovery of the luminosity-width relation (Phillips, 1993)
to the range of subclasses proposed in the last few years, including SN
2002cx-like, or SNe Iax (Jha et al., 2006; Foley et al., 2013), underluminous
or SN1991bg-like (Li et al., 2001), and SN 2002ic-like or SN Ia-CSM (Silverman
et al., 2013), among others. One promising avenue to explain this diversity,
especially of spectral properties near maximum light, is viewing-angle
dependence of the appearance of asymmetric explosions (Maeda et al., 2010).
Asymmetry in explosions can result from off-center ignition (Kasen et al.,
2009; Maeda et al., 2010) or for instance by gravitationally confined
detonation (GCD; Plewa et al., 2004).

Asymmetries in the supernova events themselves produce various signatures,
such as expansion velocity gradients (Maeda et al., 2010) or nonzero
polarization (e.g., Wang et al., 2007). These have been used to constrain
explosion models, though it is often the case that multiple models can
reproduce the same observations (Hillebrandt et al., 2013). However, one
avenue for investigation of the nature of Type Ia supernovae has been
historically underutilized: the detailed examination of Type Ia supernova
remnants (SNRs). The historical Type Ia SNRs Tycho, Kepler, and SN 1006 have
all been observed with long exposures by all functioning X-ray satellites, and
important clues to the Type Ia phenomenon have been obtained. Spatially
resolved spectroscopy provides direct information on anisotropies obtained
only with difficulty, if at all, from observations of the supernovae
themselves. Tycho has a light-echo spectrum indicating a completely normal SN
Ia (Krause et al., 2008), and its X-ray spectrum shows substantial radial and
azimuthal asymmetries that can be traced to the initial SN ejecta distribution
(Miceli et al., 2015). SN 1006, far above the Galactic plane, is expanding
into quite low-density material and is consequently less evolved than Tycho or
Kepler. One manifestation of this is that very little shocked iron is observed
in X-rays (Yamaguchi et al., 2008), indicating that large-scale overturn has
not occurred to move iron to large radii. However, even when unshocked iron
seen in absorption in the UV (Winkler et al., 2005) is included, the total
iron mass seems anomalously low. Kepler’s SNR is sufficiently peculiar that
its classification was still in doubt until quite recently, although evidence
has mounted for years pointing to its thermonuclear nature (see Reynolds et
al. (2007) for a summary of the arguments). But Kepler also shows unmistakable
signs of interaction with a circumstellar medium (CSM) modified by the
progenitor system (Burkey et al., 2013; Chiotellis et al., 2013; Kasuga et
al., 2021), indicating a single-degenerate origin. One additional remnant of
probable Type Ia origin is RCW 86 (Williams et al., 2011), which shows
evidence that the explosion took place off-center in a wind-blown cavity. More
recently, Seitenzahl et al. (2019) detected coronal emission lines from iron
in the young Type Ia SNRs 0509$-$67.5 and 0519$-$69.0 in the LMC, originating
from the nonradiative reverse shock. These observations allow discrimination
among models of differing progenitor mass.

Multidimensional models of SNe Ia are now within reach of modern computational
resources (Hillebrandt et al., 2013), and modelers have produced families of
such models under various assumptions about number and location of ignition
points and other variables (Kasen et al., 2009; Seitenzahl et al., 2013).
These explosion models are sometimes used to generate predicted light curves
and spectra (e.g., Sim et al., 2013; Fink et al., 2014; Noebauer et al.,
2017). Since SN Ia progenitors have such low masses, the explosions become
ballistic (that is, unaffected by pressure forces) in times of order 10 – 100
s, and typical explosion models easily reach this stage. This is important,
because subsequent evolution can then be described with these models as
initial conditions for pure hydrodynamic codes, following the evolution for
hundreds or thousands of years. Such calculations then open a whole new area
of potential confrontation between models and observations: the detailed
comparison of supernova-remnant properties with theoretical predictions. This
area holds great promise for advancing our understanding of the Type Ia
phenomenon.

In recent years, several groups have considered the supernova-to-supernova-
remnant transition using hydrodynamic modeling. 1-D SN models were used as
input to study two core-collapse remnants, SN 1987A (Orlando et al., 2015) and
Cas A (Orlando et al., 2016). Full evolution of 3D Type Ia supernova models to
SNR stage is relatively new. Orlando et al. (2020) presented a full 3D MHD
simulation of SN 1987A, and Orlando et al. (2021) performed a similar
calculation aimed at Cassiopeia A. Ferrand et al. (2019) and Ferrand et al.
(2021) evolved some of the models of Seitenzahl et al. (2013) to an age of 500
years, focusing on the morphology of the forward and reverse shocks, and the
contact discontinuity. Tycho’s SNR appears to have been a fairly normal SN Ia
(Krause et al., 2008), but symmetric initial conditions apparently cannot
account for some asymmetry as deduced from spatial power spectra (Ferrand et
al., 2019). Ferrand et al. (2021) evolved a quite asymmetric model also from
Seitenzahl et al. (2013) (model N5ddt; see below) again focusing on shock
morphology. Here we describe a similar evolutionary calculation, to explain a
highly unusual young supernova remnant, and requiring the most asymmetric of
all the models of Seitenzahl et al. (2013). We consider both morphological and
abundance structure, as the target of our study is remarkable in both
dimensions.

##  2 A new test case: G1.9+0.3

One major difficulty afflicts the study of SNRs to learn about their
supernovae: separating ejected material from swept-up surrounding unmodified
interstellar medium (ISM) or modified CSM. Abundance clues are powerful, but
have limitations. Ideally one would observe the youngest possible remnant, but
large enough for adequate spatial resolution. That remnant appears to be the
youngest Galactic SNR, G1.9+0.3 (Reynolds et al., 2008, see Fig. 1). This
object is about $100^{\prime\prime}$ in diameter, the smallest angular size of
any confirmed Galactic SNR. Unfortunately, it is very highly absorbed, with an
X-ray column density of about $5\times 10^{22}$ cm-2 (Reynolds et al., 2009),
implying $A_{V}\sim 23^{m}$, so radio and X-rays are the only useful
observational channels. The angular expansion rate of 0.64 arcsec yr-1
obtained from comparing X-ray images from 2007 and 2009 (Carlton et al., 2011)
gives an upper limit for the age of about 160 yr, less if (as is almost
certainly the case) deceleration has occurred; spatial variations in expansion
rate (Borkowski et al., 2014) are consistent with an age of about 100 yr, or a
date of around 1900. The high extinction would have rendered it unobservable
in optical telescopes of that era. Furthermore, its X-ray spectrum is almost
entirely synchrotron emission, making it a member of the small class of X-ray-
synchrotron-dominated SNRs. However, long observations with Chandra have
allowed the detection of thermal emission from small regions (Borkowski et
al., 2011, 2013b), with spectroscopic widths $\sim 14,000$ km s-1 confirming
the large expansion proper motion, refined with a second Chandra observation
(Carlton et al., 2011). The distance is still uncertain; the high column,
higher than the entire Galactic column along nearby sight-lines, suggests an
association with the Galactic Center, and a provisional distance of order 8.5
kpc has been assumed. Nearer would be very unlikely in view of the high
absorption, but too much farther would make the expansion proper motion
unreasonably large. An H I observation with the Giant Metrewave Radio
Telescope (Roy & Pal, 2014) has been used to set a lower limit of 10 kpc,
certainly consistent with the known properties of G1.9+0.3.

While the SN type of G1.9+0.3 is not absolutely confirmed as Ia, most
indicators point in that direction. The very high expansion speeds even after
$\sim 100$ yr; the clear presence of supersolar iron (Fe K$\alpha$) in several
regions (Borkowski et al., 2013b); the absence of any pulsar-wind nebula in
the center (although a neutron star itself would be too highly absorbed to be
detected); and the bilateral symmetry of the synchrotron X-rays, suggesting a
parallel with SN 1006, all point to a Ia origin. A core-collapse (CC) scenario
can be constructed with difficulty, but it requires a highly unusual event,
with a low mass but very high energy at explosion, and a low-density
surrounding medium. We shall assume for the remainder of this work that
G1.9+0.3 originated in a thermonuclear event.

Careful analysis of the total current Chandra exposure of about 1.7 Ms has
shown clear evidence for Si and S K$\alpha$ emission, as well as Fe
K${\alpha}$ (Borkowski et al., 2011, 2013b). Remarkably, Fe is found at quite
large radii, and the Si/Fe ratio varies considerably. There is tentative
evidence for a broad feature centered at 4.1 keV from the remnant center,
attributed to an electron capture in 44Sc, formed in the decay chain of 44Ti
and indicating a 44Ti mass of about $1\times 10^{-5}\ M_{\odot}$ (Borkowski et
al., 2010, 2013a). The Fe lines are quite broad, consistent with the presence
of Fe at large projected radii from the remnant center; Borkowski et al.
(2013b) find FWHM values of 15,000 km s-1.

The high observed velocities and proper motions of G1.9+0.3 indicate that it
is at quite an early evolutionary stage. Defining the deceleration parameter
$m$ of a region at radius $r$ by $r\propto t^{m}$, Borkowski et al. (2014)
compared the 2007, 2009, and 2011 Chandra observations to obtain $m$ values
varying spatially, from $\mathrel{\mathchoice{\lower
3.1pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.1pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.1pt\vbox{\halign{$\m@th\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
<EMAIL_ADDRESS>for the outer blast wave to $\mathrel{\mathchoice{\lower
3.1pt\vbox{\halign{$\m@th\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.1pt\vbox{\halign{$\m@th\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
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<EMAIL_ADDRESS>for inner features identified with the reverse shock. In the Sedov self-
similar phase, we have $m=0.4$ throughout, so the expansion is considerably
less decelerated. However, the expansion velocities in G1.9+0.3 vary by a
large factor. Borkowski et al. (2017) used a nonparametric method to measure
the expansion throughout the remnant and found velocities varying strongly
with position, over a factor of 5 in magnitude (Fig. 2). In particular,
expansion to the north was found to be several times slower than in other
directions, strongly suggesting a recent encounter with denser material. Such
an encounter could explain the peak in radio intensity found there, as a
higher density of accelerated electrons would be expected, but the slower
shock velocity would result in a lower maximum accelerated electron energy,
explaining the lower X-ray brightness to the north. Such denser material
almost certainly originated in asymmetric pre-SN mass loss rather than in the
dynamics of the explosion itself, although such a possibility cannot be ruled
out.

The total mass of shocked ejecta in G1.9+0.3 is still quite small. An analysis
based on the average expansion rate found 0.033 $M_{\odot}$ (Carlton et al.,
2011), while one based on scaling an energetic explosion model WS15DD3
(Hachinger et al., 2013) produced about twice as high a value, 0.064
$M_{\odot}$ (Borkowski et al., 2013b). While these values are much less than
the total ejecta mass, we observe Fe to make up a significant fraction of
material already shocked.

Figure 1: G1.9+0.3 in X-rays in 2011 (based on the deep Chandra observations
described by Borkowski et al. (2013b)). Red, 1 – 3 keV; green, 3 – 4.5 keV;
blue, 4.5 – 7.5 keV.  Figure 2: Proper motion vectors between 2011 and 2015
(Borkowski et al., 2017), color-coded according to deviations from radial
directions, as measured with respect to the geometrical center (indicated with
the red X). The vertical scale shows angles in degrees. The white arrow
represents an expansion rate of $0.25^{\prime\prime}$ yr-1.

The presence of high-velocity shocked Fe is especially problematic in G1.9+0.3
as it was presumably near the outer edge of the expanding ejecta at a very
early time. One might expect evidence of such fast Fe in observations of Type
Ia supernovae themselves; while high-velocity features are seen in a large
fraction of all SNe Ia (e.g., Maguire et al., 2014), Fe is rarely seen, and
when it is, the estimated mass is very small. Fitting of SN spectra with
stratified models (“supernova tomography,” e.g., Stehle et al., 2005; Tanaka
et al., 2008; Sasdelli et al., 2014; Mazzali et al., 2015) also show little
evidence for high-velocity Fe. This raises the question of whether current
Type Ia supernova models will evolve to resemble G1.9+0.3 in this property as
well as in the bulk expansion velocity and observed spatial heterogeneity of
Si, S, and Fe. In this paper, we address this question, using two multi-
dimensional SN Ia models as initial conditions and following their expansion
into uniform media with the VH-1 hydrodynamics code.

##  3 Simulations

The VH-1 hydrocode is a conservative, finite-volume code evolving the Euler
equations for an ideal gas, based on the Piecewise Parabolic Method with the
Lagrange-Remap implementation (Colella & Woodward, 1984). It is particularly
well suited to resolving shocks and other discontinuities, and has been
extensively used for the study of supernova remnants (see, for instance, 3D
hydrodynamic simulations of Type Ia SNRs: Warren & Blondin, 2013). It can be
used in one, two, or three dimensions; in two dimensions, either with $r,z$
cylindrical coordinates, or $r,\theta$ spherical coordinates, and in three
dimensions, using the overset spherical grids known as “Yin-Yang” (Kageyama &
Sato, 2004) that eliminate the singularities on the axis of symmetry. Here, we
shall perform 2D simulations in cylindrical $r,z$ coordinates, and 3D ones
using the Yin-Yang grids. We do not include effects of energy losses due to
cosmic rays, which can increase the compressibility of the gas (Blondin &
Ellison, 2001), but use a uniform adiabatic index of $5/3$. Our 2D cylindrical
grid has a resolution of $512\times 1024$, and expands to follow the evolution
of the SNR. The 3D simulation also expands with the blast wave. It has an
angular resolution of $0.24^{\circ}$ (384 $\times$ 1152 angular zones in each
of the Yin and Yang grids). For the radial coordinate, we cover the inner half
of the grid with 84 zones and the outer half with 300, to maximize resolution
in the region between the shocks. This gives a maximum fractional resolution
of $1.65\times 10^{-3}$ of the maximum grid radius for the outer 50% of that
maximum radius.

We use two initial Type Ia supernova models (F. Röpke, private communication).
Table 1 gives the compositions of both. Our 2D model comes from the family
described in Kasen et al. (2009), in which 2000 otherwise identical 1.4
$M_{\odot}$ C/O white dwarfs (50% each) were ignited with varying numbers and
locations of ignition points. The model was provided with a resolution of
$512\times 1024$. Our 3D model is model N3 from Seitenzahl et al. (2013), with
three slightly off-center ignition points. The mass is 1.40 $M_{\odot}$, the
radius $1.96\times 10^{8}$ cm, and the central density $\rho_{c}=2.9\times
10^{9}$ g cm-3. The composition is also equal parts 12C and 16O (with 2.5%
22Ne to account for the approximately solar metallicity of the ZAMS
progenitor). The white dwarf undergoes a delayed detonation; a detailed
discussion of the ignition setup is given in Seitenzahl et al. (2013). This
model was chosen as the most asymmetric of the suite of 14 models in
Seitenzahl et al. (2013). We describe each simulation in more detail below.

###  3.1 2D Simulation

Our 2D simulation was conducted at the same resolution as that of the initial
model, $(r,z)=(512,1024)$. The initial model at an age of 100 s is shown in
Figure 3, showing the location and iron content of ejecta, as well as the
location of the contact discontinuity. The composition is given in Table 1;
the total asymptotic kinetic energy is $1.45\times 10^{51}$ erg, and the total
mass 1.408 $M_{\odot}$. The model initially expands into an unrealistic number
density of $10^{6}$ cm-3, but it is straightforward to scale this to an
arbitrary value. (Figure 4 shows that the outermost ejecta density is more
than three orders of magnitude larger; the ram pressure $\rho v_{\rm sh}^{2}$
is about $10^{5}$ dyn, for an initial expansion velocity of $10,000$ km s-1,
while the unrealistically high-density initial CSM still has a pressure of
only $nkT\sim 10^{-4}$ dyn, even for a temperature of $10^{6}$ K.) The radial
distribution of elements is also shown in Figure 3, where we combine elements
into intermediate-mass elements (IME) and iron-group elements (IGE). CSM
represents all surrounding material. The azimuthally averaged density profile
is fairly well fit by an exponential (Dwarkadas & Chevalier, 1998), as shown
in Figure 4.

Table 1: Initial Model Compositions Element | 2D Model Mass ($M_{\odot}$) | 3D Model Mass ($M_{\odot}$)
---|---|---
C+O | 0.0853 | 0.0624
IME | 0.378 | 0.121
Iron-group | 0.945 | 1.22
56Ni | 0.698 | 1.10
Total | 1.41 | 1.40

Figure 3: 2D model. Left: 2-D distribution. Black contour: contact
discontinuity between CSM and ejecta. Purple contour: IME’s. Color scale: Mass
fraction of iron-group elements. Right: Initial mass fractions, azimuthally
averaged.

Figure 4: Azimuthally averaged density of 2D model, with superimposed least-
squares fit to an exponential, which describes the data fairly well.

Figure 5: Forward and reverse shock locations. Points: Azimuthal averages from
2D simulation. Curves: Results from a 1D simulation using the exponential fit
to the initial density profile (Fig. 4). The time and radius are scaled to the
present age of G1.9+0.3. At these early times, the shock positions in the
exponential model are quite close to those from the 2D calculation.

The model was then evolved for two different values of constant upstream
density, one chosen to match the value of 0.022 cm-3 found from the expansion
measurements of Carlton et al. (2011), and the other a value of 1.08 cm-3,
calculated to bring the reverse shock farther in (see below). Figure 5 shows
the angle-averaged forward and reverse shock locations for the high-density
run, compared to a 1D simulation using the exponential fit to the initial
density profile shown in Fig. 4.

Figure 6: 2D model at the size of G1.9+0.3 ($r=2.0$ pc), for $n_{0}=0.022$
cm-3. Left: Iron-group elements. Center: Intermediate-mass elements. Right:
Circumstellar material. All are given as mass fractions.

Figure 6 shows the distribution of elements at the current size of G1.9+0.3
($r=2.0$ pc), as mass fractions, in the low-density calculation. The contact
discontinuity is visible as the bound of IME’s; iron-group elements are well
inside at all radii (excepting an artifact along the polar axis). The
relatively low resolution of the initial 2D model results in a slight shape
distortion in the outermost contour visible in Fig. 3. In the low-density
case, this slightly rectangular shape persists to the age of G1.9+0.3, but in
the more highly evolved high-density case, it has evolved away by this time.
The remnant age is 104 yr for the low ambient density; the total swept-up mass
is 0.026 $M_{\odot}$. About 0.06 $M_{\odot}$ of ejecta have been shocked,
quite close to one of our estimates based on the observed expansion rate.
However, only a minuscule amount, $7\times 10^{-5}\ M_{\odot}$, of that is
iron-group elements – clearly inconsistent with the observations.

Figure 7: 2D model at the size of G1.9+0.3 ($r=2.0$ pc), for $n_{0}=1.08$ cm-3
(but at a much later age than appropriate for G1.9+0.3). Left: Iron-group
elements. Center: Intermediate-mass elements. Right: Circumstellar material.
All are given as mass fractions.

Figure 7 shows the elemental distribution for the high-density model, at an
age of 173 yr. This age is of course inconsistent with the observational data,
and the shock velocity and expansion rate are far too low, but some such
combination of unrealistic parameters is required to obtain a significant
fraction of shocked iron from this initial model. Iron-rich material is
nearing the reverse shock. (Note the much more developed Rayleigh-Taylor
structures beyond the contact discontinuity). Figure 8 compares the
azimuthally averaged mass fractions for the two models. In the low-density
case that matches the size and expansion rate of G1.9+0.3, very little iron
($\sim 7\times 10^{-5}\ M_{\odot}$) has passed through the reverse shock to
become heated and visible in X-rays. The very high density of the other case
is required so that at an age of a few hundred years, a significant (but still
small) amount of iron has been shocked, as the observations require.

Figure 8: Mass fractions at the size of G1.9+0.3, for the two values of
ambient density in the 2D simulation. Left: $n_{0}=0.022$ cm-3. Right:
$n_{0}=1.08$ cm-3. Vertical lines indicate the average positions of forward
and reverse shocks. Only material at larger radii than the reverse shock would
be visible in X-rays.

###  3.2 3D simulation

Our 3D model, model N3 from Seitenzahl et al. (2013), has a composition given
in Table 1. The total kinetic energy is $1.61\times 10^{51}$ erg, and the
total mass 1.40 $M_{\odot}$. We illustrate the initial spatial distribution of
elements and of kinetic energy using Mollweide (equal-area) projections of the
entire surface. Figure 9 shows the sky distribution of kinetic energy (KE) and
ejecta composition (masses) in three bins (unburned C-O, IMEs, and Fe-group).
Scales refer to masses in g pix-1, and KE in erg pix-1, where one pixel
subtends $1.72\times 10^{-5}$ sr. Substantial asymmetries are evident. A ring
of enhanced IMEs correlates with an absence of iron-group elements, which are
enhanced in the opposite hemisphere. Histograms show the distribution of
values in each plot. The strong variation in distribution of Fe-group elements
reflects the substantial anisotropy of the initial model. Angle-averaged total
density and mass fractions are shown in Fig. 10, along with an exponential and
power-law (not fit) for comparison. The exponential shown has an $e$-folding
radius of $3.3\times 10^{10}$ cm, and the power-law index is 5.5.

The model was evolved to an age of about 2000 yr, assuming a uniform ambient
medium of density $n_{0}=0.022$ cm-3 ($\rho_{0}=5.08\times 10^{-25}$ g cm-3)
(Carlton et al., 2011). Frames were extracted from the simulation for ages of
13, 33, 75, 100, 200, 400, 800, and 1800 yr; the fourth is about the estimated
age of G1.9+0.3. Montages of Mollweide projections for various quantities are
shown in Figures 15 through 22.

Figure 9: Mollweide (equal-area) projections of the 3D model N3, for (left to
right, top to bottom): kinetic energy, and total (shocked and unshocked)
masses of C/O, IMEs, and IGEs. Histograms of values show the distributions.
Subsequent composition images show shocked masses only.

Figure 10: Left: Angle-averaged density of the initial 3D model. Also shown
are exponential and power-law curves for comparison (not fits). Scaling radius
for exponential: $3.3\times 10^{10}$ cm. Power-law index: 5.5. Right: Angle-
averaged mass fractions of different species.

The evolution of the angle-averaged forward and reverse shock radii is shown
in Figure 11. The current observed shock radius is about 2 pc $\cong 6\times
10^{18}$ cm, consistent with our age estimate of 100 yr. Figure 12 shows the
forward-shock deceleration parameter $m\equiv d\log R/d\log t\equiv vt/R$;
while it shows variations, the deceleration of the average shock radius
evolves fairly smoothly toward the Sedov value of 0.4. A 1D thin-shell
analytic model fit to the remnant’s current size and mean expansion rate
(Carlton et al., 2011) gave $m=0.69$, between the average and maximum values
we find here. At the end of the simulation, the total swept-up mass is about
12 $M_{\odot}$, by which time Sedov dynamics should be a good approximation.
The reverse-shock deceleration drops to zero and becomes negative as the
reverse shock begins to move inward at an age of around 2000 years. It cannot
be observed directly.

Shocked masses and emission measures (EMs) of the different composition
categories are shown in Figs. 13 and 14. Here EM$=\int\rho f_{i}n_{e}dV$,
where $\rho$ is the total density, $f_{i}$ are the mass fractions of different
species, and $n_{e}$ is the electron density, calculated by approximating
11/13 electrons per amu of CSM and 1/2 electron per amu of all other species.
We have assumed full ionization of all species; observed EMs will reflect
various confusing effects, such as time-dependent ionization. IME’s in the
simulation are fully shocked by about $10^{10}$ s, while iron-group elements
approach being fully shocked only near the end of the simulation (Fig. 13). Of
course, the shocked CSM mass continues to climb. The shocked-CSM emission
measure also continues to climb, but ejecta EMs peak once all the masses are
shocked, and then drop as $R_{s}^{-3}$.

Figure 11: Forward and reverse shock radii as a function of time, averaged
over angles. The current observed shock radius, shown by the grey horizontal
bar, is about 2 pc.

Figure 12: Forward and reverse shock deceleration parameters as a function of
time, averaged over angles.

Figure 13: Shocked masses of the various composition categories.

Figure 14: Emission measures of the various composition categories (see text
for definitions).

Figure 15: Mollweide images of (left to right, top to bottom): kinetic energy,
shocked CO mass, shocked IMEs, shocked Fe-group elements, and forward and
reverse-shock radii, all at an age of 13 years. Histograms show the
distribution of values in each case. Energy and masses are in units of log(erg
pix-1) and log(g pix-1), respectively, where one pixel subtends $1.718\times
10^{-5}$ sr. Shock radii are given as log($r/r_{\rm av}$), where $r_{\rm av}$
is the mean radius. (NOTE: The online article when published will contain
animations of the forward and reverse shocks and the three categories of
shocked material.)

Figure 16: As in Fig 15, at an age of 33 years.

Figure 17: As in Fig 15, at an age of 75 years.

Figure 18: As in Fig 15, at an age of 100 years. This is about the age of
G1.9+0.3.

Figure 19: As in Fig 15, at an age of 200 years.

Figure 20: As in Fig 15, at an age of 400 years, roughly that of the remnants
of Kepler’s and Tycho’s supernovae.

Figure 21: As in Fig 15, at an age of 800 years.

Figure 22: As in Fig 15, at an age of 1800 years, roughly the age of the
likely SNR Ia RCW 86.

The sequence of equal-area Mollweide plots (Figs. 15 through 22) shows the
development of hydrodynamic instabilities, most obvious near the end of the
simulation. At early stages, the shocked masses are all quite small and show
very pronounced spatial variations, but as more material is shocked, these
variations decrease in amplitude. Near the age of G1.9+0.3 (about 100 years;
Figure 18), however, substantial asymmetry remains, on the relatively large
scale with which it was imprinted in the explosion. Significant variations in
kinetic energy in different directions, still apparent out to 75 years, have
mostly been smoothed away by 100 years, and are thereafter dominated by
hydrodynamic instabilities. Comparing the sequence of images shows where
material is shocked. Note particularly the concentration of shocked iron in
one hemisphere, surrounded by a deficit.

Figure 23: The structure of the SNR at ages of 25, 50, 100, 200, and 400 years
(by row). First column: surface of the forward shock colored by the deviation
of the radial extent of the shock, with red protruding out beyond the average
radius and blue lagging behind. Second column: three-dimensional surface of
the reverse shock colored by mass fraction of iron from zero (black) to 100%
(red). Light blue: gas pressure in a two-dimensional slice perpendicular to
the view, illustrating the relationship between the reverse shock and the
forward shock. Third column: mass fraction in a two-dimensional slice through
the center of the remnant. IGEs in blue, IMEs in red, and C+O in green. Black
contours: forward and reverse shocks. Fourth column: mass fraction as three-
dimensional surfaces. IMEs (red) and IGEs (blue) shown as isosurfaces at 50%
mass fraction. Green surface: reverse shock colored by mass fraction of C+O
from zero (black) to 100% (bright green). The pressure in the central plane
shown as in Column 2.

Fig. 23 shows perspective views at ages of 25 to 400 yr, during which time the
increasing smoothness of forward and reverse shock surfaces can be observed.
Fe-group ejecta (blue in the last column of Fig. 23) are dominated by unburned
C+O and IME’s for the viewing angle shown, at the earlier times, but become
more prominent with time.

##  4 Discussion

###  4.1 2D results

The 2D simulations of the evolution of the model from Kasen et al. (2009) show
that for an ambient density that can reproduce the observed size and expansion
rate of G1.9+0.3, almost no iron-group elements will have been shocked after
100 years (Fig. 8), contrary to observations. The asymmetries in the initial
explosion model are substantial, but not sufficient to give rise to the
overturn (iron beyond silicon and sulfur) observed in G1.9+0.3. Far greater
initial asymmetries will be required to reproduce these observed features. The
high-density model illustrates what is required to obtain sufficient shocked
iron; it is completely inconsistent with the observed size and expansion rate
of G1.9+0.3.

###  4.2 3D results

Our 3D simulation begins with the most anisotropic of the suite of models of
Seitenzahl et al. (2013). With only three ignition points, all in the same
hemisphere, the N3 model might be expected to show strong variations, and Fig.
9 substantiates this expectation. While we focus on the evolution to SNR
stages, the appearance of such a supernova would clearly have a substantial
dependence on viewing angle. Most iron is ejected preferentially in a
relatively small solid angle, roughly on the opposite hemisphere from IMEs.
The hemisphere opposite to the direction of most iron injection is relatively
smooth. The viewing-angle dependence of the calculated spectra and lightcurves
for the N3 model is discussed in some detail in Sim et al. (2013).

As time advances (Figs. 15 to 17), the amount of ejecta passing through the
reverse shock increases. At an age of 13 years, relatively little iron is
shocked. The histogram shows that most pixels have a small amount of iron, but
a few reach much higher values. At 75 years, most pixels now have larger
amounts of shocked iron. (The overall mass of iron that is shocked is still
relatively small, about 0.01 $M_{\odot}$; see Fig. 13.)

At the age of G1.9+0.3, Fig. 13 shows comparable masses in all shocked
components: CSM, ejecta C and O, IMEs, and Fe-group elements. In particular,
only about $0.02\ M_{\odot}$ of 56Fe (originally 56Ni) has been shocked,
though this is far larger than the amount of shocked 56Fe found by an age of
100 yr in the 2D simulation. The total shocked mass is about 0.1 $M_{\odot}$,
somewhat larger than in the 2D case. This is also about three times larger
than that estimated from observations of G1.9+0.3 as described in Section 2
above. The preferential ejection of iron in some direction could be identified
with the stronger Fe features in the north rim of G1.9+0.3 (Borkowski et al.,
2013b). By the age of G1.9+0.3 (Fig. 18), a significant amount of shocked Fe
can be seen to the right of center in the images.

As the remnant continues to evolve, the spread in all quantities decreases, as
shown by the histograms. At our earliest epoch of 13 years, different pixels
vary in amount of iron by 8 orders of magnitude, while kinetic energy varies
by a factor of about 30. At 100 years, the spread in KE is less than one order
of magnitude. The nature of asymmetries varies as well. Large-scale
asymmetries visible in the initial model dominate to an age of 100 – 200
years, at which point smaller-scale irregularities ascribable to Rayleigh-
Taylor instabilities become apparent (see Fig. 19). By an age of 800 years,
kinetic-energy values spread by only about a factor of 4, while shocked iron
ranges over only about 20. The reverse shock has become very smooth at late
times (a small range of values), so the finite extent of radial zones results
in artifacts visible in the lower-right panels of Figures 21 and 22. By the
end of our simulation, roughly the age of the likely Type Ia SNR RCW 86
(Williams et al., 2011), unburned C and O still have a spatial distribution
reminiscent of the initial model, while IMEs and IGEs are fairly smooth.

An alternative view of the evolution of asymmetries is presented in Figure 23,
showing isosurfaces of various quantities for different ages: forward and
reverse shock surfaces and mass fractions, and 2D slices through the remnant
center of mass fractions, for one particular viewing angle. The same trends
apparent in the Mollweide projections can be seen here.

Our simulations show that the imprint of a highly asymmetric thermonuclear
explosion is clearly detectable in the remnant for at least 500–1000 years,
though it decreases in prominence as evolutionary asymmetries due to
instabilities take over. Of course, we have assumed a uniform external medium;
when strongly asymmetric ISM or CSM is present, morphological changes can be
expected, though compositional asymmetries should not be affected. The north
rim of G1.9+0.3 expands five times more slowly than other parts of the remnant
(Borkowski et al., 2017), a circumstance likely due to asymmetric
surroundings.

A similar simulation, focusing on the morphology of the forward and reverse
shocks and contact discontinuity, was recently presented by Ferrand et al.
(2021), who evolved two models from the same suite of Seitenzahl et al.
(2013), N5 and N100 (that is, with five and one hundred ignition points,
respectively). The results of their N5 simulation bear resemblances to our N3
results, but the N3 model is even more asymmetric. The shock locations we find
do show a strong initial imprint of the asymmetries, though in percentage
terms these are less dramatic than the compositional asymmetries (not
presented in Ferrand et al. 2021). Only three ignition points produce clear
global-scale asymmetries, already distinct from the pattern (in the contact
discontinuity) seen at an age of one year in the N5 simulation of Ferrand et
al. (2021). Both models evolve toward greater symmetry with smaller-amplitude
variations, as might be expected. The interpretation of supernova-remnant
observations can profit greatly from the investigation of initial supernova
models in this way.

##  5 Conclusions

The youngest supernova remnants can be expected to exhibit the dynamics of
their birth event most clearly. The youngest Galactic SNR, G1.9+0.3 shows
remarkable levels of asymmetry in the location of ejecta of different
composition, with shocked iron at large radii, sometimes beyond IMEs such as
silicon and sulfur. Our hydrodynamic simulations of the evolution of two
different asymmetric SN Ia models show that such pronounced asymmetries at an
age of about 100 years are not easily produced. Our 2D model produces far too
little shocked iron, and at too small radii, to reproduce the Chandra
observations of G1.9+0.3. This is perhaps not surprising in view of the fairly
symmetric distribution of iron in the initial model (Fig. 3), but it is not
obvious that evolutionary effects are unable to produce adequate shocked iron,
as our calculation demonstrates. The most extreme 3D model, N3 from Seitenzahl
et al. (2013), shows that somewhat larger ejecta masses are shocked by this
age, possibly consistent with the observations. It must be borne in mind that
ejecta are detected at all only in a few regions in G1.9+0.3 whose X-ray
spectrum is dominated by nonthermal synchrotron radiation.

The strong asymmetries observed in G1.9+0.3 also argue against currently
popular models of sub-Chandrasekhar (0.8 – 1.1 $M_{\odot}$) detonations for
SNe Ia, since pure detonations tend to produce little in the way of asymmetry
(e.g., Sim et al., 2010; Chamulak et al., 2012), unless the companion also
explodes (e.g., Pakmor et al., 2012). In particular the high-velocity Fe
required is difficult to produce unless progenitors possess a high-mass He
layer, as in some double-detonation models (e.g., Woosley & Weaver, 1994; Sim
et al., 2012). However, these models also produce amounts of 44Ti in conflict
with observations (Borkowski et al., 2010; Weinberger et al., 2020).

We exhibit the evolution of the initial asymmetry of our model in a series of
equal-area projections of the entire surface, which illustrate the character
of the asymmetries of model N3. In particular, a large plume of IME-rich
material can be seen, directed approximately opposite to iron-group material,
though the angular variation in initial expansion velocity and kinetic energy
is less pronounced. After a few hundred years, the imprint of the initial
explosion asymmetry has faded, being replaced by the development of
hydrodynamic instabilities on smaller scales, and lessening, though not
eliminating, compositional traces of the explosion asymmetry.

For the particular case of G1.9+0.3, explosion asymmetries should still
dominate, although confused by apparently strongly asymmetric ambient medium.
However, changes on timescales of decades can be seen in Figs. 17 and 18. The
most recent Chandra observations were done in 2019-2020, 13 years after the
initial discovery. We may be able to observe the fading of the explosion
imprint as time goes on, though the dominant nonthermal component of the X-ray
spectrum, which is still increasing (Borkowski et al., 2017), continues to
impede more detailed compositional studies. In any case, G1.9+0.3 challenges
models of Type Ia supernovae and its continuing study should be highly
informative for the development of such models. In general, our results
illustrate the value of detailed observations of the dynamics of the youngest
SNRs and their power to constrain supernova explosion models.

This work was supported by the National Science Foundation through grant
AST-1062736 to NCSU’s REU program, and by NASA through the Chandra Guest
Observer program (GO1-12098A and B). A preliminary version of this work was
presented at the 2011 meeting of the High-Energy Astrophysics Division of the
American Astronomical Society (Griffeth et al., 2011).

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