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Jinan, Shandong Province, China

# Palatini Higgs inflation and the refined dS conjecture

Yang Liu<EMAIL_ADDRESS>
###### Abstract

The refined de Sitter derivative conjecture provides constraints to potentials
that are low energy effective theories of quantum gravity. It can give direct
bounds on inflationary scenarios and determine whether the theory is in the
Landscape or the Swampland [1]. Any infationary model can be checked by these
conditions and non-minimally coupled scalar field theory is not an exception.
We consider the “Palatini Higgs inflation” scenario taking the refined de
Sitter derivative conjecture into account. According to the latest
cosmological observations from Planck 2018, BICEP2+$Keck$ [2,3,4] and the
bound of non-minimal coupling parameter $\xi$ [5,6], we suggest that if the
refined dS conjecture does indeed hold, then the Palatini Higgs inflation
model cannot be the low energy effective theory of a consistent quantum
gravity theory since the two parameters $c_{1}$ and $c_{2}$ are much smaller
than order 1, which is inconsistent with the refined dS conjecture. Therefore
we suggest that it is in the “Swampland”.

##  1 Introduction

Motivated by a number of string theoretical constructions with controlled
approximations, it has recently been proposed in ref.[7] that the potential
for scalar fields in string theory satisfies the bound [7,8],

$|\nabla V|\geq\frac{c_{1}}{M_{p}}\cdot V,$ (1)

where $c_{1}$ is a positive constant of the order of 1, and $M_{p}$ is the
Planck mass.
The conjecture, which corresponds to the initial "dS derivative conjecture"
had severe tension with regard to numerous phenomenological models, both in
particle physics and inflation [1]. In particular, the conjecture rules out
the de Sitter (dS) vacua of a consistent theory since $|\nabla V|=0$ but $V>0$
for a dS vacuum, even if the dS vacuum is the vacuum energy solution of
Einstein’s equation with a positive cosmological constant. Therefore, the
cosmological constant scenario is excluded if this conjecture is assumed to be
true [1]. In order to explain the currently observed universe, the
quintessence field with an exponentially decaying potential can be adopted,
namely [1],

$V_{Q}(Q)=\Lambda^{4}_{Q}e^{-c_{Q}Q},$ (2)

which allows the dynamical vacuum energy with the present values of the
quintessence field, the scale parameter as $Q\sim 0$ and $\Lambda_{Q}$ of
order $O(1)$ meV [1]. There have been many follow-up papers considering
various implications of the conjecture, such as discussed in ref.[9-13].
Following the earlier version of the conjecture proposed by Obied, Ooguri,
Spondyneikeo and Vafa [7], i.e., “dS derivative conjecture”, the latest
version of the Swampland conjecture was recently proposed by Ooguri, Palti,
Shiu and Vafa [8], namely, “the refined de Sitter (dS) conjecture”. This
conjecture states that any scalar potential $V(\phi)$ for scalar fields in a
low energy effective theory of a consistent quantum gravity must satisfy at
least one of the following conditions:

$|\nabla V|\geq\frac{c_{1}}{M_{p}}\cdot V,$ (3)

or

$min(\nabla_{i}\nabla_{j}V)\leq-\frac{c_{2}}{M^{2}_{p}}\cdot V,$ (4)

where $c_{1}$ and $c_{2}$ are both positive constants of the order of 1 and
the left-hand side of eq.$(1.4)$ is the minimum eigenvalue of the Hessian
$\nabla_{i}\nabla_{j}V$ in an orthonormal frame [8].
Let us review the two swampland conjectures briefly. We live in a universe
where the vacuum energy is positive [7]. This can be realized by having a
scalar field potential $V$ with a local minimum at a positive value, leading
to a meta-stable or stable de Sitter (dS) vacuum [7]. Despite many efforts in
string theory community have been devoted to constructing de Sitter (dS)
vacuum, there still exist a number of issues [7]. Therefore, it is fair to say
that these scenarios have not yet been rigorously shown to be realized in
string theory [7].
Considering the difficulties in obtaining dS-like vacua in string theory, it
is natural to consider the alternative possibility where no dS vacua, not even
meta-stable dS vacua, can exist in a consistent quantum theory of gravity [7].
One may conjecture $|\nabla V|>A$ for some constant $A$, which can lead to
this exclusion [7]. However, supersymmetric vacua with flat directions provide
counter-examples [7]. To avoid these counter-examples, one may consider
allowing $A$ to depend on scalar fields $\phi$ in such a way that $A(\phi)\leq
0$ for supersymmetric vacua [7]. A simple choice would be $A(\phi)=c\cdot
V(\phi)$ with positive constant $c$, since $V\leq 0$ in supersymmetric vacua
[7]. Therefore, in ref.[7], the authors proposed that $|\nabla V|\geq c\cdot
V$ ($M_{p}=1$), namely, eq.$(1.3)$ as a swampland criterion in any low energy
theory of a consistent quantum theory of gravity [7]. This conjecture is the
so-called “dS swampland conjecture”. The norm $|\nabla V|$ of the potential
gradient is defined using the metric on field space in the kinetic term of the
scalar fields [7].
Furthermore, in order to connect the “dS swampland conjecture” to another
better established swampland condition, which is known as the distance
conjecture [8], the authors of ref.[8] proposed the “refined dS swampland
conjecture” which we have stated in the third paragraph. If $V$ is non-
positive or $M_{p}\rightarrow\infty$, then the conjecture becomes trivial.
This refined version still excludes (meta-)stable de Sitter vacua [8].
A multitude of coupling constants, such as the string coupling or the volume
of extra dimensions can control effective descriptions of string theory [8].
All such couplings are dependent on field, and in particular, any weak
couplings are associated with large distances in field space. The distance
conjecture states that a scalar field can always be possible to change its
value from its original point with an arbitrarily large amount of geodesic
distance and that towers of light states with masses,

$m\sim e^{-a\Delta\phi}$ (5)

descend from the ultraviolet if the change $\Delta\phi$ is trans-Planckian
[8].
The newly added bound is rather easily satisfied for a generic potential in
low scale, $\triangle\phi\ll M_{p}$, since [1,14]

$M^{2}_{p}\frac{\nabla_{i}\nabla_{j}V}{V}\sim-\frac{M^{2}_{p}}{(\Delta\phi)^{2}}\ll-
c_{2}\sim O(1),$ (6)

where $\Delta\phi$ is the change of field $\phi$. For convenience, we will
call eq.$(1.3)$ and $(1.4)$ “Condition 1” and “Condition 2”, respectively. In
this article, we will focus our interest on a non-minimally coupled scalar
field theory known as the Palatini Higgs inflation scenario [5,6,15,16], which
has not been examined so far. The goal is to find the constraints for
constants $c_{1}$, $c_{2}$ which result from the Palatini Higgs inflation
potential and to test the inflationary condition of this model.
The rest of this article is structured as follows: In section 2, we will
briefly review the Palatini Higgs inflation scenario. In section 3, the range
of swampland parameters $c_{1}$ and $c_{2}$ of Palatini Higgs inflation will
be obtained. In section 4, we will reexamine the requirements of inflation for
the refined dS conjecture. In section 5 we will present the conclusions.

##  2 The Palatini Higgs inflation model

In the usual formulation of Higgs inflation, the action is minimized with
respect to the metric [16]. This procedure implicitly assumes the existence of
a Levi-Civita connection depending on the metric tensor and the inclusion of a
York-Hawking-Gibbons term ensuring the cancellation of a total derivative term
with no-vanishing variation at the boundary [16,17,18]. One could
alternatively consider a Palatini formulation of gravity in which the metric
tensor and the connection are treated as independent variables and no
additional boundary term is required to obtain the equations of motion
[16,19]. Roughly speaking, this formulation corresponds to assuming an
ultraviolet completion involving different gravitational degrees of freedom
[16].
In this article, we neglect the running of the parameter $\xi$ and apply
unitary gauge for the Higgs field, $H=(0,h)^{T}/\sqrt{2}$, the relevant part
of the Lagrangian then reads [5]:

$L=-\frac{M^{2}_{p}+\xi
h^{2}}{2}R+\frac{1}{2}(\partial_{\mu}h)^{2}-\frac{\lambda}{4}h^{4},$ (7)

where $\xi>0$ is the strength of the non-minimal coupling of the Lagrangian to
gravity.
We can perform a Weyl transformation of the metric to make the kinetic term of
$h$ canonical, i.e. [5],

$\hat{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu},$ (8)

where

$\Omega^{2}=1+\frac{\xi h^{2}}{M^{2}_{p}},$ (9)

and the field can be redefined [5,20]

$h=\frac{M_{p}}{\sqrt{\xi}}\sinh(\frac{\sqrt{\xi}\chi}{M_{p}}).$ (10)

Then the Lagrangian then becomes [5]

$L=-\frac{M^{2}_{p}}{2}\hat{R}+\frac{1}{2}(\partial_{\mu}\chi)^{2}-U(\chi),$
(11)

where the scalar potential energy is given by

$U(\chi)=\frac{\lambda
M^{4}_{p}}{4\xi^{2}}\left(\tanh(\frac{\sqrt{\xi}\chi}{M_{p}})\right)^{4}.$
(12)

Note that if we had used in the metric formulation of the theory, we would
have obtained the same form of the Lagrangian but with a different potential
$U(\chi)$ [5,20]. The potential of eq.$(2.6)$ gives rise to inflation at the
field values $\sqrt{\xi}\chi\gg M_{p}$ [5,20]. More details can be found in
ref.[16].
In the following sections, we will take $M_{p}=1$. In other words, the
inflationary regime is $\sqrt{\xi}\chi\gg 1$ and eqs.$(1.3)$ and $(1.4)$
become

$|\nabla V|\geq c_{1}\cdot V,$ (13) $min(\nabla_{i}\nabla_{j}V)\leq-c_{2}\cdot
V.$ (14)

##  3 Swampland parameters of the Palatini Higgs inflation model

Since there is only one scalar field $\chi$ in the Lagrangian $(2.5)$, it is
convenient to define two functions:

$F_{1}(\chi)=\frac{|dU(\chi)/d\chi|}{U(\chi)},$ (15)

and

$F_{2}(\chi)=\frac{d^{2}U(\chi)/d\chi^{2}}{U(\chi)}.$ (16)

Then considering eq.$(2.6)$, we obtain

$F_{1}(\chi)=\frac{4[1-\tanh^{2}(\sqrt{\xi}\chi)]\sqrt{\xi}}{\tanh(\sqrt{\xi}\chi)},$
(17)

and

$F_{2}(\chi)=\frac{4\xi[1-\tanh^{2}(\sqrt{\xi}\chi)][3-5\tanh^{2}(\sqrt{\xi}\chi)]}{\tanh^{2}(\sqrt{\xi}\chi)}.$
(18)

In the inflationary regime, $\sqrt{\xi}\chi\gg 1$, the value of
$\tanh(\sqrt{\xi}\chi)$ is very close to 1 but less than 1, therefore
$F_{2}(\chi)$ must be less than zero.
Next, we need to introduce two “slow-roll” field inflation indices [15,21,22]:

$\epsilon_{V}=\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2},$ (19)
$\eta_{V}=\frac{V^{\prime\prime}}{V},$ (20)

where $V$ is the potential energy in the Lagrangian. For the Palatini Higgs
inflation model, we have

$\epsilon_{V}=\frac{1}{2}F^{2}_{1},$ (21) $\eta_{V}=F_{2}.$ (22)

In the slow-roll region, the tensor-to-scalar ratio $r$ and the spectral index
$n_{s}$ can be expressed as [1]:

$r=16\epsilon_{V}=8F^{2}_{1},$ (23)
$n_{s}=1-6\epsilon_{V}+2\eta_{V}=1-3F^{2}_{1}+2F_{2},$ (24)

or

$F_{1}=\sqrt{2\epsilon_{V}}=\sqrt{\frac{r}{8}},$ (25)
$F_{2}=\eta_{V}=\frac{n_{s}-1+3r/8}{2}.$ (26)

According to ref.[15], up to $O(1/\xi)$ corrections, the slow-roll parameters
in the Palatini Higgs inflation take the form:

$\epsilon_{V}\approx\frac{1}{8\xi N^{2}_{V}},$ (27)
$\eta_{V}\approx-\frac{1}{N_{V}},$ (28)

with

$N_{V}=\int_{i}^{\chi}\frac{d\chi}{\sqrt{2\epsilon_{V}}}$ (29)

defining the number of $e$-folds and $\chi$ the corresponding background field
value.
In addition, up to order of $O(1/N^{2}_{V})$, we have [15]:

$n_{s}\approx 1-\frac{2}{N_{V}}+\frac{2.8}{N^{2}_{V}}.$ (30)

Here we will use the recent inflationary data of Cosmic Microwave Background
radiation (CMB) from the Planck observatory [1,3,4] and also the data obtained
by the BICEP2+$Keck$ CMB polarization experiments [1,2]:

$n_{s}\simeq 0.965\pm 0.004,$ (31) $r\lesssim 0.06.$ (32)

According to refs.[5,6], the bound of $\xi$ is

$1.0\times 10^{6}<\xi<6.8\times 10^{7}.$ (33)

In this article, we take $N_{V}=50.9$ corresponding to $\xi=10^{7}$ [5]. In
fact, as can be seen from the following calculated results, the orders of
magnitude of the results are more important than the specific values.
Inserting $N_{V}=50.9$ into eq.$(3.16)$, we obtain $n_{s}=0.962$, which
satisfies the lower bound of eq.$(3.17)$.
Combining eq.$(3.13)$, $(3.19)$ and using $N_{V}=50.9$, we then obtain the
bounds of $\epsilon_{V}$

$7.095\times 10^{-13}<\epsilon_{V}<4.825\times 10^{-11},$ (34)

From eq.$(3.9)$, $(3.11)$ and $(3.12)$, we can obtain the bounds of $r$ and
$F_{1}$ as

$1.135\times 10^{-11}<r<7.720\times 10^{-10},$ (35) $1.191\times
10^{-6}<F_{1}<9.823\times 10^{-6},$ (36)

and that

$F_{2}\approx-0.019.$ (37)

The bounds of $r$ eq.$(3.21)$ satisfy the observatory data eq.$(3.18)$,
namely, $r\lesssim 0.06$.
If the Palatini Higgs inflation model satisfies the refined dS conjecture,
then

$c_{1}\leq 9.823\times 10^{-6},$ (38)

or

$c_{2}\leq 0.019.$ (39)

However, neither $c_{1}$ nor $c_{2}$ is positive constant of the order of 1
[1,8,14,21,22]. Therefore, we suggest that the Palatini Higgs inflation model
cannot satisfy the refined dS conjecture. In other words, if the refined dS
conjecture holds, then the Palatini Higgs inflation model cannot be the low
energy effective theory of a consistent quantum gravity theory. In our
analysis, the RG running of the non-minimal coupling $\xi$ has been neglected.
In fact, this approximation is justified [5,6].

##  4 Testing inflationary condition of the Palatini Higgs inflation model

In this section, we will determine the value of $\sqrt{\xi}\chi$ and check if
its value satisfies the inflationary condition, $\sqrt{\xi}\chi\gg 1$.
Firstly, we will use $F_{1}$ to check the condition. If $\xi=6.8\times 10^{7}$
and $F_{1}=1.191\times 10^{-6}$ these values are inserted into eq.$(3.3)$, we
then have

$32984.845m^{2}+1.191\times 10^{-6}-32984.845=0,$ (40)

where we have defined $m=\tanh(\sqrt{\xi}\chi)>0$. We then derive that
$m=\tanh(\sqrt{\xi}\chi)\approx 1$. Therefore, $\sqrt{\xi}\chi\gg 1$.
Similarly, if $\xi=1.0\times 10^{6}$ and $F_{1}=9.823\times 10^{-6}$ and these
values are inserted into eq.$(3.3)$, we then also have $\sqrt{\xi}\chi\gg 1$.
Secondly, we will use $F_{2}$ to check the condition. Based on eq.$(3.23)$, in
the range $1.0\times 10^{6}<\xi<6.8\times 10^{7}$, the value of $F_{2}$ is
always approximately -0.019.
If $\xi=6.8\times 10^{7}$ and these values are inserted into eq.$(3.4)$, we
then have

$m^{2}_{1}\approx 1,$ (41)

namely, $\sqrt{\xi}\chi\gg 1$. Or

$m^{2}_{2}\approx\frac{3}{5},$ (42)

namely, $\sqrt{\xi}\chi\approx 1.032$, which cannot satisfy the inflationary
condition, $\sqrt{\xi}\chi\gg 1$. Therefore, for $\xi=6.8\times 10^{7}$, there
is only one reasonable value of $\sqrt{\xi}\chi$.
Similarly, if $\xi=1.0\times 10^{6}$ and these values are inserted into
eq.$(3.4)$, we have the only one reasonable solution $m\approx 1$, namely,
$\sqrt{\xi}\chi\gg 1$.
Therefore, according to observational data, we can determine that the value of
$\sqrt{\xi}\chi$ satisfies the inflationary condition, $\sqrt{\xi}\chi\gg 1$.
In other words, the model can be consistent with observations. However, as we
have pointed out in section 3, if the refined dS conjecture holds, then the
Palatini Higgs inflation model cannot be the low energy effective theory of a
consistent quantum gravity theory.

##  5 Conclusions and discussions

The refined dS conjecture could provide important implications for the low
energy effective theory models which may or may not be consistent with the
quantum gravity theory [1].The conjecture suggests two related but independent
conditions for any scalar potential $V(\phi)$ of a low energy effective theory
for a consistent quantum gravity [7,8]:

$|\nabla V|\geq c_{1}\cdot V,$ (43)

or

$min(\nabla_{i}\nabla_{j}V)\leq-c_{2}\cdot V,$ (44)

which we have respectively called Condition 1 and Condition 2 in this article.
The parameters $c_{1}$ and $c_{2}$ are both positive constants of the order of
1. In this paper, we closely examined a non-minimally coupled scalar field
theory known as the Palatini Higgs inflation model as a concrete example of a
potentially realistic inflationary model and applied the refined dS conjecture
to evaluate its consistency with quantum gravity theory.
At first, we review the conclusions in section 3. On the one hand, we have
defined two functions $F_{1}$ and $F_{2}$ to determine
$\frac{|dU(\chi)/d\chi|}{U(\chi)}$ and
$\frac{d^{2}U(\chi)/d\chi^{2}}{U(\chi)}$ for Palatini Higgs inflation model,
respectively. Then, we express $F_{1}$ and $F_{2}$ using the tensor-to-scalar
ratio $r$ and the spectral index $n_{s}$ in the “slow-roll” regime. On the
other hand, according to ref.[15], the spectral index $n_{s}$ in Palatini
Higgs inflation model can be reexpressed by the number of $e$-folds $N_{V}$,
i.e., eq.$(3.16)$. According to ref.[5], the bound of the coupling parameter
$\xi$ in Palatini Higgs inflation model can be obtained. In this article, we
take $N_{V}=50.9$ corresponding to $\xi=10^{7}$ and obtain that $n_{s}=0.962$
which is consistent with the CMB observational data [1,2]. In fact, as can be
seen from the following calculated results, the orders of magnitude of the
results are more important than the specific values. Furthermore, we have
applied the latest cosmological observations from Planck 2018, from
BICEP2+$Keck$ [2,3,4] and the bound of $\xi$ [5,6] to determine the range of
two functions $F_{1}$ and $F_{2}$. If the refined dS conjecture was indeed to
hold, we derived the upper bounds of $c_{1}$ and $c_{2}$, namely, $c_{1}\leq
9.823\times 10^{-6}$ and $c_{2}\leq 0.019$. However, the values of $c_{1}$ and
$c_{2}$ are much smaller than $O(1)$. We theorefore suggest that if the
refined dS conjecture was to hold, then the Palatini Higgs inflation model
cannot be the low energy effective theory for a consistent quantum gravity
theory.
We have also determined the value of $\sqrt{\xi}\chi$ using observational data
in section 4. Using the upper and lower bound of $\xi$ and the corresponding
$F_{1}$, we can find that $\sqrt{\xi}\chi\gg 1$. Since the value of
$\sqrt{\xi}\chi$ satisfies the inflationary condition of Palatini Higgs
inflation model, i.e., $\sqrt{\xi}\chi\gg 1$ [5,20], the model is self-
consistent. We therefore suggest that the “Palatini Higgs inflation” scenario
is in the “Swampland” if the refined dS swampland conjecture holds.
Our understanding of the “Swampland” is still very much in flux. Since it is
not a mature field there are no underlying principles from which results can
be rigorously derived. Nevertheless, the past few years have seen significant
interest and work on the “Swampland” and as a result several ideas and
proposals have emerged as strong candidates framed within a more coherent
structure. This is encouraging and suggests that we are on the right track
towards uncovering some interesting and possibly impactful new physics. Of
course, it is highly likely that the various conjectures will be modified,
some perhaps will be ruled out altogether in view of counter-examples and
others will be refined. There is also increasing evidence that there are
strong links between conjectures which were initially considered to be
independent of each other, suggesting that we may eventually obtain a picture
in which the conjectures are all consequences of a single underlying newly
discovered principle [23].
In the future work, we will apply double field theory in order to investigate
the “Swampland” scenario. Double field theory formalism allows a description
of the fundamental string in a way that is manifestly symmetric under
$T$-duality [24,25]. The cosmological consequences of the resulting low energy
vacua have not yet been fully explored. The quest to understand how de Sitter
space can emerge from both supergravity and string theory is also of crucial
importance and touches on aspects of the “Swampland” scenario.

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# Hiding Signatures of Gravitational Instability in Protoplanetary Discs with
Planets

Sahl Rowther Centre for Exoplanets and Habitability, University of Warwick,
Coventry CV4 7AL, UK  Department of Physics, University of Warwick, Coventry
CV4 7AL, UK  Farzana Meru Centre for Exoplanets and Habitability, University
of Warwick, Coventry CV4 7AL, UK  Department of Physics, University of
Warwick, Coventry CV4 7AL, UK  Grant M. Kennedy Centre for Exoplanets and
Habitability, University of Warwick, Coventry CV4 7AL, UK  Department of
Physics, University of Warwick, Coventry CV4 7AL, UK  Rebecca Nealon
Department of Physics and Astronomy, University of Leicester, Leicester, LE1
7RH, United Kingdom  Christophe Pinte School of Physics and Astronomy, Monash
University, Clayton Vic 3800, Australia

(Received date; Revised date; Accepted date)

###### Abstract

We carry out three dimensional SPH simulations to show that a migrating giant
planet strongly suppresses the spiral structure in self-gravitating discs. We
present mock ALMA continuum observations which show that in the absence of a
planet, spiral arms due to gravitational instability are easily observed.
Whereas in the presence of a giant planet, the spiral structures are
suppressed by the migrating planet resulting in a largely axisymmetric disc
with a ring and gap structure. Our modelling of the gas kinematics shows that
the planet’s presence could be inferred, for example, using optically thin
13C16O. Our results show that it is not necessary to limit the gas mass of
discs by assuming high dust-to-gas mass ratios in order to explain a lack of
spiral features that would otherwise be expected in high mass discs.

protoplanetary discs — planet-disc interactions

††journal: ApJL††software: Matplotlib (Hunter, 2007), numpy (van der Walt et
al., 2011), astropy (The Astropy Collaboration et al., 2018), phantom (Price
et al., 2018), splash (Price, 2007), mcfost (Pinte et al., 2006, 2009)

##  1 Introduction

In the last few years, a large number of discs have revealed substructure in
the form of rings and gaps when observed at millimeter wavelengths with ALMA
(ALMA Partnership et al., 2015; Andrews et al., 2016; Fedele et al., 2018;
Andrews et al., 2018; Huang et al., 2018; Dipierro et al., 2018; Booth & Ilee,
2020). A few of these discs are thought to be very young ($<1$Myr) (ALMA
Partnership et al., 2015; Fedele et al., 2018; Dipierro et al., 2018). Ring
and gap structures have also been observed in even younger ($\,\lesssim
0.5\,$Myrs) Class 1 discs (Sheehan & Eisner, 2018; Segura-Cox et al., 2020).
Young discs are thought to be massive and could potentially be gravitationally
unstable. Such discs are expected to harbour spiral arms. There is evidence
that discs with spiral arms in the midplane exist (Pérez et al., 2016; Huang
et al., 2018), although they seem to be quite rare. Is the observed rarity due
to young discs being less massive or can spiral structures in massive discs be
hidden by other processes?

The goal of this Letter is to investigate the impact that a migrating giant
planet, irrespective of how it formed, has on the structure of a
gravitationally unstable disc and the resulting implications for observations
of such discs. We find that a migrating giant planet is able to suppress
spiral structures yielding discs that appear axisymmetric with rings and gaps.

##  2 Method

###  2.1 Hydrodynamical simulations & initial conditions

We perform scale free 3D gas hydrodynamic simulations using phantom, a
smoothed particle hydrodynamics (SPH) code developed by Price et al. (2018).

The disc setup is identical to that in Rowther & Meru (2020). We model a disc
using 2 million particles between $R_{in}=1$ and $R_{out}=25$ in code units
with a disc-to-star mass ratio of 0.1. The central star, and the planet are
modelled as sink particles (Bate et al., 1995). The accretion radius of the
central star is set to be equal to the disc inner boundary, $R_{in}$. To
maintain a roughly constant planet mass throughout the simulation, the
accretion radius of the planet is limited by setting it to 0.001 in code
units. This is ${\sim}10$ times smaller than the minimum Hill radius for a
$3M_{Jup}$ planet at $R_{in}$. The initial surface mass density is set as a
smoothed power law and is given by

$\Sigma=\Sigma_{0}\left(\frac{R}{R_{0}}\right)^{-1}f_{s},$ (1)

where $\Sigma_{0}$ is the surface mass density at $R=R_{0}=1$ and
${f_{s}=1-\sqrt{R_{in}/R}}$ is the factor used to smooth the surface density
at the inner boundary of the disc. The initial temperature profile is
expressed as a power law

$T=T_{0}\left(\frac{R}{R_{0}}\right)^{-0.5},$ (2)

where $T_{0}$ is set such that the disc aspect ratio ${H/R=0.05}$ at
$R=R_{0}$. The energy equation is

$\frac{\mathrm{d}u}{\mathrm{d}t}=-\frac{P}{\rho}\left(\nabla\cdot\mathbf{v}\right)+\Lambda_{\mathrm{shock}}-\frac{\Lambda_{\mathrm{cool}}}{\rho}$
(3)

where we assume an adiabatic equation of state, $u$ is the specific internal
energy, the first term on the RHS is the $P\mathrm{d}V$ work,
$\Lambda_{\mathrm{shock}}$ is a heating term that is due to the artificial
viscosity used to correctly deal with shock fronts, and

$\Lambda_{\mathrm{cool}}=\frac{\rho u}{t_{\mathrm{cool}}}$ (4)

controls the cooling in the disc. Here we use a simple implementation of the
cooling time which is proportional to the dynamical time by a factor of
$\beta$,

$t_{\mathrm{cool}}=\beta(R)\Omega^{-1}=\beta_{0}\left(\frac{R}{R_{0}}\right)^{-2}\Omega^{-1},$
(5)

where $\Omega$ is the orbital frequency and we have varied $\beta$ with radius
(Rowther & Meru, 2020). This allows us to mimic a realistic self-gravitating
disc that is only gravitationally unstable in the outer regions (Rafikov,
2005; Stamatellos & Whitworth, 2009; Rice & Armitage, 2009; Clarke, 2009).

To model shocks, we use an artificial viscosity switch that utilises the time
derivative of the velocity divergence introduced by Cullen & Dehnen (2010).
The artificial viscosity parameter $\alpha_{v}$ has a maximum of
$\alpha_{\text{max}}=1$ near the shock and a minimum of
$\alpha_{\text{min}}=0$ far away from the shock. The artificial viscosity
coefficient $\beta_{v}$ is set to 2 (see Price et al. 2018).

####  2.1.1 Embedding the planet

The simulation is first run for 10 orbits to allow spiral structure in the
disc to develop. We then split it into two simulations, one with an embedded
planet and one which continues to evolve without a planet. The latter is a
control simulation to compare the impact a migrating giant planet has on the
spiral structure in a gravitationally unstable disc. A planet with a planet-
to-star mass ratio of ${q=2.9\times 10^{-3}}$, equivalent to
$3M_{\mathrm{Jup}}$ in a $0.1M_{\odot}$ disc around a $1M_{\odot}$ central
star is added at $R_{p}=20$.

The simulations are run for another 8 orbits, with a total simulation time of
18 orbits.

###  2.2 Post processing of simulations

The raw synthetic continuum images at $1.3$mm are created using mcfost (Pinte
et al., 2006, 2009). We first scale the simulations such that $R_{out}=200$au,
and initial ${R_{p}=160}$au. We use $10^{8}$ photon packets on a Voronoi
tesselation where each mcfost cell corresponds to an SPH particle. The
luminosity of the star is calculated using an assumed mass of $1M_{\odot}$ and
a 1Myr isochrone from Siess et al. (2000) which corresponds to a temperature
of $T_{\star}=4286$K. Since the gas surface density of the disc is quite
large, the Stokes numbers in the disc are small enough ($<0.1$ for mm sized
grains) that we assume that the dust is well coupled to gas. Therefore we
assume that the dust distribution is identical to the gas distribution and a
constant dust-to-gas ratio of 0.01. The dust sizes vary between 0.3 and 1000
$\mu$m and are distributed across 100 different sizes with a power-law
exponent of -3.5. We assume all dust grains are made of astronomical
silicates, and are spherical and homogeneous. We compute the dust properties
using Mie theory. The disc is assumed to be at a distance of 140pc.

To create mock millimeter continuum observations, we use the ALMA Observation
Support Tool (Heywood et al., 2011). We use integration times of 12, 30, 60,
and 120 minutes at 230GHz (1.3mm) in Band 6 with the ALMA Cycle 8 C43-7
configuration. We assume a bandwidth of 7.5GHz and a precipitable water vapour
level of 0.913mm. CLEAN images are created using natural weights resulting in
a beam size of $0.107{}^{\prime\prime}\times 0.124{}^{\prime\prime}$, or
equivalently $15.0$au $\times 17.3$au at 140pc. As shown by Mayer et al.
(2016), different ALMA configurations can alter the detectability of features.
Hence a similar set mock observations are created with the C43-6 configuration
($0.156{}^{\prime\prime}\times 0.196{}^{\prime\prime}$) and with a disc
inclined at $40^{\circ}$ using the C43-7 configuration.

Assuming a disc inclination and position angle of $40^{\circ}$, optically
thick 12CO, and optically thinner 13C16O CO-isotopologue channel maps are
generated for the ${J=3-2}$ transition using a velocity resolution of $0.1$
km/s. We choose this transition as it can be observed with a good compromise
between observation time and signal-to-noise ratio. The abundances for 12CO
and 13C16O are assumed to be a fraction $1\times 10^{-4}$ and $2\times
10^{-7}$ of the total disc mass (i.e. relative to $H_{2}$) respectively. We
account for CO freeze-out at $T<20$K, and photo-dissociation and photo-
desorption in regions of high UV radiation (see Appendix B of Pinte et al.
2018). We then convolve the resulting images with a beam size of
$0.05{}^{\prime\prime}\times 0.05{}^{\prime\prime}$; the maximum resolution
expected for CO line observations.

##  3 Results

###  3.1 Impact on spiral structure

As shown in Rowther & Meru (2020), planets migrate inwards rapidly until they
reach the gravitationally stable inner disc. For the simulations presented
here, the gravitationally stable region is inside $R\approx 120$au. The impact
of the planet on the disc as it migrates is compared to the control simulation
using the Toomre parameter (Toomre, 1964),

$Q=\frac{c_{s}\Omega}{\pi G\Sigma},$ (6)

which gives a measure of how gravitationally unstable the disc is, where
$\Sigma$ and $c_{s}$ are the disc surface density and sound speed,
respectively.

Figure 1: Surface density (left) and 2D Toomre parameter (right) plots of a
disc without a planet (top) and with a $3M_{\mathrm{Jup}}$ planet (bottom) at
the end of the simulation. The 2D Toomre plots show how gravitationally
unstable the disc is. The critical value for non-axisymmetric instabilities is
when $Q\lesssim 1.7$ (Durisen et al., 2007), shown in red. The planet erases
the spiral structure and is massive enough to carve out a gap. This results in
a gravitationally stable axisymmetric disc.

Figure 1 shows the surface density (left) and the 2D Toomre parameter $Q$
(right) of the simulations without a planet (top) and with a
$3M_{\mathrm{Jup}}$ planet (bottom) at the end of the simulation. It is clear
that the presence of the planet significantly impacts the structure. The
planet suppresses the spiral structure and opens up a gap. Despite both discs
retaining the same disc mass (${\sim}0.09M_{\odot}$) within 200au, the 2D $Q$
plots show that the presence of the planet results in the disc becoming mostly
gravitationally stable with $Q>1.7$ throughout the disc. Without a planet,
strong gravitationally unstable ($Q<1.7$) arms are present.

Figure 2: Azimuthally averaged Toomre parameter, $Q$ (top), surface density,
$\Sigma$ (middle), and the sound speed, $c_{s}$ (bottom) for both the disc
without a planet (red lines) and with a $3M_{\mathrm{Jup}}$ planet (blue
lines) for the first 2 orbits after the planet is added. The darker shades
represent later times. The increase in $c_{s}$ due to the planet is the
dominant reason for the increase in $Q$ in the outer regions of the disc
(beyond $R\approx 120$au), causing the disc to become gravitationally stable.

The dominant reason for the increase in $Q$ is due to the planet’s influence
on the disc temperature, or equivalently the sound speed $c_{s}$, in the disc.
The spiral wakes generated by the planet as it migrates are regions of
relative overdensities with respect to the disc background. The radially
propagating spiral wakes can evolve into shocks. The exchange of momentum and
energy between the density wakes and the disc at the shocks influence the
global properties of the disc, heating it up (Goodman & Rafikov, 2001;
Rafikov, 2016; Ziampras et al., 2020).

The azimuthally averaged Toomre parameter (top), surface density (middle), and
sound speed (bottom) is plotted in Figure 2 for the first 2 orbits after the
$3M_{\mathrm{Jup}}$ planet (blue lines) is embedded. The red lines represent
the disc without the planet at the same times. It can be seen, particularly in
the regions of the disc beyond $R\approx 120$au, that there is a quick and
significant increase in the sound speed after the planet has been added,
whereas, the change in $\Sigma$ is relatively small. This increase in
temperature causes the disc to become gravitationally stable, resulting in a
higher value of the Toomre parameter.

To ensure the sudden inclusion of the $3M_{\mathrm{Jup}}$ planet did not
trigger artificial excess heating, we compared the artificial viscosity to the
gravitational stress parameter immediately after the planet was embedded. The
increase in artificial viscosity due to the planet is negligible compared to
the magnitude of the gravitational stress parameter.

###  3.2 Continuum images

Figure 3: Mock observation (left) using the C43-7 configuration with an
integration time of 30 minutes, axisymmetric flux map (middle), and residual
flux (right) of a disc without a planet (top) and with a $3M_{\mathrm{Jup}}$
planet (bottom). The white ellipse in the bottom left of each mock observation
represents the beam size. The presence of the planet results in a disc that no
longer consists of spiral structure due to gravitational instability and
appears to be more axisymmetric. Figure 4: Top subset of panels show the mock
observations (top half) and residual plots (bottom half) of a disc with a
$3M_{\mathrm{Jup}}$ planet with integration times of 12, 30, 60, and 120
minutes using the higher resolution C43-7 configuration. The bottom left
subset of panels is similar but only for integrations times of 12 and 30
minutes using the C43-6 configuration. At high resolution the disc appears
completely axisymmetric with short integration times. Whereas with longer
integration times (over 60 minutes), spiral arms caused by the planet can be
seen in the residual plots. At lower resolutions, the gap carved by the planet
appears shallower, but the planet’s spiral arms are easier to detect even with
short integration times. The bottom right subset of panels show that the
planet’s spiral arms are also detectable for moderately inclined discs. This
is shown for a disc inclined by $40^{\circ}$.

The left column of Figure 3 compares the mock observations of a disc with a
$3M_{\mathrm{Jup}}$ planet (bottom) and without a planet (top) with an
integration time of 30 minutes using the C43-7 configuration. An axisymmetric
flux map is produced as the azimuthally averaged flux of the mock observations
and is shown in the middle column. Finally to highlight non-axisymmetric
features, the residual flux is plotted in the right column by subtracting the
axisymmetric map from the mock observation. The top subset of panels in Figure
4 show the mock observations (top half) and residuals (bottom half) using the
higher resolution C43-7 configuration. The bottom left subset of panels show
the same with the lower resolution C43-6 configuration. The bottom right
subset of panels are for a disc inclined by $40^{\circ}$ using the C43-7
configuration.

From Figure 3, it can be seen that without a planet, spiral arms due to
gravitational instability would be readily apparent. Whereas with the planet,
the only major non-axisymmetric feature that remains are the spiral arms
caused by the planet as shown in Figures 3 and 4. The higher resolution mock
observations are able to better resolve the gap, however the planet induced
spiral arms are only visible with integration times $>1$hr. This is true for
both the non-inclined and moderately inclined disc. Using a lower resolution,
which increases the signal-to-noise, allows the spiral arms to be easily seen
in the residuals at lower integration times. The tradeoff is a gap that is
less resolved and shallower in the continuum image. Despite the lower
resolution images showing the spiral arms of the planet more easily, higher
resolution images would still be favourable as a less resolved and shallower
gap can lead to underestimates of the planet’s mass. This could in turn lead
to predictions with other methods of estimating the planet’s mass, such as
with the CO kinematics, that are inconsistent.

###  3.3 CO kinematics

Figure 5: Synthetic continuum subtracted channel maps ($\mathrm{J}=3-2$
transitions) for a) 12CO and b) 13C16O at $\Delta v=\pm 0.5\,\mathrm{km\
s}^{-1}$ from the systemic velocity of the disc with and without the planet. A
kink is not detected in the optically thick 12CO, but is visible in the
optically thin 13C16O in the negative velocity channel within the vicinity of
the planet, see bottom right inset in Fig 5b. The bend in the velocity profile
in the bottom right inset is referred to as the kink. This is contrasted with
the smooth velocity profile in the bottom left inset. The disc inclination and
position angle are both $40^{\circ}$.

The spiral waves generated by the planet can cause localised deviations in the
Keplerian flow of the disc as shown recently by Pinte et al. (2019, 2020).
These deviations can be detected as kinks in the gas channel maps.

Figure 5 shows the channel maps for 12CO and 13C16O at $\Delta v=\pm
0.5\,\mathrm{km\ s}^{-1}$ from the systemic velocity with the continuum
subtracted. The optically thick 12CO does not reveal any signs of a planet,
but a kink is noticeable in the optically thinner 13C16O in the
$-0.5\,\mathrm{km\ s}^{-1}$ channel within the vicinity of the planet. We can
exclude large scale perturbations or any azimuthally symmetric mechanisms
since the kink is not seen in the opposite velocity channel where the profile
remains smooth. The kink is shown in the bottom right panel of Fig 5b in the
inset. The kink is also seen in the channels from $v=-0.3\mathrm{\ to\
}{-1.2}\,\mathrm{km\ s}^{-1}$.

Although the channel maps without the planet are not perfectly smooth as would
be expected from a Keplerian disc, this is likely to be due to the
gravitational instabilities. Hall et al. (2020) show that for a
gravitationally unstable disc the perturbations due to gravitational
instabilities are seen in all azimuths and in both the positive and negative
velocity channels. However in our simulations with a planet, the kink is more
localised appearing in only the negative velocity channel. To ensure that the
observed kink is due to the planet, we re-calculated the channel maps with the
planet at slightly different times with the planet being in different
azimuthal locations. Although the kink is not detectable at all times,
whenever the kink is detected it is in a different location determined by the
position of the planet. Additionally, in some cases the kink was only detected
for a narrower range of $\Delta v$.

##  4 Discussion

###  4.1 Implications for dust-to-gas mass ratios

In the early stages of evolution, discs are expected to be massive and
gravitationally unstable (e.g. Bate, 2018). A characteristic feature of such
discs are their spiral arms. However, from observations so far (Pérez et al.,
2016; Huang et al., 2018), these discs appear to be quite rare. Due to the
difficulty in directly measuring the gas mass of the disc, the disc mass is
often inferred via the dust mass using a fixed dust-to-gas mass ratio. In some
scenarios, such as with MWC 480 (Liu et al., 2018), the observed dust mass can
be high enough such that inferring the gas mass via the canonical dust-to-gas
mass ratio of 0.01 results in a disc that is massive enough to be
gravitationally unstable. Thus in their models, they assume a higher dust-to-
gas mass ratio to ensure that the disc is gravitationally stable. Facchini et
al. (2020) show that LkCa-15 retains a significant amount of dust mass in its
rings. Consequently, when modelling LkCa-15, they assume an upper limit on the
disc gas mass (and hence a higher dust-to-gas mass ratio) to ensure the disc
is not locally gravitationally unstable.

In both cases this assumption was deemed necessary to explain the lack of
spiral features in the observations, which would be expected from more massive
discs. However, we show that a migrating $3M_{\mathrm{Jup}}$ planet can cause
a massive disc to become gravitationally stable and suppress any spiral
structure that would otherwise be present. Hence, large dust-to-gas mass
ratios are not necessarily required to explain a lack of spiral features.

###  4.2 Caveats

A caveat to the results presented here is the long term gravitational
stability of the discs due to the cooling mechanism used. Although it mimics
the characteristics of a realistic self-gravitating disc, i.e. a disc which is
only gravitationally unstable in the outer regions, it is still a straight-
forward implementation where the cooling time is a simple expression
determined by the location in the disc.

The actual cooling in the disc is likely to be more complex and evolve over
time. Recently Mercer et al. (2018) calculated an effective $\beta$ from their
radiative transfer calculations, which was found to vary both spatially and
with time. Hence, it is unknown what impact using more complex cooling methods
such as radiative transfer to cool the disc will have on the long term
gravitational stability of the disc with a migrating planet. Whilst we show
that a migrating giant planet can erase the spiral structure in a self-
gravitating disc, it remains to be investigated whether the loss of spiral
structure remains for a significant amount of time.

The results presented here initially appear contrary to Meru (2015) where it
was shown that a fragment formed by gravitational instability in the outer
disc could trigger subsequent fragmentation in the inner disc. The main
difference is that the Toomre profiles are quite different. In Meru (2015),
the inner disc was on the edge of fragmentation ($Q\sim 1$). Hence the
increase in the surface density as a result of the fragment driving material
inwards towards the disc caused the disc to fragment. This behaviour is not
seen or expected in this work as the inner disc remains well above the
gravitationally stable regime ($Q\gtrsim 1.7)$.

In this work, we have only presented results for one planet and disc mass.
Lower mass planets would be less likely to impact self-gravitating spiral
structures. On the other hand, lower mass discs, which have weaker
gravitational instabilities, would be more susceptible to having their spiral
structures suppressed by planets. Although we find that a planet that is large
enough to open up a gap also suppresses the spiral structure, we cannot
conclude that non-gap-opening planets are unable to affect the disc structure.
We will present a follow up study that considers a variety of planet and disc
properties, as well as ALMA configurations, to investigate the conditions at
which spiral suppression is likely to be observed.

##  5 Conclusion

We perform 3D SPH simulations to investigate the impact a migrating giant
planet has on the structure of a gravitationally unstable disc. Our work shows
that the presence of the planet suppresses the spiral structure in the disc
and causes the disc to become gravitationally stable because it alters the
temperature structure. This interaction between the planet and the disc causes
the self-gravitating phase of the disc to be shortened, while retaining the
same disc mass. The planet is able to carve open a gap resulting in an
axisymmetric disc.

The mock ALMA observations of the continuum presented here show that the disc
can appear completely axisymmetric for higher resolution ALMA configurations.
However, with longer integration times or by sacrificing resolution, spiral
arms from the planet become observable. In the latter case, the gap carved by
the planet will be less resolved and appear shallower. We also show that the
planet can be detected with high resolution kinematics using optically thin
CO-isotopologues like 13C16O. Our results show it is possible to explain a
lack of spiral structure in high mass discs without requiring high dust-to-gas
mass ratios to limit the gas mass.

## acknowledgments

We thank the anonymous referee for their useful comments which benefited this
work. SR acknowledges support from the Royal Society Enhancement Award. FM
acknowledges support from the Royal Society Dorothy Hodgkin Fellowship. GMK is
supported by the Royal Society as a Royal Society University Research Fellow.
RN acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 681601). This work was performed using Orac and Tinis HPC
clusters at the University of Warwick.

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