[a]Jong-Hyun Yoon

# Inflaton Dark Matter

###### Abstract

We discuss a minimal extension of the Standard Model (SM) where a single real
scalar field serves as both inflaton and dark matter. The corresponding
Lagrangian contains the renormalizable interactions of the inflaton field.
Quantum effects generally induce a non–minimal coupling to gravity which
facilitates inflation consistent with the PLANCK constraints. A large fraction
of the inflaton quanta produced after inflation must be converted into the SM
radiation reheating the Universe and the rest remains dark matter today. We
consider thermal and non–thermal production of inflaton dark matter. In the
non–thermal case, we take into account collective effects with the help of
lattice simulations. Combining analytic and numerical results with the
unitarity consideration, we find that the inflaton dark matter model is viable
only in the thermal case where the inflaton mass is near half the Higgs mass.

##  1 Introduction

Dark matter (DM) is one of the deepest mysteries in the history of physics,
yet no convincing signatures of dark matter have been detected. Motivated by
the inflationary Big Bang cosmology, one may consider an economical
possibility that a single scalar field plays the role of both the inflaton
field and dark matter [1]. Non–minimal couplings to gravity induced by the
quantum effects [2] may be used to fit the inflationary dynamics while the
renormalizable interactions with the Higgs explain the reheating of the
Universe. The simplest thermal DM model based on a non–minimal scalar coupling
to curvature was studied in [3]. The non–thermal case often requires one to
perform lattice simulations [4, 5, 6, 7, 8]. Taking into account the unitarity
of the system, we discuss the viability of the inflaton dark matter model and
find the allowed parameter space [9].

##  2 Singlet–driven inflation

Let us study general renormalizable interactions of the real singlet scalar
inflaton field $\phi$ and the Higgs field $H$ [10, 11]

${\cal L}_{J}=\sqrt{-\hat{g}}\left(-{1\over 2}\Omega\hat{R}\,+{1\over
2}\,\partial_{\mu}\phi\partial^{\mu}\phi+\,(D_{\mu}H)^{\dagger}D^{\mu}H-{V(\phi,H)}\right)\;,$
(1)

where $\hat{g}^{\mu\nu}$ is the Jordan frame metric and $\hat{R}$ is the
corresponding scalar curvature. In the unitary gauge,

$H(x)={1\over\sqrt{2}}\left(\begin{matrix}0\\\ h(x)\end{matrix}\right)\;,$ (2)

the $Z_{2}$–symmetric potential has been assumed to stabilize $\phi$

$V(\phi,h)={1\over 4}\lambda_{h}h^{4}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}+{1\over 4}\lambda_{\phi}\phi^{4}+{1\over
2}m_{h}^{2}h^{2}+{1\over 2}m_{\phi}^{2}\phi^{2}\;.$ (3)

In Planck units ($M_{\rm Pl}=1$), the function $\Omega$ at the lowest order of
non–minimal scalar–gravity couplings is given by

$\Omega=1+\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\;.$ (4)

The non–minimal coupling to gravity term can be eliminated by a conformal
transformation $g_{\mu\nu}=\Omega\,\hat{g}_{\mu\nu}$ from the Jordan frame to
the Einstein Frame. The Einstein frame Lagrangian reads

${\cal L}={3\over
4}\Bigl{(}\partial_{\mu}\ln(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})\Bigr{)}^{2}+{1\over
2}{1\over\xi_{h}h^{2}+\xi_{\phi}\phi^{2}}\Bigl{(}(\partial_{\mu}h)^{2}+(\partial_{\mu}\phi)^{2}\Bigr{)}-{V\over(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})^{2}}\;.$
(5)

Introducing the variables and normalizing them canonically

$\displaystyle\chi=\sqrt{3\over
2}\ln(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})~{}~{},~{}~{}\tau={h\over\phi}\;,~{}~{}~{}~{}\chi^{\prime}={\chi\;\sqrt{1+{1\over
6\xi_{\phi}}}}~{}~{},~{}~{}\tau^{\prime}={\tau\over\sqrt{\xi_{\phi}}}\;,$ (6)

one may obtain the potential of the new inflaton field $\chi^{\prime}$

$V_{E}={\lambda_{\phi}\over
4\xi_{\phi}^{2}}\left(1+\exp\left(-{2\gamma\chi^{\prime}\over\sqrt{6}}\right)\right)^{-2}\;,$
(7)

where $\gamma=\sqrt{6\xi_{\phi}\over 6\xi_{\phi}+1}\;$. The slow roll
parameters are expressed in terms of the new inflaton field $\chi^{\prime}$,

$\displaystyle\epsilon={1\over 2}\left({\partial
V_{E}/\partial\chi^{\prime}\over
V_{E}}\right)^{2}~{}~{},~{}~{}\eta={\partial^{2}V_{E}/\partial\chi^{\prime\,2}\over
V_{E}}\;.$ (8)

Since inflation ends when $\epsilon\simeq 1$, it determines
$\chi^{\prime}_{\rm end}$. The number of $e$–folds is given by

$N=\int_{\rm in}^{\rm end}H\,dt=-\int^{\rm end}_{\rm in}{V_{E}\over\partial
V_{E}/\partial\chi^{\prime}}\,d\chi^{\prime}\;.$ (9)

That is, a fixed $N$ determines the initial $\chi^{\prime}_{\rm in}$. The COBE
constraint on inflationary perturbations requires $V_{E}/\epsilon\simeq
0.027^{4}$ at $\chi^{\prime}_{\rm in}$. Therefore, one finds

${\lambda_{\phi}\over 4\xi_{\phi}^{2}}=4\times
10^{-7}\,{1\over\gamma^{2}N^{2}}\;.$ (10)

The spectral index $n$ and the tensor to scalar ratio $r$ are

$\displaystyle n=1-6\epsilon+2\eta\simeq 1-{2\over N}-{9\over
2\gamma^{2}N^{2}}~{}~{}~{}~{},~{}~{}~{}~{}r=16\epsilon\simeq{12\over\gamma^{2}N^{2}}\;.$
(11)

Our set–up $\gamma\sim 1$ fits well the PLANCK observation ($n\sim 1$ and
$r\ll 1$) for $N=50$ to 60 [12].

Since a non–minimal coupling term corresponds to a non–renormalizable dim–5
operator, the system is meaningful up to a cutoff scale, $\Lambda\sim
1/\xi_{\phi}$. That is, the inflationary scale
$(\lambda_{\phi}/4\xi_{\phi}^{2})^{1/4}$ must be below the cutoff. Combined
with (10), this requires at the inflationary scale

$\lambda_{\phi}(H)<4\times 10^{-5}$ (12)

and $\xi_{\phi}(H)<300$, where we have set $\gamma\sim 1$ for large
$\xi_{\phi}$.

Our generalized framework allows one to reproduce the Higgs inflation model,
where $\xi_{h}$ must be large such that the corresponding $\gamma\sim 1$ [13].
However, low $\xi$ is also possible in the present model and it may lead to
small deviations in the inflationary prediction (11). The cutoff scale
$\Lambda$ depends on the inflaton background value and therefore the Higgs
inflation model survives the unitarity constraint during inflation [14]. At
the end of inflation and the beginning of reheating, however, this background
becomes less important, so the unitarity issue remains in the Higgs inflation
model.

##  3 Non–thermal inflaton dark matter and reheating

Since the inflaton amplitude decreases such that
$\xi_{\phi}\phi^{2},\xi^{2}_{\phi}\phi^{2}<1$ after inflation, one finds

$\chi\simeq\phi~{}~{},~{}~{}V_{E}(\phi)\simeq{1\over
4}\lambda_{\phi}\,\phi^{4}\;.$ (13)

Therefore, the equation of motion (EOM) for the inflaton background field is

$\ddot{\phi}+3H\dot{\phi}+\lambda_{\phi}\,\phi^{3}=0\;.$ (14)

After a few oscillations, the solution is approximated by a Jacobi cosine
function,

$\phi(t)={\Phi_{0}\over a(t)}\;{\rm cn}\left(x,{1\over\sqrt{2}}\right)\;,$
(15)

where $\Phi_{0}$ is the initial amplitude of the inflaton background field and
$x=(48\lambda_{\phi})^{1/4}\sqrt{t}$ is the conformal time. The EOM for the
Higgs in momentum space takes the form of the Lamé equation,

$X_{k}^{\prime\prime}+\left[\kappa^{2}+{\lambda_{\phi h}\over
2\lambda_{\phi}}\;{\rm
cn}^{2}\left(x,{1\over\sqrt{2}}\right)\right]\,X_{k}=0\;,$ (16)

where $X_{k}(t)\equiv a(t)h_{k}(t)$,
$\kappa^{2}\equiv{k^{2}/(\lambda_{\phi}\Phi_{0}^{2})}$, and the prime denotes
differentiation with respect to $x$.

Figure 1:  Left: Instability bands of the Lamé equation with $q=\lambda_{\phi
h}/(2\lambda_{\phi})$. The Floquet exponent is constant along the contours and
the innermost corresponds to the strongest resonance. Right: Decay of the
classical inflaton background $\langle\varphi\rangle$ in the comoving frame
($\varphi=a\,\phi$) due to rescattering of the inflaton quanta. The conformal
time $z$ is defined by $dz=\sqrt{\lambda_{\phi}}\,\Phi_{0}\,dt/a(t)$;
$\lambda_{\phi}=10^{-13}$ and $\Phi_{0}=1.7$ in Planck units. The amplitude is
normalized to 1 at the initial point.

The ratio $q=\lambda_{\phi h}/(2\lambda_{\phi})$ determines the behavior of
the solution. The momentum modes inside instable regions shown in Fig. 1 have
exponential increase in $X_{k}$. The exponential growth due to the parameteric
band structure of the Lamé equation is called parameteric resonance. Namely,
it leads to resonance particle production since the occupation number gets
amplified accordingly

$n_{k}={\omega_{k}\over
2}\,\left({|X_{k}^{\prime}|^{2}\over\omega_{k}^{2}}+|X_{k}|^{2}\right)-{1\over
2}\;,$ (17)

where the prime represents the conformal time derivative $a(d/dt)$ and
$\omega_{k}^{2}=k^{2}+{\lambda_{\phi h}\over 2}\Phi_{0}^{2}\;{\rm
cn}^{2}\left(x,{1\over\sqrt{2}}\right)$.

However, the linear analysis above breaks down when the interaction term
neglected in (14) becomes significant and thereby invalidates (15) and (16).
Besides such a backreaction effect, the generated particle can also rescatter
the inflaton background field. Even in the absence of interactions with other
fields, self–interaction of the inflaton fragments the homogeneous background
field by rescattering, as illustrated in Fig. 1. To take into account the
non–linear and non–perturbative effects, one has to resort to lattice
simulations. We have obtained the results with LATTICEEASY [15].

Figure 2:  The energy fraction of 4 Higgs d.o.f. and the inflaton as a
function of the conformal time $z$,
$dz=\sqrt{\lambda_{\phi}}\,\Phi_{0}\,dt/a(t)$ and $\Phi_{0}=1.7$ in Planck
units.

Motivated by the simulation results presented in Fig. 2, we expect the Higgs
field to attain the quasi–equilibrium with the SM fields, where the energy is
distributed almost democratically among the SM relativistic degrees of
freedom,

${\rho_{\phi}\over\rho_{\rm tot}}\sim{1\over\\#\,{\rm d.o.f.}}$ (18)

When the preheating stage is over, the total number of the inflaton dark
matter quanta remains approximately constant. This allows us to set the
$lower$ bound on the dark matter abundance $Y$, which is also invariant from
this point. The relic abundance $Y$ is defined by $n_{\phi}/s_{\rm SM}$, where
$n_{\phi}$ is the inflaton number density and $s_{\rm SM}$ is the entropy
density of the SM fields. $s_{\rm SM}$ is close to the number density of the
SM quanta $n_{\rm SM}$, up to a factor of a few. For $107$ SM d.o.f. at high
temperature, we find

$\displaystyle{\rho_{\phi}\over\rho_{\rm tot}}\sim{n_{\phi}\over n_{\rm
SM}}\sim{Y\over(n_{\rm SM}/s_{\rm SM})}\gtrsim{1\over 107}\;,$ (19)
$\displaystyle Y\gtrsim 10^{-3}\;.$ (20)

This number is far above the observed DM abundance $Y_{\rm obs}=4.4\times
10^{-10}\;{\rm GeV}/m_{\phi}$ for $m_{\phi}\gtrsim 10$ keV required by the
structure formation constraints. Therefore, the resulting Universe is too
dark.

##  4 Thermal inflaton dark matter

If $\lambda_{\phi h}$ is sufficiently large, the system may reach thermal
equilibrium via processes like $\phi h\rightarrow\phi h,\phi\phi\rightarrow
hh$, etc. Let us consider constraints away from the narrow resonance region
$m_{\phi}\simeq 62$ GeV for efficient DM annihilation
$\phi\phi\rightarrow\;$SM combined with the XENON1T bound [16]. Then, one
reads off

$\lambda_{\phi h}(1\,{\rm TeV})\gtrsim 0.25\;.$ (21)

This bound applies at the TeV scale. The renormalization group running
generates the inflaton self–coupling at least of the size $\lambda_{\phi
h}^{2}/(8\pi^{2})$ (ignoring a large $log$),

$\lambda_{\phi}(H)\gtrsim 10^{-3}\;.$ (22)

Clearly, the generated coupling $\lambda_{\phi}$ is inconsistent with the
unitarity bound (12).

One may evade the above conclusion by assuming the Higgs resonance,

$m_{\phi}\simeq m_{h_{0}}/2\;,$ (23)

where $m_{h_{0}}=125$ GeV is the physical Higgs mass. In this case, resonant
DM annihilation $\phi\phi\rightarrow h\rightarrow{\rm SM}$ is efficient even
for small couplings $\lambda_{\phi h}\gtrsim 10^{-4}$ [16]. The correction to
the inflaton self–coupling is insignificant and all of the constraints can be
avoided. We note however that $|m_{\phi}-m_{h}/2|$ must be below a few GeV,
which is not impossible yet rather unnatural.

##  5 Conclusion

The inflaton dark matter model is an economical and intriguing possibility
that a single scalar field is responsible for both inflation and dark matter.
Based on minimalism, we have examined thermal and non–thermal production of
the inflaton dark matter quanta. It is important to take into account
non–linear and non–perturbative effects in analyzing non–thermal production of
particles during preheating, for which we have performed lattice simulations.
We find that at weak couplings only a small fraction of the inflaton energy is
converted into the SM radiation, while at large couplings the system reaches
quasi–equilibrium. In both cases, dark matter is overabundant. Thermal
inflaton dark matter is viable but only within the narrow range of the
inflaton mass, such that the inflaton quanta annihilate through the Higgs
resonance. The conclusion of the inflaton dark matter model may change in less
minimalistic set–ups.

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#  MRRT: Multiple Rapidly-Exploring Random Trees for Fast Online Replanning in
Dynamic Environments

Zongyuan Shen†⋆ James P. Wilson† Ryan Harvey† Shalabh Gupta† Abstract †Dept.
of Electrical and Computer Engineering, University of Connecticut, Storrs, CT
06269, USA.⋆Corresponding author (email<EMAIL_ADDRESS>
Motion planning is a fundamental task of determining a collision-free
trajectory that drives the robot to a target state in the configuration space,
minimizing a user-defined cost function such as trajectory length[1]. It has a
wide range of applications for autonomous underwater vehicles (AUVs), such as
seabed mapping[2], structural inspection[3], oil spill cleaning[4], mine
hunting[5], and other underwater tasks[6]. In order to ensure that the AUV
successfully completes its mission, the path planner must be adaptive to
previously-unknown obstacles in the environment as well as moving obstacles
(e.g., other marine vessels) [7]. As such, it is critical to ensure rapid
replanning of the AUV paths online to facilitate safe operation as new
information becomes available.

A variety of methods have been developed to solve the motion planning problem;
a review is presented in [8]. In general, motion planning methods can be
broadly classified into two main categories: grid-based and sample-based. The
grid-based methods, such as A⋆[9], discretize the configuration space and
search for the optimal solution; however, these approaches suffer from the
curse of dimensionality and their accuracy depends on the grid resolution. On
the other hand, the sampling-based methods generate samples randomly in the
configuration space, construct a graph structure to capture connectivity
between different configurations, and search a solution on it [10]. Therefore,
sample-based motion planning approaches are becoming increasingly popular
since they can find feasible solutions quickly in high-dimensional spaces
while asymptotically approaching the optimal one. Specifically, they are very
useful for online planning in dynamic environments. Online methods are
characterized as active or reactive. Active strategies predict the future
trajectory of a moving obstacle for a fixed time duration and generate a
collision-free trajectory for the robot; however, the performance can degrade
if the predicted information is incomplete or incorrect. In contrast, the
reactive strategies plan the path based only on the obstacle information at
the current time, and replan the path whenever the obstacle information
changes.

Recently, several variants of the RRT/RRT⋆ have been presented for online
motion planning in dynamic environments. Otte et al. [11] proposed
RRT${}^{\text{X}}$ algorithm which utilizes a rapidly-exploring random graph
(RRG)[12] to explore the search area. This method maintains a well-defined and
connected structure over the entire explored region; however, re-optimizing
the connections in the entire graph when the environment changes makes it
inefficient in highly dynamic settings. Chen et al.[13] presented Horizon-
based Lazy RRT⋆ (HL-RRT⋆) algorithm in the context of RRT⋆ algorithm[12]. In
this method, invalid parts of the search tree due to moving obstacles are
pruned, and new samples are drawn to find a new path using a trained Gaussian
mixture model. While this can find a new solution quickly, the machine
learning based sampler cannot guarantee the performance. Furthermore, this
method removes the invalid parts which are disconnected from the main tree to
maintain a single tree structure, resulting in repeated exploration of the
same area. Finally, Yuan et al.[14] proposed Efficient Bias-goal Factor RRT
(EBG-RRT), where invalid parts of the tree are pruned if it does not contain
the current position or destination. Then, leaf nodes are identified in the
main tree that are near the destination tree. Finally, samples are drawn using
heuristics or a uniform distribution to attempt to reconnect the trees. This
method is easy to implement; however, it prunes the tree portions which do not
contain the current node or goal, resulting in repeated exploration of the
same area.

(a) Obstacle information is updated; infeasible edges are pruned from the
tree.

(b) By removing infeasible edges multiple disjoint trees are created.

(c) A random sample is generated; it is joined to all disjoint trees in the
local neighborhood by connecting to the nearest node.

(d) A random sample is generated; it is joined to all disjoint trees in the
local neighborhood by connecting to the nearest node.

(e) A random sample is generated; it is joined to all disjoint trees in the
local neighborhood by connecting to the nearest node.

(f) Once the full tree is obtained, a new feasible solution is searched on it.

Figure 1: An illustrative example of MRRT algorithm.

We present a novel online reactive sampling-based motion planning algorithm
for dynamic environments, called Multiple Rapidly Exploring Random Trees
(MRRT), to address the limitations of existing approaches. The proposed
algorithm is built upon the RRT algorithm and multi-tree structure. At the
beginning, RRT algorithm is applied to find the initial solution based on the
partial knowledge of the environment. Then, the robot starts to execute this
path. At each iteration, the new obstacle configurations are collected by the
robot’s sensor and used to replan the path. This new information can come from
unknown static obstacles (e.g., seafloor layout) as well as moving obstacles.
Then, to accommodate the environmental changes, two procedures are adopted: 1)
edge pruning, and 2) tree regrowing. Specifically, the edge pruning procedure
checks the collision status through the tree and only removes the invalid
edges while maintaining the tree structure of already-explored regions. Due to
removal of invalid edges, the tree could be broken into multiple disjoint
trees. As such, the RRT algorithm is applied to regrow the trees.
Specifically, a sample is created randomly and joined to all the disjoint
trees in its local neighborhood by connecting to the nearest nodes. Finally, a
new solution is found for the robot. Figure 1 shows an execution example of
the online motion planning algorithm in a dynamic environment with moving
obstacles.

The advantages of the proposed algorithm are as follows: i) retains the
maximal tree structure by only pruning the edges which collide with the
obstacles, ii) guarantees probabilistic completeness, and iii) is
computational efficient for fast replanning since all disjoint trees are
maintained for future connections and expanded simultaneously.

In the full paper, we will present details of the proposed MRRT algorithm and
the simulation results of rapid path replanning in an unknown static
environment as well as in a dynamic environment containing moving obstacles.

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