# Search for dark Higgs inflaton with curvature couplings at LHC experiments

Lucia Aurelia Popa

###### Abstract

We analyse the dark Higgs inflation model with curvature corrections and
explore the possibility to test its predictions by the particle physics
experiments at LHC.
We show that the dark Higgs inflation model with curvature corrections is
strongly favoured by the present cosmological observation. The cosmological
predictions of this model, including the quantum corrections of dark Higgs
coupling constants and the uncertainty in estimation of the reheating
temperature, lead to the dark Higgs mass $m_{\phi}=0.919\pm$ 0.211 GeV and the
mixing angle $\theta=1.492\pm$ 0.045 (at 68% CL).
We evaluate the FASER and MAPP-1 experiments reach for dark Higgs inflaton
mass and mixing angle in the 95% CL cosmological confidence region for an
integrated luminosity of 3ab-1 at 13 TeV LHC, assuming 100% detection
efficiency.
We conclude that the dark Higgs inflation model with curvature corrections is
a compelling inflation scenario based on particle physics theory favoured by
the present cosmological measurements that leaves imprints in the dark Higgs
boson searchers at LHC.

##  1 Introduction

The precise Cosmic Microwave Background (CMB) properties reported by the
Planck experiment [1, 2, 3] and the discovery by LHC of the Higgs boson [4, 5]
increased the interest in so called Higgs portal interactions that connect the
hidden (dark) sector and the visible sector of the Standard Model (SM), with
expected imprints on collider experiments [6]. Scenarios beyond-the-SM (BSM),
that introduce a dark sector in addition to the visible SM sector are required
to explain a number of observed phenomena in particle physics, astrophysics
and cosmology such as the non-zero neutrino masses and oscillations, the Dark
Matter (DM), baryon asymmetry of the universe, the cosmological inflation.

It is usual to assume that cosmic inflation is decoupled from the SM at
energies lower than the inflationary scale since the slow-roll conditions for
inflation generally permit only tiny couplings of the inflaton field to other
fields. This assumption prevents the direct investigation of inflation
mechanism in particle physics experiments. Consequently, there are little
compelling scenarios of inflation based on particle physics theory.

Since the only known fundamental scalar quantum field is the SM Higgs field,
the inflation models using the SM Higgs boson as inflaton attained great
attention over the past years. A number of Higgs inflation models, mostly with
non-canonical action, have been proposed. They include models with Higgs
scalar field non-minimaly coupled to gravity [7, 8, 9], non-minimal derivative
coupling to the Einstein tensor [10, 11, 12, 13], scalar-tensor models [14,
15], Galileon models [16, 17, 18, 19], quartic hilltop models [24, 25].
The viability of these models is already substantially limited mostly because
they predict tensor-to-scalar ratios larger than the upper bound set by the
combined analysis of Planck and BICEP-Keck Array data (hereafter Planck+BK15)
that constrain the energy scale of inflation to [2, 3]:

${V}^{1/4}_{*}=\left({\frac{3\pi^{2}A_{s}^{*}}{2}r_{*}}\right)^{1/4}M_{pl}<1.6\times
10^{16}{\rm GeV}\hskip 5.69046pt(95\%{\rm CL})\,.$ (1.1)

Here the quantities with $(^{*})$ are evaluated at the pivot scale
$k_{*}=0.002$, $r_{*}$ is the ratio of tensor-to-scalar amplitudes,
$A_{s}^{*}$ is the amplitude of the curvature density perturbations and
$M_{pl}$ is the reduced Planck mass. This imply an upper bound for the Hubble
expansion rate during inflation:

$H_{*}<2.5\times 10^{-5}M_{pl}\,\hskip 5.69046pt(95\%{\rm CL})\,.$ (1.2)

The above bound selects the viable Higgs inflation models from the requirement
$H_{*}\ll\Lambda_{c}$, where $\Lambda_{c}$ is the unitary bound of each
underlying theory, defined as the scale below which the quantum gravitational
corrections are sub-leading [20, 21, 22].
It worths to mention that the chaotic inflation model with quartic potential
is excluded by the data at more than 95% confidence level [23].

Among the models used to lower the predictions for tensor-to-scalar ratio, the
most studied is the Higgs inflation with non-minimal coupling to gravity [7].
At tree level and for large non-minimal coupling $\xi\sim{\cal O}(10^{4})$,
this model gives a small tensor-to-scalar ratio, in agreement with the
Planck+BK15 data. However, for such large values of $\xi$ the unitary bound
scale, $\Lambda_{c}=M_{pl}/\xi$, could be close or below the energy scale of
inflation [20, 26].
An interesting framework for Higgs inflation is provided the scalar-tensor
models including the non-minimal kinetic coupling to the Einstein tensor and
to the Gauss-Bonnet invariant. These models can produce inflation
simultaneously satisfying the present inflationary observational constrains
and the unitary bound constraints [14, 15].

Higgs portal interactions via the Renormalisation Group (RG) loop
contributions can also lower the predictions of Higgs inflation models for the
tensor-to-scalar ratio. The price to pay in these models is the electroweak
(EW) vacuum metastability issue. The actual values of Higgs boson and top
quark masses imply that the EW vacuum is metastable at energies larger than
$\Lambda_{I}\sim 10^{11}$ GeV, where Higgs quartic coupling turns
negative111The actual value of EW vacuum metastability scale is defined for
the top quark mass $m_{t}=173.15$ GeV and Higgs boson mass $m_{H}=125.10$ GeV
[31] as the value of the Higgs field at which the Higgs quartic coupling,
$\lambda_{h}$, becomes negative due to radiative corrections.[27, 28, 29, 30].
However, it is found that a small admixture of the Higgs field with a SM
scalar singlet with non-zero vacuum expectation value (vev) can make the EW
completely stable due to a tree-level effect on the Higgs quartic coupling
which may be enough to guarantees the stability at large Higgs field values
[32, 33, 34].

An appealing scenario in the presence of Higgs portal interactions is given by
a SM singlet scalar field with non-zero vev mixed with the SM Higgs boson,
often called dark Higgs boson. The dark Higgs mixing with the SM Higgs boson
make possible the direct search of the dark Higgs inflaton at collider
experiments. The mixing guarantee that dark Higgs can be produced in the same
channels as the SM Higgs boson if its mass would be the same as that of the
dark Higgs boson. Through the same mixing the dark Higgs boson inherits the SM
Higgs boson couplings to SM fermions via the Yukawa interaction term:

$L\supset\theta\frac{m_{f}}{v}\phi{\bar{f}}{f}\,,$ (1.3)

where: $\phi$ is the dark Higgs field, $\theta$ is its mixing angle with SM
Higgs boson and $m_{f}$ is the fermion mass.

Dark Higgs bosons can be produced at LHC in rare heavy meson decays (such as K
and B mesons). They are highly collimated, with characteristic angles
$\alpha=M/E$ relative to the parent meson’s direction ($M$ is the meson mass
and $E$ is the dark Higgs energy). For $E\sim$ 1 TeV the light dark Higgs
decay lengths are of ${\cal O}$($10^{3}\,m$). Therefore a significant number
of dark Higgs bosons can be detected in faraway detectors of the LHC
experiments [6]. Thus, present and future experimental sensitivity to the
light dark Higgs boson decay crucially depends on its production and decay
rates and on detector’s location and acceptance.

The light dark Higgs boson as inflaton (rather than the Higgs boson) has been
first implemented in Ref. [37], extending the $\nu$MSM model [38, 39] to
simultaneously explain the cosmological inflation, the DM sterile neutrino
masses and the baryon asymmetry of the universe [37, 40]. The light dark Higgs
inflaton properties has been mostly studied in the frame of dark Higgs
inflation with non-minimal coupling to gravity [41, 42, 53, 44]. Refs.[45, 46]
present a detailed analysis on the possibility to explore this model in the
particle physics experiments.
This possibility has been also investigated in the frame of low-scale
inflation models, such the quartic hilltop model [24] that predicting a very
small value for tensor-to scalar ratio, beyond the sensitivity of the CMB
experiments. Thus, the dark Higgs searchers at LHC could experimentally test
the low-scale of inflation.

In this paper we analyse the dark Higgs inflation model with curvature
corrections given by the kinetic term non-minimally coupled to the Einstein
tensor and the coupling to the Gauss-Bonnet (GB) 4-dimensional invariant
(hereafter EGB dark Higgs inflation) and explore the possibility to test its
predictions by the particle physics experiments.
In this model, the non-minimal kinetic coupling to the Einstein tensor causes
the inflaton field to roll slower, avoiding the problem of large fields
present in chaotic inflation [11]. On the other hand, the second-order
curvature corrections represented by the scalar field coupled to the GB term
can increment or suppress (depending on the sign) the tensor-to-scalar ratio
[47, 48, 49]. The dynamics of the slow-roll inflation by combining both
corrections has been proposed in context of the SM Higgs inflation in
Refs.[14, 15].
The possibility to explore this model by the dark Higgs searchers at LHC could
provide connections between fundamental theories like supergravity and string
theories where these couplings are expected to arise, and the Higgs portal
interactions.

The paper is organised as follows. In the next section we discuss the dark
Higgs inflaton properties. In Section 3 we introduce the EGB dark Higgs
inflation model. In Section 4 we analyse the cosmological consistency of the
EGB dark Higgs inflation predictions. Section 5 discuss the possibility to
test the EGB dark Higgs inflation predictions by some representative particle
physics experiments at LHC. In Section 6 we draw our conclusions.
Throughout the paper we consider an homogeneous and isotropic flat background
described by the Friedmann-Robertson-Walker (FRW) metric:

${\rm d}s^{2}=g_{\mu,\nu}{\rm d}x_{\mu}{\rm d}x^{\nu}=-{\rm
d}t^{2}+a^{2}(t)dx^{2}\,,$ (1.4)

where $a$ is the cosmological scale factor ($a_{0}$=1 today). Also, we use the
overdot to denote the time derivative and $(^{\prime})$ to denote the
derivative with respect to the scalar field.

##  2 Dark Higgs inflaton properties

###  2.1 Dark Higgs inflaton parameters

We consider the extension of the SM canonical action with the dark Higgs
inflaton field, as introduced in Ref. [37]:

$\displaystyle S=\int{{\rm d}^{4}\,x}\sqrt{-g}\,\left[\frac{\cal
R}{2\kappa^{2}}+\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\right]\,,$ (2.1)

where ${\cal R}$ is the Ricci scalar, $\kappa^{2}=M^{-2}_{pl}$, $\phi$ is the
dark Higgs inflaton field with the potential $V(\phi)$ defined as:

$V(\phi)=-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\frac{\beta}{4}\phi^{4}+\lambda\left({\cal
H}^{\dagger}{\cal H}-\frac{\alpha}{\lambda}\phi^{2}\right)^{2}\,.$ (2.2)

In the above equation $\lambda$ is the SM Higgs field self coupling,
$m_{\phi}$ is the dark Higgs mass, $\beta$ is the dark Higgs quartic coupling
and $\alpha$ is the coupling between the SM Higgs field ${\cal H}$ and the
dark Higgs inflaton. For $\alpha,\beta\ll\lambda$, inflation is driven along a
flat direction of the scalar potential given by:

${\cal H}^{\dagger}{\cal H}\simeq\frac{\alpha}{\lambda}\phi^{2}\,.$ (2.3)

Along this direction the dark Higgs potential is $V(\phi)=\beta\phi^{4}/4$ and
the coupling constant $\beta$ can be fixed from the requirement to obtain the
correct the amplitude of the curvature density perturbations. This condition
leads to $\beta\sim 1.3\times 10^{-13}$ [50].
The negative sign of the quadratic term in (2.2) ensures that the scale
invariance is explicitly broken on the classical level in the inflaton sector,
leading to non-zero vev for the dark Higgs inflaton after reheating. Then, the
condition (2.3) gives rise to EW spontaneous symmetry breaking and the SM
Higgs field gains non-zero vev  too.

We remind that the SM Higgs boson mass is given by $m_{H}=\sqrt{2\lambda}\,v$,
where the SM Higgs vev is fixed at $v\equiv(\sqrt{2}G_{F})^{1/2}=246.22$ GeV
by the Fermi coupling constant $G_{F}$, and the experimentally measured Higgs
boson mass is $m_{H}=125.10\pm 0.14$ GeV [31].
In the gauge base $(\sqrt{2}{\cal H}-{\it v},\phi)$ the dark Higgs field
expectation value, $<\phi>$, its mass $m_{\phi}$ and mixing angle $\theta$,
are given by:

$\displaystyle<\phi>=\frac{m_{H}}{2\sqrt{\alpha}}\,,\hskip
14.22636ptm_{\phi}=m_{H}\sqrt{\frac{\beta}{2\alpha}}\,,\hskip
14.22636pt\theta=\sqrt{\frac{2\alpha}{\lambda}}\,.$ (2.4)

For the purpose of this work we choose $\alpha>\beta/2$, therefore the dark
Higgs inflaton is lighter than the SM Higgs boson, $m_{\phi}<m_{H}$.

Figure 1: The evolution with the scale dependent variable $t={\rm
ln}(\phi/m_{t})$ of the running of $\lambda$, $\beta$ and $\alpha$ coupling
constants normalised to their initial values chosen at $t=0$ as:
$\lambda(0)=0.129$, $\beta(0)=1.3\times 10^{-13}$ and $\alpha(0)=3\times
10^{-7}$. The SM Higgs mass is fixed at $m_{H}=125.09$ GeV. The right-hand
blue region indicates the slow-roll inflationary regime.

.

The upper bound on the coupling constant $\alpha$ cames from the requirement
that the quantum corrections do not upset the flatness of the inflation
potential. This constrain leads to $\alpha<3\times 10^{-7}$ at the tree level
[45] and corresponds to the lower bound of the dark Higgs inflaton mass:

$m_{{\phi}}\geqslant 0.058\left(\frac{\beta}{1.3\times
10^{-13}}\right)^{1/2}\,\,{\rm GeV}.$ (2.5)

The lower bound on $\alpha$ cames from the requirement to have an efficient
conversion of the lepton asymmetry to baryon asymmetry during baryogengesis.
This requirement leads to to $\alpha>\beta\sim 10^{-13}$ [37]. A stronger
lower bound on $\alpha$ is placed by the estimate of the reheating
temperature. For the inflaton particles in thermal equilibrium the reheating
temperature is given by [40]:

$T_{r}\simeq\frac{\zeta(3)\alpha^{2}}{4\pi^{2}}\sqrt{\frac{90}{g_{r}}}\,{\rm
M_{pl}}\,,$ (2.6)

where $g_{r}=106.75$ is the SM effective number of relativistic degrees of
freedom at reheating and $\zeta(3)=1.202$ is the Reimann zeta function. The
requirement that $T_{r}>150$ GeV ($T\simeq 150$ GeV is the temperature of the
EW symmetry breaking), leads to $\alpha>7.3\times 10^{-8}$.
For a non-thermal distribution of the inflaton field the estimate of the
reheating temperature is $\sim 10^{5}T_{r}$ [51, 52], leading to
$\alpha>7\times 10^{-10}$.
These constraints are consistent to the upper bounds of the dark Higgs
inflaton mass [40]:

$m_{{\phi}}\leqslant(0.116-1.166)\left(\frac{\beta}{1.3\times
10^{-13}}\right)^{1/2}\,\,{\rm GeV}\,,$ (2.7)

where the range corresponds to the thermal or non-thermal estimates.

The above bounds of the inflaton mass may be changed if the quantum
corrections of the coupling constants are take into account.
Figure 1 presents the evolution with the scale dependent variable $t={\rm
ln}(\phi/m_{t})$ of the running of $\lambda(t)$, $\beta(t)$ and $\alpha(t)$
coupling constants normalised to their initial values $\lambda(0)$, $\beta(0)$
and $\alpha(0)$, obtained by the integration the corresponding beta functions
[29, 41, 44, 53].
As the SM Higgs mass is fixed at $m_{H}=125.09$ GeV we take
$\lambda(0)=0.129\,$. We also fix $\alpha(0)=3\times 10^{-7}$ to the $\alpha$
upper bound and take $\beta(0)=1.3\times 10^{-13}$.
One should note that the correction to $\beta$ from the coupling of the dark
Higgs inflaton to the SM Higgs boson is $\delta\beta\sim\alpha^{2}$ and
therefore the evolution of $\beta$ is dominated by the $\alpha^{2}$
contribution. Figure 1 shows that all coupling constants remain positive at
the inflationary scale while the flatness of the inflationary potential is
preserved.

###  2.2 Reheating and horizon crossing

The reheating proceeds by the energy transfer from the dark Higgs inflaton
field to the SM Higgs particles through a regime of parametric resonance [51,
52]. At early stages the entire energy is in the inflaton zero-mode and all
other modes are absent. The inflaton zero-mode oscillations excite the non-
zero modes of the inflaton and of the SM Higgs particles. This parametric
resonance regime ends before a significant part of the inflaton zero-mode
energy is depleted [40]. The reason is the SM Higgs re-scattering process that
become important quite early because of the large SM Higgs sef-coupling
($\lambda\sim 0.1$).
After the end of the parametric resonance regime, the fluctuations of the
inflaton field continue to grow exponentially while the energy transferred to
the SM Higgs field is negligible small. The SM Higgs re-scattering processes
bring the inflaton particles in the thermal equilibrium and the reheating
proceeds through the decay of the dark Higgs inflaton into the SM Higgs
particles.

The inflationary observables are evaluated at the epoch of the Hubble crossing
scale $k_{*}$ (pivot scale) quantified by the number of e-folds ${\cal N}$
before the end of the inflation. Therefore, the uncertainties in the
determination of $\cal N$ translates into theoretical uncertainties in
determination of the inflationary observables [54, 41]. Assuming that the
ratio of the today entropy per co-moving volume to that after reheating is
negligible, the main error $\Delta{\cal N}$ in the determination of ${\cal N}$
is given by the uncertainty in the determination of the reheating temperature
${T_{r}}$. The number of e-foldings at Hubble crossing scale $k_{*}$ is
related to $T_{r}$ through:

${\cal N}={\rm
log}\left[\left(\frac{\rho_{r}}{\rho_{e}}\right)^{1/4}\left(\frac{g_{0}T^{3}_{0}}{g_{r}T^{3}_{r}}\right)^{1/3}\left(\frac{k_{*}}{a_{0}H_{0}}\right)\right]\,,$
(2.8)

where $\rho_{r}$ and $\rho_{e}$ refer to the densities at reheating and at end
of inflation, $T_{0}$ is the present photon temperature, $H_{*}$ is the Hubble
parameter at $k_{*}$, $g_{r}=106.75$ and $g_{0}=43/11$ are the effective
number of relativistic degrees of freedom at reheating and at present time.
From (2.6) and (2.8) we get $\Delta{\cal N}\simeq 3$ corresponding to the
uncertainty in determination of $T_{r}$ for to a thermal distribution of the
inflaton. This uncertainty is four times higher, $\Delta{\cal N}\simeq 12$, in
the case of non-thermal distribution.

##  3 Dark Higgs inflation with curvature corrections

Closely following [15, 14], in this section we introduce the inflation model
assuming non-minimal coupling of the dark Higgs field with the Einstein tensor
and to the Gauss-Bonnet (GB) 4 - dimensional invariant (the EGB dark Higgs
inflation model), derive the background field equations, the slow-roll
parameters and evaluate the primordial power spectra the the scalar and tensor
perturbations.
The action of this model is:

$S_{E}=\int{{\rm d}^{4}\,x}\sqrt{-g}\,\left[\frac{\cal
R}{2\kappa^{2}}+X-V(\phi)+F_{1}(\phi)G_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi-
F_{2}(\phi){\cal G}\right]\,,$ (3.1)

where $V(\phi)$ is the dark Higgs potential given in (2.2), $F_{1}(\phi)$ and
$F_{2}(\phi)$ are coupling functions, $G_{\mu\nu}$ is the Einstein tensor,
$\cal G$ is the GB 4-dimensional invariant:

${\cal G}={\cal R}^{2}-4{\cal R}_{\mu\nu}{\cal R}^{\mu\nu}+{\cal
R}_{\mu\nu\delta\rho}{\cal R}^{\mu\nu\delta\rho}\,.$ (3.2)

The field equations in a spatially flat background described by the FRW metric
(1.4) are of the form (see Appendix B from [15]) :

$H^{2}=\frac{\kappa^{2}}{3}\left(\frac{1}{2}+V(\phi)+9H^{2}F_{1}{\dot{\phi}}^{2}+24H^{3}{\dot{F}_{2}}\right)\,,$
(3.3) $\displaystyle{\ddot{\phi}}+3H{\dot{\phi}}$ $\displaystyle+$
$\displaystyle
V^{{}^{\prime}}+24H^{2}(H^{2}+{\dot{H}})F^{{}^{\prime}}_{2}+18H^{3}F_{1}{\dot{\phi}}$
$\displaystyle+$ $\displaystyle
12H{\dot{H}}F_{1}{\dot{\phi}}+6*H^{2}F_{1}{\ddot{\phi}}+3H^{2}F^{{}^{\prime}}_{1}+{\dot{\phi}}^{2}=0\,.$

The slow-roll parameters are defined as:

$\displaystyle\epsilon_{0}=-\frac{{\dot{H}}}{H^{2}}\,,\hskip
5.69046pt\epsilon_{1}=\frac{{\dot{\epsilon}_{0}}}{H\epsilon_{0}}\,,\hskip
5.69046ptk_{0}=3F_{1}{\dot{\phi}}^{2}\,,\hskip
5.69046ptk_{1}=\frac{{\dot{k}_{0}}}{Hk_{0}}\,,\hskip
8.5359pt\Delta_{0}=8H{\dot{F}_{2}}\,,\hskip
5.69046pt\Delta_{1}=\frac{{\dot{\Delta}_{0}}}{H\Delta_{0}}.$ (3.5)

Under the slow-roll conditions ${\ddot{\phi}}\ll 3H{\dot{\phi}}$ and
$|\epsilon_{0}|\,,|\epsilon_{1}|\,,...|\Delta_{1}|\,\ll 1$ the potential and
field equations take the form:

$\displaystyle H^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{\kappa^{2}}{3}V(\phi)\,$ (3.6) $\displaystyle
3H{\dot{\phi}}$ $\displaystyle\simeq$
$\displaystyle-V^{{}^{\prime}}-18H^{3}F_{1}{\dot{\phi}}-24H^{4}F^{{}^{\prime}}_{2}\,.$
(3.7)

The number of e-folds before the end of inflation expressed in terms of the
inflaton field is given by:

${\cal N}=\int^{\phi_{E}}_{\phi_{I}}\frac{H}{{\dot{\phi}}}{\rm
d}\phi=\int^{\phi_{e}}_{\phi_{\cal
N}}\frac{H^{2}+6H^{4}F_{1}}{-8H^{4}F^{{}^{\prime}}_{2}-\frac{1}{3}V^{{}^{\prime}}}{\rm
d}\phi\,,$ (3.8)

where $\phi_{I}$ and $\phi_{E}$ are the values of the inflaton field at the
begging and at the end of inflation.
The power spectra of the primordial scalar and tensor perturbations,
$\cal{P}_{R}$ and ${\cal P}_{T}$, are computed as:

$\displaystyle{\cal P}_{R}$ $\displaystyle=$ $\displaystyle
A_{S}\frac{H^{2}}{2\pi^{2}}\frac{{\cal G_{S}}^{1/2}}{{\cal
F_{S}}^{3/2}}\,,\hskip 2.84544pt\hskip
11.38092ptA_{S}=\frac{1}{2}2^{2\mu_{S}-3}\left|\frac{\Gamma(\mu_{S})}{\Gamma(3/2)}\right|^{2},\hskip
2.84544pt\mu^{2}_{S}=\frac{9}{4}\left[1+\frac{4}{3}\epsilon_{0}+\frac{2}{3}\frac{2\epsilon_{0}\epsilon_{1}-\Delta_{0}\Delta_{1}}{2\epsilon_{0}-\Delta_{0}}\right]$
$\displaystyle{\cal P}_{T}$ $\displaystyle=$ $\displaystyle
16A_{T}\frac{H^{2}}{2\pi^{2}}\frac{{\cal G_{T}}^{1/2}}{{\cal
F_{T}}^{3/2}},\hskip
5.69046ptA_{T}=\frac{1}{2}2^{2\mu_{T}-3}\left|\frac{\Gamma(\mu_{T})}{\Gamma(3/2)}\right|^{2},\hskip
5.69046pt\mu_{T}=\frac{3}{2}+\epsilon_{0}\,,$ (3.9) $\displaystyle{\cal
F_{S}}=c^{2}_{S}{\cal G_{S}}\,,\hskip 5.69046pt{\cal G_{S}}$ $\displaystyle=$
$\displaystyle\epsilon_{0}-\frac{1}{2}\Delta_{0}\,,\hskip
34.14322ptc^{2}_{S}=1-\frac{\frac{4}{3}k_{0}(\Delta_{0}+\frac{4}{3}k_{0})+\frac{4}{3}k_{0}\epsilon_{0}}{2\epsilon_{0}-\Delta_{0}}\,,$
$\displaystyle{\cal F_{T}}=c^{2}_{T}{\cal G_{T}}\,,\hskip 5.69046pt{\cal
G_{T}}$ $\displaystyle=$ $\displaystyle 1-\frac{1}{3}k_{0}-\Delta_{0}\,,\hskip
11.38092ptc^{2}_{T}=\frac{3+k_{0}-3\Delta_{0}(\epsilon_{0}+\Delta_{1})}{3-k_{0}-3\Delta_{0}}\,,$
(3.10)

where $c_{S}$ and $c_{T}$ are the sound speeds of scalar and tensor density
perturbations.
The spectral index of scalar density perturbations $n_{S}$ and the tensor-to-
scalar ratio expressed in terms of slow-roll parameters are given by:

$n_{S}=-2\epsilon_{0}-\frac{2\epsilon_{0}\epsilon_{1}-\Delta_{0}\Delta_{1}}{2\epsilon_{0}-\Delta_{0}}\,.$
(3.11)
$r=8\left(\frac{2\epsilon_{0}-\Delta_{0}}{1-\frac{1}{3}k_{0}-\Delta_{0}}\right)\,.$
(3.12)

Hereafter we take $F_{1}(\phi)$ and $F_{2}(\phi)$ power-law functions of the
form:

$\displaystyle F_{1}(\phi)=\frac{\gamma}{\phi^{4}}\,,\hskip
14.22636ptF_{2}(\phi)=\frac{\eta}{\phi^{4}}\,,$ (3.13)

where $\gamma$ and $\eta$ are positive constants with the dimension
$[\gamma]=M_{pl}^{2}$ and $[\eta]=M^{4}_{pl}$.
For this setup, the first slow-roll parameter $\epsilon_{0}$ reads as:

$\epsilon_{0}=\frac{16}{3}\frac{(3-2\eta\beta)}{(2+\gamma\beta)\phi^{2}}\,.$
(3.14)

From (3.8) one gets the number of e-folds before the end of inflation:

${\cal
N}=-\frac{3(2+\gamma\beta)}{16(3-2\eta\beta)}\phi^{2}\bigg{|}^{\phi_{E}}_{\phi_{I}}\,.$
(3.15)

The value of the scalar field at the end of inflation, $\phi_{E}$, is obtained
from the requirement $\epsilon_{0}=1$, while (3.15) allows the determination
of the inflaton field value $\phi_{I}$ at ${\cal N}$e-folds before the end of
inflation:

$\displaystyle\phi_{E}=\frac{4\sqrt{3-2\eta\beta}}{\sqrt{3(2+\gamma\beta)}}\,,\hskip
14.22636pt\phi_{I}=\sqrt{{\cal N}+1}\phi_{E}\,.$ (3.16)

##  4 Cosmological constraints

###  4.1 Parameterisation and methods

The dark Higgs baseline cosmological model is described by the following
parameters:

${\bf
P}=\left\\{\Omega_{b}h^{2}\,,\,\Omega_{c}h^{2}\,,\,\theta_{s}\,,\,\tau\,,\,{\rm
log}(10^{10}A_{s})\,,\,n_{s}\,,\,{\cal N}\,,\,\beta\,,\alpha\right\\}\,,$
(4.1)

where: $\Omega_{b}h^{2}$ is the present baryon energy density,
$\Omega_{c}h^{2}$ is the present CDM energy density, $\theta_{s}$ is the ratio
of sound horizon to angular diameter distance at decoupling, $\tau$ is the
optical depth at reionization, $A_{s}$ and $n_{s}$ are the amplitude and the
spectral index of the primordial curvature perturbations, ${\cal N}$ is the
number of e-folds introduced to account for the uncertainty in the
determination of the reheating temperature, $\beta$ is the dark Higgs quartic
coupling and $\alpha$ is the dark Higgs - SM Higgs coupling constant.
The EGB dark Higgs inflation model extends the dark Higgs baseline model by
including the following parameters:

${\bf P}_{\rm EGB}=\left\\{\gamma\beta\,,\eta\beta\right\\}\,,$ (4.2)

where the coupling constants $\gamma$ and $\beta$ are defined in (3.13).

We compute the dependence on the scaling variable $t={\rm ln}(\phi/m_{t})$ of
the running of various coupling constants by integrating the corresponding
beta functions:

$\displaystyle Y(t)=\int^{t}_{0}{{\bf\beta}}_{Y}{\rm d}t\,,\hskip
28.45274ptY=\\{g,\,g^{{}^{\prime}}\,,g_{s}\,,y_{t}\,,\beta\,,\alpha\\}\,,$
(4.3)

where $g,\,g^{{}^{\prime}}\,,g_{s}$ are the gauge couplings, $y_{t}$ is the
Yukawa coupling, $\beta$ is the dark Higgs quartic coupling and $\alpha$ is
the dark Higgs - SM Higgs coupling (for the relevant beta functions see
Appendix A from [44] and references therein).
At $t=0$ the SM Higgs self coupling $\lambda(0)=0.129$ and the top Yukawa
coupling $y_{t}(0)=0.976$ are fixed by the SM Higgs and top quark pol masses
[29]. For the gauge couplings at $t=0$ we take $g^{{}^{\prime}}(0)=0.364$,
$g(0)=0.64$ and $g_{s}(0)=1.161$ [55]. The priors for $\beta(0)$ and
$\alpha(0)$ are given in Table 1 (see below).

We modify the standard Boltzmann code camb222http://camb.info [56] to
calculate the primordial power spectra of scalar ${\cal P}_{R}(k)$ and tensor
${\cal P}_{T}(k)$ density perturbations for the dark Higgs inflation model
with curvature corrections presented in the previous section. The code evolves
the coupled dark Higgs field equations (3.3) and (3) with respect to the
conformal time for wave numbers in the range $5\times 10^{6}$ \- $5$ Mpc-1 and
evaluate the RG corrections to the coupling constants as presented before. The
value of the inflaton field $\phi_{I}$ and $\phi_{E}$ at the beginning and at
the end of inflation are obtained from (3.16). The primordial power spectra
${\cal P}_{R}(k)$ and ${\cal P}_{T}(k)$ are obtained from (3) with the slow-
roll parameters defined in (3.5).
The scalar spectral index of the curvature perturbations $n_{S}$ and the ratio
of tensor-to-scalar amplitudes $r$ are then evaluated at the pivot scale
$k_{*}=0.002$Mpc-1 as:

$\displaystyle n_{S}=\frac{{\rm d\,ln}{\cal P}_{R}(k)}{{\rm
d\,ln}k}\bigg{|}_{k_{*}}\,,\hskip 14.22636ptr=\frac{{\cal P}_{T}(k)}{{\cal
P}_{R}(k)}\bigg{|}_{k_{*}}\,.$ (4.4)

The extraction of parameters from the cosmological dataset is based on Monte-
Carlo Markov Chains (MCMC) technique. We modify the publicly available version
of the package CosmoMC333http://cosmologist.info/cosmomc/ [57] to sample from
the space of dark Higgs inflation model parameters and generate estimates of
their posterior mean and confidence intervals.
We made some test runs to optimise the parameters prior intervals and
sampling. The final run is based on 120 independent channels, reaching the
convergence criterion $(R-1)\simeq 0.01$. The $(R-1)$ criterion is defined as
the ratio between the variance of the means and the mean of variances for the
second half of chains [57].
We assume a flat universe and uniform priors for all parameters adopted in the
analysis in the intervals listed in Table 1. The Hubble expansion rate $H_{0}$
is a derived parameter in our analysis. We constrained $H_{0}$ values to
reject the extreme models.

For the cosmological analysis we use the CMB temperature (TT), polarization
(EE,TE) and lensing angular power spectra from Planck 2018 release [1] and the
likelihood codes corresponding to different multipole ranges
[58]444http://pla.esac.esa.int/pla/cosmology. The Planck data currently
provide the best characterisation of the primordial density perturbations [2],
constraining the cosmological parameters at the sub-percent level [1].

We use the following combinations of TT, TE, EE and lensing Planck likelihoods
[2]:
(i) Planck TT+lowE: the combination of high-l TT likelihood at multipoles l
$\geq$ 30, the Commander likelihood for low-l temperature-only and the SimAll
low-l EE likelihood in the range 2 $<$ l $<$ 29; (ii) Planck TE and Planck EE:
the combination of TE and EE likelihoods at l $\geq$ 30; (iii) Planck
TT,TE,EE+lowE: the combination of Commander likelihood using TT, TE, and EE
spectra at $\it l$ $\geq$30, the low-l temperature, and the low- SimAll EE
likelihood; (iv) Planck TT,TE,EE+lowP: the combination of the likelihoods
using TT, TE, and EE spectra at l $>$ 30; (v) Planck high-l and Planck low-l
polarization: the Plik likelihood; (vi) Planck CMB lensing: the CMB lensing
likelihood [59] for lensing multipoles 8 $<$ l $<$ 400.

We also consider the measurement of the CMB B-mode polarization angular power
spectrum by the BICEP2/Keck Array collaboration [3]. The BK15 likelihood
B-mode polarization only leads to an upper limit of tensor-to-scalar ratio
amplitudes $r<$ 0.07 (95% CL) [3].

We will refer to the combination of these datasets as Planck
TT,TE,EE+lowE+lensing+BK15.

Table 1:  Priors and constraints on EGB dark Higgs inflation model parameters adopted in the analysis. All priors are uniform in the listed intervals. We assume a flat universe. Parameter | Prior
---|---
$\Omega_{b}h^{2}$ | [0.005, 0.1]
$\Omega_{c}h^{2}$ | [0.001, 0.5 ]
$100\theta_{s}$ | [0.5, 10]
$\tau$ | [0.01, 0.9]
${\rm log}(10^{10}A_{s})$ | [2.5, 5]
$n_{s}$ | [0.5, 1.5]
${\cal N}$ | [54, 64]
$\alpha\times 10^{7}$ | [0.007, 3]
$\beta\times 10^{13}$ | [1, 5]
$\gamma\beta$ | [0, 3]
$\eta\beta$ | [0 ,3]
$H_{0}({\rm km\,s}^{-1}{\rm Mpc}^{-1})$ | [20, 100]

###  4.2 Analysis

Left panel from Figure 2 presents the marginalised likelihood probability
distributions of the inflationary parameters, $A_{s}$, $n_{s}$, $r$ and ${\cal
N}$ from the fit of the EGB dark Higgs inflation model with the Planck
TT,TE,EE+lowE+lensing+BK15 dataset. These predictions are computed at pivot
scale $k_{*}$=0.002 Mpc-1 and include the uncertainty in the number of
e-folds. For comparison, we also show the corresponding 65% and 95% limits
from the fit of $\Lambda$CDM model with the same dataset [2]. The right panel
from the same figure presents the joint confidence regions (68% and 95% CL) of
$n_{s}$ and $r$.
The mean values and the errors for all parameters are presented in Table 2.
We find that the EGB dark Higgs inflation model is strongly favoured by the
Planck+BK15 data [2].

We test the consistency of the EGB dark Higgs inflation model predictions for
the mean values of $\gamma\beta$, $\eta\beta$ and ${\cal N}$ given in Table 2.
From (3.16) we evaluate the dark Higgs field at the beginnig of inflation,
$\phi_{I}=10.36M_{pl}$. The slow-roll parameters at $\phi_{I}$ defined in
(3.14) are given by:

$\displaystyle\epsilon_{0}$ $\displaystyle=$
$\displaystyle\epsilon_{1}=0.0163,\hskip 8.5359ptk_{0}=0.0017\,,\hskip
8.5359ptk_{1}=0.0165\,,\hskip 8.5359pt\Delta_{0}=0.0249\,,\hskip
8.5359pt\Delta_{1}=0.0165\,,$ (4.5)

while the tensor-to-scalar ratio (3.12) and the amplitude of the curvature
perturbations (3) at $\phi_{I}$ are:

$\displaystyle r=0.065\,,\hskip 14.22636pt{\cal P}_{R}=1.472\times 10^{-9}\,.$
(4.6)

The inflation potential (1.1) at $\phi_{I}$ is obtained as:

$\displaystyle V^{1/4}(\phi_{I})=\left(\frac{3\pi^{2}{\cal
P}_{R}}{2}r\right)^{1/4}M_{pl}\simeq 6.11\times 10^{-3}M_{pl}\simeq 1.46\times
10^{16}{\rm GeV}\,.$ (4.7)

This constraint applied to the dark Higgs potential $V=\beta\phi_{I}^{4}/4$
leads to quartic coupling $\beta<3.38\times 10^{-13}$, value consistent with
the inflationary observables and with the dark Higgs parameters. Under the
slow-roll conditions, we get from (4.7) the Hubble parameter at $\phi_{I}$:

$\displaystyle H(\phi_{I})\simeq 1.51\times 10^{-5}M_{pl}\simeq 3.63\times
10^{13}{\rm GeV}\,.$ (4.8)

From (4.8) it follows that the curvature scale at $\phi_{I}$ satisfy the
condition $R\simeq 12H^{2}\ll M^{2}_{pl}$ and therefore the unitarity bound of
dark Higgs inflation model with curvature couplings is not exceeded.

Figure 2: Left: Marginalised likelihood probability distributions of the main
inflationary parameters from the fit of the EGB dark Higgs inflation model
with the Planck TT,TE,EE+lowE+lensing+BK15 dataset. The distributions are
obtained at $k_{*}$=0.002 Mpc-1 and include the uncertainty in the number of
e-folds. For comparison we also show the corresponding 65% (dark blue) and 95%
(light blue) limits from the fit of $\Lambda$CDM model with the same dataset
[2]. Right: Marginalised joint 68% and 95% CL regions for $n_{s}$ and $r$
distributions presented in the left panel. Figure 3: Left: Marginalised
likelihood probability distributions of the dark Higgs parameters from the fit
of the EGB dark Higgs inflation model with the Planck
TT,TE,EE+lowE+lensing+BK15 dataset, obtained with (red) and without (dashed
black) quantum corrections. Right: Marginalised joint 68% and 95% CL regions
for $m_{\phi}$ and $\theta$ obtained for EGB dark Higgs inflation model.

Left panel from Figure 3 presents the likelihood probability distributions of
the dark Higgs parameters $\beta$, $\alpha$, $m_{\phi}$ and $\theta$ obtained
from the fit of the EGB dark Higgs inflation model with the Planck
TT,TE,EE+lowE+lensing+BK15 dataset. These predictions are computed at
$k_{*}$=0.002 Mpc-1 and include the quantum corrections of the coupling
constants. The mean values and the errors of these parameters are given in
Table 2.
For comparison we plot the same distributions without quantum corrections.

The dark Higgs mass $m_{\phi}$ and mixing angle $\theta$ are derived
parameters in our analysis and are obtained from (2.4). The predictions of the
EGB dark Higgs inflation model for the joint confidence regions (68% and 95%
CL) of $m_{\phi}$ and $\theta$ are shown in the right panel of Figure 3.
We find for the dark Higgs - SM Higgs coupling, $7\times
10^{-10}<\alpha<3\times 10^{-8}$. These bounds are in agreement with the
estimate of the reheating temperature for a non-thermal distribution of the
inflaton field [40].
The bounds on the dark Higgs mass and mixing angle are found to be:

$\displaystyle 0.49\,\,{\rm GeV}$ $\displaystyle<$ $\displaystyle
m_{\phi}<1.43\,\,{\rm GeV}\,,\hskip 28.45274pt(95\%\,\,{\rm CL})$ (4.9)
$\displaystyle 4.48\times 10^{-5}$ $\displaystyle<$
$\displaystyle\theta<1.88\times 10^{-4}\,.$ (4.10)

##  5 Dark Higgs inflaton at LHC experiments

###  5.1 Dark Higgs inflaton decay inside detector

Figure 4: The evolution with $m_{\phi}$ of dark Higgs inflaton decay length,
$d=c\tau_{\phi}\gamma\beta$, for various dark Higgs energies $E_{\phi}$ and
$\theta=10^{-4}$.

The dark Higgs decay widths are suppressed by $\theta^{2}$ relative to those
of the SM Higgs boson if it would have the some mass as the dark Higgs. For
$m_{\phi}<2m_{\pi}$ the inflaton mostly decays in $e^{+}e^{-}$,
$\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$ with decay width given by:

$\displaystyle\Gamma(\phi\rightarrow{\bar{l}}l)=G_{F}\frac{m^{2}_{l}m_{\phi}}{4\sqrt{2}\pi}\beta^{3}_{l}\theta^{2}\hskip
14.22636pt(l=e\,,\mu\,,\tau)\,,$ (5.1)

where $G_{F}$ is the Fermi constant and
$\beta_{l}=\sqrt{1-m^{2}_{l}/m^{2}_{\phi}}$ is the lepton velocity.
For inflaton masses in the range $2m_{\pi}<m_{\phi}<2.5$ GeV the dominant
decay modes are to $\pi^{+}\pi^{-}$, $k^{+}k^{-}$ and other hadrons.
The dark Higgs hadronic decay modes suffers from theoretical uncertainties
since the chiral expansion breaks down above $2m_{\pi}$ while the perturbative
QCD calculation are reliable for masses of few GeV [60, 61].
For the inflaton mass range (4.9) we adopt the numerical results from [61]
that uses the dispersive analysis for $2m_{\pi}<m_{\phi}<1.3$ GeV [62], the
perturbative spectator model [63, 64] for $m_{\phi}\,>\,$ 2GeV and interpolate
between these two for 1.3 GeV$<m_{\phi}<$2 GeV.
Figure 4 presents the dependence on $E_{\phi}$ of the dark Higgs decay length:

$\displaystyle d=c\tau_{\phi}\gamma\beta\,$ (5.2)

where $\tau_{\phi}=1/\Gamma(\phi\rightarrow ll,hh)$ is the dark Higgs
lifetime, $\Gamma(\phi\rightarrow ll,hh)$ is the decay width scaled with
$\theta^{2}$, $\gamma=E_{\phi}/m_{\phi}$ and $\beta=\sqrt{1-1/\gamma^{2}}$.
The decay length scales as $d\sim E_{\phi}$ for large $E_{\phi}$. For
$E_{\phi}\sim{\cal O}(10^{3})$ GeV the decay lengths are d$\sim{\cal O}(1)$ km
and therefore a significant number of dark Higgs inflatons can decay within
the detector volume.

To determine the number of dark Higgs inflatons that decay inside the detector
volume, we must specify the size, shape, and location of the detector relative
to the LHC collider interaction point (IP).
We consider two representative experiments, FASER (the ForwArd Search
ExpeRiment) [36] and MAPP-1(the MoEDAL Apparatus for Penetrating Particles)
([65]).
FASER detectors have cylindrical shape and are centred on the LHC beam
collision axis. The detectors have the radius R and the depth
$\Delta=L_{max}-L_{min}$, where $L_{max}$ and $L_{min}$ are the distances from
the IP to the far and near edge of detectors along the beam axis. The location
of FASER detectors is:

$\displaystyle{\rm FASER\,\,\,far\,\,\,location}$ $\displaystyle:$
$\displaystyle\hskip 14.22636ptL_{max}=400\,m\,,\,\Delta=10\,m,\,R=1\,m\,,$
(5.3) $\displaystyle{\rm FASER\,\,\,near\,\,\,location}$ $\displaystyle:$
$\displaystyle\hskip 14.22636ptL_{max}=150\,m\,,\,\Delta=5\,m\,,\,R=4\,{\rm
cm}\,.$ (5.4)

MAPP detector is a parallelepiped at approximately $5^{0}$ from the beam
collision axis with the following location:

$\displaystyle\hskip 5.69046pt{\rm MAPP-1}:\hskip
14.22636ptL_{max}=55\,m\,,\Delta=3\,m\,,H=1\,{\rm m}\,,$ (5.5)

where H is the is the parallelepiped height.
The probability of the dark Higgs boson to decay inside the detector volume is
given by:

${\cal
P}^{det}(E_{\phi},\theta_{\alpha})=\left(e^{-L_{min}/d}-e^{-L_{max}/d}\right)\Theta(R,\tan(\theta_{\alpha})L_{max})\,$
(5.6)

where $E_{\phi}$ is the dark Higgs energy, $d$ is its decay length,
$\theta_{\alpha}$ is the angular acceptance of the detector,
$\tan(\theta_{\alpha})=R/L_{max}$, and $\Theta$ is the Heaviside step
function. For MAPP-1 we take $R=H\pi^{-1/2}$ in (5.6) to conserve the
effective acceptance area.
In Figure 5 we present the dependence on $E_{\phi}$ of the normalised
detection probability corresponding to the different experimental
configurations obtained for cosmological best fit solution for $m_{\phi}$ and
$\theta$. The figure shows that the experimental configurations are sensitive
to complementar ranges of the dark Higgs energy.

Figure 5:  The dependence on the dark Higgs energy $E_{\phi}$ of the
normalised detection probability corresponding to different experimental
configurations obtained for cosmological best fit solution for $m_{\phi}$ and
$\theta$.

###  5.2 Dark Higgs inflaton production at LHC

The dark Higgs bosons are mainly produced through K and B meson decays. As
$m_{\phi}>m_{K}$ ($m_{K}=0.494$ GeV) for the inflaton mass range (4.9), in the
following we consider the dark Higgs production only through the B-meson
decay. The B-meson branching fraction is given by [36]:

$Br(B\rightarrow
X_{s}\phi)=5.7\left(1-\frac{m^{2}_{\phi}}{m^{2}_{B}}\right)^{2}\theta^{2}\,,$
(5.7)

where $X_{s}$ denotes any strange hadronic state and $m_{B}$ is the B-meson
mass ($m_{B}=5.28$ GeV).

Figure 6: LHC experiments reach for dark Higgs inflaton in the cosmological
confidence region (4.9) for an integrated luminosity of 3ab-1 at 13 TeV LHC
assuming 100% detection efficiency.

The dark Higgs production cross section at LHC energies can be estimated as
[45]:

$\displaystyle\frac{\sigma_{\phi}}{\sigma_{inel}}=M_{pp}Br(B\rightarrow
X_{s}\phi)\,,$ (5.8)

where $M_{pp}$ is the proton multiplicity and $\sigma_{inel}$ is the $pp$
inelastic cross section.

###  5.3 LHC experiments reach for dark Higgs inflaton

The total number of dark Higgs bosons that decay inside detector are then
given by:

$N_{sig}(m_{\phi},\theta)=N_{inel}\frac{\sigma_{\phi}}{\sigma_{inel}}\,Br(\phi\rightarrow
KK)\,Br(\phi\rightarrow\pi\pi)\int{\cal P}^{det}(E_{\phi},\theta_{\alpha}){\rm
d\,}\theta_{\alpha}{\rm d}\,E_{\phi}\,,$ (5.9)

where $N_{inel}$ is the total number of inelastic $pp$ scatering events.
Throughout we assume an integrated luminosity of 3 a$b^{-1}$ at the 13 TeV
LHC, implying $N_{inel}\simeq 1.1\times 10^{16}$. We also take
$\sigma_{inel}$(13 TeV) $\simeq$ 75 mb and $M_{pp}$(13 TeV) $\simeq$ 66 [31].

Figure 6 shows the predicted number of dark Higgs inflaton decays in the
cosmological confidence region (4.9) obtained for the experimental
configurations discussed for an integrated luminosity of 3ab-1 at 13 TeV LHC
assuming 100% detection efficiency.
In our computation we take the dark Higgs energy in the range 100 GeV
$<E_{\phi}<10{{}^{6}}$ GeV, imposed by the requirement that the dark Higgs
inflaton propagate to the detector locations, as shown in Figure 5.
For comparison, in Figure 7 we present the FASER reach [36] and the MAPP-1
reach [65] for the dark Higgs boson for an integrated luminosity of 3ab-1 at
13 TeV LHC. The cosmological dark Higgs inflaton confidence region (4.9) is
also shown.

Figure 7:  FASER far location, FASER near location and MAPP-1 reach for dark Higgs boson for an integrated luminosity of 3ab-1 at13 TeV LHC. The cosmological dark Higgs inflaton confidence region (4.9) is also shown.  Table 2: The mean values and the absolute errors of the main parameters obtained from the fit of the EGB dark Higgs inflation model with Planck TT,TE,EE+lowE+lensing+BK15 dataset. The errors are quoted at 68% CL. The upper limits are quoted at 95% CL. The first group of parameters are the base cosmological parameters sampled in the Monte-Carlo Markov Chains analysis with uniform priors.The others are derived parameters. Parameter |
---|---
$\Omega_{b}h^{2}$ | 0.0223 $\pm$ 0.0002
$\Omega_{c}h^{2}$ | 0.1194 $\pm$ 0.0011
$\theta_{s}$ | 1.0410 $\pm$ 0.0004
$\tau$ | 0.050 $\pm$ 0.009
${\rm ln}(10^{10}A_{s})$ | 3.050 $\pm$ 0.008
$n_{s}$ | 0.967 $\pm$ 0.0044
$r_{0.002}$ |  $<$ 0.059
${\cal N}$ | 59.4 $\pm$ 1.210
$10^{13}\times\beta$ | 0.892 $\pm$ 0.051
$10^{9}\times\alpha$ | 1.021 $\pm$0.219
$\gamma\beta$ | 0.218 $\pm$ 0.015
$\eta\beta$ | 1.129 $\pm$ 0.067
$H_{0}({\rm km\,s}^{-1}{\rm Mpc}^{-1})$ | 67.729 $\pm$ 0.641
$m_{\phi}$ (GeV) | 0.919 $\pm$ 0.211
$10^{4}\times\theta$ | 1.492 $\pm$ 0.045

##  6 Conclusion

In this paper we analyse the dark Higgs inflation model with curvature
corrections given by the kinetic term non-minimally coupled to the Einstein
tensor and the coupling to the Gauss-Bonnet (GB) 4-dimensional invariant (EGB
dark Higgs inflation) and explore the possibility to test its predictions by
particle physics experiments at LHC.
The dynamics of the slow-roll inflation with curvature corrctions has been
proposed in context of the SM Higgs inflation in Refs [14, 15].

We show that the EGB dark Higgs inflation model is strongly favoured by
Planck+BK15 data [2]. The cosmological predictions of this model for dark
Higgs inflaton mass $m_{\phi}$ and mixing angle $\theta$, including the RG
quantum corrections of dark Higgs coupling constants and the uncertainty in
estimation of the reheating temperature, are found to be:

$\displaystyle 0.49\,\,{\rm GeV}$ $\displaystyle<$ $\displaystyle
m_{\phi}<1.34\,\,{\rm GeV}\,,\hskip 28.45274pt(95\%\,\,{\rm CL})$
$\displaystyle 4.48\times 10^{-5}$ $\displaystyle<$
$\displaystyle\theta<1.88\times 10^{-4}\,.$

The consistency test of the EGB dark Higgs inflation model predictions leads
to a lower bound of dark Higgs inflaton quartic coupling $\beta<$ 3.38 $\times
10^{-13}$, value consistent with the inflationary observables and with the
dark Higgs parameters.
We find the dark Higgs inflaton - SM Higgs boson coupling constant
$\alpha>7\times 10^{-10}$, in agreement with the estimate of the reheating
temperature for a non-thermal distribution of the inflaton field [40].

We evaluate the FASER and MAPP-1 experiments reach for dark Higgs inflaton
parameters $m_{\phi}$ and $\theta$ in the 95% CL cosmological confidence
region, for an integrated luminosity of 3ab-1 at 13 TeV LHC assuming 100%
detection efficiency.
We conclude that the dark Higgs inflation model with curvature corrections is
a compelling inflation scenario based on particle physics theory favoured by
the present cosmological measurements that leaves imprints in the dark Higgs
boson searchers at LHC.

## Acknowledgments

The author would like to thank to Vlad Popa for helpful discussions and
acknowledge the use of the GRID system computing facility at the Institute of
Space Science.
This work was partially supported by ESA/PRODEX Contract 4000124902.

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# Honest Confidence Bands for Isotonic Quantile Curves

Lutz Dümbgen, Lukas Lüthi
University of Bern

### Abstract.

We provide confidence bands for isotonic quantile curves in nonparametric
univariate regression with guaranteed given coverage probability. The method
is an adaptation of the confidence bands of Dümbgen and Johns (2004) for
isotonic median curves.

### AMS subject classification:

63G08, 62G15, 62G20

### Key words:

Binomial distribution, union-intersection test.

##  1 Introduction

Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots,(X_{n},Y_{n})$ be independent random
pairs consisting of covariate values $X_{i}$ in some real interval
$\mathcal{X}$ and response values $Y_{i}\in\mathbb{R}$. We assume that
$\operatorname{\mathbb{P}}(Y_{i}\leq y\,|\,X_{i})=F(y\,|\,X_{i})$ with unknown
conditional distribution functions $F(\cdot\,|\,x)$, $x\in\mathcal{X}$. For
any fixed $\gamma\in(0,1)$ and $x\in\mathcal{X}$, let $Q_{\gamma}(x)$ be a
$\gamma$-quantile of $F(\cdot\,|\,x)$, that is,

$F(Q_{\gamma}(x)-|\,x)\ \leq\ \gamma\ \leq\ F(Q_{\gamma}(x)\,|\,x).$

Our goal is to compute a $(1-\alpha)$-confidence band $(L,U)$ for $Q_{\gamma}$
under the sole assumption that $Q_{\gamma}:\mathcal{X}\to\mathbb{R}$ is
isotonic, i.e. non-decreasing. Precisely, we want to determine functions
$L=L(\cdot,\mathrm{data})$ and $U=U(\cdot,\mathrm{data})$ on $\mathcal{X}$
with values in $[-\infty,\infty]$ such that

$\operatorname{\mathbb{P}}(L\leq Q_{\gamma}\leq U\ \text{on}\ \mathcal{X})\
\geq\ 1-\alpha$

whenever $Q_{\gamma}$ is isotonic. In Section 2 we describe the construction
of the bands. It is a simplified and generalized version of the confidence
bands of Dümbgen and Johns (2004) who treated the case $\gamma=0.5$ only. The
bands are also similar in spirit to the confidence bands of Dimitriadis et al.
(2022) for univariate binary regression or the confidence bands of Yang and
Barber (2019) for isotonic regression with subgaussian errors. An important
property of the bands is that their computation requires only $O(n^{2})$
steps. An alternative method to construct confidence bands for isotonic
quantile curves has been proposed recently by Chatterjee and Sen (2021), but
it requires certain a priori knowlegde about the conditional distribution
functions $F(\cdot\,|\,x)$, $x\in\mathcal{X}$.

Section 3 presents some asymptotic properties of the bands as the sample size
$n$ tends to infinity. The results are similar to the results of Dümbgen and
Johns (2004) and Mösching and Dümbgen (2020). The latter paper treats point
estimation of the conditional distribution functions $F(\cdot\,|\,x)$ under
the assumption that they are isotonic in $x\in\mathcal{X}$ with respect to the
usual stochastic order. Section 4 illustrates the bands with some numerical
examples. There it is also illustrated what happens if the quantile functions
are even assumed to be S-shaped, that is, isotonic and convex-concave, and if
we replace our confidence band with the hull of all S-shaped functions fitting
within the band. This stronger shape-constraint is plausible in many
applications, see Feng et al. (2021) and the references cited therein. The
latter paper treats least squares estimation of S-shaped mean functions.

##  2 Construction of the bands

We condition on the observed values $X_{1},\ldots,X_{n}$, so they become fixed
real numbers, and the observations $Y_{i}$ are independent with distribution
functions $F_{i}:=F(\cdot\,|\,X_{i})$. For an interval $B\subset\mathcal{X}$,
let

$N(B)\ :=\ \\#\\{i\colon X_{i}\in B\\}.$

Suppose that $g:\mathcal{X}\to\mathbb{R}$ is a candidate for $Q_{\gamma}$. If
$N(B)>0$ and $g=Q_{\gamma}$, then the distribution of the random sum

$S_{\ell}(B,g)\ :=\ \\#\\{i\colon X_{i}\in B,Y_{i}\leq g(X_{i})\\}$

is stochastically greater than or equal to a binomial distribution with
parameters $N(B)$ and $\gamma$. That is, for any integer $k\geq 0$,

$\operatorname{\mathbb{P}}\bigl{(}S_{\ell}(B,g)\leq k\bigr{)}\ \leq\
F_{N(B),\gamma}(k),$

where $F_{m,p}$ denotes the distribution function of the binomial distribution
with parameters $m\in\mathbb{N}$ and $p\in[0,1]$. Analogously, the
distribution of the random sum

$S_{u}(B,g)\ :=\ \\#\\{i\colon X_{i}\in B,Y_{i}\geq g(X_{i})\\}$

is stochastically greater than or equal to a binomial distribution with
parameters $N(B)$ and $1-\gamma$. For our confidence bands, we need an
analogous statement simultaneously for all intervals $B\subset\mathcal{X}$.

######  Lemma 1.

There exists a coupling between the observations $Y_{1},\ldots,Y_{n}$ and
stochastically independent Bernoulli variables
$\xi_{1},\ldots,\xi_{n}\in\\{0,1\\}$ with
$\operatorname{\mathbb{E}}(\xi_{i})=\gamma$, such that for any version $g$ of
$Q_{\gamma}$ and all intervals $B\subset\mathcal{X}$,

$S_{\ell}(B,g)\ \geq\ T_{\ell}(B):=\sum_{i\colon X_{i}\in
B}\xi_{i}\quad\text{and}\quad S_{u}(B,g)\ \geq\ T_{u}(B):=\sum_{i\colon
X_{i}\in B}(1-\xi_{i}).$

###### Proof of Lemma 1.

Let $U_{1},\ldots,U_{n}$ be independent random variables with uniform
distribution on $(0,1)$. Then the quantile transformations

$Y_{i}\ :=\ F_{i}^{-1}(U_{i})=\min\\{y\in\mathbb{R}\colon F_{i}(y)\geq
U_{i}\\}$

yield independent random variables $Y_{1},\ldots,Y_{n}$ with distribution
functions $F_{1},\ldots,F_{n}$, respectively. For any version $g$ of
$Q_{\gamma}$ and $1\leq i\leq n$, it follows from $F_{i}(g(X_{i}))\geq\gamma$
that

$1_{[Y_{i}\leq g(X_{i})]}\ =\ 1_{[F_{i}(g(X_{i}))\geq U_{i}]}\ \geq\
1_{[U_{i}\leq\gamma]}.$

Similarly, the inequality $F_{i}(g(X_{i})-)\leq\gamma$ implies that

$1_{[Y_{i}\geq g(X_{i})]}\ =\ 1_{[F_{i}(y)<U_{i}\ \text{for all}\
y<g(X_{i})]}\ \geq\ 1_{[F_{i}(g(X_{i})-)<U_{i}]}\ \geq\
1_{[U_{i}>\gamma]}=1-1_{[U_{i}\leq\gamma]}.$

Consequently, $\xi_{i}:=1_{[U_{i}\leq\gamma]}$ defines Bernoulli variables
$\xi_{1},\ldots,\xi_{n}$ with the desired properties. ∎

The confidence band can be constructed as follows. Let $\mathcal{B}$ be a
given family of compact intervals $B\subset\mathcal{X}$ with endpoints in
$\\{X_{1},\ldots,X_{n}\\}$. Suppose that for any integer
$m\in\\{1,\ldots,n\\}$, we have specified integers
$c_{\ell}(m),c_{u}(m)\in\\{0,1,\ldots,m\\}$ such that

$\operatorname{\mathbb{P}}\bigl{(}T_{\ell}(B)\geq c_{\ell}(N(B))\ \text{and}\
T_{u}(B)\geq c_{u}(N(B))\ \ \text{for all}\ B\in\mathcal{B}\bigr{)}\ \geq\
1-\alpha.$ (1)

Then we may claim with confidence $1-\alpha$ that for any version $g$ of
$Q_{\gamma}$ and all intervals $B\in\mathcal{B}$,

$S_{\ell}(B,g)\ \geq\ c_{\ell}(N(B))\quad\text{and}\quad S_{u}(B,g)\ \geq\
c_{u}(N(B)).$ (2)

Let $Y_{B:1}\leq Y_{B:2}\leq\cdots\leq Y_{B:N(B)}$ be the order statistics of
the observations $Y_{i}$ with $X_{i}\in B$, and let $Y_{B:0}:=-\infty$,
$Y_{B:N(B)+1}:=\infty$. Then it follows from isotonicity of $g$ that

$S_{\ell}(B,g)\ \geq\ c_{\ell}(N(B))\quad\text{implies that}\quad g(\max(B))\
\geq\ Y_{B:c_{\ell}(N(B))},$

because $S_{\ell}(B,g)\leq\\#\bigl{\\{}i\colon X_{i}\in B,Y_{i}\leq
g(\max(B))\bigr{\\}}$. Likewise,

$S_{u}(B,g)\ \geq\ c_{u}(N(B))\quad\text{implies that}\quad g(\min(B))\ \leq\
Y_{B:N(B)+1-c_{u}(N(B))}.$

Assuming that $Q_{\gamma}$ is isotonic, $Y_{B:c_{\ell}(N(B))}$ is a lower
bound for all values $Q_{\gamma}(x)$, $x\geq\max(B)$, and
$Y_{B:N(B)+1-c_{i}(N(B))}$ is an upper bound for all values $Q_{\gamma}(x)$,
$x\leq\min(B)$. These considerations lead to the following
$(1-\alpha)$-confidence band $(L,U)$ for $Q_{\gamma}$:

$\displaystyle L(x)\ :=\ $
$\displaystyle\sup_{B\in\mathcal{B}\colon\max(B)\leq
x}\,Y_{B:c_{\ell}(N(B))},$ (3) $\displaystyle=\ $
$\displaystyle\sup_{B\in\mathcal{B}\colon\max(B)\leq
x}\,\min\bigl{\\{}r\in[-\infty,\infty)\colon S_{\ell}(B,r)\geq
c_{\ell}(N(B))\bigr{\\}}$ $\displaystyle U(x)\ :=\ $
$\displaystyle\inf_{B\in\mathcal{B}\colon\min(B)\geq
x}\,Y_{B:N(B)+1-c_{u}(N(B))}$ (4) $\displaystyle=\ $
$\displaystyle\inf_{B\in\mathcal{B}\colon\min(B)\geq
x}\,\max\bigl{\\{}r\in(-\infty,\infty]:S_{u}(B,r)\geq c_{u}(N(B))\bigr{\\}}$

with the conventions that $\sup(\emptyset):=-\infty$ and
$\inf(\emptyset):=\infty$, where $S_{\ell}(B,r):=\\#\\{i\colon X_{i}\in
B,Y_{i}\leq r\\}$ and $S_{u}(B,r):=\\#\\{i\colon X_{i}\in B,Y_{i}\geq r\\}$
for $r\in[-\infty,\infty]$.

### Finding the critical values $c_{\ell}(m)$ and $c_{u}(m)$.

Let $\tilde{n}\leq n$ be the number of elements of the set
$\\{X_{1},\ldots,X_{n}\\}$. Then $\mathcal{B}$ consists of at most
$\tilde{n}(\tilde{n}+1)/2$ different intervals. By Bonferroni’s inequality,
for any choice of the bounds $c_{\ell}(m)$ and $c_{u}(m)$,

$\displaystyle\operatorname{\mathbb{P}}\bigl{(}$ $\displaystyle
T_{\ell}(B)<c_{\ell}(N(B))\ \text{or}\ T_{u}(B)<c_{u}(N(B))\ \text{for some}\
B\in\mathcal{B}\bigr{)}$ $\displaystyle\leq\
\sum_{m=1}^{n}h_{m}\bigl{[}F_{m,\gamma}(c_{\ell}(m)-1)+F_{m,1-\gamma}(c_{u}(m)-1)\bigr{]},$
(5)

where $h_{m}$ is the number of intervals $B\in\mathcal{B}$ such that $N(B)=m$.
Now let

$c_{\ell}(m)\ :=\ F_{m,\gamma}^{-1}(\kappa)\quad\text{and}\quad c_{u}(m)\ :=\
F_{m,1-\gamma}^{-1}(\kappa)$ (6)

for some $\kappa\in(0,1]$. With a simple bisection search, one can find a
maximal value of $\kappa$ such that the bound (5) is not larger than $\alpha$.
Since $\sum_{m=1}^{n}H(m)\leq\tilde{n}(\tilde{n}+1)/2$ and
$F_{m,p}(F_{m,p}^{-1}(\kappa)-1)\leq\kappa$, we know that

$\kappa\ \geq\ \frac{\alpha}{2\\#\mathcal{B}}\ \geq\
\frac{\alpha}{\tilde{n}(\tilde{n}+1)}.$

The simple choice $\kappa=\alpha/(\tilde{n}(\tilde{n}+1))$ would be similar to
the approach of Yang and Barber (2019) or Dimitriadis et al. (2022).

Alternatively, we propose to use the critical values in (6) with $\kappa$
equal to a Monte-Carlo estimate of the $\alpha$-quantile of the random
variable

$\min_{B\in\mathcal{B}}\
\min\bigl{\\{}F_{N(B),\gamma}(T_{\ell}(B)),F_{N(B),1-\gamma}(T_{u}(B))\bigr{\\}}.$
(7)

Both proposals correspond to a union-intersection test: For each interval
$B\in\mathcal{B}$ and any candidate $g$ for $Q_{\gamma}$, one may interpret

$F_{N(B),\gamma}(S_{\ell}(B,g))$

as a p-value of the null hypothesis that $Q_{\gamma}\leq g$ on $B$, and

$F_{N(B),1-\gamma}(S_{u}(B,g))$

is a p-value of the null hypothesis that $Q_{\gamma}\geq g$ on $B$.

### Choosing $\mathcal{B}$.

In principle, one could take $\mathcal{B}$ to be the set of all
$\tilde{n}(\tilde{n}+1)/2$ compact intervals $B\subset\mathcal{X}$ with
endpoints in $\\{X_{1},\ldots,X_{n}\\}$. But for very large values
$\tilde{n}$, the Bonferroni version of our critical values $c_{\ell}(m)$ and
$c_{u}(m)$ may become rather conservative, while Monte-Carlo simulation of (7)
may be too time-consuming. Alternatively, we propose to specify a subset
$\mathcal{D}$ of $\\{1,\ldots,\lceil\tilde{n}/2\rceil\\}$ and to restrict
ourselves to intervals $B\subset\mathcal{X}$ such that the cardinality of
$B\cap\\{x_{1},\ldots,x_{n}\\}$ is in $\mathcal{D}$. Specific choices are the
intersection of $\\{1,\ldots,\lceil\tilde{n}/2\rceil\\}$ with the set of
numbers $a_{\ell}:=1+\ell(\ell-1)/2$, $\ell\geq 1$. That is, $a_{1}=1$ and
$a_{\ell+1}-a_{\ell}=\ell$. Here, $\\#\mathcal{D}=O(\tilde{n}^{1/2})$ and
thus, $\\#\mathcal{B}=O(\tilde{n}^{3/2})$. Alternatively, one could intersect
$\\{1,\ldots,\lceil\tilde{n}/2\rceil\\}$ with the set of Fibonacci numbers
$1,2,3,5,8,13,\ldots$ or the powers $2^{\ell}$, $\ell\geq 0$. In both cases,
$\\#\mathcal{D}=O(\log\tilde{n})$ and
$\\#\mathcal{B}=O(\tilde{n}\log\tilde{n})$.

### Explicit computation of $L$ and $U$.

Let $z_{1}<\cdots<z_{\tilde{n}}$ be the elements of
$\\{X_{1},\ldots,X_{n}\\}$. Note first that $L\equiv-\infty$ on
$(-\infty,z_{1})$, and $L$ is constant on all intervals $[z_{j},z_{j+1})$,
$1\leq j\leq\tilde{n}$, where $z_{\tilde{n}+1}:=\infty$. Similarly,
$U\equiv\infty$ on $(z_{\tilde{n}},\infty)$, and $U$ is constant on all
intervals $(z_{j-1},z_{j}]$, $1\leq j\leq\tilde{n}$, where $z_{0}:=-\infty$.
Thus it suffices to compute $\boldsymbol{L}=(L(z_{j}))_{j=1}^{\tilde{n}}$ and
$\boldsymbol{U}=(U(z_{j})_{j=1}^{\tilde{n}}$.

Now we describe an algorithm to compute $\boldsymbol{L}$ in $O(n^{2})$ steps.
The same algorithm can be applied to compute
$\boldsymbol{U}=(U(z_{j}))_{j=1}^{\tilde{n}}$ by replacing temporarily the
observations $(X_{i},Y_{i})$ with $(-X_{i},-Y_{i})$ and $\gamma$ with
$1-\gamma$. This is possible because our methods of setting $\kappa$ for the
critical values in (6) are invariant under these replacements.

In what follows, let $N_{j}:=\\#\\{i\colon X_{i}\leq z_{j}\\}$ for
$j\in\\{0,1,\ldots,\tilde{n}\\}$. Suppose that for some
$k\in\\{1,\ldots,\tilde{n}\\}$, the value $L(z_{k-1})$ is known already, where
$L(z_{0}):=-\infty$. Then $L(z_{k})$ is the smallest number
$r\in[L(z_{k-1}),\infty)$ such that

$S([z_{j},z_{k}],r)\ \geq\ c_{\ell}(N_{k}-N_{j-1})\quad\text{for all}\
[z_{j},z_{k}]\in\mathcal{B}.$

To determine this number $r$, we start with $r=L(z_{k-1})$ and check for
$j=k,k-1,k-2,\ldots$ whether $[z_{j},z_{k}]$ belongs to $\mathcal{B}$ and, if
yes, satisfies $S([z_{j},z_{k}],r)\geq c_{\ell}(N_{k}-N_{j-1})$. While doing
this, we determine the next candidate

$r_{\rm new}\ :=\ \min\bigl{(}\\{\infty\\}\cup\\{Y_{i}\colon
X_{i}\in[z_{j},z_{k}],Y_{i}>r\bigr{)}$

for $r$. As soon as $[z_{j},z_{k}]\in\mathcal{B}$ but
$S([z_{j},z_{k}],r)<c_{\ell}(N_{k}-N_{j-1})$, we replace $r$ with $r_{\rm
new}$ and reset $j$ to $k$. This leads to the algorithm described in Table 1.

$\begin{array}[]{|l|}\hline\cr r\leftarrow-\infty\\\ \text{for}\
k=1,2,\ldots,n\ \text{do}\\\ \quad j\leftarrow k+1\\\ \quad S\leftarrow 0\\\
\quad r_{\rm new}\leftarrow\infty\\\ \quad\text{while}\ j>1\ \text{do}\\\
\quad\quad j\leftarrow j-1\\\ \quad\quad S\leftarrow S+\sum_{i\colon
X_{i}=z_{j}}1_{[Y_{i}\leq r]}\\\ \quad\quad r_{\rm
new}\leftarrow\min\bigl{(}\\{r_{\rm new}\\}\cup\\{Y_{i}\colon
X_{i}=z_{j},Y_{i}>r\\}\bigr{)}\\\ \quad\quad\text{if}\
S<c_{\ell}(N_{k}-N_{j-1})\ \text{then}\\\ \quad\quad\quad r\leftarrow r_{\rm
new}\\\ \quad\quad\quad j\leftarrow k+1\\\ \quad\quad\quad S\leftarrow 0\\\
\quad\quad\quad r_{\rm new}\leftarrow\infty\\\ \quad\quad\text{end if}\\\
\quad\text{end while}\\\ \quad L(z_{k})\leftarrow r\\\ \text{end for}\\\
\hline\cr\end{array}$ Table 1: Algorithm to compute the lower confidence
bounds $L(\cdot)$.

That this algorithm has running time $O(n^{2})$ can be verified as follows:
For a given pair $(k,r)$ in
$\\{1,\ldots,\tilde{n}\\}\times(\\{-\infty\\}\cup\\{Y_{1},\ldots,Y_{n}\\})$,
we initialise and update $S$ and $r_{\rm new}$ for $j=k,k-1,k-2,\ldots$ until
$j=1$ or $[z_{j},z_{k}]\in\mathcal{B}$ but
$S([z_{j},z_{k}])<c_{\ell}(N_{k}-N_{j-1})$. This requires $O(n)$ steps. The
next pair $(k^{\prime},r^{\prime})$ satisfies either $k^{\prime}>k$ or
$r^{\prime}>r$. Consequently, from the starting point $(k,r)=(1,-\infty)$, we
arrive at the point $(k,r)=(\tilde{n},L(z_{\tilde{n}}))$ in at most
$\tilde{n}-1+n=O(n)$ steps. Hence, the total running time is of order
$O(n^{2})$.

##  3 Asymptotics

Similarly as in Dümbgen and Johns (2004), Mösching and Dümbgen (2020) and
Dimitriadis et al. (2022), we consider a triangular scheme of observations.
For each sample size $n\geq 1$, we observe
$(X_{n1},Y_{n1}),\ldots,(X_{nn},Y_{nn})$ with fixed numbers
$X_{n1}\leq\ldots\leq X_{nn}$ in $\mathcal{X}$ and independent random
variables $Y_{n1},\ldots,Y_{nn}$, where $Y_{ni}$ has distribution function
$F(\cdot\,|\,X_{ni})$. Let $(L_{n},U_{n})$ be the confidence band described
before for sample size $n$, based on a family
$\mathcal{B}_{n}\subset\mathcal{B}_{n}^{*}$, where $\mathcal{B}_{n}^{*}$ is
the family of all compact intervals with endpoints in
$\\{X_{n1},\ldots,X_{nn}\\}$. We analyze asymptotic properties of
$(L_{n},U_{n})$ on a bounded open interval $(a_{0},b_{o})\subset\mathcal{X}$
under the following three assumptions.

(A) For any set $B_{*}\in\mathcal{B}_{n}^{*}$, there exists a set
$B\in\mathcal{B}_{n}$ such that $B\subset B^{*}$ and $N_{n}(B)\geq
N_{n}(B^{*})/2$.

(B) Let $\mathrm{Leb}(\cdot)$ denote Lebesgue measure, and let
$N_{n}(B)=\\#\\{i\colon X_{ni}\in B\\}$ for $B\subset\mathcal{X}$. There exist
constants $C_{1},C_{2}>0$ such that for sufficiently large $n$,

$N_{n}(B)\ \geq\ C_{1}n\,\mathrm{Leb}(B)$

for arbitrary intervals $B\subset(a_{o},b_{o})$ such that $\mathrm{Leb}(B)\geq
C_{2}\log(n)/n$.

(C) There exist constants $D_{1},D_{2}>0$ such that for all
$x\in(a_{o},b_{o})$, the $\gamma$-quantile $Q_{\gamma}(x)$ is unique, and

$\frac{F(Q_{\gamma}(x)+h\,|\,x)-\gamma}{h}\ \geq\ D_{1}$

for all nonzero $h\in[-D_{2},D_{2}]$.

One can easily show that assumption (A) is satisfied for all proposals of
$\mathcal{B}_{n}$ in the previous section. Assumption (B) is satisfied, for
instance, in a deterministic setting with fixed design points
$X_{ni}=G^{-1}(i/n)$ for some distribution function $G$ on $\mathcal{X}$ such
that $G^{\prime}$ exists and is bounded away from $0$ on $(a_{o},b_{o})$. If
the $X_{ni}$ are the order statistics of $n$ independent random variables with
such a distribution function $G$, the arguments in Section 4.3 of Mösching and
Dümbgen (2020) can be modified to show that Condition (A) is satisfied almost
surely, provided that $C_{1},C_{2}>0$ are chosen appropriately.

######  Theorem 2.

Suppose that assumptions (A,B,C) are satisfied. Let $\rho_{n}=\log(n)/n$.
There exist constants $C,C^{\prime}>0$ depending only on
$C_{1},C_{2},D_{1},D_{2}$ with the following properties:

(i) Suppose that $Q_{\gamma}$ is constant on $(a_{o},b_{o})$. Then with
asymptotic probability one,

$\displaystyle U_{n}(x)\ $ $\displaystyle\leq\
p(x)+C\sqrt{\rho_{n}/(b_{o}-x)}\quad\text{for all}\
x\in(a_{o},b_{o}-C^{\prime}\rho_{n}],$ $\displaystyle L_{n}(x)\ $
$\displaystyle\geq\ p(x)-C\sqrt{\rho_{n}/(b_{o}-x)}\quad\text{for all}\
x\in[a_{o}+C^{\prime}\rho_{n},b_{o}).$

(ii) Suppose that $p$ is Hölder-continuous with exponent $\beta\in(0,1]$ and
constant $\lambda>0$ on $(a_{o},b_{o})$, that is,
$\bigl{|}Q_{\gamma}(x)-Q_{\gamma}(x^{\prime})\bigr{|}\leq\lambda|x-x^{\prime}|$
for all $x,x^{\prime}\in(a_{o},b_{o})$. Then for
$\delta_{n}:=(\lambda^{-2}\rho_{n})^{1/(2\beta+1)}$,

$\sup_{x\in(a_{o},b_{o}-\delta_{n}]}\,\bigl{(}U_{n}(x)-Q_{\gamma}(x)\bigr{)}^{+},\,\sup_{x\in(a_{o}+\delta_{n},b_{o})}\,\bigl{(}Q_{\gamma}(x)-L_{n}(x)\bigr{)}^{+}\
\leq\ C\lambda^{1/(2\beta+1)}\rho_{n}^{\beta/(2\beta+1)}.$

(iii) Suppose that $Q_{\gamma}$ is discontinuous at some point
$x_{o}\in(a_{o},b_{o})$. Then with asymptotic probability one,

$\displaystyle U_{n}(x)\ $ $\displaystyle\leq\
Q_{\gamma}(x_{o}-)+C\sqrt{\rho_{n}/(x_{o}-x)}\quad\text{for all}\
x\in(a_{o},x_{o}-C^{\prime}\rho_{n}],$ $\displaystyle L_{n}(x)\ $
$\displaystyle\geq\
Q_{\gamma}(x_{o}+)-C\sqrt{\rho_{n}/(x-x_{o})}\quad\text{for all}\
x\in[a_{o}+C^{\prime}\rho_{n},b_{o}).$

Part (i) shows that for arbitrary fixed numbers $a_{o}<a<b<b_{o}$,

$\sup_{x\in(a_{o},b]}\,(U_{n}-p)^{+}(x)+\sup_{x\in[a,b_{o})}\,(p-L_{n})^{+}(x)\
=\ O_{p}(\rho_{n}^{1/2})$

Part (iii) shows that for with asymptotic probability one,

$U_{n}(x_{\ell})\ <\ \frac{Q_{\gamma}(x_{o}-)+Q_{\gamma}(x_{o}+)}{2}\ <\
L_{n}(x_{r})$

for $a_{o}<x_{\ell}<x_{o}-D\rho_{n}$ and $x_{o}+D\rho_{n}<x_{r}<b_{o}$, where
$D$ is the maximum of $C^{\prime}$ and
$4C^{2}\bigl{(}Q_{\gamma}(x_{o}+)-Q_{\gamma}(x_{o}-)\bigr{)}^{-2}$.
Consequently, whenever $Q_{\gamma}$ has a discontinuity at some point in
$(a_{o},b_{o})$, the confidence band $(L_{n},U_{n})$ crosses a horizontal line
on an interval of length $O(\rho_{n})$.

###### Proof of Theorem 2.

For symmetry reasons, it suffices to prove the claims about the lower bound
function $L_{n}(\cdot)$. For notational convenience, the additional subscript
$n$ for the triangular scheme is often dropped. In what follows, $C_{o}$
denotes a generic universal constant, and $C^{\prime}$ denotes a generic
constant depending possibly on $C_{1},D_{1}$.

Applying Hoeffding’s inequality (Hoeffding, 1963) to binomial distributions
leads to the inequalities

$c_{\ell}(m)\ \geq\ m\gamma-\sqrt{m\log(1/\kappa)/2},$ (8)

where $c_{\ell}(m)=F_{m,\gamma}^{-1}(\kappa)$. Since
$\kappa\geq\alpha/[n(n+1)]$, we may conclude from (9) that for $1\leq m\leq n$
and sufficiently large $n$,

$c_{\ell}(m)\ \geq\ m\gamma-C_{o}\sqrt{m\log(n)},$ (9)

provided that $C_{o}>1$. For $B\in\mathcal{B}$ and $r\in\mathbb{R}$, let

$\Sigma_{\ell}(B,r)\ :=\ \operatorname{\mathbb{E}}S_{\ell}(B,r).$

Then it follows from Mösching and Dümbgen (2020, Corollary 4.7) that with
asymptotic probability one,

$\sup_{B\in\mathcal{B},r\in\mathbb{R}}\,N(B)^{-1/2}\bigl{|}(S_{\ell}-\Sigma_{\ell})(B,r)\bigr{|}\
\leq\ C_{o}\sqrt{\log(n)},$ (10)

provided that $C_{o}>1$.

From now on, we always assume that $n$ is large enough such that all
inequalities for $N(\cdot)$ in assumption (A) are satisfied, and we tacitly
assume that inequalities (9) and (10) hold true as well. For $B\in\mathcal{B}$
and $j\in\\{0,1,\ldots,N(B)\\}$, it follows from (10) that

$j\ \leq\ S_{\ell}(B,Y_{B:j})\ \leq\
\Sigma_{\ell}(B,Y_{B:j})+C_{o}\sqrt{N(B)\log(n)}.$

Plugging in $j=c_{\ell}(N(B))$ and using (9), we may conclude that for all
$B\in\mathcal{B}$,

$\Sigma_{\ell}(B,Y_{B:c_{\ell}(N(B))})\ \geq\ N(B)\gamma-
C_{o}\sqrt{N(B)\log(n)},$

provided that $C_{o}>2$. Combining this inequality with the definition (3) of
$L$ shows that $L$ is bounded from below by the following deterministic
function $\tilde{L}$:

$\tilde{L}(x)\ :=\ \sup_{B\in\mathcal{B}\colon\max(B)\leq
x}\,\inf\bigl{\\{}r\in\mathbb{R}\colon\Sigma_{\ell}(B,r)\geq N(B)\gamma-
C_{o}\sqrt{N(B)\log(n)}\bigr{\\}}.$

Consequently, it suffices to verify the assertions for $\tilde{L}$ instead of
$L$.

As to part (i), suppose that $Q_{\gamma}\equiv q$ on $(a_{o},b_{o})$. For
$x\in[a_{o}+C_{2}\rho_{n},b_{o})$, let $B(x)$ be the largest interval in
$\mathcal{B}_{n}$ which is contained in $(a_{o},x]$. By assumptions (A-B),
$N(B(x))\geq 2^{-1}C_{1}n(x-a_{o})$, and assumption (C) implies that for
$h\in(0,D_{2}]$,

$\Sigma_{\ell}(B(x),q-h)\ =\ \sum_{i\colon X_{i}\in
B(x)}F(Q_{\gamma}(X_{i})-h\,|\,X_{i})\ \leq\ N(B(x))\gamma-D_{1}N(B(x))h.$

Consequently, $\Sigma_{\ell}(B(x))\geq N(B(x))\gamma-
C_{o}\sqrt{N(B(x))\log(n)}$ implies that

$h\ \leq\ C_{o}D_{1}^{-1}\sqrt{\log(n)/N(B(x))}\ \leq\
h_{n}(x):=2^{1/2}C_{o}D_{1}^{-1}C_{1}^{-1/2}\sqrt{\rho_{n}/(x-a_{o})}.$

If we restrict $x$ to $[a_{0}+C^{\prime}\rho_{n},b_{o})$ with sufficiently
large $C^{\prime}\geq C_{2}$, the number $h_{n}(x)$ is no larger than $D_{2}$,
and we may conclude that

$\tilde{L}(x)\ \geq\
\inf\bigl{\\{}r\in\mathbb{R}\colon\Sigma_{\ell}(B(x),r)\geq N(B(x))\gamma-
C_{o}\sqrt{N(B(x))\log(n)}\bigr{\\}}\ \geq\ q-h_{n}(x),$

which proves part (i).

Concerning part (ii), suppose that $Q_{\gamma}$ is Hölder-continuous on
$(a_{o},b_{o})$ with exponent $\beta\in(0,1]$ and constant $\lambda>0$. Since
$\delta_{n}/\rho_{n}\to\infty$, we may assume that $\delta_{n}\geq
C_{2}\rho_{n}$. For $x\in[a_{o}+\delta_{n},b_{o})$, let $B(x)$ be the largest
interval in $\mathcal{B}_{n}$ which is contained in $(x-\delta_{n},x]$. Then
assumptions (A-B) imply that $N(B(x))\geq 2^{-1}C_{1}n\delta_{n}$. For
$h\in(0,D_{2}]$, it follows from assumption (C) that

$\displaystyle\Sigma_{\ell}(B(x),Q_{\gamma}(x)-h)\ $ $\displaystyle=\
\sum_{i\colon X_{i}\in B(x)}F(Q_{\gamma}(X_{i})-h\,|\,X_{i})$
$\displaystyle\leq\ \sum_{i\colon X_{i}\in
B(x)}F(Q_{\gamma}(x)+L\delta_{n}^{\beta}-h\,|\,X_{i})$ $\displaystyle\leq\
N(B(x))-D_{1}N(B(x))(h-L\delta_{n}^{\beta})^{+}.$

Consequently, $\Sigma_{\ell}(B(x),Q_{\gamma}(x)-h)\geq N(B(x))\gamma-
C_{o}\sqrt{N(B(x))\log(n)}$ implies that

$h\ \leq\ L\delta_{n}^{\beta}+C_{o}\sqrt{N(B(x))\log(n)}\ \leq h_{n},$

where

$h_{n}\ :=\
L\delta_{n}^{\beta}+2^{1/2}C_{o}C_{1}^{-1/2}\sqrt{\rho_{n}/\delta_{n}}\ =\
(1+2^{1/2}C_{o}C_{1}^{-1/2})\lambda^{1/(2\beta+1)}\rho_{n}^{\beta/(2\beta+1)}.$

Since $h_{n}\to 0$, the requirement $h_{n}\leq D_{2}$ is satisfied for
sufficiently large $n$, and then for all $x\in[a_{o}+\delta_{n},b_{o})$,

$\tilde{L}(x)\ \geq\
\inf\bigl{\\{}r\in\mathbb{R}\colon\Sigma_{\ell}(B(x),r)\geq N(B(x))\gamma-
C_{o}\sqrt{N(B(x))\log(n)}\bigr{\\}}\ \geq\ Q_{\gamma}(x)-h_{n},$

which proves part (ii).

Finally, suppose that $Q_{\gamma}$ is discontinuous at
$x_{o}\in(a_{o},b_{o})$. For $x\in(a_{o},x_{o}-C_{2}\rho_{n}]$, let $B(x)$ be
the largest interval in $\mathcal{B}_{n}$ which is contained in $[x,x_{o})$.
By assumptions (A-B), $N(B(x))\geq 2^{-1}C_{1}n(x_{o}-x)$. For
$h\in(0,D_{2}]$, it follows from assumption (C) that

$\Sigma_{\ell}(B(x),Q_{\gamma}(x_{o}-)-h)\ \leq\ \sum_{i\colon X_{i}\in
B(x)}F(Q_{\gamma}(X_{i})-h\,|\,X_{i})\ \leq\ N(B(x))\gamma-D_{1}N(B(x))h.$

Consequently, $\Sigma_{\ell}(B(x))\geq N(B(x))\gamma-
C_{o}\sqrt{N(B(x))\log(n)}$ implies that

$h\ \leq\ C_{o}D_{1}^{-1}\sqrt{\log(n)/N(B(x))}\ \leq\
h_{n}(x):=2^{1/2}C_{o}D_{1}^{-1}C_{1}^{-1/2}\sqrt{\rho_{n}/(x_{o}-x)}.$

If we restrict $x$ to $(a_{o},x_{o}-C^{\prime}\rho_{n}]$ for sufficiently
large $C^{\prime}\geq C_{2}$, then $h_{n}(x)\leq D_{2}$, and we may conclude
that

$\tilde{L}(x)\ \geq\
\inf\bigl{\\{}r\in\mathbb{R}\colon\Sigma_{\ell}(B(x),r)\geq N(B(x))\gamma-
C_{o}\sqrt{N(B(x))\log(n)}\bigr{\\}}\ \geq\ q-h_{n}(x),$

which proves part (iii). ∎

##  4 Numerical examples

Figure 1 shows a simulated data set of $n=500$ pairs $(X_{i},Y_{i})$, $1\leq
i\leq n$, where $X_{i}\in(0,50)$. To computed $95\%$-confidence bands for a
corresponding quantile function $Q_{\gamma}$, we used the family $\mathcal{B}$
of all compact intervals $B$ with endpoints in $\\{X_{1},\ldots,X_{n}\\}$ such
that $\\#B$ is in $\\{1+\ell(\ell-1)/2\colon\ell\geq 1\\}$.

For $\gamma=0.5$ and $\alpha=0.05$, the Bonferroni bound for the critical
value $\kappa$ was $1.041119\cdot 10^{-5}$, but by means of $199999$ Monte
Carlo simulations of (7) we obtained the more accurate and substantially
larger value $\hat{\kappa}=9.901973\cdot 10^{-4}$. Figure 2 shows the data
with the resulting $95\%$-confidence band for $Q_{0.5}$; the latter is shown
as well.

For $\gamma\in\\{0.25,0.75\\}$ and $\alpha=0.05$, the Bonferroni bound for the
critical value $\kappa$ was $1.152103\cdot 10^{-5}$, but with $199999$ Monte
Carlo simulations of (7) we obtained the value $\hat{\kappa}=1.07446\cdot
10^{-4}$. Figures 3 and 4 show the data with the resulting $95\%$-confidence
band for $Q_{0.25}$ and $Q_{0.75}$, respectively.

Figures 5, 6, 7 and 8 show analogous pictures for a data set of size $n=2500$.

Figure 1: An example data set of size $n=500$. Figure 2: $95\%$-Confidence
band for $Q_{0.5}$ for the data in Figure 1, together with the true quantile
function $Q_{0.5}$ (green, dotted). Figure 3: $95\%$-Confidence band for
$Q_{0.25}$ for the data in Figure 1, together with the true quantile function
$Q_{0.25}$ (green, dotted). Figure 4: $95\%$-Confidence band for $Q_{0.75}$
for the data in Figure 1, together with the true quantile function $Q_{0.75}$
(green, dotted). Figure 5: An example data set of size $n=2500$. Figure 6:
$95\%$-Confidence band for $Q_{0.5}$ for the data in Figure 5, together with
the true quantile function $Q_{0.5}$ (green, dotted). Figure 7:
$95\%$-Confidence band for $Q_{0.25}$ for the data in Figure 5, together with
the true quantile function $Q_{0.25}$ (green, dotted). Figure 8:
$95\%$-Confidence band for $Q_{0.75}$ for the data in Figure 5, together with
the true quantile function $Q_{0.75}$ (green, dotted).

### An stronger shape constraint.

In some applications, it is plausible that the quantile functions should be
isotonic and convex or isotonic and concave. Both constraints are special
cases of the more general assumption that the quantile functions are S-shaped.
That means, they are isotonic on $\mathcal{X}$, and for some inflection point
$\mu\in[-\infty,\infty]$, they are convex on $\mathcal{X}\cap(-\infty,\mu]$
and concave on $[\mu,\infty)\cap\mathcal{X}$. Since S-shapedness implies
isotonicity, assuming the quantile function $Q_{\gamma}$ is S-shaped allows
for the following refinement of our $(1-\alpha)$-confidence band $(L,U)$ for
$Q_{\gamma}$. We compute $(\tilde{L},\tilde{U})$ with

$\displaystyle\tilde{L}(x)\ $ $\displaystyle:=\ \inf\bigl{\\{}S(x)\colon S\
\text{S-shaped},L\leq S\leq U\bigr{\\}},$ $\displaystyle\tilde{U}(x)\ $
$\displaystyle:=\ \sup\bigl{\\{}S(x)\colon S\ \text{S-shaped},L\leq S\leq
U\bigr{\\}}.$

Precisely, we choose a fine grid $M$ of potential values for the inflection
point $\mu$ of $Q_{\gamma}$ and determine

$\displaystyle\tilde{L}(x,\mu)\ $ $\displaystyle:=\ \inf\bigl{\\{}S(x)\colon
S\ \text{S-shaped with inflection point}\ \mu,\,L\leq S\leq U\bigr{\\}},$
$\displaystyle\tilde{U}(x,\mu)\ $ $\displaystyle:=\ \sup\bigl{\\{}S(x)\colon
S\ \text{S-shaped with inflection point}\ \mu,\,L\leq S\leq U\bigr{\\}}.$

These functions $\tilde{L}(\cdot,\mu)$ and $\tilde{U}(\cdot,\mu)$ are easily
determined, and then we approximate $\tilde{L}(x)$ and $\tilde{U}(x)$ with
$\min_{\mu\in M}\tilde{L}(x,\mu)$ and $\max_{\mu\in M}\tilde{U}(x,\mu)$,
respectively.

For our numerical example with sample size $n=500$, the bands
$(\tilde{L},\tilde{U})$ differ only little from $(L,U)$, but for sample size
$n=2500$, the improvement is substantial; see Figures 9, 10 and 11. The
original band $(L,U)$ is depicted with thin black lines, and the refined band
$(\tilde{L},\tilde{U})$ is depicted with blue lines.

Figure 9: $95\%$-Confidence band for S-shaped $Q_{0.5}$ for the data in Figure
5, together with the true quantile function $Q_{0.5}$ (green, dotted). Figure
10: $95\%$-Confidence band for S-schaped $Q_{0.25}$ for the data in Figure 5,
together with the true quantile function $Q_{0.25}$ (green, dotted). Figure
11: $95\%$-Confidence band for S-shaped $Q_{0.75}$ for the data in Figure 5,
together with the true quantile function $Q_{0.75}$ (green, dotted).

## References

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