\section{Introduction}\label{intro}

Nuclei in the nobelium region represent the heaviest systems for which
detailed spectroscopic information on the structure is
available~\cite{herzberg08}. Hence, they provide an excellent testing
ground for various theoretical models that aim at reliable
predictions of properties of superheavy elements. Although
significant progress has been made both experimentally and
theoretically, there are still challenges for self-consistent
mean-field calculations (see Ref.~\refcite{bender12} for a recent
review). One of the challenges is the correct description of deformed shell gaps.
Macroscopic-microscopic calculations predict shell gaps at $N=152$
and $Z=100$, whereas typical self-consistent mean-field calculations give
shell gaps at $N=150$, and $Z=98$ and $Z=104$. The experimental findings
are consistent with the macroscopic-microscopic shell gaps.

Current advanced readjustements~\cite{kortelainen12} of coupling
constants of the Skyrme functional do not cure this deficiency,
although they point to a large underdetermination of the spin-orbit
properties. To study detailed spectroscopic properties of heavy and
superheavy nuclei, one thus may attempt a fine tuning of the
spin-orbit coupling constants, so as to bring the deformed shell gaps
in closer agreement with data. For this purpose, we recently
readjusted~\cite{shi13} the spin-orbit coupling constants,
$C_0^{\nabla J}$ and $C_1^{\nabla J}$, and pairing strengths,
$V_0^{\nu}$ and $V_0^{\pi}$, of the Skyrme UNEDF1
functional~\cite{kortelainen12} to match the experimental excitation
energies of the $11/2^-$ state in $^{251}$Cf and $7/2^+$ state in
$^{249}$Bk.
For the time-odd coupling constants of the functional, we adopted the
prescription based on the Landau parameters, as defined in Ref.~\refcite{[Ben02]}.
The remaining parameters of UNEDF1 were kept unchanged.
We call this new parameter set UNEDF1$^{\textrm{SO}}_L$.
Details of the adjustment procedures will be presented in
a forth-coming publication~\cite{shi13}. Here we only quote the obtained values,
which read,
\begin{eqnarray}
(C_0^{\nabla J}, C_1^{\nabla J}) &=& (-88.05040, 8.45838)\,\mbox{MeV\,fm$^5$},\\
(V_0^{\nu}, V_0^{\pi}) &=& (-205.05, -252.48)\,\mbox{MeV}.
\end{eqnarray}

As a result, in Ref.~\refcite{shi13} we found improvements of shell
gaps, quasiparticle energies, and moments of inertia (MoI) in nuclei
around $^{254}$No. Also a good agreement with experiment has been
obtained for the quasiparticle energies in $^{249}$Bk and $^{251}$Cf.
The single-particle spectra more closely resemble those obtained with
the WS potential, and shell gaps are opened at $Z=100$ and $N=152$ in $^{254}$No.
In the present work, we extended the calculations to other
nuclei in the nobelium region. We investigated the self-consistent
deformations as a function of frequency and we performed blocked
calculations for odd-$A$ $N=151$ isotones and Es ($Z=99$) isotopes.

\section{The model}\label{model}

In this work, all the calculations were performed by using the
symmetry-unrestricted solver HFODD~\cite{schunck12} (v2.52j).
For rotational calculations, the standard cranking term $-\omega J_y$ was added to the
Hamiltonian. The effect of an odd nucleon was treated by employing the blocking
approximation, that is, by using the density matrix and
pairing tensor in the form~\cite{ring80},
\begin{eqnarray}
\rho_{mn}^{\mu} &=& (V^*V^T)_{mn} + U_{m\mu}U^*_{n\mu} - V^*_{m\mu}V_{n\mu},\\
\kappa_{mn}^{\mu} &=& (V^*U^T)_{mn} + U_{m\mu}V^*_{n\mu} - V^*_{m\mu}U_{n\mu}.
\end{eqnarray}
We refer the reader to Ref.~\refcite{schunck10} for details.

In the cranking-plus-blocking calculations, the time-reversal symmetry is broken, so
the time-odd terms of the Skyrme density functional are present.
The importance of time-odd terms in describing collective rotational bands has
been emphasized in Ref.~\refcite{dobaczewski95}.
The advantage of using a symmetry-unrestricted solver is that the effects of non-zero
time-odd potentials can be taken into account in a strict manner.

We used 680 deformed harmonic-oscillator basis states, with the
basis-deformation parameters of $\hbar \omega_x$ = $\hbar \omega_y$ =
8.5049\,MeV and $\hbar \omega_z$ = 6.4823\,MeV. The cutoff energy in
the quasiparticle spectrum was $E_c=60$\,MeV. In the pairing channel,
to describe the approximate particle-number projection, we adopted
the Lipkin-Nogami formalism.

\section{Results}

In the nobelium region, high-order deformations of the WS
potential were found to play important roles in explaining many
high-spin phenomena. For example, the decrease of $E_x(2^+)$ in
$^{254}$No~\cite{sobiczewski01} and the delayed upbending of MoI in
$^{254}$No as compared to $^{252}$No~\cite{liu12} were attributed to
the effect of $\beta_6$ deformation on the relevant single-particle levels.

We performed cranking calculations for the 6 nuclei, in which the
ground-state rotational bands are observed, namely, for
$^{246,248,250}$Fm, $^{252,254}$No, and $^{256}$Rf. We determined the
MoIs, proton and neutron pairing gaps, and self-consistent
deformations as functions of the rotational frequency~\cite{shi13}.
The experimental kinematic and dynamic MoIs were well reproduced by
our calculations. Deformations obtained within our fully
self-consistent calculations provide a reasonable description of
polarization effects exerted by the occupations of individual
deformed orbitals. They were determined as the Bohr deformation
parameters $\beta_\lambda$ of sharp-edge mass distributions that have
multipole moments $Q_{\lambda}$, for $\lambda=2$--8, identical to
those calculated microscopically, see Ref.~\refcite{dobaczewski09}.
All nuclei considered here have axial ground-state shapes.

Figure~\ref{fig1} shows deformations $\beta_{2,4,6,8}$ as a function
of rotational frequency $\omega$. At $\omega=0$,
deformations $\beta_2$ are systematically higher than those obtained by
macroscopic-microscopic WS calculations~\cite{sobiczewski01,liu12},
and, in general, lower than those obtained within other relativistic and
non-relativistic mean-field calculations~\cite{afanasjev03,bender03}. It
is interesting to note that the high-order deformations presented in
Figure~\ref{fig1} are very close to those of
Refs.~\refcite{sobiczewski01} and~\refcite{liu12}. With increasing rotational
frequency, the deformations stay almost constant. Only at $\omega \sim
0.3$~MeV, deformations $\beta_2$ begin to decrease. This behavior
is consistent with that of Ref.~\refcite{liu12}.

\begin{figure}
\begin{center}
\psfig{file=B2468.eps,width=4.5in}
\end{center}
\caption{Deformations $\beta_{2,4,6,8}$ as functions of the
rotational frequency, calculated self-consistently for six nuclei,
$^{246,248,250}$Fm, $^{252,254}$No, and $^{256}$Rf.}
\label{fig1}
\end{figure}

Figure~\ref{fig2} (left) shows the calculated quasiparticle spectra
for $N=151$ isotones from $Z=94$ to 104. For these nuclei,
experimental data are available in this mass region~\cite{asai11}.
The observed ground states are dominated by the 9/2[734] orbital,
with the 5/2[622] state located at $\sim$200\,keV. In $^{245}$Pu and
$^{247}$Cm, our calculations predict ground states of 9/2[734]. For
heavier isotones, the 7/2[624] state becomes lower than the
9/2[734] state, but
only by several tens of keV. These two levels are right below the
$N=152$ shell gap. They are very close to each other, and change
order with increasing proton number. This indicates that (i)
there is a significant $N$-dependence of the single-particle and
quasiparticle energies and (ii) adjusting the spin-orbit force to obtain
better agreement for the $11/2^-$ state in $^{251}$Cf results in the $j_{15/2}$ and $g_{9/2}$
spherical shells being too close together.
The present calculations predict a 5/2[622] state which is too high
and underestimate the energies of the 7/2[624] and 1/2[620] states.

\begin{figure}
\begin{center}
\psfig{file=pu245-rf255.eps,width=2.2in}~
\psfig{file=es243-es253.eps,width=1.945in}
\end{center}
\caption{Quasiparticle energies calculated for the
UNEDF1$^{\textrm{SO}}_L$ parameter set\protect\cite{shi13},
and for the $N=151$ isotones (left) and Es ($Z=99$) isotopes (right).}
\label{fig2}
\end{figure}

Comparing our results with those of Ref.~\refcite{parkhomenko05}, we
note the presence of low-lying 7/2[624] states there too. We also
note an overestimation of their 1/2[620] energies. The excitation
energy of the 1/2[620] orbital is related to the shell gap at
$N=152$. Its isospin dependence is very interesting -- we note that
our results reproduce the observed increase of excitation energy of
1/2[620] with increasing proton number~\cite{asai11}.

Experimental information on quasi-proton spectra are relatively
scarce\cite{herzberg08}. Close to $Z=100$, systematic spectra are
only observed for Es ($Z=99$) isotopes~\cite{hessberger05}.
Figure~\ref{fig2} (right) displays the calculated quasiparticle
spectra for $Z=99$ isotopes from $N=144$ to 154. One can see that for
the ground states the calculations predict two very close states,
3/2[521] and 7/2[633], which is consistent with experimental
observations~\cite{hessberger05}. Compared to the calculations
presented in Ref.~\refcite{hessberger05}, our results show a much
lower 3/2[521] orbital. This is because our calculations predict very
close 7/2[633] and 3/2[521] levels below the $Z=100$ shell opening,
whereas the WS calculations give them $\sim$200\,keV apart. The
7/2[514] state is also systematically observed in the Es isotopes.
However, our calculation fails to reproduce the increase of excitation
energy of this orbital with increasing neutron number. We also note
that in our calculations, the 5/2[523] states exhibit an increase
similar to that observed in experiment for the state attributed to
the 7/2[514] orbital. Further experimental and theoretical work is
needed for these odd-$Z$ nuclei.

\section{Conclusion}

We performed self-consistent Skyrme Hartree-Fock-Bogolyubov
calculations for nuclei in nobelium region. Deformations
$\beta_{2,4,6,8}$ were studied for even-even nuclei
$^{246,248,250}$Fm, $^{252,254}$No, and $^{256}$Rf. The quasiparticle
spectra for $N=151$ isotones and $Z=99$ Es isotopes were
systematically calculated by using the blocking approximation within
the Hartree-Fock-Bogolyubov method. Comparisons were made with the
available experimental data and WS calculations. It was found that
our results give similar deformations to those obtained within the WS
calculations. The observed quasi-particle energies are reasonably well
reproduced, especially for proton quasiparticle states.

\section*{Acknowledgments}
This work was supported in part by the Academy of Finland and
University of Jyv\"askyl\"a within the FIDIPRO programme, the
Centre of Excellence Programme 2012--2017 (Nuclear and Accelerator
Based Physics Programme at JYFL), the European Research Council through the SHESTRUCT project (grant agreement number 203481), and by the Office of Nuclear
Physics,  U.S. Department of Energy under Contract
No. DE-FG02-96ER40963.
We acknowledge the CSC - IT Center
for Science Ltd, Finland, for the allocation of computational
resources.

\bibliographystyle{ws-procs9x6}


\section{Swampland Conjectures}

During the Moriond electroweak session we have seen many examples of fruitful interplays between theory and experiment---theorists try to explain experimental findings, while experimentalists turn to theorists for useful guidelines for what to look for.

From a theorist's viewpoint, coming up a ``theory'' often involves picking up a certain low-energy quantum field theory (QFT).
One can then discuss experimental constraints on the parameters of the theory, possible new signatures predicted by the theory, and so on.
The big problem, however, is that there are tremendous numbers of possible quantum field theories you can write down, and hence one has to face with huge numbers of possibilities.

One might therefore wish that there are new ingredients in theorist's toolkit.
I am going to argue that such a new tool comes from quantum gravity, albeit in somewhat unconventional forms (Fig.~\ref{topdown}).

\begin{figure}[htbp]
\centering\includegraphics[scale=0.45]{swampland_topdown.eps}
\caption{A phenomenologist often comes up with a specific low-energy QFT and works out its experimental consequences. It is often not clear, however, which QFT one should start with, since there are simply too many possibilities.
If one starts with quantum gravity, one sometimes finds good evidence that certain low-energy QFTs are \textrm{not possible}.
This is the content of the swampland conjectures. Such conjectures help to narrow down possibilities, and could reduce the the difficult job of a ``low-energy'' phenomenologist.}
\label{topdown}
\end{figure}

Quantum gravity in itself is typically believed to be located at an
extremely high energy scale. We can, however, ask whether or not a given low-energy effective field theories can be reproduced from a theory of quantum gravity.

Traditionally, this involves a specific string theory setup---this can be heterotic string theory on Calabi-Yau three-fold, F-theory on elliptically-fibered Calabi-Yau four-fold, M-theory on $G_2$ holonomy manifolds, etc.
By contrast we are asking a more {\it universal} question, at least in spirit: does the low-energy effective field theory
have a UV completion in quantum gravity? If the answer is yes, we say that the theory is in the \textrm{landscape}, while if no
we say that the theory is in the \textrm{swampland}.\cite{Vafa:2005ui,Ooguri:2006in} In this terminology (by C.~Vafa),
if you are a ``low-energy'' phenomenologist you need to make sure that your favorite theory is in the landscape, and not in the swampland.

\begin{figure}[htbp]
\centering\includegraphics[scale=0.45]{swampland_map.eps}
\caption{Inside the space of possible low-energy QFTs, some are realized by specific string theory constructions,
such as Calabi-Yau three-fold compactifications of the Heterotic theory. The consequences from these specific constructions,
may well be setup-dependent, and might not be obeyed in other string theory constructions. In the swampland conjectures,
by contrast, one tries to formulate \textit{universal} consequences from the existence of UV completion with gravity.
If such conjectures are true, that has some interesting implications to low-energy model building.
There are indications that the constraints from swampland conjectures are so strong that
``landscape'' is actually tiny inside the space of all possible low-energy QFTs.}
\end{figure}

One might ask ``isn't quantum gravity notoriously difficult''? For proper treatment of quantum gravity,
you have to wait for your quantum gravity colleagues to solve it, and as of this writing there is no indication
that this will happen in anytime soon. Does this mean that you can forget about this altogether?

Despite the lack of full understanding of quantum gravity, we do have some useful clues about quantum gravity.
First, we know about semiclassical description of black holes, such as that black holes have Bekenstein-Hawking entropy,
that black holes evaporate with Hawking radiation.\cite{Hawking:1974sw}
We can devise many gedanken-experiments on
black holes, and think about the consequences, loopholes, and so on.
Second, there are many examples/data from string theory,\footnote{String theory is a rather promising theory of quantum gravity, and makes it possible to do many quantitative computations, thus providing invaluable data for swampland constraints. For this reason we often use string theory in swampland discussions. One should note, however, that the arguments from semiclassical black holes, as mentioned previously, are independent of string theory. Consistency between black-hole-inspired conjectures and string-theory-inspired conjectures have also been discussed.} e.g.\ results based on
a specific class of compactification manifolds.

The way to proceed is look at various data available, and formulate \emph{swampland conjectures},
a set of necessary conditions for the existence of UV completion. Once we formulate these conjectures,
we can set these conjectures by looking many more examples, this often leads to quantitative tests.
We can also try to verify the mutual consistency between different conjectures.

There are many swampland conjectures in the market, indeed too many to be listed here and readers are referred to comprehensive reviews.\cite{Brennan:2017rbf,Palti:2019pca} Here, to illustrate the idea let me mention one particular conjecture by M.~Reece \cite{Reece:2018zvv} (see Hebecker et al.\cite{Hebecker:2017uix} for a closely related work).

Consider a $U(1)$ gauge field $A$, namely a photon, with
Su\"uckelberg mass $m$.\footnote{This is a gauge-invariant mass of the form $\mathcal{L}\supset m^2 (A_{\mu}- \partial_{\mu} \theta)^2$ for a scalar field $\theta$, with gauge transformation $\delta A_{\mu}=\partial_{\mu} \chi, \delta\theta=\chi$.}
Let us denote the gauge coupling constant by $e$.
Then the conjecture states that  the UV-cutoff $\Lambda_{\rm UV}$ of the theory
should satisfy the inequality
\begin{equation}
\Lambda_{\rm UV} \lesssim \textrm{min} \left( (m M_{\rm Pl}/e)^{\frac{1}{2}},\,  e^{\frac{1}{3}} M_{\rm Pl} \right) \;.
\label{Lambda.conjecture}
\end{equation}
This places interesting constraints on light dark photons,\cite{Reece:2018zvv}; note scenarios involving such light dark photons were discussed during the Moriond conference.\footnote{See, however, Craig et al.\cite{Craig:2018yld} for discussion on possible loopholes.}

This conjecture is motivated by many other swampland conjectures, in particular the distance conjecture\cite{Ooguri:2006in,Klaewer:2016kiy}
and the weak gravity conjecture\cite{ArkaniHamed:2006dz}, where the latter
is an upgrade of the old conjecture that there is no global symmetry in quantum gravity.\cite{Misner:1957mt,Polchinski:2003bq,Banks:2010zn,Harlow:2018tng}
In this respect, the conjecture \eqref{Lambda.conjecture} is an excellent example for an outcome
from interrelations between several different approaches to the swampland program.

\section{De Sitter Swampland Conjecture}

While the swampland program by now has history of more than $10$ years,
recently there has been renewal of interest in this topic.  One of the reasons for this
is the paper\cite{Obied:2018sgi}  on the ``de Sitter swampland conjecture'' (hereafter dS conjecture)  in June 2018. This paper immediately generated interest and  and controversies in the community.

The dS conjecture states that the total scalar potential $V$ of any effective field theory\footnote{We choose canonical normalization for scalar fields. We assume there are only finitely many scalar fields in the Lagrangian.}
satisfies an inequality
\begin{align}
M_{\rm Pl}\,  | \nabla V | \ge c \,V \;.
\label{dS}
\end{align}
Here $M_{\rm Pl}\simeq 2\times 10^{18} \textrm{GeV}$ is the Planck scale,
$c$ is an $O(1)$ positive constant.

After the proposal of the conjecture, problems of this conjecture has been pointed out---the dS conjecture excludes local maximum with positive energy
\begin{align}
\nabla V=0 \;, \quad \nabla^2 V<0 \;, \quad V>0 \;.
\end{align}
Such local maximum, however, clearly exists in the potential of the Higgs field,
and of the axion\cite{Peccei:1977ur,Weinberg:1977ma,Wilczek:1977pj}
(if an axion exists).

This in itself does not exclude the dS conjecture, since one can think of various loopholes---for example, we can
enlarge the field space by coupling the Higgs/axion to the quintessence field.
However, after several independent analysis (involving many fun ingredients,
such as no-go theorem for electroweak modification, fifth-force constraints, etc.), one comes to the conclusion that most such loopholes can be eliminated, except for some rather exotic scenarios.\cite{Denef:2018etk,Murayama:2018lie,Choi:2018rze,Hamaguchi:2018vtv}

In view of these results, a number of refinements of the dS conjecture have been proposed.\cite{Andriot:2018wzk,Garg:2018reu,Murayama:2018lie,Ooguri:2018wrx,Garg:2018zdg,Andriot:2018mav}
For example, one possible modification\cite{Garg:2018zdg,Ooguri:2018wrx} states that
\begin{align}
M_{\rm Pl} | \nabla V | \ge c \, V \quad \textrm{or } \quad \textrm{min}(\nabla^2 V) \le -c' \, V \;,
\label{RdS}
\end{align}
where $c$ and $c'$ are $O(1)$ positive constants. This is slightly stronger than the another conjecture,\cite{Murayama:2018lie}
which corresponds to the special value $c'=0$ of the conjecture above. We here call these conjectures
the refined de Sitter conjectures (hereafter RdS conjectures).

While the RdS conjectures evades the problems with the local maxima of the Higgs or axion potential,
they still have an important consequence---the conjectures forbid the stable de Sitter vacua:
\begin{align}
\nabla V=0 \quad \;, \quad \nabla^2 V > 0 \;,  \quad V>0 \;.
\end{align}

This conjecture is therefore tension with the literature (e.g.\ KKLT scenario\cite{Kachru:2003aw}) for de Sitter vacua in string theory (see also recent articles\cite{Sethi:2017phn,Akrami:2018ylq,Kachru:2018aqn}). It seems fair to say that the RdS conjectures are speculative, at least as general statements, and might hold only in the asymptotic region of the moduli space where we have parametric control.\footnote{It would be interesting to fully justify the RdS conjectures even in such weakly-coupled regions of the parameter space (there has recently been attempts in this direction\cite{Ooguri:2018wrx,Hebecker:2018vxz}; note that the classic argument by Dine and Seiberg\cite{Dine:1985he} in itself
does not automatically guarantee this). As we emphasized before, the point of the swampland conjectures
is to work out the consequences which holds irrespective of specific string theory setups, and even the ``weakly-coupled RdS conjecture'' is universal in this sense.} In the following we will assume the RdS conjectures and
work out the consequences.
This will lead us to interesting scenarios,
and in my opinion it is of value to explore these possibilities, irrespective of the validity of the RdS conjectures---ultimately the finally verdict is up to the Nature.

\subsection{Dark Energy}

If RdS conjectures are correct, dark energy cannot be the cosmological constant.
One possibility is then to consider a dynamical scalar field, the quintessence.\cite{Ratra:1987rm,Wetterich:1987fm,Zlatev:1998tr}

There are enormous challenges in quintessence model building.
First of all, the quintessence potential should be
extremely flat, to avoid rapid change of the size of the dark energy.
Second, the potential in itself should have a correct size to
explain the present-day energy scale of dark energy $\Lambda^4 \simeq  10^{-120} M_{\rm Pl}^4  \ll M_{\rm Pl}^4$.

In our recent paper\cite{Ibe:2018ffn}, we looked into the possibility of the
electroweak quintessence axion, where the quintessence is
the axion field\cite{Fukugita:1994hq,Frieman:1995pm,Choi:1999xn}
for the electroweak $SU(2)$ gauge group.\cite{Fukugita:1994hq,Nomura:2000yk,McLerran:2012mm}\footnote{See e.g.\ \cite{Agrawal:2018own,Heisenberg:2018yae,Cicoli:2018kdo,Akrami:2018ylq,Heisenberg:2018rdu,Marsh:2018kub,DAmico:2018mnx,Han:2018yrk,Brandenberger:2018xnf,Yang:2018xah,Olguin-Tejo:2018pfq,Agrawal:2018rcg,Heckman:2018mxl,Chiang:2018lqx,Thompson:2018ifr,Elizalde:2018dvw,Ibe:2018ffn,Tosone:2018qei,Emelin:2018igk,Acharya:2018deu,Hertzberg:2018suv,Raveri:2018ddi,Heckman:2019dsj,Heisenberg:2019qxz,Brahma:2019kch,Kaloper:2019lpl}
for other recent discussion of quintessence and swampland conjectures.}

If such an electroweak axion exists, it can explain the flatness of the potential
since the shift symmetry of the axion is broken only by non-perturbative instanton effects.
Moreover, the energy scale of the axion potential is given by the dynamical scale,
which is estimated to be\cite{Nomura:2000yk,McLerran:2012mm}
\begin{align}
\Lambda^4&=M_{\rm Pl}^4 \, e^{-\frac{2\pi}{\alpha_2(M_{\rm Pl})}} \simeq 10^{-130} M_{\rm Pl}^4  \;,
\label{Lambda_a}
\end{align}
where we use the value of the electroweak coupling constant $\alpha_2=g_2^2/(4 \pi)$ at the Planck scale.
This is very close to the present-day scale of the dark energy. Since the axions are often generated by string theory, one hope
such electroweak quintessence axions arise from string theory, and further motivations for the existence of such axion are provided by the RdS conjectures.\cite{Ibe:2018ffn,Dvali:2018dce}

Of course, irrespective of the validity of the RdS conjectures,
the existence of the quintessence can be probed by observation,
from the measurement of the equation-of-state parameter $w=p/\rho$
for various values of redshift. While the current results show no sign of deviation from
$w=-1$,\cite{Aghanim:2018eyx} it is important to continue observational searches in higher precision.

\subsection{Inflation}

What are the consequences of the RdS conjecture for the early universe, namely inflation?

If we insist on the two $O(1)$ parameters $c, c'$ in Eq.~\eqref{RdS} to be $c,c'\sim 1$
(and not $c,c'\sim 0.01$, for example),
then we found\cite{Fukuda:2018haz} that the simplest type of inflationary models,
namely single-field inflationary models with canonical kinetic terms,
can generate inflation with sufficient number of e-foldings, but
have trouble reproducing the observed value\cite{Akrami:2018odb} of the spectral index.\footnote{Note that
this conclusion is different from earlier papers (see e.g.\  \cite{Achucarro:2018vey,Kehagias:2018uem,Matsui:2018bsy,Garg:2018reu,Ben-Dayan:2018mhe,Kinney:2018nny,Brahma:2018hrd}) which discussed the consequences of the condition \eqref{dS} (as opposed to the refined version \eqref{RdS})---under \eqref{dS}
single-field inflationary models with canonical kinetic terms have trouble generating a sufficient number of e-foldings.}
The problem can be evaded in multi-field inflationary models,
e.g.\ by the curvaton scenario.\cite{Linde:1996gt,Enqvist:2001zp,Lyth:2001nq,Moroi:2001ct}
This could lead to interesting signatures, such as primordial non-Gaussianities,\cite{Kawasaki:2011pd}
which can be tested in future improved observations of the
cosmic microwave background radiation.

\section{Summary}

In this paper
we have introduced the swampland program, which is an attempt
to extract {\it universal} consequences of quantum gravity
in low-energy effective field theory. The swampland conjectures, if correct, have
many implications to physics in vastly different energy scales,
ranging from the energy scale of dark energy all the way up to the
energy scale of inflation. Conversely, we have presented examples where
the bottom-up phenomenological constraints
affect the on-going discussion about
swampland conjectures.

\begin{figure}[htbp]
\centering\includegraphics[scale=0.5]{swampland_scale.eps}
\caption{While quantum gravity in itself might be associated with very high energy scale (such as the Planck scale),
swampland conjectures originating from quantum gravity have implications phenomena in vastly different energy scales,
as explained in this paper. }
\end{figure}

In this sense, the search for quantum gravity
is tied with the study of ``low-energy physics'', either in theory and experiment (as expressed in Fig.~\ref{topdown}).
In order to ``make sure that no stones are left unturned''\footnote{Comment from
the theory summary talk by Gudrun Hiller in Moriond EW 2019.},
swampland program is a useful approach to keep in mind
for any particle phenomenologist and cosmologist.

\section*{Acknowledgments}

MY would like to thank the organizers of the 54th Rencontres de Moriond
for invitation and for providing stimulating atmosphere. He is partially supported by WPI program (MEXT, Japan)
and by JSPS KAKENHI Grant No.\ 17KK0087, No.\ 19K03820 and No.\ 19H00689.

\bibliographystyle{nb}


\section{Introduction}
\label{sec:intro}
If globular clusters follow the same ``$M_{BH}-\sigma$'' relation
between central black hole mass and effective velocity dispersion as
is observed in galaxies (Tremaine et al. 2002), many clusters should
host intermediate-mass black holes (IMBHs) with masses in the range
$10^2 \ltorder M_{BH}/M_{\odot} \ltorder 10^4$.  Unfortunately, hard
observational evidence for IMBHs in globular clusters has proved
elusive.  Currently the strongest case is the massive cluster G1 in
M31, where dynamical, X-ray, and radio studies are consistent with a
mass $M_{BH}\sim2\times10^4~$ (Gebhardt et al. 2005, Ulvestad et
al. 2007). In our own Galaxy, a measurement of an IMBH mass $M_{BH}
\sim 4\times10^4 M_{\odot}$ has been reported for $\omega$ Centauri
(Noyola et al. 2008), although this result has been questioned in a
recent study (Anderson \& van der Marel 2010, van der Marel \&
Anderson 2010) suggesting that the IMBH signature becomes much weaker
when a more accurate estimate of the cluster center is used.  Recently
Ibata et al. (2009) have reported evidence of a density and kinematic
cusp in the core of M54, a cluster located at the center of the
Sagittarius dwarf galaxy; their models suggest that the cusp could be
due to a $\sim 10^4 M_{\odot}$ IMBH.

For most Galactic globular clusters, kinematic and structural data
currently provide only upper limits on possible IMBH masses
(McLaughlin et al. 2006, Pasquato et al. 2009, van der Marel \&
Anderson 2009). However, these upper limits are larger than the values
of $M_{BH}$ expected from the $M_{BH}-\sigma$ relation, and do not
exclude the possibility that many globular clusters host IMBHs with
masses in the above range (van der Marel \& Anderson 2009).

The presence of an IMBH in a globular cluster has important
consequences for the cluster's structure, kinematics, internal
energetics and long-term dynamical evolution (see e.g. Baumgardt et
al. 2004, Heggie et al. 2007, Trenti et al. 2007, Miocchi 2007;
see also McMillan 2008 for a review).  In
addition, Eddington accretion onto IMBHs has been proposed as the
mechanism powering some ultraluminous X-ray sources (e.g. Farrell et
al. 2009), and IMBHs have long been recognized as potential rich
sources of gravitational waves detectable by LISA (Miller 2009). For
these reasons, despite the lack of strong observational constraints,
the formation and consequent properties of IMBHs have become a topic of
considerable theoretical and observational interest.

One possible IMBH formation mechanism is direct collapse of a very
massive ($\geq 250 M_{\odot}$) Population III star (see e.g. Bond et
al. 1984, Madau \& Rees 2001 and references therein; but see Abel et
al. 2002 for hydrodynamical simulations suggesting that these stars
would form in isolation).
For non-primordial or more massive IMBHs, the leading formation
mechanisms center on dynamical processes in dense star clusters.
These include runaway mergers of massive stars in clusters with high
central densities (Portegies Zwart \& McMillan 2002, G\"urkan et
al. 2004, Portegies Zwart et al. 2004), and merging of black holes in
binaries (see O'Leary et al. 2006 and references therein).  However,
numerous potential problems affecting these models have been pointed
out in the recent literature, suggesting that the net growth rate and
hence the final mass of the resulting IMBH may be much lower than
suggested by earlier dynamical simulations (Yungelson et al. 2008,
Glebbeek et al. 2009).

Thus, while there appear to be numerous plausible stellar, binary, and
dynamical pathways to the formation of relatively ``low-mass''
($\sim100-200 M_{\odot}$) IMBHs in clusters, there is currently no
clear consensus on a mechanism for the production of high-mass
($10^3-10^4 M_{\odot}$) IMBHs.

In this Letter we propose a new scenario for the growth of IMBHs in
early globular clusters.  It is a natural consequence of the physical
conditions that may have obtained in cluster cores during the
formation of the second generation (hereafter SG) stars now observed
in many globular clusters.  Specifically, we show that accretion by a
pre-existing seed black hole of just a small fraction of the gas from
which the SG stars formed can lead to black holes in the high-mass
($10^3-10^4 M_{\odot}$) IMBH range.

 The structure of this Letter is as follows.
In Section \ref{sec:multip} we briefly review the observational
evidence for multiple stellar populations in globular clusters, and
summarize the main elements of our model for the formation and
evolution of multiple populations.  In Section \ref{sec:bh}, we
present our scenario for IMBH growth.  We discuss some consequences of
this scenario in Section \ref{sec:concl}.

\section{Multiple stellar populations in globular clusters}
\label{sec:multip}
Spectroscopic and photometric observations provide strong evidence
for multiple stellar generations in globular clusters.

Spectroscopic studies have shown that, even in clusters whose stars
have a common Fe abundance, there are substantial star-to-star
differences in the abundances of lighter elements (see Gratton et
al. 2004 and references therein, and Carretta et al. 2009a,b for a
recent spectroscopic study of 19 Galactic clusters).  All the Galactic
globular clusters for which spectroscopic studies have been carried
out show evidence of multiple stellar populations, and in all cases SG
stars make up a significant fraction ($50-80\%$) of the total
(D'Antona \& Caloi 2008, Carretta et al. 2009a,b).

Several recent photometric studies have provided further support for
these results by showing the photometric fingerprints of multiple
populations (multiple main sequences, significant main sequence
broadening, or subgiant branch splits) in the color-magnitude diagrams
both of ``normal'' clusters in the Galaxy (Piotto et al. 2007, Marino
et al. 2008, Milone et al. 2008, Anderson et al. 2009), and of more
complex systems such as $\omega$ Cen (Lee et al. 1999, Pancino et
al. 2000, Bedin et al. 2004, Villanova et al. 2007), M22 (Marino et
al. 2009, Lee et al. 2009) and probably Terzan 5 (Ferraro et
al. 2009).

The differences in light-element abundances suggest that SG stars
formed out of gas containing matter processed through high-temperature
CNO cycle reactions in first-generation (FG) stars.  The main
candidates currently suggested as possible sources of gas for SG
formation are massive AGB stars (Ventura et al. 2001) and rapidly
rotating massive stars (Decressin et al. 2007). Although many aspects
of these two scenarios are still under investigation, in order to form
the large mass of SG stars suggested by observations, both scenarios
require that either the IMF of the FG stars was highly anomalous, with
an unusually large fraction of massive stars, or the FG population had
a normal IMF but was initially at least ten times more massive than is
now observed.

We have recently carried out hydrodynamical and N-body simulations to
explore the formation and dynamical evolution of multiple population
clusters formed out of AGB star ejecta (D'Ercole et al. 2008).  Our
simulations show that the gas lost by FG AGB stars collects in a
cooling flow in the cluster core, where most SG stars are formed.  We
have further demonstrated a dynamical mechanism whereby, after the
centrally concentrated SG stars have formed, the
cluster's response to early mass loss (of intracluster gas and/or
supernova ejecta) leads to the escape of a substantial fraction of FG
stars.  During the subsequent long-term evolution, the SG and FG
populations mix by two-body relaxation, plausibly leading to the
multiple-population clusters observed today.

We now explore the implications for the growth of a pre-existing seed
black hole of the physical conditions in the central regions of the
cluster during the SG formation phase.

\section{Growth of a seed black hole}
\label{sec:bh}

In the hydrodynamical simulations presented in D'Ercole et al. (2008),
we assumed that the cluster FG stars followed a King (1962) density
profile
\begin{equation}
    \rho=\rho_0 \left[1+\left({r \over r_c}\right)^2\right]^{-3/2}
\end{equation}
out to a truncation radius $R_t$, with stellar masses distributed in
the range $0.1<m/m_{\odot}<100$ according to a ``Kroupa'' initial mass function (Kroupa
et al. 1993).  We adopted a concentration
$c=\log(R_t/r_c)=1.5$ and followed the evolution of the AGB ejecta in
clusters having different values of the total FG mass $M$ and
truncation radius $R_t$.

Figure 1 shows the time evolution of the AGB ejecta density and sound
speed within 0.1 pc of the cluster center during the SG formation
phase, for four clusters spanning broad ranges in initial mass and
tidal radius.  In all cases, if the cluster already hosts a seed black
hole with mass $M_{BH} \sim 10^2~M_{\odot}$---the result of one of the
``prompt'' low-mass IMBH formation mechanisms discussed previously,
operating in the FG population---the gas conditions shown in Figure 1
will lead to accretion.  The Bondi accretion rate onto a black hole of
mass $M_{BH}$ is
\begin{eqnarray}
\dot M &\simeq& 3 \times 10^{-4} \left(M_{BH}\over 100
       M_{\odot}\right)^2\left(\rho\over 10^{-18} \hbox{g
       cm}^{-3}\right) \nonumber\\
       &~&~~~~~~~~~~~~~~~~~~~~~
       \times \left({3 \hbox{km s}^{-1}\over c_s}\right)^3
       ~\hbox{M}_{\odot}\,\hbox{yr}^{-1}
\end{eqnarray}
where the density $\rho$ and the sound speed $c_s$ have been
normalized to the typical values found in our simulations and shown in
Figure 1. For the physical conditions in our four simulations, the Bondi
accretion rate is $\sim 10^{-4}-10^{-3} ~\hbox{M}_{\odot}/\hbox{yr}$,
much larger than the Eddington rate,$~~$
$\sim3\times10^{-9}\,(M_{BH}/M_{\odot})/\eta\,\,\hbox{M}_{\odot}$/yr (Krolik
1999), where $\eta$ is the radiative efficiency, to be discussed
further below.  The black hole will therefore accrete gas at the
Eddington rate.

\begin{figure}
\begin{center}
 \includegraphics[width=3.5in]{f1.eps}
 \caption{Time evolution of the AGB ejecta density (lower panel) and
   sound speed (upper panel) within 0.1 pc of the cluster center
   during the SG formation phase for a cluster with FG total mass $M$
   and tidal radius $R_t$ equal to $(10^7~M_{\odot}, 200~pc)$ (solid
   line), $(10^7~M_{\odot}, 40~pc)$ (dashed line), $(10^6~M_{\odot},
   90~pc)$ (long-dashed line), $(10^6~M_{\odot}, 18~pc)$ (dot-dashed
   line).}
\end{center}
\end{figure}

Under these circumstances, the black hole mass will grow on a Salpeter
time scale
\begin{equation}
    T_S \simeq 3.9\times 10^8 \,\eta\, L_E/L ~\hbox{yr},
\end{equation}
where $L_E \sim 1.45 \times 10^{38} M_{BH}/M_{\odot}$ erg/s is the
Eddington luminosity and a value of the mass per electron $\mu_e\sim
1.15$ has been adopted (see e.g. Krolik 1999).  The value of $\eta$ is
uncertain and depends on the properties of the accretion disk and on
the spin of the accreting black hole.  Here we adopt $\eta\sim
0.05-0.1$, a range commonly adopted in studies of black hole growth in
galactic nuclei.  We note that values of $\eta$ larger than 0.1,
appropriate for rapidly rotating black holes, would reduce the growth
of the IMBH, while much smaller values ($\eta \ll 1$ ) typical of
thick accretion disks would significantly increase the IMBH growth
rate.  For accretion at the Eddington rate with $\eta\sim 0.05-0.1$,
$T_S\sim 20-40$ Myr.

The duration of the SG star formation episode in the models of
D'Ercole et al. (2008) is $\Delta T \approx 10^2$ Myr.  This time
scale is constrained by spectroscopic observations showing little
variation in the total CNO of FG and SG stars, implying that only AGB
progenitors with $M>4-5 M_{\odot}$ must have contributed gas to the SG
formation process.  Thus, if the seed black hole accretes at close to
the Eddington rate during this time, its mass will grow by a factor of
$\sim 10-100$, leading, by the end of the SG formation phase, to a
final black hole mass of $M_{BH} \sim 10^3-10^4~M_{\odot}$.
$~~$\\

\section{Discussion}
\label{sec:concl}

Several aspects of the scenario just presented merit further
discussion and exploration.

We consider first the effect on the cooling flow of the energy
radiated by the accreting black hole.
To address this issue we follow
the analysis of black hole outflow presented by King (2003, 2005), in
which it is shown that the wind produced by the black hole sweeps up
the surrounding gas into an expanding shell.  Because of effective
 energy losses due to inverse Compton
scattering, the shell expands in the momentum-conserving regime.
Under these conditions (see King
2003, 2005), in order for the momentum flux from a black hole accreting at
the Eddington rate to overcome gravity and expel the shell from the
cluster, the black hole mass must be larger than
\begin{equation}
    M_{BH,crit} = \frac{f_g\kappa\sigma^4}{\pi G^2}.
\end{equation}
Here, $\kappa$ is the electron scattering opacity, $f_g$ is the
fraction of the total cluster mass in the form of gas, and it has been
assumed that the gas is embedded in an isothermal system with velocity
dispersion $\sigma$.

Assuming $\sigma \gtorder 20~\hbox{km/s}$ (the smallest value for
which the AGB ejecta can be retained) and $f_g\approx0.05$ (D'Ercole
et al. 2008), we find $M_{BH,crit} \gtorder 10^4 M_{\odot}$, much
larger than the assumed mass of our initial seed black hole.  Thus
feedback from the accreting black hole is {\em not} sufficient to
alter significantly the global dynamics of the cooling flow or the SG
star formation process.

On the other hand, the effects of feedback may significantly alter the
local gas dynamics in the vicinity of the black hole.  For example,
recent detailed 2-D hydrodynamical models of accretion onto a 100
$M_{\odot}$ seed black hole in a dense protogalactic cloud
(Milosavljevic et al. 2009) show intermittent accretion, reducing the
net accretion rate to about $1/3$ of Eddington.  Such a reduction in
the mean accretion rate in our case would reduce the growth of the
seed black hole to a factor $\sim2-5$ for the range of $\eta$ values
adopted in \S\ref{sec:bh}.

We note here that during the accretion phase the black hole will be
very luminous and might be observed as a (possibly intermittent)
ultraluminous X-ray source (ULX).  We caution however that, before
linking this scenario to the ULXs observed in nearby young massive
clusters (see e.g. Portegies Zwart et al. 2010), one must first verify
whether these clusters (1) meet the conditions to form a seed black
hole, (2) are sufficiently massive and (3) have the structural properties
required for SG star formation.

A second important issue is the possible relation between the IMBH
mass and the current structural properties of the parent cluster.

Any correlation between the mass of the seed black hole and the early
properties of the cluster should be preserved during the Eddington
growth phase, provided that the duration $\Delta T$ of the SG star
formation episode is approximately independent of the cluster
environment. The mass of the seed black hole depends on its formation
mechanism.  If it is the result of stellar evolution in a massive
Pop. III progenitor, a roughly universal mass seems the most likely
result (see e.g. Madau \& Rees 2001).
Early dynamical simulations of the runaway merger scenario
suggested that the mass of the runaway should scale with the mass of
the cluster (or the cluster core) in which it forms (G\"urkan et
al. 2004, Portegies Zwart et al. 2004).  However, as mentioned in
\S\ref{sec:intro}, strong stellar winds may well limit the actual mass
attained, possibly to as little as a few hundred solar masses, again
largely independent of the initial cluster mass.

For the black hole growth phase to occur, the cluster must {\em
  initially} have been massive and/or compact enough to retain the AGB
ejecta that subsequently flowed into the cluster core and formed the
SG population.  As discussed in D'Ercole et al. (2008), multiple
population clusters must have undergone an early phase of strong FG
mass loss and structural evolution; it is therefore not
straightforward to connect the initial properties of a cluster during
the growth of the seed black hole with the current properties of the
cluster hosting an IMBH.  However, given the loss of most of the FG
population and the possible near-universality of the seed black hole
mass, the most likely outcome of the scenario described here seems to
be one in which the present IMBH mass has only a weak dependence on
current cluster properties.
$~~~~$\\
{\bf Acknowledgments.}
EV and SM were supported in part by NASA grants NNX07AG95G,
NNX08AH15G, and NNX10AD86G and by NSF grant AST-0708299.
AD acknowledges financial support from italian MIUR through grant PRIN 2007
(prot. 2007JJC53X). FD was  supported by PRIN MIUR 2007 'Multiple
Stellar Populations in Globular Clusters: Census, Characterization
and Origin' (prot. n. 20075TP5K9).