aainstitutetext: Rudolf Peierls Centre for Theoretical Physics
Beecroft Building, Clarendon Laboratory, Parks Road, University of Oxford, OX1
3PU, UK

# Catch-Me-If-You-Can: The Overshoot Problem and the Weak/Inflation Hierarchy

Joseph P. Conlon a Filippo Revello<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the overshoot problem in the context of post-inflationary string
cosmology (in particular LVS). LVS cosmology features a long kination epoch as
the volume modulus rolls down the exponential slope towards the final minimum,
with an energy density that scales as $m_{s}^{4}$. It is a known fact that
such a roll admits attractor tracker solutions, and if these are located the
overshoot problem is solved. We show that, provided a sufficiently large
hierarchy exists between the inflationary scale and the weak scale, this will
always occur in LVS as initial seed radiation grows into the tracker solution.
The consistency requirement of ending in a stable vacuum containing the weak
hierarchy therefore gives a preference for high inflationary scales – an
anthropic argument, if one likes, for a large inflation/weak hierarchy. We
discuss various origins, both universal and model-dependent, of the initial
seed radiation (or matter). One particularly interesting case is that of a
fundamental string network arising from brane inflation – this may lead to an
early epoch in which the universe energy density principally consists of
gravitational waves, while an LVS fundamental string network survives into the
present universe.

##  1 Introduction and the Overshoot Problem

Inflation is (justly) widely believed to be the physics underlying the origin
of structure in the universe, through quantum fluctuations in a primordial
inflaton field during a quasi-de Sitter phase in the early universe. While the
precise energy scales involved in inflation are not known, in most models
these are high. $V_{inf}\sim\left(10^{15}{\rm GeV}\right)^{4}$ is typical,
while even larger values arise in high-scale inflation models associated to
observable levels of primordial gravitational waves. For a review of the
physics of inflation, see BaumannBook . We shall assume without further
comment that the universe did indeed go through an inflationary phase in its
earliest moment.

Whatever the universe was, today its vacuum state involves an electroweak
symmetry broken at a characteristic energy scale $E_{weak}\sim 100\,{\rm
GeV}$. Despite the discovery of the Higgs boson in 2012, the origin of this
scale is not understood. It appears unstable to quantum corrections, requiring
either fine tuning or the appearance of additional sectors or symmetries (for
example, supersymmetry) to control radiative corrections and maintain a Higgs
mass and vev sixteen orders of magnitude below the Planck scale.

String theory is our leading candidate for a fundamental theory of nature that
includes quantum treatments of both gauge and gravitational interactions. If
string theory is the theory of _this_ world, it must account for a
cosmological evolution that started with an inflationary phase and ends in a
vacuum state today characterised by $E_{weak}\ll E_{inflation}$ which also
includes the many other hierarchies of particle physics (for example, the
Yukawa couplings).

To even talk about string vacua, we are required to confront the physics of
moduli stabilisation and construct models which both stabilise moduli and
exhibit a stable (ideally de Sitter) minimum. Although many aspects of our
discussion extend more generally, here we will mostly consider type IIB
compactifications for which the most developed models are KKLT and LVS kklt ;
bbcq ; cqs . However, all string theories have a vanishing energy in the
decompactification limit ($\mathcal{V}\to\infty$ or $g_{s}\to 0$). String
models therefore have a natural tendency to exhibit runaway behaviour from the
interior of moduli space towards the boundary, and so all vacua with
(effectively) zero vacuum energy will contain a barrier height to
decompactification.

Although precise details are model-dependent, the barrier height is normally
much smaller than the inflationary energy scales (as the barrier is associated
to the vacuum and _not_ the physics during inflation). In a typical model with
$m_{3/2}\sim 100\,{\rm TeV}$, the barrier height may be $V_{barrier}\sim
m_{3/2}^{2}M_{P}^{2}\sim 10^{-26}M_{P}^{4}$ Kallosh:2004yh ; Blanco-
Pillado:2005arr ; Kallosh:2007wm  (or $V_{barrier}\sim m_{3/2}^{3}M_{P}$ in
the case of LVS Conlon:2008cj ). For an inflationary scale
$V_{inf}\sim\left(10^{15}{\rm GeV}\right)^{4}$, this gives

$\frac{V_{barrier}}{V_{inf}}\sim 10^{-13}.$ (1)

This leads to the _overshoot_ problem BrusteinSteinhardt  \- how can the
minimum ‘trap’ the post-inflationary theory in the vacuum? To make the problem
more graphic: suppose a ball rolls from the top of Mount Everest towards a
hole with vertical sides of one nanometer. How do we trap the ball in the
hole?

Neglecting other sources of energy density, the equations of motion for a
scalar field $\phi$ in an expanding universe are

$\displaystyle\ddot{\phi}+3H\dot{\phi}$ $\displaystyle=$
$\displaystyle-\frac{\partial V}{\partial\phi},$ (2) $\displaystyle H^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{3}\left(V(\phi)+\frac{\dot{\phi}^{2}}{2}\right).$ (3)

After inflation occurs at a scale $\Lambda_{\rm{inf}}$, the field starts
rolling down its potential. String theory potentials are naturally steep.
Potentials that are power-law in a volume are exponential in the canonical
field (e.g. LVS, with $V(\Phi)\sim\exp\left(-\sqrt{\frac{27}{2}}\Phi\right)$)
while non-perturbative potentials (as in KKLT) are double exponentials,
$V(\Phi)\sim\exp\left(-\alpha\exp\left(\beta\Phi\right)\right)$. The initial
potential energy is converted into kinetic energy of the scalar field and the
universe enters a kination phase, where $\dot{\Phi}^{2}\gg V(\Phi)$ and energy
density scales as $\rho_{\Phi}\sim a^{-6}$. If no other sources of energy
density are present, the field will approach the minimum of the potential
still in the kination phase. In this case, the field kinetic energy will be
much greater than the barrier height and the field will overshoot the barrier
and run away to the decompactification limit at infinity. This is the
_overshoot_ problem.

Although it appears obvious, we note explicitly here that in this conventional
formulation of the problem, the overshoot problem is made worse either by (a)
making the inflationary scale larger or (b) reducing the barrier height. We
emphasise this explicitly because one of the key points of this paper is that
this conventional wisdom should be inverted in LVS.

The overshoot problem has been periodically studied in the literature on
string cosmology, and solutions to it can be placed in two classes. In the
first, arguably less interesting, class the problem is solved by removing any
hierarchy between the inflationary scale and the barrier height. This can be
done either by keeping the barrier height low and associated to the physics of
the weak scale but working with models of very low-scale inflation (for
example, see German:2001tz ), or alternatively by continuing with high-scale
inflation and bringing the barrier up to a similar scale (for example, see
Kallosh:2004yh ). This latter approach is associated with models that regard
the weak scale as an accidental feature generated by fine-tuning, when there
is no _dynamical_ reason for the hierarchical smallness of the weak scale.

The second, arguably far more interesting, class of solutions uses dynamical
effects to avoid overshooting ( used, for example, in Barreiro:1998aj ;
hepth0001112 ; hepph0010102 ; hepth0408160 ; hepph0506045 ; Conlon:2008cj ).
These exploit the fact that, as $\rho_{kination}\sim a^{-6}$, almost any other
early source of energy $\rho_{extra}$ will eventually catch up with the
kinetic energy of the scalar (as e.g. $\rho_{m}\sim a^{-3}$ and
$\rho_{\gamma}\sim a^{-4}$).

$H^{2}=\frac{1}{3M_{P}^{2}}\Big{(}\rho_{extra}+\frac{1}{2}\dot{\Phi}^{2}+V(\Phi)\Big{)},$
(4)

The $\rho_{extra}$ term increases the Hubble friction to a level where it can
halt the runaway behaviour. Provided $\rho_{extra}$ catches up with the
kinetic energy before the field reaches the barrier, the modulus will settle
into the minimum and avoid both overshooting and decompactification.

In the next section (Section 2) we review the dynamics of a kination epoch
(while the field is fast-rolling down the exponential slope) in a language
adapted to string moduli with runaway exponential potentials. Whereas previous
studies have already exploited the presence of initial radiation to solve the
overshoot problem, less attention has been devoted to the possible origins of
the latter. In Section 3, we discuss some model-independent ways to generate
initial radiation in a typical compactification scenario (with LVS as a
benchmark example), along with some phenomenological implications. These are
further developed in Section 4, where we argue that that if these generic
sources of radiation are to solve the overshoot problem in LVS, a large
hierarchy must exist between the inflationary and electroweak111Taken to be
close to the scale of susy breaking. scales.

##  2 Kination

Soon after inflation ends, the field starts rolling down from the high
inflationary scale and enters the (LVS) exponential slope. It rapidly converts
potential into kinetic energy and the universe enters a period of kination
domination. In this epoch, the energy density of the Universe is dominated by
the kinetic energy of the rolling scalar and so the potential energy is, to
first approximation, irrelevant.

###  2.1 Pure Kination Epoch

In a pure kination epoch, where we neglect all other sources of energy, the
scalar field equations of motion

$\displaystyle\ddot{\phi}+3H\dot{\phi}$ $\displaystyle=$
$\displaystyle-\frac{\partial V}{\partial\phi},$ (5) $\displaystyle H^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{3M_{P}^{2}}\left(V(\phi)+\frac{\dot{\phi}^{2}}{2}\right),$
(6)

reduce to

$\displaystyle\ddot{\phi}+3H\dot{\phi}$ $\displaystyle=$ $\displaystyle 0,$
(7) $\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{6M_{P}^{2}}.$ (8)

The resulting equation for $\phi$,

$M_{P}\,\ddot{\phi}+\sqrt{\frac{3}{2}}\dot{\phi}^{2}=0,$ (9)

is solved by (denoting the initial condition $\phi(t_{0}))=\phi_{0}$.

$\phi=\phi_{0}+\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t}{t_{0}}\right).$ (10)

One other integration constant has been removed by requiring that, as
conventional, a time coordinate of $t=0$ corresponds to a (formal) initial
singularity where energy densities diverge. The initial time $t_{0}$, assumed
to be the beginning of the kination epoch, is determined by requiring the
kinetic energy to dominate over the potential one:

$\frac{M_{P}^{2}}{3t_{0}^{2}}\gtrsim V(\phi_{0})\sim\Lambda^{4}_{\rm{inf}}.$
(11)

For $t<t_{0}$, the approximations (7)-(8) are no longer justified. The scale
factor behaves as

$a(t)\propto t^{1/3},$ (12)

which follows immediately from
$H^{2}\equiv\frac{\dot{a}(t)^{2}}{a(t)^{2}}=\frac{\dot{\phi}^{2}}{6M_{P}^{2}}$.
The energy density during a kination epoch therefore drops off as

$\rho_{kination}(t)\propto\frac{1}{a(t)^{6}}.$ (13)

Note that in a kination epoch, the field evolves through approximately one
Planckian distance each Hubble time. From a stringy perspective, this is
automatically interesting: transPlanckian field excursions require a theory of
quantum gravity in order to ensure control of the effective field theory over
such large displacements. While this result is long-standing, it has received
more recent attention following the Swampland Distance Conjecture hep-
th/0605264 ; 181005506 . It automatically follows that during an extended
kination epoch lasting many Hubble times, the field must traverse a
substantially transPlanckian distance. Assuming the initial inflationary epoch
occurred towards the centre of moduli space, any long kination epoch propels
the system towards the boundary.

In general, there are control issues for field excursions $\Delta\Phi\gg
M_{P}$. Fortunately, in the LVS context we are primarily considering, the
interpretation of the field excursion is straightforward - the field $\Phi$
controls the overall volume, and the transPlanckian excursion corresponds to a
growth in the size of the extra dimensions as the field rolls towards the
decompactification limit. While, as per the distance conjecture, there is a
tower of states that becomes ‘light’ (here, the KK modes) this does not affect
control – which actually improves as the volume increases, as the scale of the
potential drops even faster and we move deeper into the supergravity limit.
While the above is true in a static limit, slightly more care should be taken
in a time dependent background, where energy is not conserved and the cutoff
can vary with time. In general, an EFT with a time dependent cutoff
$\Lambda(t)$ is still well defined if the mixing between the high and low
(with respect to the cutoff) energy modes can be neglected. Quantitatively,
this amounts to the requirement that variations in the cutoff scale be
adiabatic Baumann:2014nda , _i.e_

$\left|\frac{d\Lambda(t)}{dt}\right|\ll\Lambda(t)^{2}.$ (14)

If we take the cutoff to be the KK scale,

$\Lambda(t)=\frac{M_{P}}{\mathcal{V}_{0}^{2/3}}\left(\frac{t_{0}}{t}\right)^{2/3},\quad\quad\frac{\lvert\dot{\Lambda}\rvert}{\Lambda^{2}}=\frac{2}{3M_{P}}\left(\frac{\mathcal{V}_{0}^{2}}{t_{0}^{2}\,t}\right)^{1/3}\ll
1\quad\text{for}\quad t>t_{0},$ (15)

so that in our example (14) is satisfied at any time during the
evolution.222Already for $t=t_{0}$,
$\lvert\dot{\Lambda}/\Lambda^{2}\rvert\sim(\Lambda_{\rm{inf}}/M_{P})^{10/9}\ll
1$. Moreover, the dependence of the various mass scales on the volume is such
that there are no instances of _level-crossing_ , where states either appear
or disappear from the EFT as the cutoff varies.

Let us relate this kination epoch to the compactification parameters. The
basic kinetic term in IIB compactifications originates from

$K=-3\ln\left(T+\bar{T}\right),$ (16)

where $T=\tau+ic$ is a Kähler modulus whose real part corresponds to the
volume of a 4-cycle, $\tau\sim\mathcal{V}^{2/3}$, where $\mathcal{V}$ is the
overall Calabi-Yau volume, and all volumes are measured in appropriate powers
of $l_{s}=2\pi\sqrt{\alpha^{\prime}}$. The imaginary part $c$ is an axion from
an RR 4-form. The canonical field $\Phi$ is

$\Phi=M_{P}\sqrt{\frac{3}{2}}\ln\tau\equiv
M_{P}\sqrt{\frac{2}{3}}\ln\mathcal{V},$ (17)

It follows that for a field excursion
$\Delta\Phi=\langle\Phi\rangle-\langle\Phi_{0}\rangle$, we can write

$\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t}{t_{0}}\right)=\Delta\Phi=\langle\Phi\rangle-\langle\Phi_{0}\rangle=\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{\mathcal{V}}{\mathcal{V}_{0}}\right).$
(18)

In this stringy kination epoch, it follows that the evolution of the internal
compactification volume is linear in cosmic time,

$\mathcal{V}=\mathcal{V}_{0}\left(\frac{t}{t_{0}}\right).$ (19)

In the context of LVS, as the terms contributing to the potential energy in
LVS all behave as $V\sim\mathcal{V}^{-3}$, this shows that during the
kination-dominated roll down the exponential slope, the magnitude of the LVS
potential energy behaves as

$V_{LVS,kination}(t)\sim t^{-3}.$ (20)

We note that (as must be the case) this also follows from using the
exponential potential $V\propto\exp\left(-\sqrt{\frac{27}{2}}\Phi\right)$ and
the time-evolution of the canonical field $\Phi$,

$V_{LVS}(\Phi)\sim\exp\left(-3\ln t\right)\sim t^{-3}.$ (21)

As the overall energy density $V_{total}\sim t^{-2}$, the potential energy in
this epoch is therefore suppressed compared to the overall energy by a factor
$\frac{t_{0}}{t}$. This self-consistently justifies our decision to drop the
potential in the analysis of the kination epoch. As a last important point,
let us notice that since the string scale is $m_{s}\sim
M_{P}/\sqrt{\mathcal{V}}$,

$\rho_{\rm{kin}}=\frac{M_{P}^{2}}{3t^{2}}\sim m_{s}^{4}$ (22)

for the whole of the kination epoch.

The appearance of such a high energy density appears striking as it appears to
raise potential questions about the validity of the EFT (given the overall
energy density is higher than $m_{KK}^{4}$, and KK modes are excluded from the
4d effective field theory). However, we note that the relevant quantity
controlling the validity of EFTs in a cosmological setting is normally the
Hubble scale ($H\sim\left(\rho/M_{P}^{2}\right)^{1/2}$ and not simply
$\rho^{1/4}$), as this relates to the scales of actual dynamical processes
within the cosmological background (we are not studying, for example, particle
scattering at energies $E>M_{KK}$). Indeed, energy densities comparable to (or
larger than) $\rho\sim m_{s}^{4}$ are common in various well-studied scenarios
of stringy inflation, for example axion monodromy inflation Silverstein:2008sg
; McAllister:2008hb  or brane inflation Dvali:1998pa ; Burgess:2001fx ;
Dvali:2001fw ; Sarangi:2002yt ; Kachru:2003sx ; Baumann:2007ah . In our
example, the Hubble and cutoff scales (which we take to be the KK scale) are
proportional to

$H(t)\sim\frac{m_{s}^{2}}{M_{P}}\sim\frac{1}{t},\quad\quad\text{and}\quad\quad
m_{KK}(t)\sim\frac{1}{t^{2/3}},$ (23)

and so $m_{KK}(t)\gg H(t)$, suggesting that the EFT should be under good
control.

Nonetheless, this argument notwithstanding, we do note that with any energy
density greater than the KK scale, one can reasonably be concerned that there
could exist some instability or way in which the KK (or string) modes manifest
themselves and either modify or lead to novel phenomena inside the 4d low-
energy effective field theory. Any such effects would have the potential to
invalidate the standard understanding of reheating in string cosmology (both
in this scenario, and also in other scenarios such as brane inflation or axion
monodromy).

As well as these possible downsides, given the universal coupling of the
volume, it is also tempting to speculate whether the presence of such a high
energy density might be able to excite a (very small number) of KK or stringy
modes, and whether this could carry any observable consequences. A possible
mechanism could involve cosmological particle production in a time-varying
background, but we leave such an investigation to future work.

###  2.2 The Tracker Solution

We here review the well-known existence of an attractor tracker solution for
exponential potentials Wetterich:1987fm ; Copeland:1997et ; Ferreira:1997hj ,
which implies that, in the presence of sufficient initial radiation (or
matter), the overshoot problem can be solved Barreiro:1998aj ; hepth0001112 ;
hepph0010102 ; hepth0408160 ; hepph0506045 ; Conlon:2008cj .

The tracker solution relies on additional contributions that redshift slower
than kinetic energy. For a generic cosmic fluid with equation of state
$P=(\gamma-1)\rho$, $\rho\sim a^{-3\gamma}$, and so this condition is
equivalent to $\gamma<2$. Both matter and radiation satisfy this condition.
Given the high inflationary scales, the presence of stable matter at the end
of inflation appears unlikely (although we consider primordial black holes
later). We mostly consider just the case of initial radiation, where
$\rho_{extra}=\rho_{\gamma}$ (note we use $\rho_{\gamma}$ to denote any form
of radiation, not just photons).

For this system, the Friedmann equations are

$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2M_{P}^{2}}\Big{(}\rho_{\gamma}+P_{\gamma}+\dot{\Phi}^{2}\Big{)}=-\frac{1}{2M_{P}^{2}}\Big{(}\gamma\rho_{\gamma}+\dot{\Phi}^{2}\Big{)},$
(24) $\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3M_{P}^{2}}\Big{(}\rho_{\gamma}+\frac{1}{2}\dot{\Phi}^{2}+V(\Phi)\Big{)},$
(25)

while the energy conservation equation is

$\dot{\rho}_{\gamma}=-3H\big{(}\rho_{\gamma}+P_{\gamma}\big{)}=-3H\gamma\rho_{\gamma}.$
(26)

Alternatively, one can switch to the variables (see Copeland:1997et )

$x=\frac{\dot{\Phi}}{M_{P}}\frac{1}{\sqrt{6}H},\quad\quad
y=\sqrt{\frac{V(\Phi)}{3}}\frac{1}{M_{P}H},$ (27)

which encode the fractional energy densities in kinetic and potential energy
respectively, $\Omega_{k}=x^{2},\Omega_{p}=y^{2}$, with a radiation energy
density $\Omega_{\gamma}=1-x^{2}-y^{2}$. The same dynamical evolution can be
recast as the system

$\left\\{\begin{aligned}
x^{\prime}(N)&=-3x-\frac{V^{\prime}(\Phi)}{V(\Phi)}\sqrt{\frac{3}{2}}y^{2}+\frac{3}{2}x\big{[}2x^{2}+\gamma(1-x^{2}-y^{2})\big{]}\\\
y^{\prime}(N)&=\frac{V^{\prime}(\Phi)}{V(\Phi)}\sqrt{\frac{3}{2}}xy+\frac{3}{2}y\big{[}2x^{2}+\gamma(1-x^{2}-y^{2})\big{]}\\\
H^{\prime}(N)&=-\frac{3}{2}H(2x^{2}+\gamma(1-x^{2}-y^{2}))\\\
\Phi^{\prime}(N)&=\sqrt{6}x}{\end{aligned}\right.$ (28)

where the time variable is $N=\log a$.

A simple case (which holds in LVS) is where the potential can be approximated
by a single (steep) exponential,333This expression is valid far away from the
minimum, where the uplifting term can be neglected. Including the uplift, the
full potential takes the form
$V(\Phi)=V_{0}\big{(}(1-\varepsilon\Phi^{3/2})e^{-\lambda\Phi}+\delta
e^{-\sqrt{6}\Phi}\big{)}.$ (29) The parameter $\delta$ needs to be fine-tuned
to achieve a dS vacuum at $\Phi\sim\varepsilon^{-2/3}$.

$V=V_{0}\exp\left(-\lambda\frac{\Phi}{M_{P}}\right),$ (30)

so that $V^{\prime}(\Phi)/V(\Phi)=-\frac{\lambda}{M_{P}}$. For LVS,
$\lambda=\sqrt{\frac{27}{2}}$. The precise value of $V_{0}$ will depend on the
details of the compactification, but for reasonable values of $W_{0}$ we
expect it to be of order $M_{P}^{4}$. In this regime, the system (28) is known
to have a stable attractor solution where the scalar field and radiation have
a fixed ratio of energy densities. The fixed point is characterised by

$\Omega_{k}=x^{2}=\frac{3}{2}\frac{\gamma^{2}}{\lambda^{2}}\quad\quad\Omega_{p}=y^{2}=\frac{3(2-\gamma)\gamma}{2\lambda^{2}}\quad\quad\Omega_{\gamma}=1-x^{2}-y^{2}=1-\frac{3\gamma}{\lambda^{2}}.$
(31)

If the attractor solution is obtained before the rolling field reaches the
barrier, it will not overshoot.

###  2.3 Picking up the Tracker Solution

As a solution to the overshoot problem, the tracker solution relies on other
sources of radiation or matter ‘catching up’ with the kinetic energy through
their slower redshift. One aspect with this is that this takes (a lot of) time
and so it may appear impractical for small seed amounts of radiation to grow
sufficiently.

A distinctive, and almost unique, feature of LVS (which makes it highly
appealing for this purpose) is that the minimum of the potential is located a
long way (in principle, many Planckian distances in field space) from the
centre of moduli space where we assume inflation originally happened. While
this feature has been touched on in Conlon:2008cj , here we develop this
aspect, and its phenomenological implications, significantly.

As $a(t)\propto t^{1/3}$ and $\rho\propto a(t)^{-6}$ during the kination
epoch, the relative proportion of initial radiation $\rho_{\gamma}$ grows as

$\left(\frac{\rho_{\gamma}(t)}{\rho_{KE}(t)}\right)=\left(\frac{t}{t_{0}}\right)^{2/3}\left(\frac{\rho_{\gamma}(t_{0})}{\rho_{KE}(t_{0})}\right),$
(32)

or equivalently,

$\ln\left(\frac{\rho_{\gamma}(t)}{\rho_{KE}(t)}\right)=\frac{2}{3}\ln\left(\frac{t}{t_{0}}\right)+\ln\left(\frac{\rho_{\gamma}(t_{0})}{\rho_{KE}(t_{0})}\right).$
(33)

If kination-radiation equality is obtained, the system will find its way to
the tracker solution as subsequent expansion sees radiation dominate over the
kinetic energy; Hubble friction then effectively halts the field on its
potential until the system evolves into the tracker solution. Kination-
radiation equality occurs at

$\ln\left(\frac{t}{t_{0}}\right)=-\frac{3}{2}\ln\left(\frac{\rho_{\gamma}(t_{0})}{\rho_{KE}(t_{0})}\right).$
(34)

As we would expect, the smaller the initial fraction of radiation, the longer
it takes for this to be obtained. Using the field evolution Eq. (10) during
kination, the field displacement prior to reaching the tracker solution is

$\Delta\Phi=\langle\Phi\rangle-\langle\Phi_{0}\rangle=\sqrt{\frac{3}{2}}\ln\left(\frac{\rho_{KE}(t_{0})}{\rho_{\gamma}(t_{0})}\right).$
(35)

Expressed in terms of the physical compactification volume, this gives

$\ln\left(\frac{{\mathcal{V}}_{f}}{{\mathcal{V}}_{0}}\right)=\frac{3}{2}\ln\left(\frac{\rho_{KE}(t_{0})}{\rho_{\gamma}(t_{0})}\right).$
(36)

The physical implications of Eqs. (35) and (36) are striking. If we assume a
small fraction of initial radiation
$\ln\left(\frac{\rho_{\gamma}(t_{0})}{\rho_{KE}(t_{0})}\right)$, the field
evolution is able to avoid overshooting and settle in the stable vacuum _only
provided the vacuum is sufficiently far away in field space_. The _smaller_
the initial fraction of radiation, the _further_ away the vacuum has to be in
order for it to be accessible. As exponential potentials lead to a rapid
decrease in vacuum energy along the roll, we also see that, somewhat
paradoxically, vacua associated to much lower energy scales (and so with
smaller barriers) become _more_ accessible not less.

This is one part of the inversion of conventional wisdom on overshoot within
LVS. We now discuss the possible seeds for $\rho_{\gamma}(t_{0})$ before
returning to a discussion of physical and model-building implications.

##  3 Seeds of radiation

Previous discussions of the use of radiation or matter to guide moduli towards
the tracker solution have gone into relatively little detail on the origin and
magnitude of the seed radiation. Here we discuss the possible sources of such
seed radiation and their resulting phenomenological implications, with a
particular focus on LVS. While everything is model-dependent to some extent,
there are generic reasons why radiation should be present.

###  3.1 The Thermal de Sitter bath

The presumed dynamics is a primordial epoch of inflation exiting into a
kination phase (although we wish to be agnostic about the actual model of
inflation). If we suppose, consistent with the breakdown of slow-roll
conditions at the end of inflation, that this transition is more or less
instantaneous, any radiation present during inflation will still be present at
the start of the kination phase.

While radiation is constantly diluted during inflation, this is balanced by
the continual production of particles coming from the thermal de Sitter bath
at temperature

$T_{dS}=\frac{H_{\rm{inf}}}{2\pi}.$ (37)

This provides a natural source of a radiation energy density

$\rho_{\gamma,dS}=\frac{\pi^{2}}{30}g_{*}\left(\frac{H_{\rm{inf}}}{2\pi}\right)^{4},$
(38)

where $H$ is the Hubble scale during inflation and $g^{*}$ is the effective
number of massless degrees of freedom. There is no reason that $g^{*}$ should
be small; both chiral fermions and gauge vector bosons naturally remain light
during inflation. Even with scalars, although the $\eta$ problem implies that
it can be difficult to hold them lighter than the inflationary Hubble scale
during inflation, this is only true if their mass is unprotected against
quantum corrections. Axion fields – which are generic in string theory – are
protected by their shift symmetry and remain naturally light during inflation.
In a string theory context, $g_{*}$ could easily be rather large and
$\mathcal{O}(100-1000)$, as the multiplicity of axions or gauge bosons may be
set by the topology of the internal space and by Hodge numbers $h^{1,1}$ or
$h^{2,1}$.

Setting $\rho_{KE}(t_{0})=3H^{2}M_{P}^{2}$, the thermal de Sitter bath would
then lead to

$\frac{\rho_{\gamma}(t_{0})}{\rho_{KE}(t_{0})}=\frac{\pi^{2}}{90}g^{*}\left(\frac{H_{\rm{inf}}^{2}}{(2\pi)M_{P}^{2}}\right).$
(39)

###  3.2 Perturbative Modulus ‘Decays’

The thermal de Sitter bath has the advantage of universality: it treats all
fields democratically. However, we expect there also to be radiation whose
origin is specific to the fact that it is the volume modulus rolling down the
exponential slope (analogous to the more conventional conversion of energy in
the volume field to radiation at reheating). Although this may be small in
_overall_ magnitude, it is competing with the similarly small thermal de
Sitter bath (and the subsequent enhancement of radiation during the kination
epoch makes even small initial quantities of radiation important).

We consider a perturbative origin for radiation from ‘decays’ of a field
rolling on a potential $V=V_{0}\exp\left(-\lambda\frac{\Phi}{M_{P}}\right)$.
Our approach is to model the potential as an (instantaneous) quadratic
potential, and argue that for infinitesimal time periods it is reasonable to
treat the field as equivalent to a conventional massive scalar particle within
a quadratic potential, coupled to external degrees of freedom to which it can
decay.

Without loss of generality, we can consider the field starting at a value
$\Phi=0$ on the exponential potential
$V=V_{0}\exp\left(-\lambda\frac{\Phi}{M_{P}}\right)$.

At this time, we have

$\displaystyle V$ $\displaystyle=$ $\displaystyle V_{0},$ (40)
$\displaystyle\frac{dV}{d\Phi}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{M_{P}}V_{0},$ (41)
$\displaystyle\frac{d^{2}V}{d\Phi^{2}}$ $\displaystyle=$
$\displaystyle\left(\frac{\lambda}{M_{P}}\right)^{2}V_{0}.$ (42)

We wish to consider ‘instantaneous’ dynamics, i.e. those over a time period
$\Delta t\ll\frac{1}{H}$ for a field released at $\Phi=0$ (from Eq. (8), it
follows that over such a period the field displacement is $\Delta\Phi\ll
M_{P}$). To do so, we approximate the form of the potential in the immediate
vicinity of $\Phi=0$ as

$U(\Phi)=U_{0}+\frac{1}{2}m^{2}\left(\Phi-\Phi_{0}\right)^{2}.$ (43)

This is satisfied with $\Phi_{0}=\frac{M_{P}}{\lambda}$,
$U_{0}=\frac{V_{0}}{2}$ and
$m^{2}=\left(\frac{\lambda}{M_{P}}\right)^{2}V_{0}$.

We now imagine, at the transition from the inflationary epoch, an
approximately stationary field coming onto this potential at $\Phi=0$. For a
short period of time, we expect the field to ‘decay’ to radiation in a similar
fashion as if it were on the quadratic potential (instantaneously, it cannot
‘know the difference’). To model this source of initial radiation, we use the
behaviour of a field starting at $\Phi=0$ in the quadratic potential above.
How long does ‘instantaneous’ last for? To give an estimate, we take
approximately one Hubble time
$\tau=\left(\frac{1}{H_{0}}\right)=\frac{\sqrt{3}M_{P}}{V_{0}^{1/2}}$.

We then compute the seed radiation content by regarding this field as a
massive particle with

$m_{\Phi}=\left(\frac{\lambda}{M_{P}}\right)\sqrt{V_{0}},$

which decays via couplings to axions, photons or other light massless degrees
of freedom. The precise couplings of LVS moduli to both matter and other dark
particles, in the context of reheating, have been discussed in Cicoli:2012aq ;
Higaki:2012ar ; Higaki:2012ba ; Hebecker:2013lha ; Higaki:2013lra ;
Angus:2014bia ; Hebecker:2014gka ; Cicoli:2015bpq ; Hebecker:2022fcx ;
Baer:2022fou . We expect a broadly similar analysis to hold here (although
notice that these papers are working around the ultimate minimum of the scalar
potential, while we are interested in couplings _away_ from the minimum).

While the numerical coefficients of the exact decay rate will be model-
dependent, on general grounds we can write the decay rate $\Gamma_{\Phi\to
X_{i}X_{i}}$ of any individual channel as

$\Gamma_{i}\equiv\Gamma_{\Phi\to
X_{i}X_{i}}=\frac{\alpha_{i}}{16\pi}\frac{m_{\Phi}^{3}}{M_{P}^{2}}=\frac{\alpha_{i}}{16\pi}\left(\frac{\lambda}{M_{P}}\right)^{3}\frac{V_{0}^{3/2}}{M_{P}^{2}}.$

In the absence of any additional suppression, $\alpha\sim 1$. As the energy in
the quadratic part of the potential (which we re-interpret as particles) is
$V_{0}/2$ , it follows that the _overall_ fractional conversion rate of energy
to radiation is $\Gamma_{i}/2$. As a naive estimate for the overall fraction
of energy converted to radiation, we then have

$\displaystyle\sum_{i}\frac{\Gamma}{2}\tau$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\sum_{i}\alpha_{i}\left(\frac{\lambda^{3}}{16\pi}\right)\frac{V_{0}}{M_{P}^{4}}$
(44) $\displaystyle=$
$\displaystyle\frac{3\sqrt{3}}{2}\sum_{i}\alpha_{i}\left(\frac{\lambda^{3}}{16\pi}\right)\frac{H^{2}}{M_{P}^{2}},$
(45)

where the index $i$ runs over all decay channels. It is easily seen that this
may be dominant over the thermal bath contribution Eq. (39).

This formalism and in particular Eq. (45) also makes it clear that the
dominant contribution to radiation comes from the immediate post-inflation
period (due to the $\frac{H^{2}}{M_{P}^{2}}$ factor). As the field rolls down
the exponential slope and picks up speed, the Hubble scale decreases and the
rate at which seed radiation is generated falls off rapidly.

###  3.3 Gravitational Waves from Cosmic String Networks

We now consider possibly the most interesting scenario for the generation of
seed radiation.

This involves the radiation seed coming from gravitational waves emitted by
cosmic string networks that were formed at the end of (brane) inflation. Such
cosmic string networks can arise naturally in scenarios involving brane-
antibrane inflation Dvali:1998pa ; Burgess:2001fx ; Dvali:2001fw ;
Sarangi:2002yt ; Kachru:2003sx ; Baumann:2007ah  (see Baumann:2014nda  for a
comprehensive review). In these models, the end of inflation is coincident
with brane-antibrane annihilation producing a large number of both fundamental
and D-strings (D1-branes). In terms of their cosmic evolution, the most
important quantity is the tension of such a string network, characterised by
$G\mu$ (where $G$ is Newton’s constant, $8\pi G=\frac{1}{M_{P}^{2}}$). Reviews
of cosmic strings can be found in Hindmarsh:1994re ; Copeland:2009ga ;
Vachaspati:2015cma .

Such string networks exhibit a scaling behaviour in which their energy density
remains at a constant fraction of the overall energy density of the universe,

$\rho_{string}\sim\mu H^{2}.$ (46)

This implies that as the universe expands, this energy in the string network
decreases. This is not simply associated to a reduction in number density; the
energy in the string network is lost via gravitational radiation emitted
through string reconnections and from cusps. For this reason, cosmic string
networks are regarded as one of the plausible sources of gravitational waves
from the early universe (for example, see LIGOScientific:2017ikf  for a search
and Caprini:2018mtu  for a review).

Such cosmic string networks can survive to today, where their tension is
bounded via CMB constraints to be $G\mu\lesssim 10^{-7}$. In the context of
either fundamental or D-strings, the string tension $\mu\sim m_{s}^{2}$ and so
observational consistency implies any fundamental string network would require
either warping or a relatively low string scale.

We suppose a brane/antibrane inflationary model ending with the formation of a
cosmic string network, which for simplicity we assume consists of vanilla
fundamental strings (i.e. no warping). We also assume that the cosmic string
scaling regime is attained rapidly and so soon after inflation the energy in
the network is $\rho_{string}\sim\mu H_{\rm{inf}}^{2}$. As the Hubble scale
decreases the string network rapidly loses its energy via radiation from
string cusps or reconnection. We assume an $\mathcal{O}(1)$ fraction of the
string emission is to gravitational waves, implying an early presence of
radiation

$\rho_{GW}\sim\mu H_{\rm{inf}}^{2}.$

As $\mu\sim m_{s}^{2}$, we see that for a string scale $m_{s}\sim 10^{16}{\rm
GeV}$ during inflation, giving $\mu\sim 10^{-4}M_{P}^{2}$, the resulting early
fraction of injected radiation energy can be significant, e.g.

$\frac{\rho_{GW}}{\rho_{total}}\sim 10^{-4}.$

This can easily dominate over either of the two previously identified sources
of seed radiation Eqs. (39) and (45).

Normally, such values of $G\mu$ would be immediately ruled out by the CMB
constraint $G\mu\lesssim 10^{-7}$. However, here one appealing feature of LVS
cosmology comes into play. During the long kination roll, the volume increases
and so – as $m_{s}^{2}\sim\frac{M_{P}^{2}}{\mathcal{V}}$ – the cosmic string
tension also decreases by a similar factor of
$\frac{\mathcal{V}}{\mathcal{V}_{0}}$. The CMB constraints apply to the cosmic
string tension today, not at the time of inflation – and this increase in the
volume reduces the string tension to a level entirely compatible with current
observations.

In this scenario, the magnitude of the _initial injection_ of radiation as a
proportion of the total energy density is set by $G\mu_{early}$, while the
_current_ energy density of the cosmic string network is set by $G\mu_{today}$
– and these two can be very different.

This scenario also offers the possibility of a very intriguing aspect of the
cosmological history of the universe. Suppose the initial injection of
radiation from the cosmic string network was indeed dominantly in the form of
gravitational waves. Then, as the universe approaches the tracker solution
during the kination role, the radiation component consists of gravitational
waves. While the universe is on the tracker solution, the radiation component
makes up the largest individual part – and so in fact, the dominant energy
density of the universe is in the form of gravitational waves (for LVS
parameters, $\Omega_{GW}=19/27$)!

This would hold during the tracker epoch. Once the modulus reaches the minimum
and oscillates about it, the universe becomes matter-dominated and so this
energy would then be redshifted away. Nonetheless, it would still be extremely
striking if the universe did actually go through such a ‘graviation’ epoch,
where its energy was primarily in the form of gravitational waves. The
possibility of such an epoch does not appear to have been considered very much
in the literature, and it would be interesting to study its properties
further.

###  3.4 Primordial Black Holes

Another interesting scenario for the initial seed radiation can occur if the
dynamics at the end of inflation results in the production of primordial black
holes (PBHs). Such black holes redshift as matter for as long as they remain
black holes (i.e. before they evaporate). As $\rho_{m}\sim a(t)^{-3}$, if such
PBHs were present the universe would evolves towards a matter tracker
solution.

As with cosmic strings, the production and initial density of black holes is
highly model-dependent, as it crucially involves the spectrum of density
perturbations in the universe and the details of the particular inflationary
model (reviews of PBH can be found in e.g. Sasaki:2018dmp ; Green:2020jor ;
Byrnes:2021jka ). However, if PBHs are produced, their mass when produced is
given by the horizon mass,

$M_{BH}\sim\frac{M_{P}^{2}}{H}.$ (47)

From the point of production, the relative energy fraction in PBHs would
increase throughout any kination epoch. The subsequent evolution would depend
on the mass of the primordial black hole. If the black holes all evaporated
while still in the kination epoch, prior to the modulus reaching the LVS
minimum, their mass-energy is converted to thermal radiation. In such a
scenario, the role of the PBH has been solely to act as a radiation seed.

A more interesting scenario is when the PBHs live long enough that the
universe reaches the ‘matter tracker’. At this point
$\Omega_{m}=\frac{21}{27}$, and so the majority of the energy density of the
universe would be in the form of primordial black holes. This will remain the
case while the field settles into the minimum of the potential – as the
modulus itself also redshifts as matter, it cannot displace the black holes.
Such an epoch, with energy density dominated by the mass of primordial black
holes, would also be striking in itself as a feature of the (string)
cosmological history of the universe.

If the PBHs were able to outlive the volume modulus, then reheating would end
up proceeding via evaporation of the PBHs. Such an epoch would offer a
potential solution to the problem of excessive dark radiation production in
modulus decay. When the volume modulus decays, the branching ratio to axions
appears to be $\mathcal{O}(1)$ as many of the Standard Model decay modes are
suppressed Cicoli:2012aq ; Higaki:2012ar . However, for evaporation of a
primordial black hole, modes should be more democratically accessible,
potentially reducing the dark radiation fraction (depending on the number of
hidden sectors).

However – we do not expect this scenario to be viable, and would instead
expect any PBHs to evaporate prior to the decay of the volume modulus. To see
this, note that a black hole’s lifetime behaves as
$\tau_{BH}\propto\frac{M_{BH}^{3}}{M_{P}^{4}}$ whereas the lifetime of a
modulus is $\tau_{\Phi}\sim\frac{M_{P}^{2}}{m_{\Phi^{3}}}$. We can therefore
write the lifetime of a primordial black hole formed during kination as

$\tau_{BH}\sim\frac{M_{P}^{2}}{H_{form}^{3}},$ (48)

where $H_{form}$ is the Hubble scale at the time of formation. However, as
during the kination epoch $H>m_{\Phi}$ (as equality occurs when the field
reaches the minimum of the potential), we find that

$\tau_{BH}<\tau_{\Phi}$ (49)

and so the black hole evaporates prior to modulus decay. It therefore appears
unlikely that any black holes formed during the kination epoch could outlive
the volume modulus.

##  4 Model-Building Implications

Assuming that the kination phase starts directly at the end of inflation, one
can use an assumption about the source of the seed radiation to relate the
inflationary scale $\Lambda_{\rm{inf}}$ to the minimal field displacement
required to settle into the minimum without overshooting.

From (36), kination-radiation equality is reached for

$\mathcal{V}_{\rm{eq}}=\Bigg{(}\frac{V_{0}}{\Lambda_{\rm{inf}}^{4}}\Bigg{)}^{\sqrt{\frac{3}{2\lambda^{2}}}}\Omega_{\gamma,0}^{-3/2},$
(50)

where the initial volume has been rewritten in terms of $\Lambda_{\rm{inf}}$
and the parameters appearing in the potential. To avoid overshooting, we
require the volume at the minimum to satisfy
$\mathcal{V}_{\rm{min}}\gtrsim\mathcal{V}_{\rm{eq}}$.

In the most minimal realisation (see section 3.1), the initial radiation will
come from the thermal de Sitter radiation, viewed as a bath with
$T_{\rm{dS}}=\frac{H}{2\pi}$. All relativistic species with
$m\ll\Lambda_{\rm{inf}}$ will contribute to a normalised energy density of

$\Omega^{0}_{\rm{th}}=g^{*}\frac{\pi^{2}}{30}\Bigg{(}\frac{H_{\rm{inf}}}{2\pi}\Bigg{)}^{4}\frac{1}{3H_{\rm{inf}}^{2}M_{P}^{2}}=\frac{g^{*}}{270\times
16\pi^{2}}\Bigg{(}\frac{\Lambda_{\rm{inf}}}{M_{P}}\Bigg{)}^{4}.$ (51)

Under the conservative assumption that no new physics appears below the
inflationary scale, the number of degrees of freedom would be
$g^{*}=g^{*}_{\rm{SM}}\sim\mathcal{O}(10^{2})$. Eq. (50) implies

$\mathcal{V}_{\rm{min}}\gtrsim 8.0\times
10^{17}\,\,\bigg{(}\frac{g^{*}}{100}\bigg{)}^{-3/2}\Bigg{(}\frac{V_{0}}{M_{P}^{4}}\Bigg{)}^{1/3}\Bigg{(}\frac{\Lambda_{\rm{inf}}}{3\cdot
10^{16}\,{\rm{GeV}}}\Bigg{)}^{-22/3},$ (52)

where we have used the LVS value of $\lambda=\sqrt{\frac{27}{2}}$. It would be
entirely reasonable, however, to have $g^{*}=\mathcal{O}(10^{3}-10^{4})$,
potentially reducing the estimate (52) by up to 3 orders of magnitude. As
mentioned earlier, such a large number could easily arise from the high
multiplicity of axions and gauge bosons in a typical compactification or from
hidden sectors decoupled from the Standard Model.

Pushing this reasoning to its limits, one could imagine using the largest
conceivable number of degrees of freedom in the UV, i.e. saturating the
species bound of Dvali:2007hz ; Dvali:2007wp . If one assumes the inflationary
scale to lie below the UV cutoff, this gives an absolute upper bound on the
number of the EFT light degrees of freedom

$g^{*}\lesssim\frac{M_{P}^{2}}{\Lambda_{\rm{inf}}^{2}}.$ (53)

Substituting in (53), this turns into the laxer constraint

$\mathcal{V}_{\rm{min}}\gtrsim 1.6\times
10^{15}\,\,\Bigg{(}\frac{V_{0}}{M_{P}^{4}}\Bigg{)}^{1/3}\Bigg{(}\frac{\Lambda_{\rm{inf}}}{3\cdot
10^{16}\,{\rm{GeV}}}\Bigg{)}^{-13/3}.$ (54)

While values of the volume appearing in (54) for realistic inflationary scales
are certainly closer to what one would require for phenomenology, it is not
clear in practice how and if such a high number of light states could arise.
Therefore, we only interpret (54) as an absolute lower bound on the volume of
vacua which can avoid overshooting through thermal dS radiation. In both
cases, typical values for the volumes as a function of the inflationary scale
are reported in Table 1.

|  | $\Lambda_{\rm{inf}}({\rm{GeV}})$ | $3\cdot 10^{16}$ | $3\cdot 10^{15}$ | $3\cdot 10^{14}$
---|---|---|---|---|---
| $g^{*}=10^{2}$ | $\mathcal{V}_{\rm{min}}$ | $8.0\cdot 10^{17}$ | $1.7\cdot 10^{25}$ | $3.7\cdot 10^{32}$
| $g^{*}=10^{4}$ | $\mathcal{V}_{\rm{min}}$ | $8.0\cdot 10^{14}$ | $1.7\cdot 10^{22}$ | $3.7\cdot 10^{29}$
| $g^{*}=M_{P}^{2}/\Lambda_{\rm{inf}}^{2}$ | $\mathcal{V}_{\rm{min}}$ | $1.6\cdot 10^{15}$ | $3.3\cdot 10^{19}$ | $7.2\cdot 10^{23}$
Table 1: Lower bound for a stable volume (measured in units of
$(2\pi\sqrt{\alpha^{\prime}})^{6}$) in LVS as a function of the inflation
scale, assuming only a thermal de Sitter radiation bath. The first line
conservatively assumes the number of relativistic degrees of freedom after
inflation to be $g^{*}=g_{\rm{SM}}\simeq 100$, while the second one uses the
more aggressive estimate $g^{*}=10^{4}$. The third line gives an absolute
lower bound assuming the maximum number of degrees of freedom compatible with
the species conjecture.

In the case of perturbative decays, treated in section 3.2, the parametric
scaling of the initial radiation density as a function of the Hubble scale is
the same as for thermal de Sitter radiation ($\propto H^{4}$). However, the
numerical coefficient in front of (45) gives a larger enhancement:

$\Omega^{0}_{\rm{dec}}=\frac{3\sqrt{3}}{2}\sum_{i}\alpha_{i}\left(\frac{\lambda^{3}}{16\pi}\right)\frac{H^{2}}{M_{P}^{2}}=\frac{9\sqrt{3}}{2}g_{\rm{dec}}\left(\frac{\lambda^{3}}{16\pi}\right)\Bigg{(}\frac{\Lambda_{\rm{inf}}}{M_{P}}\Bigg{)}^{4}.$
(55)

The factor $g_{\rm{dec}}$ is the effective number of species to which the
volume modulus can decay, taking a value $\alpha_{i}=1$. Then, taking again
$\lambda=\sqrt{\frac{27}{2}}$, there is a relative factor

$\frac{\Omega^{0}_{\rm{dec}}}{\Omega^{0}_{\rm{th}}}\simeq 3.3\cdot
10^{5}\frac{g_{\rm{dec}}}{g^{*}}.$ (56)

Then,

$\mathcal{V}\gtrsim 4.2\times
10^{9}\,\,\,\,\bigg{(}\frac{g_{\rm{dec}}}{100}\bigg{)}^{-3/2}\Bigg{(}\frac{V_{0}}{M_{P}^{4}}\Bigg{)}^{1/3}\Bigg{(}\frac{\Lambda_{\rm{inf}}}{3\cdot
10^{16}\,{\rm{GeV}}}\Bigg{)}^{-22/3}.$ (57)

Typical values are summarised in Table 2.

| $\Lambda_{\rm{inf}}({\rm{GeV}})$ | $3\cdot 10^{16}$ | $3\cdot 10^{15}$ | $3\cdot 10^{14}$ | $3\cdot 10^{13}$
---|---|---|---|---|---
$g_{\rm{dec}}=1$ | $\mathcal{V}_{\rm{min}}$ | $4.1\cdot 10^{12}$ | $8.9\cdot 10^{19}$ | $1.9\cdot 10^{27}$ | $4.1\cdot 10^{34}$
$g_{\rm{dec}}=10^{2}$ | $\mathcal{V}_{\rm{min}}$ | $4.1\cdot 10^{9}$ | $8.9\cdot 10^{16}$ | $1.9\cdot 10^{24}$ | $4.1\cdot 10^{31}$
$g_{\rm{dec}}=10^{4}$ | $\mathcal{V}_{\rm{min}}$ | $4.1\cdot 10^{6}$ | $8.9\cdot 10^{13}$ | $1.9\cdot 10^{21}$ | $4.1\cdot 10^{28}$
Table 2: Lower bound for the volume (measured in units of
$(2\pi\sqrt{\alpha^{\prime}})^{6}$) in LVS as a function of the inflation
scale, assuming initial radiation from perturbative decays. The first line
conservatively assumes the number of light degrees of freedom the volume can
decay to after inflation to be $g_{\rm{dec}}=1$, while the second and third
assume the more aggressive estimates $g_{\rm{dec}}=g_{\rm{SM}}\simeq 100$ and
$g_{\rm{dec}}=10^{4}$.

Perhaps the most interesting scenario of seed radiation involved the
production of gravitational waves from cosmic string scaling networks. Our
results therefore motivate the detailed study of inflationary models within
LVS which end in brane/antibrane annihilation and the production of cosmic
strings. So far, the most detailed analyses of brane inflation models have
been carried out within KKLT constructions. LVS offers a very different
phenomenology, in particular the prospect of a large reduction in the value of
the string scale between the end of inflation and the present day. The low
string scales in LVS would also make a fundamental cosmological superstring
network observationally viable, with the tension naturally satisfying
$G\mu\lesssim 10^{-7}$.

In terms of avoiding overshoot, there are two important general points to
make. The first is the fact that in all the possible radiation seeds we have
considered, the larger the initial ratio of $\frac{H_{\rm{inf}}}{M_{P}}$, the
larger the initial radiation fraction, and so the easier it is subsequently to
obtain kination-radiation equality and avoid an overshoot by locating, and
switching onto, the tracker solution. This is also true for cosmic strings and
PBHs, although estimates such as (52) or (57) would be more model dependent in
those cases.

The second point is more LVS-specific: this is that the location of the
minimum at a large distance in field space away from the ‘center’ of moduli
space is crucial for the presence of hierarchical scale separation and, when
using these hierarchies for particle physics, crucial for explaining the large
ratio between the Planck and electroweak scales. In this context, the vacuum
can only be located (avoiding overshooting) if the inflationary scale is
sufficiently high.

This inverts the usual logic of the overshoot problem – in this framing, the
problem becomes _less_ severe the higher up on the potential we start. For
those who find anthropic arguments appealing, in this context we can also turn
this into an anthropic argument for the existence of a large hierarchy between
the scale of inflation (i.e. where we start on the roll) and the electroweak
scale (characterised by the location of the minimum at a large distance in
field space from the inflationary locus). Indeed, the gravitino mass - which
could be thought to provide the scale of SUSY breaking - scales as

$m_{3/2}\simeq\frac{M_{P}|W_{0}|}{\mathcal{V}}$ (58)

in LVS. The relationship of the gravitino mass and MSSM soft terms is subtle
in LVS; while original analyses cqs  assumed these would be comparable, more
careful analyses of soft terms and sequestering Blumenhagen:2009gk ;
Reece:2015qbf  show that we would actually expect soft terms at a scale

$M_{\rm{soft}}\sim\frac{m_{3/2}^{3/2}}{M_{P}^{1/2}}.$ (59)

In any case, under the hypothesis that the solution to the hierarchy problem
is connected in some way to the presence of low energy supersymmetry, this
gives a direct link between large volumes and a low electroweak scale.
Depending on the precise model details, the appropriate LVS volume to match
onto particle physics may be $10^{6}\lesssim\mathcal{V}\lesssim 10^{13}$. From
the estimates in Tables 1 and 2, these volumes may reasonably be large enough
to avoid overshooting but with a clear requirement for relatively high-scale
inflation (not necessarily transPlanckian field excursions).

##  5 Conclusions

String theorists should feel their ears burning and exhibit a professional
interest whenever fields undergo trans-Planckian displacements
$\Delta\Phi\gtrsim M_{P}$. This fact is widely acknowledged for models of
large-field inflation. Less attention, however, has been paid to the case of a
possible kination epoch in the early universe.

The overshoot problem, and tracker solutions as a means to solve it, have over
the years received intermittent attention in string cosmology. In the context
of LVS, we have shown here that a long kination epoch is simultaneously a
natural feature of cosmology in LVS and an attractive method of solving the
overshoot problem: initial seed radiation from the immediate end of the
inflationary epoch grows in relative importance during kination, allowing the
rolling field to locate the tracker solution and avoid overshooting the
minimum.

This approach requires the initial seed radiation to grow from a (presumed)
small value to reach an order unity fraction of the overall energy density.
There are both universal and model-dependent options for the origin of the
original seed radiation, but one aspect they have in common is that the
initial radiation fraction grows with positive powers of
$\left(\frac{H_{\rm{inf}}}{M_{P}}\right)$. As it takes considerable time for
the small initial radiation fraction to ‘catch up’, the avoidance of overshoot
is therefore made _easier_ by large inflationary scales and/or a large
(significantly transPlanckian) field displacement between the inflationary
epoch and the final vacuum. This takes some time; LVS is rather distinctive
here as it naturally contains a large trans-Planckian displacement in the
field space between the locus of inflation and the final minimum. In the
context of the different scales in LVS, this offers an anthropic argument for
a large hierarchy between the weak and inflationary scales.

One particularly interesting scenario is when inflation ends with brane/anti-
brane annihilation and the creation of a cosmic superstring network. As the
string scale decreases during the kination roll, the network can be formed
with a relatively high tension (far above bounds based on the universe today).
If the network dissipates energy primarily through gravitational waves, this
can lead to the universe passing through an epoch where its energy density is
principally in the form of gravitational waves. This can also lead to a
surviving network of fundamental cosmic superstrings; as phenomenological
scenarios of LVS are expected to operate with low string scales, $10^{12}{\rm
GeV}\lesssim m_{s}\lesssim 10^{16}{\rm GeV}$, the string tension $G\mu$ can
naturally lie in interesting observational windows $10^{-7}\lesssim
G\mu\lesssim 10^{-11}$.

It will be interesting to study the physics and cosmology of these scenarios
further.

## Acknowledgements

We would like to thank Prateek Agrawal, Ed Hardy, Francesco Muia and Fernando
Quevedo for discussions, and Bruno Bento for comments on the manuscript. FR is
supported by the Dalitz Graduate Scholarship, jointly established by the
Oxford University Department of Physics and Wadham College. We also thank STFC
for funding via Consolidated Grant ST/T000864/1.

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