# The Dark Universe after Reheating in String Inflation

Michele Cicoli<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia,
Università di Bologna, via Irnerio 46, 40126 Bologna, Italy
and INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
Kuver Sinha and Robert Wiley Deal<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019,
USA

###### Abstract

We study the production of dark matter and dark radiation after reheating in
string inflation models where the Calabi-Yau has a fibred structure and the
visible sector lives on D3 branes. We show how the interplay between different
physical constraints from inflation, reheating, supersymmetry breaking and
dark radiation, leads to distinct predictions for the nature of dark matter.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II Non-thermal dark matter
  3. III Fibred Calabi-Yau Models
  4. IV Moduli decays and dark radiation
    1. IV.1 Canonical normalization
    2. IV.2 Decays to closed string axions
    3. IV.3 Decays to open string axions
    4. IV.4 Decays to Higgses
    5. IV.5 Dark radiation predictions
  5. V String inflation and dark matter
    1. V.1 Kähler moduli inflation
    2. V.2 Fibre Inflation
  6. VI Conclusions

##  I Introduction

Recent advances in string phenomenology in type IIB compactifications have
progressed in two complementary directions:

($i$) Constructing specific compactifications that are phenomenologically
promising: typically, these constructions incorporate moduli stabilization,
have chiral matter, are broadly able to accommodate the gauge groups and
matter content of the Minimal Supersymmetric Standard Model (MSSM) [1, 2, 3,
4, 5, 6], and have at least a somewhat well-defined inflationary and reheating
sector [7, 8, 9];

($ii$) Proceeding along statistical lines, by drawing statistical conclusions
about the distribution of important phenomenological quantities like the scale
of supersymmetry breaking and the axion decay constant, from the ensemble of
type IIB flux vacua [10, 11, 12, 13, 14, 15, 16, 17].

Amongst various classes of models in the first direction, type IIB Large
Volume flux compactifications are particularly well-developed. Two main
inflationary scenarios emerge in this context: Kähler moduli inflation (KMI)
[18, 19] and Fibre Inflation (FI) [20, 21, 22]. KMI is a small-field model
where inflation is driven by a blow-up mode with a non-perturbative scalar
potential. The Hubble scale during inflation is relatively low, $H_{I}\sim
5\times 10^{8}$ GeV, and the tensor-to-scalar ratio is unobservable $r\simeq
10^{-10}$. On the other hand, FI is a large field model characterized by
$H_{I}\simeq 5\times 10^{13}$ GeV and $r\simeq 0.007$ [23], where the inflaton
is a fibration bulk modulus with a perturbative scalar potential.

In these constructions the visible sector can live on either D7-branes
wrapping 4-cycles in the geometric regime, or D3-branes at singularities. In
the first case, the soft terms are around the gravitino mass, $M_{1/2}\sim
m_{0}\sim m_{3/2}$, while in the second case the visible sector can be
sequestered from the sources of supersymmetry breaking in the bulk, resulting
in soft terms which can be hierarchically smaller than the gravitino mass[24,
25]. Two limits can arise: a so-called _local limit_ with a split-SUSY
spectrum featuring $M_{1/2}\ll m_{0}\ll m_{3/2}$, and an _ultralocal limit_
with a more standard MSSM-like spectrum with $M_{1/2}\sim m_{0}\ll m_{3/2}$.

Reheating via the decay of the modulus with the smallest decay width has
already been studied in both KMI and FI for several D-brane configurations
which can realize an MSSM-like sector together with additional hidden sectors.
In particular, reheating for KMI in the simplest Swiss-cheese LVS models has
been studied in [26, 27, 28] for D3-branes in the ultralocal limit, in [29]
for D3-branes in the local limit, and in [30, 31] for D7-branes. On the other
hand, reheating for FI with the MSSM on D3-branes has been studied in [32],
and in [33] for the D7-brane case. Each of these references analyzed in detail
the constraints arising from the requirement to avoid an excessive production
of ultra-light bulk axions that behave as dark radiation. Moreover [34, 35,
36, 37] studied the implications for non-thermal neutralino dark matter for
KMI with the MSSM on D3-branes in the ultralocal limit. The constraints on the
nature of dark matter for KMI with the visible sector on D7-branes has instead
been analyzed in [38] for superheavy WIMPs and in [39] for the QCD axion
realized as a closed string mode.

The largest production of dark radiation from the decay of the lightest
modulus has been found in [32] for FI with the MSSM on D3-branes. This result
relies on a particular expression of the moduli-dependence of the Giudice-
Masiero contribution to the Kähler potential which determines an effective
decoupling of the lightest modulus from the Higgs degrees of freedom. In this
paper we will revisit this result by considering a more general moduli-
dependence of the Giudice-Masiero term that allows to considerably reduce the
production of dark radiation. In doing so, we shall follow the results of [40]
which constrained the form of the Kähler potential by analogy with explicit
toroidal computations.

We will then study the associated production of dark matter after reheating in
FI. We will find that WIMPs are always overproduced, requiring a mechanism of
R-parity breaking to make them unstable. In this case, a very promising dark
matter candidate is instead the QCD axion realized as the phase of a charged
open string field. The axion decay constant is around $f_{\rm QCD}\simeq
5\times 10^{10}$ GeV, which can avoid isocurvature bounds and lead to dark
matter in a rather natural way. We will extend our analysis also to fibred
Calabi-Yau compactifications with KMI and the MSSM on D3-branes. The best case
scenario to avoid dark radiation overproduction with the minimal tuning of the
coefficient of the Giudice-Masiero interaction among the moduli and the
Higges, is the ultralocal limit. In this case, we will find that dark matter
can be TeV-scale non-thermal neutralinos which are produced from the inflaton
decay and then undergo annihilation.

Our analysis shows how UV correlations among different observables in string
compactifications, like inflation, supersymmetry breaking and reheating, is
very powerful to obtain clear predictions for the nature of dark matter.
Moreover, we will see that avoiding dark radiation overproduction severely
constrains the moduli-dependence of the Kähler metric for matter fields.

We will also point out that, contrary to what was claimed in previous studies
[27], the moduli decay to open string axions can compete with the decay to
bulk axions, and so lead to an additional source of dark radiation.

This paper is organized as follows. In Section II and Section III, we provide
a brief review of non-thermal dark matter and fibred LVS models, respectively.
In Section IV, we first derive the moduli couplings to closed and open string
axions, as well as with other visible sector fields. We then derive the
associated predictions for dark radiation and the reheat temperature. In
Sections V.1 and V.2, we study non-thermal dark matter candidates in KMI and
FI, respectively, before presenting our conclusions in Section VI.

##  II Non-thermal dark matter

Before focusing on string models, we first briefly review non-thermal dark
matter (DM) produced by heavy scalar decay [41, 42]. The lightest modulus,
$\phi$, dominates the energy density of the universe from the onset of its
oscillations until its decay. The decay then produces a significant amount of
entropy, diluting any previously existing DM particles to a negligible
abundance. Depending on the reheat temperature of $\phi$, $T_{\text{rh}}$, the
DM particles produced by $\phi$ decay may fall into a few different scenarios
depending on the DM freeze-out temperature $T_{\rm f}\simeq m_{\rm DM}/20$
[34].

##### $T_{\text{rh}}\gtrsim T_{\text{f}}$

In this case, the DM may equilibrate. Due to the DM-DM annihilations, the DM
abundance is simply given by that of the thermal scenario.

##### $T_{\text{rh}}\lesssim T_{\text{f}}$

Here, we have two sub-cases depending on the efficiency of the DM
annihilations. More precisely, defining the critical abundance of the DM,
$Y_{\text{DM}}^{c}$, as the attractor of the relevant Boltzmann equation, we
have

$\displaystyle
Y_{\text{DM}}^{c}\simeq\frac{H}{\langle\sigma_{\text{ann}}v\rangle
s}\Big{|}_{T_{\text{rh}}}.$ (1)

If the produced DM abundance is larger than the critical abundance, i.e.
$Y_{\text{DM}}>Y_{\text{DM}}^{c}$, some annihilations still occur until the
abundance reaches $Y_{\text{DM}}^{c}$, at which point the DM becomes too
dilute to annihilate efficiently. We refer to this case as the annihilation
scenario. In the annihilation scenario, the final DM abundance can be
estimated by

$\displaystyle
Y_{\text{DM}}^{c}\simeq\left(\frac{n_{\text{DM}}}{s}\right)_{\text{obs}}\frac{\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}}{\langle\sigma_{\text{ann}}v\rangle_{\text{f}}}\left(\frac{T_{\text{f}}}{T_{\text{rh}}}\right)$
(2)

where $\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}\simeq 3\times
10^{-26}\text{ cm}^{3}s^{-1}$ is the value needed in the thermal case [43] to
match the observed DM abundance

$\displaystyle\left(\frac{n_{\text{DM}}}{s}\right)_{\text{obs}}\simeq 5\times
10^{-10}\left(\frac{1\text{ GeV}}{m_{\text{DM}}}\right).$ (3)

The DM abundance in the annihilation scenario is thus enhanced by
$T_{\text{f}}/T_{\text{rh}}$ in comparison to the thermal case, and can
accommodate thermally underproduced DM candidates, such as wino-like and
higgsino-like DM. Since $T_{\text{rh}}\lesssim T_{f}\sim m_{\text{DM}}/20$, we
must have
$\langle\sigma_{\text{ann}}v\rangle_{f}^{\text{th}}<\langle\sigma_{\text{ann}}v\rangle_{f}$
to reproduce the observed DM abundance.

If the produced DM abundance is lower than the critical abundance, i.e.
$Y_{\text{DM}}<Y_{\text{DM}}^{c}$, then annihilations cannot occur
significantly. We refer to this case as the branching scenario. The late-time
DM abundance is given simply by the fraction of the light modulus abundance,
$Y_{\phi}$, that decays to DM

$\displaystyle Y_{\text{DM}}=Y_{\phi}\,\text{Br}_{\text{DM}}.$ (4)

If the branching ratio to DM is sufficiently low, the branching scenario can
accommodate thermally overproduced DM, such as bino-like DM.

##  III Fibred Calabi-Yau Models

Fibred Calabi-Yau manifolds have been shown to be very promising for
cosmological and phenomenological applications. The simplest setup involves
three Kähler moduli. To settle notation for the rest of the paper, we briefly
review this class of compactifications.

The first Kähler modulus we consider is $T_{1}=\tau_{1}+{\rm i}c_{1}$. The
scalar $\tau_{1}$ parametrizes the dimensionless volume in string units of a
K3 or $T^{4}$ fibre over a $\mathbb{P}^{1}$ base. The axionic component
$c_{1}$ arises from the reduction of the 10D RR form $C_{4}$ on the K3 or
$T^{4}$ divisor. The scalar component of the second modulus
$T_{2}=\tau_{2}+{\rm i}c_{2}$ controls the volume of the base of the
fibration, while $c_{2}$ is the associated axion. The scalar component of the
third modulus $T_{3}=\tau_{3}+{\rm i}c_{3}$ parametrizes the size of a blow-up
mode required to stabilize the volume at leading order, and $c_{3}$ is an
additional closed string axion. The volume takes the form (for explicit
Calabi-Yau models with this volume form see [44, 7, 9])

$\mathcal{V}=\sqrt{\tau_{1}}\tau_{2}-\tau^{3/2}_{3}\,.$ (5)

The Kähler potential and the superpotential read

$\displaystyle K$ $\displaystyle=$ $\displaystyle K_{\rm
tree}+K_{\alpha^{\prime}}+K_{g_{s}}$ $\displaystyle W$ $\displaystyle=$
$\displaystyle
W_{0}+A_{1}e^{-\mathfrak{a}_{1}T_{1}}+A_{2}e^{-\mathfrak{a}_{2}T_{2}}+A_{3}e^{-\mathfrak{a}_{3}T_{3}}.$
(6)

The tree-level Kähler potential is $K_{\rm tree}=-2\ln{\mathcal{V}}$ (setting
$m_{P}=1$) and it enjoys a well-known no-scale property which makes all Kähler
moduli flat at semiclassical level. At this order of approximation, background
3-form fluxes freeze the complex structure moduli and the dilaton whose VEV
sets the string coupling $g_{s}$ which we assume to be in the weak coupling
regime, i.e. $g_{s}\lesssim\mathcal{O}(0.1)$, to trust perturbation theory.
The leading order $\alpha^{\prime 3}$ correction is given by
$K_{\alpha^{\prime}}=-\frac{\xi}{g^{3/2}_{s}\mathcal{V}}$, where $\xi$ is the
Calabi-Yau Euler number, $\xi\sim\mathcal{O}(1)$, while for string loop
corrections to $K$, given by $K_{g_{s}}$, we refer to [33]. The superpotential
contains instead the tree-level flux-generated contribution $W_{0}$ (which is
a tunable constant after complex structure and dilaton stabilization), and
non-perturbative corrections for each of the three Kähler moduli, where the
$A_{i}$’s are expected to be $\mathcal{O}(1)$ constants while
$\mathfrak{a}_{i}=2\pi/N_{i}$ with $N_{i}\in\mathbb{N}$.

In the large volume limit where the effective field theory is under control,
$\tau_{2}\gtrsim\tau_{1}\gg\tau_{3}>1$, the leading order contribution to the
potential for the Kähler moduli arises from $K_{\alpha^{\prime}}$ and the
$T_{3}$-dependent non-perturbative correction to $W$. This potential fixes
$\tau_{3}$, $c_{3}$ and the overall volume $\mathcal{V}$ at

$\langle\tau_{3}\rangle\simeq g_{s}^{-1}\qquad\langle\mathcal{V}\rangle\simeq
W_{0}\,e^{\mathfrak{a}_{3}/g_{s}}\qquad\langle
c_{3}\rangle=\pi/\mathfrak{a}_{3}.$ (7)

The minimum is AdS and breaks supersymmetry spontaneously. There are several
known mechanism to uplift it to Minkowski, among which anti-D3 branes [45],
T-branes [46], or non-zero F-terms of the complex structure moduli [47]. At
subleading order, $K_{g_{s}}$ (or higher order $\alpha^{\prime}$ corrections
[48, 22]), fix the direction orthogonal to the volume mode
$u\equiv\tau_{1}/\tau_{2}$ at [33]

$\langle u\rangle=\lambda\,g_{s}^{2}\,,$ (8)

where $\lambda$ depends on the tunable coefficients of string loop corrections
to $K$. Finally, tiny $T_{1}$\- and $T_{2}$-dependent non-perturbative effects
fix the two ultra-light axions $c_{1}$ and $c_{2}$ at $\langle
c_{i}\rangle=\pi/\mathfrak{a}_{i}$ $\forall i=1,2$.

The gravitino mass takes the form (restoring appropriate powers of $m_{P}$)

$m_{3/2}^{2}\simeq\kappa\epsilon^{2}m_{P}^{2}$ (9)

where [49]

$\epsilon\equiv W_{0}/\mathcal{V}\ll 1\quad\text{and}\quad\kappa\equiv
g_{s}/(8\pi)\ll 1$ (10)

and the moduli mass spectrum is given by [30, 26, 50]

$\displaystyle m_{\tau_{3}}^{2}$ $\displaystyle\simeq
m_{c_{3}}^{2}\simeq\left(\ln\epsilon\right)^{2}m_{3/2}^{2}>m_{3/2}^{2}$
$\displaystyle m_{\mathcal{V}}^{2}$
$\displaystyle\simeq\left(\frac{\epsilon}{g_{s}^{3/2}W_{0}|\left(\ln\epsilon\right)^{3}|}\right)m_{\tau_{3}}^{2}\ll
m_{\tau_{3}}^{2}$ $\displaystyle m_{u}^{2}$
$\displaystyle\simeq\left(\frac{\epsilon^{1/3}g_{s}^{5/6}|\ln\epsilon|}{W_{0}^{1/3}\sqrt{\lambda}}\right)m_{\mathcal{V}}^{2}<m_{\mathcal{V}}^{2}$
$\displaystyle m_{c_{i}}^{2}$
$\displaystyle\simeq\frac{\tau_{i}^{3}}{W_{0}}\,e^{-\mathfrak{a}_{i}\tau_{i}}m_{3/2}^{2}\ll
m_{u}^{2}\quad\forall i=1,2$ (11)

The visible sector lives on D3-branes at singularities which can give rise to
a scenario of sequestered SUSY breaking [24]. The common gaugino masses,
$M_{1/2}$, can then be estimated using the results of [25]

$M_{1/2}\simeq\frac{3\omega}{2}\frac{m_{3/2}}{\mathcal{V}}\,\tau_{3}^{3/2}\sim\mathcal{O}\left(\frac{m_{3/2}}{\mathcal{V}}\left(\ln\mathcal{V}\right)^{3/2}\right)$
(12)

where $\omega$ is a tunable flux-dependent parameter. The $\mu$-term, if
generated by a Giudice-Masiero contribution to the Kähler potential, is
expected to be approximately of order $M_{1/2}$ since it turns out to be
$\mu\simeq\tilde{\omega}\,M_{1/2}$ where $\tilde{\omega}$ is another tunable
flux-dependent parameter. On the other hand, the value of the common slepton
and squark mass, denoted as $m_{0}$, depends on the exact moduli-dependence of
the Kähler metric for matter fields. The so-called “local limit” is defined as
the case where the physical Yukawas are independent of the volume only to
leading order in $\mathcal{V}^{-1}$, while the “ultralocal limit” is defined
to be the case where the physical Yukawas are independent of the volume to any
order in $\mathcal{V}^{-1}$ [25]. Consequently, the local limit leads to a
split SUSY scenario, whereas the ultralocal case can give a standard MSSM-like
spectrum

$\displaystyle\text{local limit}:\qquad\quad m_{0}\simeq
M_{1/2}\sqrt{\mathcal{V}}\gg M_{1/2}$ $\displaystyle\text{ultralocal
limit}:\quad m_{0}\simeq M_{1/2}.$ (13)

##  IV Moduli decays and dark radiation

As argued above, reheating is given by the decay of the lightest modulus,
which in the case of fibred Calabi-Yaus is $u$. This Kähler modulus can decay
to visible sector fields on D3-branes at singularities, but also to the two
ultra-light closed string axions $c_{1}$ and $c_{2}$ which behave as dark
radiation.

The decay to SUSY scalars is kinematically forbidden in the local limit,
whereas it is mass-suppressed in the ultralocal limit. Similarly, the decay to
SM fermions, Higgsinos and gauginos is chirality suppressed. Moreover, the
decay to SM gauge bosons is loop suppressed since the gauge kinetic function
for D3-branes is controlled by the dilaton. The only relevant decay rates are
therefore to Higgses via a Giudice-Masiero interaction term in $K$, and to the
closed string axions $c_{1}$ and $c_{2}$. We will also argue that, if the QCD
axion is realized as the phase of a charged open string field, the lightest
modulus decay into this mode should also be taken into account since it is not
mass-suppressed.

Assembling all the results, we then provide constraints on the parameter space
coming from observed upper bounds on the effective number of neutrino species
$N_{\rm eff}$.

###  IV.1 Canonical normalization

We begin by generalizing the volume form (5) to

$\mathcal{V}=\tau_{1}^{\frac{n_{1}}{2}}\tau_{2}^{\frac{n_{2}}{2}}$ (14)

with the implicit constraint that $n_{1}+n_{2}=3$. Notice that we ignored the
blow-up mode $\tau_{3}$ given that it does not play any relevant role in
reheating. This volume corresponds to a Kähler potential of the form

$\displaystyle\frac{K}{m_{P}^{2}}$
$\displaystyle=-n_{1}\ln(T_{1}+\overline{T}_{1})-n_{2}\ln(T_{2}+\overline{T}_{2}).$
(15)

The Kähler metric,
$K_{i\overline{\jmath}}\equiv\partial_{i}\partial_{\overline{\jmath}}K$, is
then given by

$\displaystyle K_{i\overline{\jmath}}$
$\displaystyle=\frac{m_{P}^{2}}{4}\begin{pmatrix}\frac{n_{1}}{\tau_{1}^{2}}&&0\\\
0&&\frac{n_{2}}{\tau_{2}^{2}}\end{pmatrix}.$ (16)

As expected, since the Kähler potential is separable, we have already a
diagonal Kähler metric.

The kinetic term in the Lagrangian

$\displaystyle\mathcal{L}$
$\displaystyle=K_{i\overline{\jmath}}\partial_{\mu}T^{i}\partial^{\mu}\overline{T}^{\overline{\jmath}}\supset\frac{m_{P}^{2}}{4}\frac{n_{i}}{\tau_{i}^{2}}\partial_{\mu}\tau_{i}\partial^{\mu}\tau_{i}$
(17)

can then be put into canonical form with the field redefinitions

$\displaystyle\tau_{i}$
$\displaystyle=\exp\left(\sqrt{\frac{2}{n_{i}}}\frac{\phi_{i}}{m_{P}}\right)$
(18)

where $\phi_{i}$ are the new fields with canonical kinetic terms.

There is, however, still a degeneracy - the moduli $\phi_{i}$ are, in general,
not mass eigenstates. As we have seen in Section III, the Calabi-Yau volume is
fixed by the leading order dynamics, and so the mass eigenstates are the
volume mode and its transverse direction. The volume mode is given by

$\displaystyle\frac{\phi_{\mathcal{V}}}{m_{P}}$
$\displaystyle\propto\ln\mathcal{V}=\frac{1}{2}\ln\left(\tau_{1}^{n_{1}}\tau_{2}^{n_{2}}\right).$
(19)

Plugging in our field redefinitions, Eq. (18), into this expression and
multiplying by an overall normalization constant, we have the volume mode in
terms of $\phi_{i}$:

$\displaystyle\phi_{\mathcal{V}}$
$\displaystyle=\sqrt{\frac{n_{1}}{n_{1}+n_{2}}}\,\phi_{1}+\sqrt{\frac{n_{2}}{n_{1}+n_{2}}}\,\phi_{2}.$
(20)

The transverse mode, $\phi_{u}$, can then be constructed simply by
orthogonality:

$\displaystyle\phi_{u}$
$\displaystyle=-\sqrt{\frac{n_{2}}{n_{1}+n_{2}}}\,\phi_{1}+\sqrt{\frac{n_{1}}{n_{1}+n_{2}}}\,\phi_{2}.$
(21)

We will also utilize the inverse of these transformations:

$\displaystyle\phi_{1}$
$\displaystyle=\sqrt{\frac{n_{1}}{n_{1}+n_{2}}}\,\phi_{\mathcal{V}}-\sqrt{\frac{n_{2}}{n_{1}+n_{2}}}\,\phi_{u}$
$\displaystyle\phi_{2}$
$\displaystyle=\sqrt{\frac{n_{2}}{n_{1}+n_{2}}}\,\phi_{\mathcal{V}}+\sqrt{\frac{n_{1}}{n_{1}+n_{2}}}\,\phi_{u}.$
(22)

###  IV.2 Decays to closed string axions

We now discuss the moduli decays into closed string axions $c_{i}$. We start
by returning to the kinetic term (17), which also contains the terms

$\displaystyle\mathcal{L}$
$\displaystyle=K_{i\overline{\jmath}}\partial_{\mu}T^{i}\partial^{\mu}\overline{T}^{\overline{\jmath}}\supset\frac{m_{P}^{2}}{4}\frac{n_{i}}{\tau_{i}^{2}}\partial_{\mu}c_{i}\partial^{\mu}c_{i}.$
(23)

Applying the field redefinition (18), expanding the exponential, and rescaling
the axion fields by

$\displaystyle c_{i}$
$\displaystyle=\sqrt{\frac{2}{n_{i}}}\,\frac{a_{i}}{m_{P}}$ (24)

we obtain canonical kinetic terms for the axion fields $a_{i}$ in addition to
the interaction terms

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\sqrt{\frac{2}{n_{i}}}\,\frac{\phi_{i}}{m_{P}}\,\partial_{\mu}a_{i}\partial^{\mu}a_{i}.$
(25)

Applying now the transformations into the moduli mass eigenbasis, Eq. (IV.1),
we arrive at

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\sqrt{\frac{2}{3}}\frac{\phi_{\mathcal{V}}}{m_{P}}\left(\partial_{\mu}a_{1}\partial^{\mu}a_{1}+\partial_{\mu}a_{2}\partial^{\mu}a_{2}\right)$
$\displaystyle-\sqrt{\frac{2}{3}}\frac{\phi_{u}}{m_{P}}\left(\sqrt{\frac{n_{1}}{n_{2}}}\,\partial_{\mu}a_{2}\partial^{\mu}a_{2}-\sqrt{\frac{n_{2}}{n_{1}}}\,\partial_{\mu}a_{1}\partial^{\mu}a_{1}\right)$
(26)

where we have explicitly used the constraint $n_{1}+n_{2}=3$.

Focusing on the transverse mode, $\phi_{u}$, we can now write down the total
decay width to closed string axions (considering both axions to be massless)

$\displaystyle\Gamma(\phi_{u}\rightarrow aa)$
$\displaystyle=\frac{1}{48\pi}\left(\frac{n_{1}^{2}+n_{2}^{2}}{n_{1}n_{2}}\right)\frac{m_{u}^{3}}{m_{P}^{2}}\equiv
c_{\rm hid}\,\Gamma_{0}$ (27)

where we have defined

$\displaystyle c_{\rm hid}$
$\displaystyle\equiv\left(\frac{n_{1}^{2}+n_{2}^{2}}{n_{1}n_{2}}\right)$ (28)

to be the coefficient of hidden sector decays. We have also made the
definition

$\displaystyle\Gamma_{0}$
$\displaystyle\equiv\frac{1}{48\pi}\frac{m_{u}^{3}}{m_{P}^{2}}$ (29)

for future convenience.

###  IV.3 Decays to open string axions

Open string axions, which arise as the phase of charged matter fields
acquiring a non-zero VEV, are more model-dependent than the closed string bulk
axions we considered above. In [27], the modulus decay to open string axions
was considered and claimed to be negligible. Here, we demonstrate that this is
not necessarily the case - both within our fibred LVS scenario and the minimal
LVS scenario considered in [27]. We begin by considering open string axions
within the context of minimal LVS with only one bulk Kähler modulus
corresponding to the volume mode. For matter fields, collectively denoted as
$C$, the relevant term in the Kähler potential is

$\displaystyle\frac{K}{m_{P}^{2}}$
$\displaystyle\supset\frac{C\overline{C}}{T_{b}+\overline{T}_{b}}$ (30)

where $T_{b}$ is the bulk modulus, corresponding to the choice $n_{1}=3$,
$n_{2}=0$ in Eq. (15). We can then write down the Lagrangian for the canonical
moduli after using Eq. (18)

$\displaystyle\mathcal{L}$
$\displaystyle\supset\frac{m_{P}}{2\sqrt{6}}\,\phi_{\mathcal{V}}\left(C\Box\overline{C}+\overline{C}\Box
C\right).$ (31)

The matter field can then be parameterized by $C=\rho\,\exp({\rm i}\,\theta),$
where $\rho$ is the radial component which acquires a VEV
$\langle\rho\rangle\neq 0$ via D-term stabilization, while $\theta$ is the
phase field taking the role of the axion. Once $\rho$ takes on its VEV, and
after going to canonically normalized fields $\tilde{\rho}$ and
$\tilde{\theta}$, defined as
$\rho=\sqrt{\langle\tau_{b}\rangle}\tilde{\rho}/m_{P}$ and
$\tilde{\theta}=\langle\tilde{\rho}\rangle\theta$, Eq. (31) gives

$\mathcal{L}\supset-\frac{1}{\sqrt{6}}\,\frac{\phi_{\mathcal{V}}}{m_{P}}\,\partial_{\mu}\tilde{\theta}\,\partial^{\mu}\tilde{\theta}.$
(32)

Notice that the axion decay constant $f_{\theta}$ is set by
$\langle\tilde{\rho}\rangle$, i.e. $f_{\theta}=\langle\tilde{\rho}\rangle$. To
put Eq. (32) in a more illuminating form, we can integrate by parts which
gives us

$\displaystyle\mathcal{L}$
$\displaystyle\supset\frac{1}{2\sqrt{6}\,m_{P}}\left[2\phi_{\mathcal{V}}\,\tilde{\theta}\Box\tilde{\theta}-\tilde{\theta}^{2}\Box\phi_{\mathcal{V}}\right].$
(33)

The first term leads to a decay width which is mass suppressed, that was
considered in [27]. However, the second term leads to a decay width
proportional to $\Gamma_{0}$. Thus, the decay width into open string axions in
minimal LVS is given by

$\displaystyle\Gamma\left(\phi_{\mathcal{V}}\rightarrow\tilde{\theta}\tilde{\theta}\right)=\frac{1}{16}\,\Gamma_{0}\,.$
(34)

While this is not as significant as moduli decays to closed string axions for
the production of dark radiation, it slightly increases the tension for
minimal LVS models which consider open string axions as the QCD axion. Notice
moreover that the modulus decay width into the radial part is instead mass
suppressed since it arises from an interaction term of the form

$\mathcal{L}\supset\frac{1}{\sqrt{6}}\,\frac{\phi_{\mathcal{V}}}{m_{P}}\,\tilde{\rho}\Box\tilde{\rho}.$
(35)

The decay to a bulk axion and an open string axion is also possible. This
interaction comes also from expanding Eq. (30), leading to a Lagrangian of the
form

$\displaystyle\mathcal{L}$ $\displaystyle\supset\frac{{\rm
i}}{3}\frac{\phi_{\mathcal{V}}}{m_{P}^{2}}\left(C\,\partial_{\mu}\overline{C}\,\partial^{\mu}a_{b}-\overline{C}\,\partial_{\mu}C\,\partial^{\mu}a_{b}\right).$
(36)

When $\tilde{\rho}$ takes on its VEV, this becomes

$\displaystyle\mathcal{L}$
$\displaystyle\supset\frac{2}{3}\left(\frac{\langle\tilde{\rho}\rangle}{m_{P}}\right)\frac{\phi_{\mathcal{V}}}{m_{P}}\,\partial_{\mu}\tilde{\theta}\,\partial^{\mu}a_{b}$
(37)

and although this term leads to a decay width proportional to $\Gamma_{0}$, it
is also suppressed by $\langle\tilde{\rho}\rangle/m_{P}\ll 1$. Thus, we find
the contributions from the $\phi_{\mathcal{V}}\rightarrow\tilde{\theta}a_{b}$
decay to be negligible.

We now consider open string axions within fibred LVS. In this scenario, we
consider a general Kähler potential for matter fields of the form

$\displaystyle\frac{K}{m_{P}^{2}}$
$\displaystyle\supset\frac{C\overline{C}}{\left(T_{1}+\overline{T}_{1}\right)^{x_{1}}\left(T_{2}+\overline{T}_{2}\right)^{x_{2}}}$
(38)

where $x_{1}$ and $x_{2}$ are constants which fix the moduli dependence of the
matter Kähler metric. These constants are not entirely arbitrary, but rather
primarily fixed by the brane configuration. Based on results from toroidal
orientifolds - which have a similar volume scaling
($\mathcal{V}=\sqrt{\tau_{1}\tau_{2}\tau_{3}}$) to the fibred Calabi-Yau case
($\mathcal{V}=\sqrt{\tau_{1}}\tau_{2}$) through the limit
$\tau_{3}\rightarrow\tau_{2}$ \- in the case at hand where the matter fields
live on D3-branes, [40] suggests two cases

  * •

$x_{1}=1$ and $x_{2}=0$

  * •

$x_{1}=0$ and $x_{2}=1$ .

Notice that both of them reproduce the scaling of Eq. (30).

Keeping for now the general form, we can utilize the field redefinitions from
Eq. (18) and have the following Kähler potential

$\displaystyle\frac{K}{m_{P}^{2}}$
$\displaystyle\supset\frac{1}{2}C\overline{C}\left(1-x_{1}\sqrt{\frac{2}{n_{1}}}\frac{\phi_{1}}{m_{P}}-x_{2}\sqrt{\frac{2}{n_{2}}}\frac{\phi_{2}}{m_{P}}\right)$
(39)

after expanding the exponentials. The relevant interaction terms in the
Lagrangian in terms of the moduli mass eigenbasis turn out to be

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{m_{P}}{2\sqrt{6}}\left(C\Box\overline{C}+\overline{C}\Box
C\right)\phi_{\mathcal{V}}$ $\displaystyle+$
$\displaystyle\frac{m_{P}}{2\sqrt{6}}\left(x_{2}\sqrt{\frac{n_{1}}{n_{2}}}-x_{1}\sqrt{\frac{n_{2}}{n_{1}}}\right)\left(C\Box\overline{C}+\overline{C}\Box
C\right)\phi_{u}$

where we have explicitly used the constraints $n_{1}+n_{2}=3$ and
$x_{1}+x_{2}=1$. Focusing on the transverse mode, $\phi_{u}$, we can write
down the interaction with the open string axion once the radial component
acquires a VEV

$\mathcal{L}\supset\frac{1}{2\sqrt{6}\,m_{P}}\left(x_{2}\sqrt{\frac{n_{1}}{n_{2}}}-x_{1}\sqrt{\frac{n_{2}}{n_{1}}}\right)\left[2\phi_{u}\,\tilde{\theta}\Box\tilde{\theta}-\tilde{\theta}^{2}\Box\phi_{u}\right].$
(41)

Once again, the normalization of the kinetic terms gives
$\langle\tilde{\rho}\rangle=f_{\theta}$. Specializing to $n_{1}=1$ and
$n_{2}=2$, which reproduces the volume form (5), the decay to open string
axions is given by

$\displaystyle\Gamma\left(\phi_{u}\rightarrow\tilde{\theta}\tilde{\theta}\right)$
$\displaystyle=\begin{cases}\Gamma_{0}/8&\text{for }\,x_{1}=1\text{ and
}x_{2}=0\\\ \Gamma_{0}/32&\text{for }x_{1}=0\text{ and }x_{2}=1\,.\end{cases}$
(42)

Thus, we see an enhancement compared to the minimal LVS case if the Kähler
metric depends on $\tau_{1}$, and a reduction if the Kähler metric depends on
$\tau_{2}$. In this work, we focus on the case where $x_{1}=0$ and $x_{2}=1$,
so that additional contributions to dark radiation are minimized.

###  IV.4 Decays to Higgses

We analyze the Giudice-Masiero terms for separable moduli Kähler potentials by
starting with the form

$\displaystyle\frac{K}{m_{P}^{2}}$ $\displaystyle\supset$
$\displaystyle\frac{H_{u}\overline{H}_{u}}{(T_{1}+\overline{T}_{1})^{y_{1}}(T_{2}+\overline{T}_{2})^{y_{2}}}+\frac{H_{d}\overline{H}_{d}}{(T_{1}+\overline{T}_{1})^{w_{1}}(T_{2}+\overline{T}_{2})^{w_{2}}}$
$\displaystyle+\frac{ZH_{u}H_{d}+\text{h.c.}}{(T_{1}+\overline{T}_{1})^{k_{1}}(T_{2}+\overline{T}_{2})^{k_{2}}}$
(43)

where we assume $Z$ is constant with respect to the $T_{i}$. Moreover, making
again an analogy with the toroidal case and following [40], we set
$y_{1}+y_{2}=w_{1}+w_{2}=1$. We also focus on cases where the Giudice-Masiero
term has a Kähler metric of product form (i.e. assume that
$K_{H_{u}H_{d}}=\sqrt{K_{H_{u}}K_{H_{d}}}$), so that $k_{i}=(y_{i}+w_{i})/2$
$\forall i=1,2$. This leads to three possibilities

  * •

$k_{1}=1$ and $k_{2}=0$

  * •

$k_{1}=0$ and $k_{2}=1$

  * •

$k_{1}=k_{2}=1/2$ .

We may now utilize the field redefinitions from Eq. (18). After expanding the
exponentials and going to canonically normalized Higgs fields defined by

$\displaystyle\tilde{H}_{u}$ $\displaystyle=$
$\displaystyle\frac{H_{u}}{\sqrt{\langle
T_{1}+\overline{T}_{1}\rangle^{y_{1}}\langle
T_{2}+\overline{T}_{2}\rangle^{y_{2}}}}$ $\displaystyle\tilde{H}_{d}$
$\displaystyle=$ $\displaystyle\frac{H_{d}}{\sqrt{\langle
T_{1}+\overline{T}_{1}\rangle^{w_{1}}\langle
T_{2}+\overline{T}_{2}\rangle^{w_{2}}}}\,,$ (44)

we obtain the following interaction terms

$\displaystyle K$ $\displaystyle\supset$
$\displaystyle-\tilde{H}_{u}\overline{\tilde{H}}_{u}\left(y_{1}\sqrt{\frac{2}{n_{1}}}\frac{\phi_{1}}{m_{P}}+y_{2}\sqrt{\frac{2}{n_{2}}}\frac{\phi_{2}}{m_{P}}\right)$
$\displaystyle-\tilde{H}_{d}\overline{\tilde{H}}_{d}\left(w_{1}\sqrt{\frac{2}{n_{1}}}\frac{\phi_{1}}{m_{P}}+w_{2}\sqrt{\frac{2}{n_{2}}}\frac{\phi_{2}}{m_{P}}\right)$
(45)
$\displaystyle-(Z\tilde{H}_{u}\tilde{H}_{d}+\text{h.c.})\left(k_{1}\sqrt{\frac{2}{n_{1}}}\frac{\phi_{1}}{m_{P}}+k_{2}\sqrt{\frac{2}{n_{2}}}\frac{\phi_{2}}{m_{P}}\right)$

which generate the following contributions to the interaction Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\left(\tilde{H}_{u}\Box\overline{\tilde{H}}_{u}+\overline{\tilde{H}}_{u}\Box\tilde{H}_{u}\right)\left(\frac{y_{1}}{\sqrt{2n_{1}}}\frac{\phi_{1}}{m_{P}}+\frac{y_{2}}{\sqrt{2n_{2}}}\frac{\phi_{2}}{m_{P}}\right)$
$\displaystyle+\left(\tilde{H}_{d}\Box\overline{\tilde{H}}_{d}+\overline{\tilde{H}}_{d}\Box\tilde{H}_{d}\right)\left(\frac{w_{1}}{\sqrt{2n_{1}}}\frac{\phi_{1}}{m_{P}}+\frac{w_{2}}{\sqrt{2n_{2}}}\frac{\phi_{2}}{m_{P}}\right)$
$\displaystyle+\frac{1}{m_{P}}\left(Z\tilde{H}_{u}\tilde{H}_{d}+\text{h.c.}\right)\left(\frac{k_{1}}{\sqrt{2n_{1}}}\Box\phi_{1}+\frac{k_{2}}{\sqrt{2n_{2}}}\Box\phi_{2}\right).$
(46)

We now move to the moduli mass eigenbasis using Eq. (IV.1)

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{1}{\sqrt{6}}\left(\tilde{H}_{u}\Box\overline{\tilde{H}}_{u}+\overline{\tilde{H}}_{u}\Box\tilde{H}_{u}\right)\frac{\phi_{\mathcal{V}}}{m_{P}}$
$\displaystyle+\frac{1}{\sqrt{6}}\left(\tilde{H}_{d}\Box\overline{\tilde{H}}_{d}+\overline{\tilde{H}}_{d}\Box\tilde{H}_{d}\right)\frac{\phi_{\mathcal{V}}}{m_{P}}$
$\displaystyle+\frac{1}{\sqrt{6}\,m_{P}}(Z\tilde{H}_{u}\tilde{H}_{d}+\text{h.c.})\,\Box\phi_{\mathcal{V}}$
$\displaystyle+\frac{\alpha}{\sqrt{6}}\left(\tilde{H}_{u}\Box\overline{\tilde{H}}_{u}+\overline{\tilde{H}}_{u}\Box\tilde{H}_{u}\right)\frac{\phi_{u}}{m_{P}}$
$\displaystyle+\frac{\beta}{\sqrt{6}}\left(\tilde{H}_{d}\Box\overline{\tilde{H}}_{d}+\overline{\tilde{H}}_{d}\Box\tilde{H}_{d}\right)\frac{\phi_{u}}{m_{P}}$
$\displaystyle+\frac{\gamma}{\sqrt{6}\,m_{P}}(Z\tilde{H}_{u}\tilde{H}_{d}+\text{h.c.})\,\Box\phi_{u}$
(47)

where we have explicitly restored the constraints
$y_{1}+y_{2}=w_{1}+w_{2}=k_{1}+k_{2}=1$ and $n_{1}+n_{2}=3$ and we made the
definitions

$\displaystyle\alpha$
$\displaystyle\equiv\left(y_{2}\sqrt{\frac{n_{1}}{n_{2}}}-y_{1}\sqrt{\frac{n_{2}}{n_{1}}}\right)$
$\displaystyle\beta$
$\displaystyle\equiv\left(w_{2}\sqrt{\frac{n_{1}}{n_{2}}}-w_{1}\sqrt{\frac{n_{2}}{n_{1}}}\right)$
$\displaystyle\gamma$
$\displaystyle\equiv\frac{1}{2}(\alpha+\beta)=\left(k_{2}\sqrt{\frac{n_{1}}{n_{2}}}-k_{1}\sqrt{\frac{n_{2}}{n_{1}}}\right).$
(48)

Focusing on the transverse mode, $\phi_{u}$, we note that the final term will
be dominant - the others will be suppressed by a factor of
$m_{H}^{2}/m_{u}^{2}$. The dominant term is therefore proportional to the
coefficient $\gamma$ which for $n_{1}=1$ and $n_{2}=2$, reproducing Eq. (5),
reduces to

$\gamma=\frac{1}{\sqrt{2}}\left(k_{2}-2k_{1}\right).$ (49)

It is worth stressing that this coefficients vanishes only for $k_{2}=2k_{1}$
which is however never the case for our options since it would be inconsistent
with the results of [40]. The fact that $\gamma\neq 0$ is crucial to avoid
dark radiation overproduction, contrary to the results of [32] which
considered $k_{1}=1/3$ and $k_{2}=2/3$. The intuitive reason why
$k_{2}=2k_{1}$ would imply no interaction between $\phi_{u}$ and the Higgses
(at least at leading order) is that $\phi_{u}$ is orthogonal to the volume
mode and in this case the Kähler moduli dependence of the Giudice-Masiero term
in Eq. (IV.4) would scale exactly as an inverse power of the overall volume
since $\tau_{1}^{-k_{1}}\tau_{2}^{-k_{2}}=\mathcal{V}^{-2k_{1}}$ for
$k_{2}=2k_{1}$.

Let us close this section by commenting briefly on a recent result. In [39],
it was argued that there should be an additional coupling between the Higgs
sector and the moduli. This coupling is induced through loop corrections to
the Higgs mass term due to the volume dependence of the running from some high
scale (e.g. the Kaluza-Klein scale) down to the SUSY breaking scale, and is of
the form

$\displaystyle\mathcal{L}$ $\displaystyle\supset c_{\rm
loop}\left(\frac{m_{u}}{m_{\mathcal{V}}}\right)^{2}\,\frac{m_{0}^{2}}{m_{P}}\,\phi_{u}h^{2}$
(50)

where $h$ is the SM Higgs field, $c_{\rm loop}\sim 1/(16\pi^{2})$ is a typical
loop coefficient and the $(m_{u}/m_{\mathcal{V}})^{2}\sim\mathcal{V}^{-1/3}$
factor is due to the mixing between $\phi_{\mathcal{V}}$ and $\phi_{u}$
induced by string loop corrections to $K$ [51]. Thus, we would expect the
decay width to be proportional to

$\Gamma\sim c_{\rm
loop}^{2}\left(\frac{m_{0}}{m_{\mathcal{V}}}\right)^{4}\frac{m_{u}^{3}}{m_{P}^{2}}.$
(51)

In the case of D3-branes at singularities, we have seen that scalar masses
scale as $m_{0}\sim M_{1/2}\sim m_{\mathcal{V}}/\sqrt{\mathcal{V}}$ in the
ultralocal limit, and so we can safely neglect this induced coupling. On the
other hand, in the local limit $m_{0}\simeq M_{1/2}\sqrt{\mathcal{V}}\simeq
m_{\mathcal{V}}$, and so a more careful study is needed. The exact ratio
between SUSY scalar masses and the mass of the volume mode has been derived in
[29] and takes the form

$\left(\frac{m_{0}}{m_{\mathcal{V}}}\right)^{4}\simeq\left(\ln\mathcal{V}\right)^{2}.$
(52)

At first sight this factor might look large but it has to be multiplied by
$c_{\rm loop}^{2}\sim 1/(16\pi^{2})^{2}$ in the prefactor in Eq. (51). For the
values of the volume which we will consider in this paper,
$\mathcal{V}\lesssim\mathcal{O}(10^{7})$ (that also keep soft terms above the
TeV-scale), one has $c_{\rm
loop}^{2}\left(m_{0}/m_{\mathcal{V}}\right)^{4}\lesssim\mathcal{O}(0.01)$,
showing that this decay channel can be neglected also in the local limit.

###  IV.5 Dark radiation predictions

Let us now derive the prediction for the production of axionic dark radiation
from the decay of the lightest modulus $\phi_{u}$. The produced dark radiation
can be parameterized by the effective number of neutrinos, $\Delta
N_{\text{eff}}$, given by [27]

$\displaystyle\Delta N_{\text{eff}}$
$\displaystyle=3\frac{\rho_{\text{hid}}}{\rho_{\text{neutrinos}}}=\frac{43}{7}\frac{\rho_{\text{hid}}}{\rho_{\text{SM}}}$
$\displaystyle=\frac{43}{7}\frac{f_{\text{hid}}}{1-f_{\text{hid}}}\left(\frac{g_{*}(T_{\text{dec}})}{g_{*}(T_{\rm
rh})}\right)^{1/3},$ (53)

where $g_{*}(T)$ is the number of relativistic degrees of freedom at a given
temperature $T$, while $f_{\rm hid}$ is the branching fraction into hidden
sector particles (bulk axions) defined as

$f_{\text{hid}}\equiv\frac{\Gamma(\phi_{u}\to aa)}{\Gamma(\phi_{u}\to
aa)+\Gamma(\phi_{u}\to H_{u}H_{d})}\,.$ (54)

The decay width into Higgses however depends on the SUSY breaking scale, and
so we study the ultralocal and local limits separately.

##### Ultralocal limit:

In the ultralocal limit, as can be seen from Eq. (13), the soft scalar masses
scale as $m_{0}\simeq M_{1/2}\sim m_{P}/\mathcal{V}^{2}$. This mass scale is
lower than the mass of $\phi_{u}$ since
$m_{0}/m_{u}\simeq\mathcal{V}^{-1/3}\ll 1$, implying that all MSSM Higgs
degrees of freedom are accessible in the decay of $\phi_{u}$. Thus, taking the
dominant decay term of Eq. (IV.4), we can approximate the decay width into the
Higgs sector as

$\displaystyle\Gamma(\phi_{u}\rightarrow H_{u}H_{d})$
$\displaystyle=\frac{2\gamma^{2}Z^{2}}{48\pi}\frac{m_{u}^{3}}{m_{P}^{2}}\equiv
c_{\text{vis}}\,\Gamma_{0}$ (55)

where we defined

$\displaystyle c_{\text{vis}}$ $\displaystyle\equiv 2Z^{2}\gamma^{2}$ (56)

as the approximate coefficient of visible sector decays. Thus the branching
fraction into hidden sector degrees of freedom takes the form

$f_{\text{hid}}\simeq\left[1+2Z^{2}\,\frac{\left(k_{2}n_{1}-k_{1}n_{2}\right)^{2}}{n_{1}^{2}+n_{2}^{2}}\right]^{-1}.$
(57)

In the upper plots of Fig. 1 and Fig. 2, we display the produced dark
radiation in the ($n_{1}$, $Z$) plane for the three different choices of
$k_{1}$ and $k_{2}$ (where we also take $n_{2}=3-n_{1}$). Blank regions are in
excess of the current $2\sigma$ bound from Planck 2018 data [52]. For
$n_{1}=1$, the best case scenario turns out to be $k_{1}=1$ and $k_{2}=0$
which requires $Z\gtrsim 2.5$ (while $k_{1}=0$ and $k_{2}=1$ would require
$Z\gtrsim 5$, and $k_{1}=k_{2}=1/2$ $Z\gtrsim 10$). Again this result can be
understood intuitively by recalling that $\phi_{u}$ is the mode orthogonal to
the volume, corresponding to the ratio of Kähler moduli $u=\tau_{1}/\tau_{2}$,
and rewriting the moduli dependence of the Giudice-Masiero term in Eq. (IV.4)
as a function of $\mathcal{V}$ and $u$. It is then easy to check that the
combination of $k_{1}$ and $k_{2}$ which gives the largest power of $u$
($u^{2/3}$), and so the strongest coupling of $\phi_{u}$ to Higgses, is
$k_{1}=1$ and $k_{2}=0$.

Figure 1: The upper plot shows allowed $\Delta N_{\text{eff}}$ for the case
$k_{1}=k_{2}=1/2$ in the ultralocal limit. The lower plot shows the
corresponding reheat temperature in GeV for the benchmark mass in Table 1.
Figure 2: The upper plot shows allowed $\Delta N_{\text{eff}}$ for the case
$k_{1}=1$ and $k_{2}=0$ in the ultralocal limit. The lower plot again shows
the corresponding reheat temperature in GeV for the benchmark mass in Table 1.
This plot also describes the case $k_{1}=0$ and $k_{2}=1$ under the
interchange $n_{1}\leftrightarrow n_{2}$.

The associated reheating temperature becomes

$\displaystyle T_{\text{rh}}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{40c_{\text{vis}}(c_{\text{vis}}+c_{\text{hid}})}{\pi^{2}g_{*}(T_{\text{rh}})}\right)^{1/4}\sqrt{\Gamma_{0}m_{P}}$
$\displaystyle=$
$\displaystyle\left(\frac{80Z^{2}}{\pi^{2}g_{*}(T_{\text{rh}})}\right)^{1/4}\left|\frac{k_{2}}{n_{2}}-\frac{k_{1}}{n_{1}}\right|^{1/2}\sqrt{\Gamma_{0}m_{P}}$
$\displaystyle\times\left[2Z^{2}(k_{2}n_{1}-k_{1}n_{2})^{2}+n_{1}^{2}+n_{2}^{2}\right]^{1/4}.$
(58)

As a benchmark example, we take $g_{*}(T_{\text{rh}})\simeq 86.25$ and a
typical value of the mass of $\phi_{u}$ for Kähler inflation models [18] which
we will study in Section V.1 obtaining $m_{u}\simeq 2.5\times 10^{7}$ GeV, as
can be seen from Table 1. The corresponding reheat temperature $T_{\text{rh}}$
in GeV is shown in the lower plots of Fig. 1 and Fig. 2 in the ($n_{1}$, $Z$)
plane. Blank regions again correspond to values which produce dark radiation
above the current bounds. As $f_{\text{hid}}$ is independent of $m_{u}$ while
$T_{\text{rh}}$ depends on $m_{u}$ only through $\Gamma_{0}$, we expect these
plots to qualitatively describe a wide range of moduli masses.

##### Local limit:

In the the local limit, as can be seen from Eq. (13), soft scalar masses scale
as $m_{0}\simeq M_{1/2}\sqrt{\mathcal{V}}\sim m_{P}/\mathcal{V}^{3/2}$.
Contrary to the ultralocal limit, in this case $m_{0}$ is above the mass of
$\phi_{u}$ since $m_{0}/m_{u}\simeq\mathcal{V}^{1/6}>1$, implying that some of
the Higgs degrees of freedom are kinematically inaccessible for the light
modulus $\phi_{u}$. We must therefore consider only the decays into the light
degrees of freedom, involving the SM Higgs and the three would-be Goldstone
bosons. Following [53], we find the final term of Eq. (IV.4) becomes

$\displaystyle\mathcal{L}$ $\displaystyle\simeq-\frac{\gamma
Z}{2\sqrt{6}\,m_{P}}(\,|G^{+}|^{2}-\,(h^{0})^{2}+\,(G^{0})^{2})\Box\phi_{u}$
(59)

where $G^{0}$ and $G^{+}$ are the Goldstone bosons. The total decay width into
light Higgs degrees of freedom is then given by

$\Gamma(\phi_{u}\to
H_{u}H_{d})=\frac{1}{48\pi}\frac{5Z^{2}\gamma^{2}}{32}\frac{m_{u}^{3}}{m_{P}^{2}}\equiv
c_{\text{vis}}\Gamma_{0}$ (60)

where we defined

$c_{\text{vis}}\equiv\frac{5Z^{2}\gamma^{2}}{32}.$ (61)

We see that the reheat temperature and the branching fraction into hidden
sector degrees of freedom can then be obtained from the ultralocal limit
expressions, with the replacement $2Z^{2}\rightarrow 5Z^{2}/32$. The minimum
value of $Z$ allowed by dark radiation constraints thus increases by a factor
of about 3.6 in comparison to the ultralocal limit. With this scaling of $Z$,
Fig. 1 and Fig. 2 describe also the local limit. Hence in the local limit the
best case scenario, corresponding to $k_{1}=1$ and $k_{2}=0$, requires
$Z\gtrsim 9$. In terms of tuning needed to avoid dark radiation
overproduction, the ultralocal limit seems therefore to be favored.

It is important to stress that the requirement to satisfy current
observational bounds on $\Delta N_{\rm eff}$ without relying on unnaturally
large values of $Z$, constrains the form of the Kähler metric for matter
fields both at leading order, selecting $k_{1}=1$ and $k_{2}=0$, and at
subleading order by focusing on the ultralocal limit which needs a
cancellation of the $\mathcal{V}$-dependence of the physical Yukawas at all
orders.

##  V String inflation and dark matter

In this section we explore different options for the origin of dark matter
depending on the value of $\mathcal{V}$ which, as can be seen from Eqs. (III)
and (12), sets all the relevant mass scales, in particular the mass of the
lightest modulus and the SUSY breaking scale. In a given string inflation
model, $\mathcal{V}$ is in general fixed by matching the observed amplitude of
density perturbations $A_{s}$. We thus focus on two different string models
where inflation is driven by a Kähler modulus: ($i$) Kähler moduli inflation
[18] which is a small-field model characterized by $\mathcal{V}\sim 10^{7}$
and a Hubble scale during inflation of order $H_{I}\sim 5\times 10^{8}$ GeV;
and ($ii$) Fibre inflation [20] that is a large-field model featuring
$\mathcal{V}\sim 10^{3}$ and $H_{I}\sim 5\times 10^{13}$ GeV.

###  V.1 Kähler moduli inflation

In Kähler moduli inflation the role of the inflaton is played by a blow-up
mode which is displaced away from the minimum where its non-perturbative
potential becomes exponentially flat. We shall therefore focus on the volume
form (5) and consider $\tau_{3}$ as the inflaton field. Explicit realizations
of Kähler moduli inflation require actually the existence of an additional
blow-up mode which we will however ignore since it acts just as a heavy
spectator field which guarantees that $\mathcal{V}$ remains approximately
constant during inflation. This is a small-field inflationary model since it
predicts a very small tensor-to-scalar ratio at CMB horizon scales, $r\simeq
10^{-10}$. The observed amplitude of CMB scalar fluctuations fixes
$\mathcal{V}\sim 10^{7}$ corresponding to a relatively low Hubble scale during
inflation, $H_{I}\simeq m_{\mathcal{V}}\sim 5\times 10^{8}$ GeV.
Interestingly, in sequestered D3-brane models, this value of $\mathcal{V}$ can
also correlate with TeV-scale soft terms for a mild tuning of background
fluxes. In Table 1 we show a benchmark example motivated by the results of
[38]. Note that no moduli suffer from the cosmological moduli problem which
requires $m_{\rm moduli}\gtrsim 7.4\times 10^{4}\text{ GeV}$.

$W_{0}$ | 40
---|---
$\mathcal{V}$ | $10^{7}$
$g_{s}$ | $0.1$
$\lambda$ | $2$
$m_{3/2}$ |  $6.1\times 10^{11}$ GeV
$M_{1/2}$ |  $1.7\times 10^{3}$ GeV
$m_{\tau_{3}}$ |  $7.5\times 10^{12}$ GeV
$m_{\mathcal{V}}$ |  $3.1\times 10^{8}$ GeV
$m_{u}$ |  $2.5\times 10^{7}$ GeV
Table 1: Benchmark spectrum for Kähler moduli inflation. The gaugino mass
$M_{1/2}$ is obtained from (12) with $\omega=0.01$ and $\tau_{3}=1.5$.

During inflation, the bulk moduli $\mathcal{V}$ and $u$ are displaced from
their minima due to inflationary dynamics [54] and are approximately frozen
due to Hubble friction. At the end of inflation, as shown in [55], strong
preheating effects lead to a violent production of inflaton quanta which
eventually decay to either hidden sector gauge bosons on D7-branes wrapping
the inflaton 4-cycle [38] or to lighter moduli and axions [39]. Later on, the
bulk moduli $\phi_{\mathcal{V}}$ and $\phi_{u}$ come to dominate the energy
density of the universe, and their decay dilutes any previous relic. On the
other hand, as can be seen from (III), the two bulk axions, $a_{1}$ and
$a_{2}$, are in practice massless for $\mathcal{V}\simeq 10^{7}$, and so
cannot behave as fuzzy dark matter. They can however be dark radiation
produced from the decay of the lightest modulus $\phi_{u}$ which determines
the final reheating. The decay width into bulk axions has been derived in Eq.
(27) and for $n_{1}$ and $n_{2}$ reduces to

$\Gamma(\phi_{u}\rightarrow aa)=\frac{5}{2}\,\Gamma_{0}.$ (62)

Using the results of Section IV.5, we focus on the best case scenario where
$k_{1}=1$ and $k_{2}=0$, so that the coupling of $\phi_{u}$ to Higgses is
maximized to avoid dark radiation overproduction. We now analyse the
ultralocal and local limits separately.

#### Ultralocal limit

The decay width for $\phi_{u}$ into Higgses in the ultralocal limit is given
by Eq. (55) which for $k_{1}=1$ and $k_{2}=0$ becomes

$\Gamma(\phi_{u}\rightarrow H_{u}H_{d})=4Z^{2}\,\Gamma_{0}\,.$ (63)

The relative fraction to hidden sector radiation is then given by

$f_{\text{hid}}\simeq\left(1+\frac{8}{5}Z^{2}\right)^{-1}$ (64)

while the reheat temperature is

$\displaystyle T_{\text{rh}}$
$\displaystyle=2\sqrt{5}\left(\frac{Z^{2}\left[1+\frac{8}{5}Z^{2}\right]}{\pi^{2}g_{*}(T_{\text{rh}})}\right)^{1/4}\sqrt{\Gamma_{0}m_{P}}.$
(65)

Taking $g_{*}(T_{\text{rh}})\simeq 86.25$ and $Z=3$ (which brings $\Delta
N_{\rm eff}$ within current observational bounds), this gives

$T_{\text{rh}}=18.7\text{ GeV}\times\left(\frac{m_{u}}{2.5\times
10^{7}\text{GeV}}\right)^{3/2}.$ (66)

##### Branching scenario:

Assuming the branching scenario occurs if $T_{\text{rh}}\lesssim 70\text{
MeV}$ [35, 34], we see from Eq. (66) that it would require $m_{u}\lesssim
6.0\times 10^{5}$ GeV, corresponding to a volume of order $\mathcal{V}\gtrsim
5\times 10^{8}$ (keeping the other benchmark values of Table 1 fixed) that in
Kähler moduli inflation would yield an amplitude of scalar fluctuations below
the observed value. Hence the branching scenario is not viable. However, if
instead one were to extend the inflationary model to include additional fields
responsible to generate the observed value of $A_{s}$ in a way compatible with
$\mathcal{V}\gtrsim 5\times 10^{8}$, the branching scenario might be
potentially viable. Here, we assume this to be the case and provide a brief
analysis of the branching scenario.

The required modulus abundance for $T_{\text{rh}}\lesssim 70\text{ MeV}$ and
$\mathcal{V}\gtrsim 5\times 10^{8}$ can be estimated as

$\displaystyle Y_{\phi_{u}}\equiv\frac{3T_{\text{rh}}}{4m_{u}}\lesssim
8.7\times 10^{-8}\,.$ (67)

Setting the DM abundance equal to the observed value, i.e. taking
$Y_{\text{DM}}=5\times 10^{-10}(1\text{ GeV}/m_{\text{DM}})$, and recalling
Eq. (4) for the prediction of the final DM abundance in the branching
scenario, we note that smaller values of $m_{\text{DM}}$ require larger values
of $m_{u}$ to match the observed DM abundance for a given branching ratio.
However, as $m_{\text{DM}}$ decreases, the freeze-out temperature
$T_{\text{f}}$ decreases, while increasing $m_{u}$ increases the reheat
temperature $T_{\text{rh}}$. It is then imperative to check that
$T_{\text{rh}}\lesssim T_{\text{f}}$, as required to have a non-thermal
abundance.

In Fig. 3, we plot contours of the DM branching ratio which reproduce the
measured abundance for given modulus and DM masses. The lower bound on $m_{u}$
is set by BBN constraints ($T_{\text{rh}}\gtrsim 3\text{ MeV}$), while the
upper bound on $m_{u}$ is set by distinguishing the thermal and non-thermal
cases. We adopt $\text{Br}_{\text{DM}}\gtrsim 5\times 10^{-3}$ as a lower
bound on the branching ratio to dark matter.

Figure 3: Constraints on the branching scenario in the ultralocal limit.
Regions in red are excluded, while gray regions cannot be accommodated in the
branching scenario.

Thus we see that the branching scenario in Kähler moduli inflation with a non-
standard mechanism for the generation of the density perturbations is only
potentially viable for DM masses between $100\text{ MeV}\lesssim
m_{\text{DM}}\lesssim 3.3\text{ GeV}$ and $80\text{ TeV}\lesssim m_{u}\lesssim
550\text{ TeV}$.

##### Annihilation scenario:

In the annihilation scenario we expect an enhancement to the DM abundance
relative to the thermal case. To analyze this scenario, we fit the upper bound
of the annihilation cross section in the $b\bar{b}$ channel to data from [43].
If we require Eq. (2) to match the DM abundance, since the annihilation
scenario requires
$\langle\sigma_{\text{ann}}v\rangle_{\text{f}}\geq\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}$,
we have an absolute lower bound on $T_{\text{rh}}$ from DM overproduction in
the case that the annihilation cross section matches the thermal value. The
constraints due to DM overproduction bounds are displayed in Fig. 4, where the
red shaded regions are excluded. We also show potentially excluded regions
depending on the annihilation cross section for values
$3\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}$ (blue region),
$10\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}$ (green region),
and $30\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}$ (yellow
region). These regions are excluded by DM overproduction if the annihilation
cross section is between the thermal value and the given reference value, i.e.
excluded if $\langle\sigma_{\text{ann}}v\rangle_{\text{f}}\leq
n\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}$ for
$n\in\\{3,10,30\\}$, but may still be allowed for larger cross sections (so
long as they are consistent with upper bounds from data). In the thermal case,
the white region may also potentially overproduce DM, depending on the
annihilation cross section. Furthermore, if one were to assume some model with
multi-component DM, more stringent overproduction constraints may be expected.

For our benchmark value, which from Eq. (66) gives $T_{\rm rh}\simeq 18.7$
GeV, we see that we can have the annihilation scenario only for
$m_{\text{DM}}\gtrsim 400$ GeV, while we are reduced to the thermal scenario
if $m_{\text{DM}}\lesssim 400$ GeV. Furthermore, potential DM overproduction
in the annihilation scenario may be slightly improved with some minor
adjustments: decreasing $\lambda$ by a factor of 3 and increasing $g_{s}\simeq
0.15$ then increases the reheat temperature by a factor of roughly $3/2$,
easing the required annihilation cross section to match the observed DM
abundance by a factor of roughly $2/3$.

Figure 4: Constraints on the annihilation scenario due to Fermi-LAT data [43].
Regions in red are excluded, with the dashed line separating the thermal and
non-thermal cases. Other shaded regions give DM overproduction if
$\langle\sigma_{\text{ann}}v\rangle_{\text{f}}\leq
n\langle\sigma_{\text{ann}}v\rangle_{\text{f}}^{\text{th}}$ for
$n\in\\{3,10,30\\}$, but may still be allowed for larger annihilation cross
sections allowed by data.

#### Local limit

In the local limit and in the best case scenario with $k_{1}=1$ and $k_{2}=0$,
the decay of $\phi_{u}$ into Higgs degrees of freedom is given by Eq. (60) and
looks like

$\Gamma(\phi_{u}\rightarrow H_{u}H_{d})\simeq\frac{5}{16}Z^{2}\,\Gamma_{0}.$
(68)

We can thus estimate $f_{\text{hidden}}$ as

$f_{\text{hidden}}\simeq\left(1+\frac{1}{8}Z^{2}\right)^{-1}$ (69)

and the reheating temperature as

$\displaystyle T_{\text{rh}}$
$\displaystyle=2\sqrt{5}\left(\frac{\frac{5}{32}Z^{2}\left[1+\frac{1}{8}Z^{2}\right]}{\pi^{2}g_{*}(T_{\text{rh}})}\right)^{1/4}\sqrt{\Gamma_{0}m_{P}}\,.$
(70)

Due to the reduction in the visible sector decay width compared to the
ultralocal scenario, we find a value of $Z\gtrsim 10$ is required in order to
avoid dark radiation overproduction. Thus, for $Z=10$ and
$g_{*}(T_{\text{rh}})\simeq 86.25$, the reheating temperature becomes

$\displaystyle T_{\text{rh}}=20.8\text{ GeV}\times\left(\frac{m_{u}}{2.5\times
10^{7}\text{ GeV}}\right)^{3/2}.$ (71)

##### Branching scenario:

In the local limit much of our analysis follows identically to the ultralocal
case. The branching scenario can occur for $m_{u}\lesssim 5.6\times 10^{5}$
GeV (assuming $T_{\text{rh}}\lesssim 70$ MeV), which gives us a corresponding
abundance

$Y_{\phi_{u}}\equiv\frac{3T_{\text{rh}}}{4m_{u}}\lesssim 9.3\times 10^{-8}$
(72)

Due to the similar modulus abundance and reheat temperature compared to the
ultralocal case, we find bounds that are quantitatively similar to those in
Fig. 3, with the maximal DM mass pushed down to $3$ GeV and the maximal
modulus mass pushed down to $500$ TeV.

##### Annihilation scenario:

For the annihilation scenario in the local limit, our analysis for the
ultralocal limit also applies. From Fig. 4, we see that we would expect the
annihilation scenario only for $m_{\text{DM}}\gtrsim 430$ GeV, and the thermal
scenario for $m_{\text{DM}}\lesssim 430$ GeV.

###  V.2 Fibre Inflation

Fibre inflation models feature a Calabi-Yau volume of the form (5) where the
role of the inflaton is played by the mode $u$ orthogonal to the overall
volume. This is a large-field inflationary model which predicts a tensor-to-
scalar ratio at the edge of detectability, $r\simeq 0.007$. Compared to Kähler
moduli inflation, the Hubble scale during inflation is higher, $H_{I}\simeq
m_{u}\sim 5\times 10^{13}$ GeV, and value of the volume needed to match
$A_{s}$ is smaller, $\mathcal{V}\sim 10^{3}$. In turn, even with sequestering,
the scale of SUSY breaking turns out to be relatively high, $M_{1/2}\sim
5\times 10^{10}$ GeV. Table 2 shows a benchmark example with values motivated
from [56] where these parameter choices were shown to reproduce the observed
amplitude of the density perturbations.

$W_{0}$ | 15
---|---
$\mathcal{V}$ | $10^{3}$
$g_{s}$ | $0.1$
$\lambda$ | $2$
$m_{3/2}$ |  $2.3\times 10^{15}$ GeV
$M_{1/2}$ |  $6.3\times 10^{10}$ GeV
$m_{\tau_{3}}$ |  $9.5\times 10^{15}$ GeV
$m_{\mathcal{V}}$ |  $2.0\times 10^{14}$ GeV
$m_{u}$ |  $4.4\times 10^{13}$ GeV
Table 2: Benchmark spectrum for Fibre inflation. The gaugino mass $M_{1/2}$ is
obtained from (12) with $\omega=0.01$ and $\tau_{3}=1.5$.

Given that in this case the inflaton is the lightest Kähler modulus, there is
no period of early matter domination and inflationary reheating transitions
directly into a radiation dominated universe via the perturbative decay of
$\phi_{u}$. The decay width into the two ultra-light axions $a_{1}$ and
$a_{2}$ is again given by Eq. (62). The decay rates into visible sector fields
takes again the form (63) and (68) for the ultralocal and local limits
respectively. However, in this case soft masses are around the intermediate
scale, $M_{1/2}\sim 10^{10}$-$10^{11}$ GeV, and so neutralino DM would be
grossly overproduced in both thermal and non-thermal scenarios since naively
$\langle\sigma v\rangle\propto m_{\rm DM}^{-2}$ for WIMPs. We need therefore
to focus on scenarios with R-parity breaking where neutralinos are unstable.

The origin of DM therefore has to be different. Ref. [57] argued that the
potential of Fibre inflation is rich enough to be able to generate primordial
black holes that can constitute all of DM in an appropriate mass range, with
the associated production of detectable secondary gravity waves [58]. Given
that this possibility involves a consistent tuning of the underlying
parameters, we instead investigate whether DM can be made up of axions
produced via the standard misalignment mechanism. These can be either the two
ultra-light axions $a_{1}$ and $a_{2}$ which for $\mathcal{V}\simeq 10^{3}$
have masses and decay constants in the right ballpark to behave as fuzzy dark
matter [59], or the QCD axion realized as the phase of a charged open string
field living on the visible sector D3-brane stack [4]. However, strong
constraints on the type of axion are imposed due to existing bounds on
isocurvature perturbations for cases where the Peccei-Quinn symmetry is broken
during inflation so that the axion is effectively massless and undergoes
quantum fluctuations of order $H_{I}/(2\pi)$. More precisely, when an axion
(which can be either the QCD axion or an ultra-light axion) saturates the DM
relic abundance for a decay constant $f_{a}>H_{I}$ and an initial misalignment
angle $\theta_{i}$, the bound on the inflationary scale from the measured
amplitude of isocurvature modes is [60, 61]

$H_{I}\lesssim 1.4\times 10^{-5}\,f_{a}\,\theta_{i}.$ (73)

For $H_{I}\simeq 5\times 10^{13}$ GeV, this bound can be translated into a
strong bound on $f_{a}$ of the form $f_{a}\,\theta_{i}\gtrsim m_{P}$, implying
that $f_{a}$ has to be at least of order the Planck scale since
$\theta_{i}\in[0,2\pi]$. As shown in [59], each of the two bulk axions could
be fuzzy dark matter with $f_{a_{1}}\sim f_{a_{2}}\sim
m_{p}/\mathcal{V}^{2/3}$ which is definitely below $m_{P}$ for
$\mathcal{V}\simeq 10^{3}$. This implies that fuzzy dark matter is not allowed
in Fibre inflation.

The other possibility is to look at the QCD axion. As shown in Section IV.3,
the QCD axion can arise from the phase of a matter field living on the
D3-brane stack. Its decay constant $f_{\rm QCD}$ is set by the VEV of the
associated radial field that is stabilized by an interplay of D- and F-terms.
As shown in [4], the value $f_{\rm QCD}$ depends on the level of sequestering
of the visible sector from the source of SUSY breaking in the bulk. In
particular, focusing on the benchmark mass spectrum of Tab. 2, we have

$\displaystyle\text{local limit}:\qquad\quad f_{\rm QCD}\simeq m_{3/2}\simeq
10^{15}\,{\rm GeV}$ $\displaystyle\text{ultralocal limit}:\quad f_{\rm
QCD}\simeq M_{1/2}\simeq 5\times 10^{10}\,{\rm GeV}.$

In the local limit, $f_{\rm QCD}$ is therefore above $H_{I}$, implying that
this case is also ruled out by isocurvature bounds. On the other hand, in the
ultralocal limit, $f_{\rm QCD}<H_{I}$, and so during inflation the axion is
heavy given that the Peccei-Quinn symmetry is not broken yet. Thus in this
case the system if free from isocurvature bounds, and the decay constant of
the open string axion is in the right energy regime to constitute all of dark
matter [62].

The ultralocal limit seems therefore favored to describe dark matter. In this
case Eq. (64) and Eq. (65) give again the branching ratio to hidden light
degrees of freedom and the reheat temperature, respectively. Taking
$g_{*}(T_{\text{rh}})=106.75$ and $Z=3$ (to avoid dark radiation
overproduction), we have a relatively high reheat temperature of order

$T_{\text{rh}}\simeq 4.07\times 10^{10}\text{
GeV}\times\left(\frac{m_{u}}{4.4\times 10^{13}\text{ GeV}}\right)^{3/2}.$ (74)

##  VI Conclusions

In this work, we have studied the production of dark matter and dark radiation
after reheating in string inflation models with a fibred Calabi-Yau structure
and the visible sector on D3-branes.

We have imposed several physical constraints, related to the need to match
observed quantities, like the amplitude of the density perturbations generated
during inflation and the dark matter density, or coming from the requirement
to avoid the overproduction of dark radiation and isocurvature modes. The
interplay between these different physical quantities is determined by the
consistency of the underlying UV model which fixes also the scale of
supersymmetry breaking, that determines the WIMP mass scale, and the decay
constant of both closed and open string axions. We have seen that this UV
correlation is very powerful in constraining the nature of dark matter.

In fact, choosing benchmark points preferred by cosmological data for each of
our string inflation scenarios, we have found distinct predictions for the
nature of dark matter which are compatible with current experimental bounds.
In the case of Kähler moduli inflation, we have found TeV-scale WIMPs which
can easily reproduce the observed DM density in both the local and ultralocal
limits. The case of Fibre inflation instead predicts open string axions as the
DM candidate within the ultralocal limit, where isocurvature constraints
exclude the local limit.

While we found that an excess of dark radiation is generic in fibred LVS
models, we have shown that this can be compatible with the latest Planck
measurements without any significant tuning. Let us also point out that Planck
bounds on $N_{\rm eff}$ can be relaxed if local measurements of $H_{0}$ are
assumed as a prior, and a small but non-zero amount of dark radiation could
also be an intriguing effect to weaken the present $H_{0}$ tension [63]. Let
us finally stress that our analysis involved KMI and FI for fibred Calabi-Yaus
with the MSSM on D3-branes. We will instead leave the study of the case where
the MSSM is on D7-branes for future work [64].

## Acknowledgments

We would like to thank Rouzbeh Allahverdi and Arthur Hebecker for helpful
discussions. The work of KS is supported in part by DOE Grant DE-SC0009956.

## References

  * Cicoli  _et al._ [2012a] M. Cicoli, C. Mayrhofer, and R. Valandro, Moduli Stabilisation for Chiral Global Models, JHEP 02, 062, arXiv:1110.3333 [hep-th] .
  * Cicoli  _et al._ [2012b] M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo, and R. Valandro, D-Branes at del Pezzo Singularities: Global Embedding and Moduli Stabilisation, JHEP 09, 019, arXiv:1206.5237 [hep-th] .
  * Cicoli  _et al._ [2013a] M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo, and R. Valandro, D3/D7 Branes at Singularities: Constraints from Global Embedding and Moduli Stabilisation, JHEP 07, 150, arXiv:1304.0022 [hep-th] .
  * Cicoli  _et al._ [2014] M. Cicoli, D. Klevers, S. Krippendorf, C. Mayrhofer, F. Quevedo, and R. Valandro, Explicit de Sitter Flux Vacua for Global String Models with Chiral Matter, JHEP 05, 001, arXiv:1312.0014 [hep-th] .
  * Cicoli  _et al._ [2021] M. Cicoli, I. n. G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla, and R. Valandro, The Standard Model quiver in de Sitter string compactifications, JHEP 08, 109, arXiv:2106.11964 [hep-th] .
  * Cvetic  _et al._ [2022] M. Cvetic, J. Halverson, G. Shiu, and W. Taylor, Snowmass White Paper: String Theory and Particle Physics, (2022), arXiv:2204.01742 [hep-th] .
  * Cicoli  _et al._ [2016a] M. Cicoli, F. Muia, and P. Shukla, Global Embedding of Fibre Inflation Models, JHEP 11, 182, arXiv:1611.04612 [hep-th] .
  * Cicoli  _et al._ [2017a] M. Cicoli, I. n. Garcìa-Etxebarria, C. Mayrhofer, F. Quevedo, P. Shukla, and R. Valandro, Global Orientifolded Quivers with Inflation, JHEP 11, 134, arXiv:1706.06128 [hep-th] .
  * Cicoli  _et al._ [2017b] M. Cicoli, D. Ciupke, V. A. Diaz, V. Guidetti, F. Muia, and P. Shukla, Chiral Global Embedding of Fibre Inflation Models, JHEP 11, 207, arXiv:1709.01518 [hep-th] .
  * Douglas [2012] M. R. Douglas, The String landscape and low energy supersymmetry, in  _Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer_, edited by A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov, and E. Scheidegger (2012) pp. 261–288, arXiv:1204.6626 [hep-th] .
  * Denef and Douglas [2007] F. Denef and M. R. Douglas, Computational complexity of the landscape. I., Annals Phys. 322, 1096 (2007), arXiv:hep-th/0602072 .
  * Denef and Douglas [2004] F. Denef and M. R. Douglas, Distributions of flux vacua, JHEP 05, 072, arXiv:hep-th/0404116 .
  * Broeckel  _et al._ [2020] I. Broeckel, M. Cicoli, A. Maharana, K. Singh, and K. Sinha, Moduli Stabilisation and the Statistics of SUSY Breaking in the Landscape, JHEP 10, 015, arXiv:2007.04327 [hep-th] .
  * Broeckel  _et al._ [2021a] I. Broeckel, M. Cicoli, A. Maharana, K. Singh, and K. Sinha, Moduli stabilisation and the statistics of axion physics in the landscape, JHEP 08, 059, [Addendum: JHEP 01, 191 (2022)], arXiv:2105.02889 [hep-th] .
  * Broeckel  _et al._ [2021b] I. Broeckel, M. Cicoli, A. Maharana, K. Singh, and K. Sinha, On the Search for Low $W_{0}$ 10.1002/prop.202200002 (2021b), arXiv:2108.04266 [hep-th] .
  * Demirtas  _et al._ [2021] M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon, Small cosmological constants in string theory, JHEP 12, 136, arXiv:2107.09064 [hep-th] .
  * Demirtas  _et al._ [2020] M. Demirtas, L. McAllister, and A. Rios-Tascon, Bounding the Kreuzer-Skarke Landscape, Fortsch. Phys. 68, 2000086 (2020), arXiv:2008.01730 [hep-th] .
  * Conlon and Quevedo [2006] J. P. Conlon and F. Quevedo, Kahler moduli inflation, JHEP 01, 146, arXiv:hep-th/0509012 .
  * Bond  _et al._ [2007] J. R. Bond, L. Kofman, S. Prokushkin, and P. M. Vaudrevange, Roulette inflation with Kahler moduli and their axions, Phys. Rev. D 75, 123511 (2007), arXiv:hep-th/0612197 .
  * Cicoli  _et al._ [2009] M. Cicoli, C. P. Burgess, and F. Quevedo, Fibre Inflation: Observable Gravity Waves from IIB String Compactifications, JCAP 03, 013, arXiv:0808.0691 [hep-th] .
  * Broy  _et al._ [2016] B. J. Broy, D. Ciupke, F. G. Pedro, and A. Westphal, Starobinsky-Type Inflation from $\alpha^{\prime}$-Corrections, JCAP 01, 001, arXiv:1509.00024 [hep-th] .
  * Cicoli  _et al._ [2016b] M. Cicoli, D. Ciupke, S. de Alwis, and F. Muia, $\alpha^{\prime}$ Inflation: moduli stabilisation and observable tensors from higher derivatives, JHEP 09, 026, arXiv:1607.01395 [hep-th] .
  * Cicoli and Di Valentino [2020] M. Cicoli and E. Di Valentino, Fitting string inflation to real cosmological data: The fiber inflation case, Phys. Rev. D 102, 043521 (2020), arXiv:2004.01210 [astro-ph.CO] .
  * Blumenhagen  _et al._ [2009] R. Blumenhagen, J. P. Conlon, S. Krippendorf, S. Moster, and F. Quevedo, SUSY Breaking in Local String/F-Theory Models, JHEP 09, 007, arXiv:0906.3297 [hep-th] .
  * Aparicio  _et al._ [2014] L. Aparicio, M. Cicoli, S. Krippendorf, A. Maharana, F. Muia, and F. Quevedo, Sequestered de Sitter String Scenarios: Soft-terms, JHEP 11, 071, arXiv:1409.1931 [hep-th] .
  * Cicoli and Mazumdar [2011] M. Cicoli and A. Mazumdar, Inflation in string theory: A Graceful exit to the real world, Phys. Rev. D 83, 063527 (2011), arXiv:1010.0941 [hep-th] .
  * Cicoli  _et al._ [2013b] M. Cicoli, J. P. Conlon, and F. Quevedo, Dark radiation in LARGE volume models, Phys. Rev. D 87, 043520 (2013b), arXiv:1208.3562 [hep-ph] .
  * Higaki and Takahashi [2012] T. Higaki and F. Takahashi, Dark Radiation and Dark Matter in Large Volume Compactifications, JHEP 11, 125, arXiv:1208.3563 [hep-ph] .
  * Cicoli and Muia [2015a] M. Cicoli and F. Muia, General Analysis of Dark Radiation in Sequestered String Models, JHEP 12, 152, arXiv:1511.05447 [hep-th] .
  * Cicoli and Mazumdar [2010] M. Cicoli and A. Mazumdar, Reheating for Closed String Inflation, JCAP 09, 025, arXiv:1005.5076 [hep-th] .
  * Hebecker  _et al._ [2014] A. Hebecker, P. Mangat, F. Rompineve, and L. T. Witkowski, Dark Radiation predictions from general Large Volume Scenarios, JHEP 09, 140, arXiv:1403.6810 [hep-ph] .
  * Angus [2014] S. Angus, Dark Radiation in Anisotropic LARGE Volume Compactifications, JHEP 10, 184, arXiv:1403.6473 [hep-ph] .
  * Cicoli and Piovano [2019a] M. Cicoli and G. A. Piovano, Reheating and Dark Radiation after Fibre Inflation, JCAP 02, 048, arXiv:1809.01159 [hep-th] .
  * Allahverdi  _et al._ [2013] R. Allahverdi, M. Cicoli, B. Dutta, and K. Sinha, Nonthermal dark matter in string compactifications, Phys. Rev. D 88, 095015 (2013), arXiv:1307.5086 [hep-ph] .
  * Allahverdi  _et al._ [2014] R. Allahverdi, M. Cicoli, B. Dutta, and K. Sinha, Correlation between Dark Matter and Dark Radiation in String Compactifications, JCAP 10, 002, arXiv:1401.4364 [hep-ph] .
  * Aparicio  _et al._ [2015] L. Aparicio, M. Cicoli, B. Dutta, S. Krippendorf, A. Maharana, F. Muia, and F. Quevedo, Non-thermal CMSSM with a 125 GeV Higgs, JHEP 05, 098, arXiv:1502.05672 [hep-ph] .
  * Aparicio  _et al._ [2016] L. Aparicio, M. Cicoli, B. Dutta, F. Muia, and F. Quevedo, Light Higgsino Dark Matter from Non-thermal Cosmology, JHEP 11, 038, arXiv:1607.00004 [hep-ph] .
  * Allahverdi  _et al._ [2021] R. Allahverdi, I. Broeckel, M. Cicoli, and J. K. Osiński, Superheavy dark matter from string theory, JHEP 02, 026, arXiv:2010.03573 [hep-ph] .
  * Hebecker  _et al._ [2022] A. Hebecker, J. Jaeckel, and M. Wittner, Axions in String Theory and the Hydra of Dark Radiation, (2022), arXiv:2203.08833 [hep-th] .
  * Aparicio  _et al._ [2008] L. Aparicio, D. G. Cerdeno, and L. E. Ibanez, Modulus-dominated SUSY-breaking soft terms in F-theory and their test at LHC, JHEP 07, 099, arXiv:0805.2943 [hep-ph] .
  * Kane  _et al._ [2015] G. Kane, K. Sinha, and S. Watson, Cosmological Moduli and the Post-Inflationary Universe: A Critical Review, Int. J. Mod. Phys. D 24, 1530022 (2015), arXiv:1502.07746 [hep-th] .
  * Allahverdi  _et al._ [2020] R. Allahverdi _et al._ , The First Three Seconds: a Review of Possible Expansion Histories of the Early Universe 10.21105/astro.2006.16182 (2020), arXiv:2006.16182 [astro-ph.CO] .
  * Albert  _et al._ [2017] A. Albert _et al._ (Fermi-LAT, DES), Searching for Dark Matter Annihilation in Recently Discovered Milky Way Satellites with Fermi-LAT, Astrophys. J. 834, 110 (2017), arXiv:1611.03184 [astro-ph.HE] .
  * Cicoli  _et al._ [2012c] M. Cicoli, M. Kreuzer, and C. Mayrhofer, Toric K3-Fibred Calabi-Yau Manifolds with del Pezzo Divisors for String Compactifications, JHEP 02, 002, arXiv:1107.0383 [hep-th] .
  * Kachru  _et al._ [2003] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68, 046005 (2003), arXiv:hep-th/0301240 .
  * Cicoli  _et al._ [2016c] M. Cicoli, F. Quevedo, and R. Valandro, De Sitter from T-branes, JHEP 03, 141, arXiv:1512.04558 [hep-th] .
  * Gallego  _et al._ [2017] D. Gallego, M. C. D. Marsh, B. Vercnocke, and T. Wrase, A New Class of de Sitter Vacua in Type IIB Large Volume Compactifications, JHEP 10, 193, arXiv:1707.01095 [hep-th] .
  * Ciupke  _et al._ [2015] D. Ciupke, J. Louis, and A. Westphal, Higher-Derivative Supergravity and Moduli Stabilization, JHEP 10, 094, arXiv:1505.03092 [hep-th] .
  * Burgess  _et al._ [2010] C. P. Burgess, M. Cicoli, M. Gomez-Reino, F. Quevedo, G. Tasinato, and I. Zavala, Non-standard primordial fluctuations and nongaussianity in string inflation, JHEP 08, 045, arXiv:1005.4840 [hep-th] .
  * Cicoli  _et al._ [2017c] M. Cicoli, V. A. Diaz, V. Guidetti, and M. Rummel, The 3.5 keV Line from Stringy Axions, JHEP 10, 192, arXiv:1707.02987 [hep-th] .
  * Cicoli  _et al._ [2012d] M. Cicoli, G. Tasinato, I. Zavala, C. P. Burgess, and F. Quevedo, Modulated Reheating and Large Non-Gaussianity in String Cosmology, JCAP 05, 039, arXiv:1202.4580 [hep-th] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Cicoli and Muia [2015b] M. Cicoli and F. Muia, General analysis of dark radiation in sequestered string models, Journal of High Energy Physics 2015, 10.1007/jhep12(2015)152 (2015b).
  * Cicoli  _et al._ [2016d] M. Cicoli, K. Dutta, A. Maharana, and F. Quevedo, Moduli Vacuum Misalignment and Precise Predictions in String Inflation, JCAP 08, 006, arXiv:1604.08512 [hep-th] .
  * Barnaby  _et al._ [2009] N. Barnaby, J. R. Bond, Z. Huang, and L. Kofman, Preheating After Modular Inflation, JCAP 12, 021, arXiv:0909.0503 [hep-th] .
  * Cicoli and Piovano [2019b] M. Cicoli and G. A. Piovano, Reheating and dark radiation after fibre inflation, Journal of Cosmology and Astroparticle Physics 2019 (02).
  * Cicoli  _et al._ [2018] M. Cicoli, V. A. Diaz, and F. G. Pedro, Primordial Black Holes from String Inflation, JCAP 06, 034, arXiv:1803.02837 [hep-th] .
  * Cicoli  _et al._ [2022a] M. Cicoli, F. G. Pedro, and N. Pedron, Secondary GWs and PBHs in string inflation: formation and detectability, (2022a), arXiv:2203.00021 [hep-th] .
  * Cicoli  _et al._ [2022b] M. Cicoli, V. Guidetti, N. Righi, and A. Westphal, Fuzzy Dark Matter candidates from string theory, JHEP 05, 107, arXiv:2110.02964 [hep-th] .
  * Visinelli and Gondolo [2009] L. Visinelli and P. Gondolo, Dark Matter Axions Revisited, Phys. Rev. D 80, 035024 (2009), arXiv:0903.4377 [astro-ph.CO] .
  * Akrami  _et al._ [2020] Y. Akrami _et al._ (Planck), Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211 [astro-ph.CO] .
  * Gorghetto  _et al._ [2021] M. Gorghetto, E. Hardy, and G. Villadoro, More axions from strings, SciPost Phys. 10, 050 (2021), arXiv:2007.04990 [hep-ph] .
  * Brinckmann  _et al._ [2021] T. Brinckmann, J. H. Chang, and M. LoVerde, Self-interacting neutrinos, the Hubble parameter tension, and the cosmic microwave background, Phys. Rev. D 104, 063523 (2021), arXiv:2012.11830 [astro-ph.CO] .
  * [64] M. Cicoli, K. Sinha, and R. Wiley Deal, Work in progress, .