# Constraining Mach’s principle with high precision astrometry

István Szapudi Institute for Astronomy
University of Hawai’i at M$\bar{a}$noa
2680 Woodlawn Drive
Honolulu, HI 96822, USA

###### Abstract

The analyses of high precision astrometric surveys, such as Gaia, implicitly
assume a modern version of Mach’s Principle: the local inertial frame of our
Solar System should be non-rotating in the frame of distant quasars. On the
contrary, Einstein’s General Relativity allows a rotating universe. Thus,
relaxing the assumption of Mach’s Principle will allow placing a constraint on
a class of rotating cosmologies by comparing high precision astrometry of
quasars with well-measured solar system orbits. Constraining global rotation
will test General Relativity, inflation, and the isotropy of cosmological
initial conditions.

Essay written for the Gravity Research Foundation 2021 Awards for Essays on
Gravitation.

Cosmology — Gödel Universe — astrometry — Gaia

##  1 Introduction

According to Mach’s Principle (MP, attributed to Ernst Mach and Bishop
Berkeley) rotation cannot be observed without distant reference objects, thus
inertial frames are at rest with respect to and somehow a consequence of
distant stars.

While MP motivated Einstein’s General Relativity (GR), it is a non-Machian
local field theory with no action at a distance. Indeed, the rotation of the
Milky Way breaks MP in the original sense. The modern view (aside from
trivially replacing galactic stars with distant quasars and galaxies) is that
inertial frames are local and not caused by distant objects. The exception is
frame dragging (See Pfister, 2007, for the history of the Lense-Thirring
effect) on non-cosmological scales. A gyroscope rotating (or Sagnac effect) in
space with respect to distant quasars signals that the universe rotates.

For a cosmologist, the non-rotating FRW solution appears unnatural, analogous
to non-rotating Schwarzschild blackholes vs. rotating Kerr blackholes. The
crucial difference is that blackholes form through collapse while the universe
expands. Collapse increases angular momentum through momentum conservation,
while cosmological angular momentum (and any vector perturbation) decreases
with $1/a$, where $a$ is the expansion factor of the universe. Thus we would
expect only a small amount of residual rotation in most theories. In
particular, in standard inflationary theories the expansion of order $a\simeq
e^{60}$ could render any residual rotation unmeasurable. Nevertheless, some
non-standard inflationary models, (e.g., the Einstein-aether theory Nakashima
& Kobayashi, 2011) seed vector perturbations. In conclusion, any observed
rotation of our inertial frame (or the lack thereof) constrains inflation, the
isotropy of cosmological initial conditions, and GR.

Rotating models have been considered early on by Gödel (1949), and later
(e.g., Silk, 1966; Hawking, 1969; Collins & Hawking, 1973). While the original
Gödel model is pathological, there are viable generalizations; for a review
see Obukhov (2000). These models display a global rotation while still
preserving a perfectly uniform Cosmic Microwave Background (CMB, e.g.,
Obukhov, 2000). Note that vector perturbations to homogeneous FRW and related
Bianchi models belong to an interesting but separate class of models (Barrow
et al., 1985; Su & Chu, 2009), tightly constrained from the CMB (McEwen et
al., 2013; Saadeh et al., 2016a, b). Our focus is on cosmologies still allowed
by CMB constraints.

Gaia used distant quasars to determine the acceleration of the solar system at
the precision of $0.35\mu as/y$, (Gaia Collaboration et al., 2020), with the
expectation of “well below $0.2\mu as/y$ after the final data release”.
However, their analyses implicitly use MP. We demonstrate next that high
precision astrometry combined with solar system observations and accurate
dynamical modeling will constrain global rotation (see also Clemence, 1957,
for a constraint of $0.1mas/y$ based on solar system data).

##  2 The effect of global rotation on solar system orbits

To first order, our Solar system can be replaced with the Sun and Jupiter
orbiting at 5AU, since the latter contains approximately $2/3$ of the total
mass in planets. Thus, initially, let’s assume solar system with a central
star and a single planet. In addition, assume that the planet is far enough
from the central star that general relativistic corrections are negligible,
and that the local inertial frame rotates with respect to the frame defined by
distant quasars. The solution in a rotating frame is a standard Keplerian
orbit in a rotating plane. We notate the projection of the global rotation
$\omega$ perpendicular to the plane of the orbit as $\omega_{\perp}$ and in
the plane as $\omega_{\parallel}=\omega-\omega_{\perp}$.

High precision astrometry of distant stars and a precision determination of
the plane of the orbit is sufficient to determine $\omega_{\parallel}$. From
$N$ measurement of the orbit, we can determine the plane of the orbit with
$\sigma/\sqrt{N}$, $\sigma$ is the fundamental astrometric precision for the
planet orbit (depends on the brightness, color, etc. of the planet) in $\mu
as$. If in a year this becomes smaller than the astrometric precision of the
quasar frame in $\sigma_{Q}$ in $\mu as/y$, then $\omega_{\parallel}$ is
measurable.

For constraining $\omega_{\perp}$, note that a rotating coordinate system
pegged to distant quasars cases deviations to Kepler’s law. For a single
planet on a circular orbit of radius $R$ and velocity $v$ and period $P=2\pi
R/v$ the observed velocity is

$\tilde{v}=v+\omega_{\perp}R.$ (1)

The measured period is modified as

$\tilde{P}=\frac{2\pi}{v+\omega_{\perp}R}=\simeq
P\left(1-\frac{\omega_{\perp}R}{v}\right),$ (2)

where we keep terms linear with the small $\omega_{\perp}$. Kepler’s law is
modified as

$\tilde{P}^{2}=\frac{4\pi^{2}R^{3}}{GM}\left(1-\frac{2R^{3/2}}{(GM)^{1/2}}\omega_{\perp}\right).$
(3)

Precise measurements of the mass $M$ of the central star, the radius of the
orbit, and the period will pinpoint a rotating frame. To appreciate the order
of magnitudes, we can estimated the expected corrections to the period as

$\frac{\Delta P}{P}\simeq 7.7\times 10^{-13}P[y]\omega_{\perp}[\mu as/y].$ (4)

The above toy model Newtonian solar system with one planet illustrates how to
constrain a rotating frame of distant quasars. In practice, we need a general
relativistic multi-body model of our solar system. Next we outline the needed
corrections to the above simple ideas.

Angular momentum conserves in a Schwartzschild metric commonly used to model
the gravitational field of the Sun, therefore the kinematic measurement of
$\omega_{\parallel}$ does not pick up GR corrections.

On the contrary, Kepler’s law is modified in a GR case (e.g., Barker et al.,
1986). To first order such corrections depend on the Schwartzschild radius of
the Sun $R_{s}\simeq 3\,km\simeq 2\times 10^{-8}AU$, thus relativistic
corrections are likely to be important for the accuracy needed to constrain at
the level Eq. 4. The PPN formalism (Will, 2014) with additional parameters to
describe rotation can account for perturbative GR effects.

The interaction between the planets has significant effect on their orbits.
For instance the semi-major axis of Mars can change order of $10^{-4}$ due to
the effect of Earth, and Jupiter. Such perturbations alter mainly the period
and semi major axis, but in principle they can tilt the instantaneous plane of
the orbit. Therefore they could affect the measurement of both
$\omega_{\parallel}$ and $\omega_{\perp}$.

Thus rotation of our rest frame with respect to the quasar frame will show up
in local observations of planetary dynamics coupled with precision astrometry.
In practice, detailed, general relativistic models including planetary
perturbations are necessary in conjunction with high precision quasar
astrometry. Nevertheless, the principle remains the same: orbital elements and
dynamics constrain the local inertial frame, our Solar System acting as a
giant gyroscope.

##  3 Gaia

At present, Gaia provides the most precise astrometry. Their Early Data
Release 3 (Gaia EDR3) determined precise astrometry for about 1.6 million
compact (QSO-like) sources, 1.2 million of which have the best-quality five-
parameter astrometric solutions. These latter have astrometric precision
$\simeq 450mas/y$. Naively, a coherent motion on the sky could be thus
detected at the level of $0.4\mu as/y$. This order of magnitude consideration
is remarkablt close to the quoted error on the detected acceleration of the
solar system $0.35\mu as/y$ (Gaia Collaboration et al., 2020), despite the
recently identified spatial correlation of the parallaxes (Zinn, 2021).

The Gaia analysis expands the proper motion of quasars into spheroidal and
toroidal vector spherical harmonics (VSHs) according to (Mignard & Klioner,
2012). Rotation would appear as linear combination of $l=1$ toroidal VSHs.
Gaia simulations predict an error of order $0.1\mu as$ for constraining
rotation, again, close to the order of magnitude estimates earlier.

The present Gaia analysis explicitly assumes that QSOs provide a fixed frame,
i.e. they presuppose MP. Individual VSH coefficients are not presented in
(Gaia Collaboration et al., 2020): the $l=1$ toroidal VSHs provide a check on
systematic errors. They use only the glide coefficients derived from the
spherical VSHs. Comparison of the measured VSH coefficients with solar system
orbits will yield the first constraint on generalized Gödel cosmologies.

##  4 Summary and Discussion

While MP has several versions and interpretations in GR, we focus on whether
our local inertial frame is rotating with respect to distant objects, QSOs and
galaxies. We demonstrated how precision astrometry combined with accurate
general relativistic multi-body model of the Solar system will constrain MP in
this sense. By analogy of blackholes, a small rotation would be natural in GR,
while a perfect synchronization with the background, MP, would be curious. In
standard inflationary models we expect that initial rotation would be subdued
by expansion. Our proposed astrometric constraints will shed light on
inflationary theories, the isotropy of cosmology and initial conditions, and
GR. Next we propose a few additional tests for global rotation.

Any satellite, such as Gaia, is a free floating laboratory in the solar
system. Therefore, up to orbital microgravity, its gyroscope determines the
local inertial frame. Thus the ephemeris of a satellite together with
gyroscopic (or Sagnac) data compared to QSO positions yield a constraint. At
the same time, Gaia is close enough to other rotating bodies, such as the Sun,
that the Lense-Thirring frame dragging could complicate interpretation.

Since angular momentum is conserved even in a GR setting, one could measure
the total angular momentum of a few interacting Solar System bodies, such as
Jupiter, Saturn, etc., to define a conserved plane. If the torque from any
neglected bodies is negligible, the conserved reference plane constrains
rotation as in Section 2.

Expansion slows any rotation by $1/a$, therefore higher redshift quasars
should spin faster. Subdividing a sample into (photometric) redshift slices
enhances the signal compared to the solar system, displays differential
rotation. High resolution CMB observations in the quasar frame would show a
factor of 1000 faster rotation compared to local quasars.

Finally, a similar argument works for galactic orbits of stars. In the
(perhaps far) future, if sufficiently accurate measurements of stellar
positions are available, stellar orbits in the quasar frame will yield
additional constraints on rotation.

##  5 Acknowledgements

IS acknowledges support from the National Science Foundation (NSF) award
1616974 and thanks Eric Baxter, Nader Haghighipour, Dan Huber, and Jeremy
Sakstein for useful comments and suggestions.

## References

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# Stringy multifield quintessence and the Swampland

Max Brinkmann<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia,
Università di Bologna, via Irnerio 46, 40126 Bologna, Italy Dipartimento di
Fisica e Astronomia, Università di Padova, via Marzolo 8, 35131 Padova, Italy
INFN, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy  Michele Cicoli
<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia, Università di
Bologna, via Irnerio 46, 40126 Bologna, Italy INFN, Sezione di Bologna, viale
Berti Pichat 6/2, 40127 Bologna, Italy  Giuseppe Dibitetto
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Roma
Tor Vergata, via della ricerca scientifica 1, 00133 Rome, Italy  INFN, Sezione
di Roma2, via della ricerca scientifica 1, 00133 Rome, Italy  Francisco G.
Pedro<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia,
Università di Bologna, via Irnerio 46, 40126 Bologna, Italy INFN, Sezione di
Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy

###### Abstract

We consider quintessence models within 4D effective descriptions of gravity
coupled to two scalar fields. These theories are known to give rise to viable
models of late-time cosmic acceleration without any need for flat potentials,
and so they are potentially in agreement with the dS Swampland conjecture. In
this paper we investigate the possibility of consistently embedding such
constructions in string theory. We identify situations where the quintessence
fields are either closed string universal moduli or non-universal moduli such
as blow-up modes. We generically show that no trajectories compatible with
today’s cosmological parameters exist, if one starts from matter-dominated
initial conditions. It is worth remarking that universal trajectories
compatible with observations do appear, provided that the starting point at
early times is a phase of kinetic domination. However, justifying this choice
of initial conditions on solid grounds is far from easy. We conclude by
studying Q-ball formation in this class of models and discuss constraints
coming from Q-ball safety in all cases analyzed here.

###### Contents

  1. 1 Introduction
  2. 2 Two-field dynamics with non-minimal coupling
  3. 3 Stringy two-field quintessence
  4. 4 Conserved currents and observables in two-field quintessence
  5. 5 Q-ball formation in two-field models with a conserved charge
  6. 6 Conclusions

##  1 Introduction

The current phase of accelerated expansion observed in our universe poses a
surprising challenge for theoretical physicists. On the one hand, the success
of simple proposals like the $\Lambda$CDM model [1, 2, 3] suggests that a
small positive and constant value for the dark energy density (cosmological
constant) driving the aforementioned expansion is the best fit [4, 5],
certainly in accord with Occam’s Razor. On the other hand, due to the global
structure of spacetime, even a free quantum field theory in such a de Sitter
(dS) background is rather different from the flat space theories we are used
to [6]. The main puzzle here being the difficulty of defining an S-matrix that
would allow us to consistently formulate a unitary theory of scattering on
such a background.

In a theory of quantum gravity like string theory, a cosmological constant
must be explained as a positive valued minimum for a scalar field potential.
However, while several string constructions which may lead to such a vacuum
have been proposed, their reliability still remains highly debated (see [7]
for a systematic analysis of pros and cons of some putative dS string vacua).
The difficulty to build explicit metastable dS solutions has even given rise
to the common lore that positive minima of scalar potentials might not even
exist in quantum gravity [8]. This statement is commonly referred to as the dS
swampland conjecture. More precisely, not only does the dS conjecture forbid
the existence of dS extrema but it even restricts the slope of the scalar
potentials arising from string compactifications as to prevent shallow
profiles in regions where they go positive [9]. Further refinements proposed
in [10, 11, 12] involve second derivatives of the scalar potential.

An alternative to the cosmological constant scenario to account for late-time
cosmic acceleration is to allow for a slowly time-varying dark energy, i.e. a
quintessence model (see [13] for a review). Similarly to inflation, the motion
of a scalar field through its potential drives a phase of accelerated
expansion of the universe, its departure from pure dS being due to a non-zero
scalar kinetic energy. For this mechanism to work for a single (real) field,
in general the corresponding potential must have a very flat profile along the
physical trajectory. However [14] showed that slow roll quintessence cannot be
achieved in any parametrically controlled regime of the moduli space of string
theory. Viable quintessence models could therefore exist only in the bulk of
moduli space where numerical, even if not parametric, control could still be
achieved. The systematic analysis of [15] has shown however that quintessence
model building in any numerically controlled regime of the moduli space of
string theory features theoretical and phenomenological challenges which are
even more severe than those typical of dS constructions. This seems to suggest
that single field quintessence, similarly to dS vacua, may be ruled out in
string theory.

Cosmic acceleration can nevertheless arise within models containing steeper
potentials, if one generalizes the prototypical single field model to include
multiple real fields [16, 17]. The key feature that opens up this possibility
turns out be the presence of non-trivial field space curvature. In [18, 19,
20, 21, 22] this mechanism was studied in order to achieve viable descriptions
of late-time acceleration within 4D effective models in compliance with the dS
swampland conjecture. Different classes of potentials and kinetic couplings
turned out to produce desirable phenomenological features and successfully fit
the current cosmological data. The present phase of our universe appears there
as a transient between a (past) phase of matter domination and a (future)
fixed point with equation of state parameter within the range $(-1,-1/3)$.

Given the abundance of scalar fields with non-minimal coupling that
generically stem from a compactification of string theory, all of this may
sound very encouraging when it comes to trying to construct an honest model of
quintessence with a proper stringy origin. The aim of this paper is assessing
to what extent the interesting phenomenology found in [22] can be achieved in
string compactifications. To this end, we analyze typical stringy scalar
potentials and try to apply the same machinery. That means searching for
trajectories which start at a phase of matter domination and undergo a phase
of accelerated expansion, with the cosmological parameters observed today
somewhere on the trajectory. We will mainly consider two different situations.
In the first case the two scalar fields are universal closed string moduli,
such as e.g. axio-dilaton or the universal Kähler modulus in a Calabi-Yau
compactification. In the other case, the scalars are instead non-universal
moduli such as blow-up modes in the internal geometry. In both situations we
find multifield trajectories which start from matter domination and evolve to
a dark energy dominated solution. However generically none of these
accelerating solutions is in agreement with current cosmological observations.
The only trajectories compatible with today’s cosmological observations are
those with initial conditions corresponding to kinetic domination, which are
hard to justify on solid grounds.

Our analysis therefore shows that multifield quintessence, while promising
from the effective 4D point of view, turns out to be hard to realize in
explicit string compactifications. This is not due to a difficulty in finding
accelerating solutions, rather the problem turns out to be reproducing the
observed values of the dark energy density and equation of state parameters.
These findings seem to corroborate the analysis of [14, 15] which suggests
that quintessence model building in string theory is at least as challenging
as the search of dS vacua.

This paper is organized as follows. In Sec. 2 we summarize the main features
of the dynamics of two-field models where the underlying field space is
curved, while in Sec. 3 we analyze the possibility to embed these models in
string compactifications. It may be worth stressing that, in contrast to a
more general class of models analyzed recently by [23], the setup we are
considering here features a conserved charge associated with shift symmetry
for one of the fields. The observational differences between these models is
discussed in Sec. 4. Besides, due to the presence of a conserved charge, semi-
stable Q-balls [24] could form and obstruct cosmic acceleration [25, 26].
Following [25], we investigate the conditions and time-scale for the formation
of Q-balls in Sec. 5. Finally, we implement the constraints coming from Q-ball
safety within our models of interest, and in particular we apply those to the
stringy trajectories found earlier. We conclude in Sec. 6 by making further
remarks and discussing possible developments of our analysis.

##  2 Two-field dynamics with non-minimal coupling

The class of models which are relevant to our analysis is the class studied in
[22], describing the coupling between Einstein gravity and two real scalar
fields. The underlying action is given by

$S[g_{\mu,\nu},\phi]\,=\,\int
d^{4}x\,\sqrt{-g}\,\left(\frac{M_{p}^{2}}{2}\,\mathcal{R}\,-\,\frac{1}{2}\left(\partial\phi_{1}\right)^{2}\,-\,\frac{1}{2}f(\phi_{1})^{2}\left(\partial\phi_{2}\right)^{2}\,-\,V(\phi_{1})\right)\,,$
(2.1)

where the crucial universal feature is the shift symmetry in the $\phi_{2}$
direction arising from the fact that the scalar potential is independent of
the corresponding scalar field. We are interested in the study of the dynamics
of these models in an FRW type background geometry

$ds_{4}^{2}=-dt^{2}+a(t)^{2}\,ds_{\mathbb{R}^{3}}^{2}\qquad\textrm{ and
}\qquad\phi_{i}\,=\,\phi_{i}(t)\ .$ (2.2)

Note that the kinetic coupling $f(\phi_{1})$, like the scalar potential
$V(\phi_{1})$, only depends on the field $\phi_{1}$. The other field
$\phi_{2}$ is therefore a flat direction which only couples kinetically to
$\phi_{1}$. This is typical of a complex field decomposed into polar
coordinates, but also more generally for saxion and axion pairs in flux
compactifications of string theory. The equations of motion are then given by

$\left\\{\begin{array}[]{ll}\ddot{\phi}_{1}+3H\dot{\phi}_{1}-f\,f_{1}\dot{\phi}_{2}^{2}+V_{1}=0\,,\\\\[5.0pt]
\ddot{\phi}_{2}+3H\dot{\phi}_{2}+2\frac{f_{1}}{f}\dot{\phi}_{2}\dot{\phi}_{1}=0\,,\end{array}\right.$
(2.3)

where $\cdot\equiv d/dt$, $H\equiv\dot{a}/a$ and $f_{1}\equiv\partial
f/\partial\phi_{1}$ (and similarly for $V$). For the late universe, in the
presence of a barotropic fluid with pressure111We shall later assume a
background of pressureless dust $\omega_{b}=0$ but keep it in the equations
here for completeness. $p_{\gamma}=\omega_{b}\rho_{\gamma}$ that evolves
according to the continuity equation

$\dot{\rho}_{b}=-3H(1+\omega_{b})\rho_{b}\,,$ (2.4)

the corresponding Friedman equation is

$3M_{p}^{2}H^{2}=\frac{1}{2}\,\dot{\phi}_{1}^{2}+\frac{1}{2}\,f^{2}\dot{\phi}_{2}^{2}+V+\rho_{b}\,.$
(2.5)

It turns out to be particularly useful to introduce the following dynamical
variables for the kinetic coupling

$k_{1}(\phi_{1})\equiv-M_{p}\frac{f_{1}}{f}\,$ (2.6)

and potential

$k_{2}(\phi_{1})\equiv-M_{p}\frac{V_{1}}{V}\,.$ (2.7)

The special case in which $k_{1}$ and $k_{2}$ are constant was already
analyzed in [21], and it corresponds to a situation where both the kinetic
coupling and the potential are chosen to be exponential functions of
$\phi_{1}$, namely $f(\phi_{1})=e^{-k_{1}\phi_{1}/M_{p}}$ and
$V(\phi_{1})=V_{0}\,e^{-k_{2}\phi_{1}/M_{p}}$. More general functions were the
subject of further investigation in [22]. It was shown that the kinetic
coupling of the massless scalar generates a novel fixed point of the dynamics,
and allows for accelerating solutions even when the scalar potential is steep
($k_{2}\gg 1$).

Following [27] we define the dimensionless variables

$x_{1}\equiv\frac{\dot{\phi}_{1}}{\sqrt{6}HM_{p}}\,,\qquad
x_{2}\equiv\frac{f\dot{\phi}_{2}}{\sqrt{6}HM_{p}}\,,\qquad
y_{1}\equiv\frac{\sqrt{V}}{\sqrt{3}HM_{p}}\,,$ (2.8)

which allow us to formulate the dynamics of the system as an autonomous system

$\displaystyle x_{1}^{\prime}$ $\displaystyle=$ $\displaystyle
3x_{1}(x_{1}^{2}+x_{2}^{2}-1)+\sqrt{\frac{3}{2}}(-2k_{1}x_{2}^{2}+k_{2}y_{1}^{2})-\frac{3}{2}\gamma
x_{1}(x_{1}^{2}+x_{2}^{2}+y_{1}^{2}-1)\,,$ (2.9) $\displaystyle
x_{2}^{\prime}$ $\displaystyle=$ $\displaystyle
3x_{2}\left(x_{1}^{2}+x_{2}^{2}-1\right)+\sqrt{6}k_{1}x_{1}x_{2}-\frac{3}{2}\gamma
x_{2}\left(x_{1}^{2}+x_{2}^{2}+y_{1}^{2}-1\right),$ (2.10) $\displaystyle
y_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{3}{2}}k_{2}x_{1}y_{1}-\frac{3}{2}\gamma
y_{1}\left(x_{1}^{2}+x_{2}^{2}+y_{1}^{2}-1\right)+3y_{1}\left(x_{1}^{2}+x_{2}^{2}\right),$
(2.11)

where $\gamma=1+\omega_{b}$ and ${}^{\prime}\equiv\frac{d}{d\ln a}$ determines
the evolution in terms of e-foldings. In the general case of field dependent
$k_{1}$ and $k_{2}$, these equations are to be supplemented by

$\phi_{1}^{\prime}=6\,x_{1}\,.$ (2.12)

Instantaneous fixed points are solutions compatible with
$x_{1}^{\prime}=x_{2}^{\prime}=y_{1}^{\prime}=0$. Note that, unless $k_{1}$
and $k_{2}$ are constant, these are not actual fixed points of the above set
of equations, but are nonetheless useful for describing the underlying
dynamics, provided that the energy densities of the system evolve faster than
the couplings $k_{1}$ and $k_{2}$. According to the analysis in [21], there
are six instantaneous fixed points, which we list in Tab. 1 for the sake of
completeness. The system is symmetric under simultaneous
$\phi_{1}\rightarrow-\phi_{1}$, $k_{1}\rightarrow-k_{1}$ and
$k_{2}\rightarrow-k_{2}$, and therefore one can focus on the $k_{2}>0$ half-
plane without loss of generality.

| description | $x_{1}$ | $x_{2}$ | $y_{1}$ | $\Omega_{\phi}$ | $\omega_{\phi}$ | stability
---|---|---|---|---|---|---|---
$\mathcal{K}_{\pm}$ | kinetic dom. | $\pm 1$ | $0$ | $0$ | 1 | 1 | unstable
$\mathcal{F}$ | fluid dom. | $0$ | $0$ | $0$ | 0 | - | unstable
$\mathcal{S}$ | scaling sol. | $\frac{\sqrt{3/2}}{k_{2}}$ | $0$ | $\frac{\sqrt{3/2}}{k_{2}}$ | $\frac{3}{k_{2}^{2}}$ | $0$ |  $k_{2}^{2}\geq 3$ and $k_{2}>2k_{1}$
$\mathcal{G}$ | geodesic | $\frac{k_{2}}{\sqrt{6}}$ | $0$ | $\sqrt{1-\frac{k_{2}^{2}}{6}}$ | $1$ | $-1+\frac{k_{2}^{2}}{3}$ | $\left\\{\begin{matrix}k_{2}<\sqrt{3}\,,\,\,k_{1}>0\,,\\\ k_{2}<\sqrt{k_{1}^{2}+6}-k_{1}\\\ \end{matrix}\right.$
$\mathcal{NG}$ | non-geodesic | $\frac{\sqrt{6}}{(2k_{1}+k_{2})}$ | $\frac{\pm\sqrt{k_{2}^{2}+2k_{2}k_{1}-6}}{2k_{1}+k_{2}}$ | $\sqrt{\frac{2k_{1}}{2k_{1}+k_{2}}}$ | $1$ | $\frac{k_{2}-2k_{1}}{k_{2}+2k_{1}}$ | $\sqrt{6+k_{1}^{2}}-k_{1}<k_{2}<2k_{1}$

Table 1: Instantaneous fixed points of the system with a pressureless
barotropic fluid $\omega_{b}=0$ and two scalar fields for $k_{2}>0$. The
stability region is given in the last column.

In order to test the model against late time observations, it may be
enlightening to directly study the time evolution of the relevant late time
observables. We denote the equation of state parameter for the quintessence
field by $\omega_{\phi}$, and the corresponding density parameter by
$\Omega_{\phi}$. Re-expressing them in terms of our dynamical variables
$\left(x_{1},x_{2},y_{1}\right)$, we find

$\omega_{\phi}\,=\,\frac{p_{\phi}}{\rho_{\phi}}\,=\,\frac{x_{1}^{2}+x_{2}^{2}-y_{1}^{2}}{x_{1}^{2}+x_{2}^{2}+y_{1}^{2}}\,,$
(2.13)

and

$\Omega_{\phi}\,=\,x_{1}^{2}+x_{2}^{2}+y_{1}^{2}\,.$ (2.14)

The evolution of these quantities obeys the differential equations

$\begin{split}\Omega_{\phi}^{\prime}&=-3\left(\Omega_{\phi}-1\right)\Omega_{\phi}(\omega_{b}-\omega_{\phi})\,,\\\
\omega_{\phi}^{\prime}&=(\omega_{\phi}-1)\left(-k_{2}\sqrt{3(\omega_{\phi}+1)\Omega_{\phi}-6x_{2}^{2}}+3(1+\omega_{\phi})\right).\end{split}$
(2.15)

It is immediately clear from Tab. 1 that the fixed points cannot describe our
current universe with the cosmological observables $\Omega_{\phi}^{(0)}\sim
0.7$ and $\omega_{\phi}^{(0)}\sim-1$. We should instead look for transients in
the $(x_{1},x_{2},y_{1})$ parameter space which start from matter domination,
exhibit a (short) period of accelerated expansion222 The number of e-foldings
from the end of matter domination until today may be roughly estimated to be
$\Delta
N=\ln\left(\nicefrac{{a_{0}}}{{a}}\right)\sim\frac{1}{3}\,\ln\left(\nicefrac{{\Omega_{\Lambda}^{(0)}}}{{\Omega_{\textrm{m}}^{(0)}}}\right)\sim
0.28$. before passing through our current universe. The universal regime of
matter domination occurs when the barotropic fluid is the dominant
contribution in the energy-momentum tensor. This implies the necessity of
choosing initial conditions close to the origin $(0,0,0)$, where both
$\omega_{\phi}$ and $\Omega_{\phi}$ are small.

Summarizing, in [22] various different choices of functional dependence for
the scalar potential and the kinetic coupling were investigated. Generically
it was observed that accessing the stability region for the non-geodesic fixed
point produces trajectories which display cosmic acceleration in the transient
path towards the fixed point itself. This was found to hold both in the case
where $k_{1}$ and $k_{2}$ are constant and when they vary with $\phi_{1}$,
without substantial difference at a qualitative level. However empirically, an
$\mathcal{O}(10)$ hierarchy between $k_{1}$ and $k_{2}$, i.e.
$k_{1}\gtrsim\mathcal{O}(10)\,k_{2}$, seemed to be needed in order to obtain
late time observables compatible with their current experimental range. This
particular feature might already intuitively generate the suspicion that a
possible compatibility of these constructions with a proper stringy embedding
may be contrived.

As we will see later on, it turns out to be difficult to shoot from the origin
and have full control over all the possible transients. An alternative
approach could be to require a transient which is compatible with current
observations in the first place, and integrate backwards to see the past
trajectory and associated initial conditions. It may be worth stressing that,
although we only have two observables to constrain the initial value of three
coordinates, eqs. (2.13) and (2.14) fully determine the observationally viable
region to be around $\left(0,0,\sqrt{\Omega_{\phi}^{(0)}}\right)$.

##  3 Stringy two-field quintessence

The goal of this section is to investigate concrete string inspired two-field
models and assess the existence of viable accelerating trajectories in
agreement with today’s observations. As we have already anticipated earlier,
the possibility of producing a hierarchy between $k_{1}$ and $k_{2}$ appears
rather unnatural within a stringy context. Indeed, we will show the absence of
quintessence models compatible with the data, both within a class of examples
where the quintessence fields are universal moduli, and within a different
class in which the same role is played by blow-up modes. Note that this result
crucially relies on the assumption of matter-dominated initial conditions
$\Omega_{\phi}\sim 0$. In contrast, we will also show that good fits for
current observations are in principle possible if such an assumption is
dropped.

### Universal closed string moduli

Effective Lagrangians of the form (2.1) can arise in the context of string
compactifications. Consider for example a situation where all but one closed
string moduli are stabilized at specific vacuum expectation values by means of
suitable internal fluxes and sources. This leaves one single complex field,
whose real and imaginary parts are then to be mapped into our real fields
$(\phi_{1},\phi_{2})$.

A particularly thorough construction of this type was presented in [28], where
type IIB compactifications on Calabi-Yau 3-folds were considered, with the
addition of non-vanishing internal R-R and NS-NS 3-form fluxes. Flux
quantization was properly discussed, and tadpole cancellation was found to
require spacetime filling O3 planes and parallel D3-branes. In this context,
the authors were able to show that, for some explicit choices of the flux
quanta, it was actually possible to stabilize all complex structure moduli and
the axio-dilaton at a non-supersymmetric locus. This generates a runaway for
the overall Kähler modulus $T={\rm Vol}(\Sigma_{4})+{\rm
i}\int_{\Sigma_{4}}C_{(4)}$ where ${\rm Vol}(\Sigma_{4})$ is the volume of the
internal overall 4-cycle $\Sigma_{4}$ in string units, while $C_{(4)}$ is the
R-R 4-form. Though the explicit solutions that they provide are for the
special choice of Calabi-Yau given by a $\mathbb{Z}_{2}$ orbifold of the six-
torus, similar examples exist for Calabi-Yau 3-folds [29]. The constructions
involve $0$, $2$ or $4$ D3 branes, a positive vacuum energy and all the modes
but $T$ have been stabilized.

The effective 4D description turns out to be given by $\mathcal{N}=1$
supergravity coupled to chiral multiplets. The corresponding flux-induced
superpotential is independent of the overall volume modulus, and hence a no-
scale structure emerges. Once all the chiral fields but the volume $T$ are
stabilized, the effective description only involves one single complex field,
for which the kinetic Lagrangian is fully specified by the following Kähler
potential

$K=-3\ln\left(T+\overline{T}\right),$ (3.1)

while the effective scalar potential reads

$V_{\rm eff}=\frac{V_{0}}{\left(T+\overline{T}\right)^{3}}\,M_{p}^{4}\,,$
(3.2)

where $V_{0}>0$ appears after evaluating all the other scalar fields in their
constant vacuum expectation values. After choosing the convenient
parametrization333The parametrization introduced here yields a canonically
normalized $\phi_{1}$, while the kinetic term for $\phi_{2}$ couples
exponentially to $\phi_{1}$.
$T\,=\,e^{\sqrt{\frac{2}{3}}\phi_{1}/M_{p}}\,+\,{\rm
i}\,\frac{\phi_{2}}{M_{p}}$, one precisely recovers exponential potential and
kinetic coupling, with

$k_{1}=\sqrt{\frac{2}{3}}\qquad\text{and}\qquad k_{2}=\sqrt{6}\,.$ (3.3)

While in [28, 29] these compactifications with run-away volume modulus were
considered with the aim of stabilizing the remaining directions by adding
higher derivative or non-perturbative effects, we want to use them to
construct two-field quintessence configurations instead.

Beyond the above stringy type IIB model, one might think of considering a
whole class of string inspired models of the same type, by simply considering
the following Kähler potential for a chiral superfield $X$

$K=-p\ln\left(X+\overline{X}\right)\,+\,\dots\,,$ (3.4)

where $X$ is no longer assumed to be identified with the volume modulus, but
is instead allowed to be any complex field appearing in the compactification
at hand, and which has not yet been stabilized. The dots in (3.4) denote $X$
independent contributions. This type of Kähler potential is quite common for
closed string moduli. Should in addition the superpotential $W$ be
independent444Note that this is an immediate generalization of what happens in
the no-scale supergravity model arising in type IIB Calabi-Yau orientifolds,
as shown in [30, 31]. of $X$, $\partial_{X}W=0$, the supergravity F-term
potential

$V_{F}=e^{K}\left(K^{X\overline{X}}D_{X}WD_{\overline{X}}\overline{W}-3|W|^{2}\,+\,\dots\right)M_{p}^{4}$
(3.5)

would then turn out to depend on $X$ only through the $e^{K}$ factor, and
hence we have

$V_{F}\propto\left(X+\overline{X}\right)^{-p}\,.$ (3.6)

Note that the above class contains the type IIB case of [28] discussed above,
for which $p=3$. Another prototypical situation within this class is given by
heterotic compactifications on $\mathrm{SU}(3)$ structure manifolds. The
associated 4D effective description is again a minimal supergravity coupled to
chiral multiplets, but this time the role of the field $X$ not appearing in
the superpotential is given by $S=e^{-\varphi}+{\rm i}\,a$ where $\varphi$ is
the 4D dilaton while its axionic counterpart $a$ is the 4D dual of $B_{(2)}$,
$\star da=dB_{(2)}$. Also in this case one could imagine stabilizing all of
the complex structure and Kähler moduli, leaving the $S$ direction as a
runaway (see [32] for a systematic analysis of heterotic moduli
stabilization). This would then, by construction, fit our discussion when
$p=1$.

The case with $p=2$ can be reproduced instead by considering type IIB
compactified on orientifolds of CY 3-folds which are K3 or $T^{4}$ fibrations
over a $\mathbb{P}^{1}$ base. In this situation the CY volume in string units
reads $\mathcal{V}={\rm Vol}(K3){\rm
Vol}(\mathbb{P}^{1})=\sqrt{\tau_{1}}\tau_{2}$ where ${\rm Vol}(K3)=\tau_{1}$
and ${\rm Vol}(\mathbb{P}^{1})=\tau_{2}/\sqrt{\tau_{1}}$ [33, 34, 35]. Clearly
$K=-2\ln\mathcal{V}$ reproduces the $p=2$ case if $X$ is identified with
$T_{2}=\tau_{2}+{\rm i}\int_{\Sigma_{4}^{(2)}}C_{(4)}$ and $T_{1}$, together
with all the other geometric moduli, is stabilized at a non-supersymmetric
locus. Finally, for other intermediate values of $p$, one could imagine
starting from the model with seven chiral multiplets arising from
compactification of M-theory on a $\mathbb{Z}_{2}^{3}$ orbifold of the seven-
torus, and start identifying some of the fields among them. The extreme $p=7$
case is obtained by identifying all of them in a single field $Z={\rm
Vol}(\Sigma_{3})+{\rm i}\int_{\Sigma_{3}}A_{(3)}$ where ${\rm
Vol}(\Sigma_{3})$ is the volume of an internal 3-cycle $\Sigma_{3}$ in 11D
units, while $A_{(3)}$ is the M-theory 3-form. A panoramic view of relevant
stringy examples yielding models within this class is offered in Tab. 2.

$p$ | $X$ | Theory | Sources | $\mathcal{M}_{\rm internal}$ | References
---|---|---|---|---|---
$1$ | $S=e^{-\varphi}+{\rm i}\,a$ | Heterotic | — |  $\mathrm{SU}(3)$ str. | [36]
$2$ | $T_{2}={\rm Vol}(\Sigma_{4}^{(2)})+{\rm i}\int_{\Sigma_{4}^{(2)}}C_{(4)}$ | Type IIB | D3/D7, O3/O7 |  K3-fibered $\mathrm{CY}_{3}$ | [33, 34, 35]
$3$ | $T={\rm Vol}(\Sigma_{4})+{\rm i}\int_{\Sigma_{4}}C_{(4)}$ | Type IIB | D3/O3 | $\mathrm{CY}_{3}$ | [28]
$7$ | $Z={\rm Vol}(\Sigma_{3})+{\rm i}\int_{\Sigma_{3}}A_{(3)}$ | M-theory | KK6/KKO6 |  $\mathrm{G}_{2}$ str. | [37]
Table 2: Examples of string compactifications yielding 4D two-field models
with exponential kinetic coupling and exponential runaway potential. The two-
field models arise after fixing the rest of the moduli present in the given
setup.

Note that $V_{F}$ as illustrated in (3.6) only depends on the real part of
$X$, leaving the imaginary part as a flat direction. This is in perfect
agreement with our construction of the two-field models described above, where
the kinetically coupled field has a flat potential. The kinetic part of the
action reads

$\mathcal{L}_{\textrm{kin}}=-\sqrt{-g}\,K_{X\overline{X}}\,\partial
X\partial\overline{X}\,M_{p}^{2}=-\frac{1}{2}\sqrt{-g}\,\left((\partial\phi_{1})^{2}+\frac{p}{2}\
e^{2\sqrt{2/p}\ \phi_{1}/M_{p}}(\partial\phi_{2})^{2}\right),$ (3.7)

where

$\frac{\phi_{1}}{M_{p}}\equiv\,\sqrt{\frac{p}{2}}\ln\left(\frac{X+\overline{X}}{2}\right)\qquad\text{and}\qquad\frac{\phi_{2}}{M_{p}}\equiv\frac{X-\overline{X}}{2\,{\rm
i}}\,.$ (3.8)

One can therefore define the kinetic coupling function as

$f(\phi_{1})=\sqrt{\frac{p}{2}}\,e^{\sqrt{2/p}\ \phi_{1}/M_{p}}\,.$ (3.9)

The potential depends on $\phi_{1}$ as

$V(\phi_{1})=\frac{V_{0}}{\left(X+\overline{X}\right)^{p}}=\frac{V_{0}}{2^{p}}\,e^{-\sqrt{2p}\
\phi_{1}/M_{p}}\,.$ (3.10)

So we see that in these simple supergravity models $k_{1}$ and $k_{2}$ are
constant and their values are expressed in terms of $p$ as

$k_{1}=\sqrt{2/p}\qquad\text{and}\qquad k_{2}=\sqrt{2p}\,.$ (3.11)

Let us now look at the qualitative and quantitative features of dynamical
trajectories. At first glance, from (3.11) we can already see that
$k_{2}/k_{1}=p$, which rules out the desired hierarchy that made it possible
in [22] to achieve phenomenologically viable transients. This confirms our
initial suspicions, at least for integer values of $p$, which are those
admitting a clear stringy interpretation. Indeed, if we first start by giving
initial conditions near matter domination, we find the trajectories displayed
in the left panel of Fig. 1. As is evident from there, though $p=1$ admits an
accelerating regime when $\Omega_{\phi}=0.7$, neither of the $p=1,2,3$ cases
allows for a viable description of the present day universe. This is due to
the fact that in regimes close to $\Omega_{\phi}=0.7$, the equation of state
parameter deviates too much from $-1$. If we considered higher values of $p$,
the associated curves would bend even more sharply down to the right, only
making the situation worse.

Figure 1: Dynamical trajectories in the $(\omega,\Omega)$ plane for stringy
models with $k_{1}$ and $k_{s}$ specified by (3.11). Left: Initial condition
are chosen to be matter domination. None of the trajectories go through the
black dot representing the correct current value for late time observables.
Right: The trajectories are required to pass through the observed current
values of $(\omega,\Omega)$.

In line with what anticipated above, in order to have a viable transient
through the observationally preferred region of the $(\omega,\Omega)$ plane,
one would need to have $p\leq 0.2$, which rules out the possibility of
$\mathcal{NG}$ dynamics playing a role in this class of models. Needless to
say, such small and non-integer values for $p$ are difficult to motivate from
a UV point of view. Note that this upper bound on $p$ corresponds to
$k_{2}<0.6$ as found in [38].

However, if we instead look for trajectories that pass through observationally
viable points, we find that these do exist. They may be found by using the
correct current values of late time observables as conditions to integrate the
equations (2.9), (2.10) and (2.11). By repeating the analysis for $p=1,2,3$,
we see that the future fixed points differ but the behavior in the past is
qualitatively similar for every $p$. This situation is shown in the right
panel of Fig. 1.

Figure 2: Dynamical trajectories in the $(x_{1},x_{2},y_{1})$ space for
stringy models with $k_{1}$ and $k_{s}$ specified by (3.11). Left: Initial
condition are chosen to be matter domination. All trajectories start from the
center of the hemisphere and approach the surface without going through the
black dot. Right: All trajectories pass through the correct today’s values of
late time observables. In this case all trajectories originate from kinetic
domination initial conditions.

In Fig. 2, we display the same solutions but in the $(x_{1},x_{2},y_{1})$
space, for matter domination initial conditions (left panel) and for
trajectories which feature transients compatible with today’s observations
(right panel). All the trajectories in the right plot feature kinetic
domination initial conditions since they move towards the observable point
from the kinetic dominated regime $\mathcal{K}_{-}$, and universally originate
from $\mathcal{K_{+}}$ in the asymptotic past, as long as $x_{2}^{\rm
today}\neq 0$. While these trajectories do not originate from matter
domination in the sense of fluid domination, there are points along the past
trajectory which could be interpreted as matter dominated. In particular, the
points on the trajectories where $\omega_{\phi}=0$ could be indistinguishable
from pure matter cosmology since the quintessence field mimics the barotropic
fluid with $\omega_{b}=0$. In this sense, one might say that in this
cosmological phase the universe would effectively behave as a matter dominated
universe, where $\phi$ counts as part of (actually roughly half of) the matter
content of the universe. While these phases are generically present, it is
hard to motivate them as initial conditions in comparison with the more
standard case of fluid domination initial conditions.

### Non-universal moduli: blow-up modes

In the context of Calabi-Yau compactifications, particularly simple examples
that one can work with are toroidal orbifolds. These geometries present
singularities at the location of the fixed points of the corresponding
orbifold action. This issue is overcome via the blow-up procedure which
provides an explicit prescription for resolving these singularities into
compact and smooth manifolds, at the price of introducing a number $N$ of non-
universal moduli controlling the size of these blow-up modes. Effective 4D
models for blow-up modes can be obtained from a class of type IIB orientifold
flux compactifications with D3/O3 as well as D7/O7 spacetime filling sources.
The corresponding internal manifold is a Calabi-Yau 3-fold of a so-called
‘weak Swiss-cheese’ form [39, 40]. In this case, the total internal volume may
be written as

$\mathcal{V}\,=\,f_{3/2}(\tau_{j})\,-\,\sum\limits_{i=1}^{N}\lambda_{i}\tau_{i}^{3/2}\,,$
(3.12)

where $\left\\{\tau_{j}\right\\}$ are the universal moduli controlling the
volume, $f_{3/2}$ is a homogeneous function of degree $3/2$, while
$\left\\{\tau_{i}\right\\}$ are blow-up moduli controlling the volume of
‘diagonal del Pezzo’ divisors [33]. Let us focus for simplicity on the
simplest case with just one universal modulus $\tau_{b}$ and one blow-up mode
$\tau_{s}$ with $\tau_{s}\ll\tau_{b}$. This hierarchy can be achieved if at
leading order $\tau_{b}$ is stabilized exponentially large by balancing
$\alpha^{\prime}$ against loop corrections as in [41, 42]. The non-universal
modulus $\tau_{s}$ can then be fixed at subleading order by the interplay
between higher order perturbative corrections as in [43]. At this level of
approximation, the axionic partner of $\tau_{s}$ is still massless due to its
perturbative shift symmetry. This situation therefore would be described by
the Lagrangian (2.1) where $\phi_{1}$ is the canonically normalized blow-up
mode and $\phi_{2}$ the corresponding axion.

In the $\tau_{s}\ll\tau_{b}$ limit, the effective Kähler potential for
$\tau_{s}$ is power-law rather than logarithmic

$K=-2\ln\mathcal{V}\simeq-3\ln\tau_{b}+2\left(\frac{\tau_{s}}{\tau_{b}}\right)^{3/2}\,,$
(3.13)

implying the following form of the kinetic terms (for
$\tau_{s}=(T_{s}+\overline{T}_{s})/2$)

$\mathcal{L}_{\textrm{kin}}=-\sqrt{-g}\,K_{T_{s}\overline{T}_{s}}\,\partial
T_{s}\partial\overline{T}_{s}\,M_{p}^{2}=-\frac{1}{2}\sqrt{-g}\,\left((\partial\phi_{1})^{2}+\left(\frac{M_{p}}{\phi_{1}}\right)^{2/3}(\partial\phi_{2})^{2}\right),$
(3.14)

where

$\frac{\phi_{1}}{M_{p}}\equiv\,\frac{2}{\sqrt{3}}\left(\frac{\tau_{s}}{\tau_{b}}\right)^{3/4}\qquad\text{and}\qquad\frac{\phi_{2}}{M_{p}}\equiv\left(\frac{3}{4}\right)^{1/3}\frac{(T_{s}-\overline{T}_{s})}{2\,{\rm
i}\,\tau_{b}}\,.$ (3.15)

Thus the kinetic coupling function is given by

$f(\phi_{1})=\left(\frac{M_{p}}{\phi_{1}}\right)^{1/3}\,.$ (3.16)

The scalar potential generated by perturbative corrections (string loops or
higher derivative effects) has been studied in concrete Calabi-Yau orientifold
examples with explicit brane setup and tadpole cancellation [44, 34, 35] and
can take slightly different forms. Its behavior is always power-law

$V(\phi_{1})=V_{0}\left(\frac{M_{p}}{\phi_{1}}\right)^{\pm
2/3}\qquad\text{or}\qquad
V(\phi_{1})=\frac{V_{0}}{C-\left(\phi_{1}/M_{p}\right)^{2/3}}$ (3.17)

where $C$ is a positive $\phi_{1}$-independent quantity which in explicit
Calabi-Yau models turns out to be of $\mathcal{O}(1)$. These models motivate
the following more general class of choices for $f$ and $V$

$f(\phi_{1})=\left(\frac{M_{p}}{\phi_{1}}\right)^{p_{1}}\qquad\text{and}\qquad
V(\phi_{1})\,=\left(\frac{M_{p}}{\phi_{1}}\right)^{p_{2}}\frac{V_{0}}{C\left(M_{p}/\phi_{1}\right)^{p_{2}}-1}\,,$
(3.18)

with $p_{1}>0$, $p_{2}>0$ for $C\neq 0$, and $p_{2}$ which can have either
sign for $C=0$. This leads to

$\frac{k_{2}}{k_{1}}=\left(\frac{p_{2}}{p_{1}}\right)\frac{1}{1-C\left(M_{p}/\phi_{1}\right)^{p_{2}}}\,.$
(3.19)

If $C=0$, accelerating multifield solutions with observationally viable
transients could in principle exist if $p_{2}>0$ but they would require
$k_{2}/k_{1}\lesssim 0.1$.555For $p_{2}<0$ the ratio $k_{2}/k_{1}$ is
negative, implying that no non-geodesic dynamics can be present according to
the analysis of [22]. This regime could be achieved for $p_{2}\ll p_{1}$ but
explicit stringy examples do not seem to reproduce this hierarchy. In fact,
the concrete Calabi-Yau models of [44, 34, 35] would lead to $k_{2}/k_{1}=2$.
On the other hand, if $C\neq 0$, the term proportional to $C$ in (3.19) would
dominate since $\phi_{1}\ll M_{p}$ in the $\tau_{s}\ll\tau_{b}$ limit, as can
be seen from (3.15). In this case (3.19) would however be negative for
$p_{2}>0$, leading to the absence of any non-geodesic dynamics as shown in
[22]. Notice that, even allowing $p_{2}<0$, the system would still not feature
any phenomenologically viable transient since $k_{2}/k_{1}$ would become:

$\frac{k_{2}}{k_{1}}\simeq\left(\frac{|p_{2}|}{Cp_{1}}\right)\left(\frac{M_{p}}{\phi_{1}}\right)^{|p_{2}|}\gg
1\qquad\text{for}\qquad\phi_{1}\ll M_{p}\,,$ (3.20)

unless $|p_{2}|\ll 1$ which is however very unnatural from the UV point of
view.

We have therefore shown that blow-up modes enrich the class of multifield
models by including power-law potentials and kinetic couplings together with a
field-dependent $k_{2}/k_{1}$ ratio. Unfortunately, they do not lead to
qualitatively new results regarding the possibility to achieve a non-geodesic
multifield dynamics which reproduces the observational features of today’s
universe.

##  4 Conserved currents and observables in two-field quintessence

In this section we clarify the relation between two-field quintessence models
that feature a conserved current like the one under consideration [45], and
those that do not [46]. In order to do so we will try to reduce the problem to
a single field one. In the former case this can be done by exploiting the
conserved charge to eliminate the light degree of freedom from the action,
thereby obtaining an action with a time-dependent effective potential. In the
latter case, whenever there is a mass hierarchy (be it at the background or
perturbation level) one can integrate out the heavy degree of freedom to find
a low energy effective action. This procedure allows us to show that, though
the two classes of models descend from a similar Lagrangian, they are in fact
observationally distinguishable.

The starting point for both models is an action which generalizes (2.1) by
allowing the potential to depend on both fields, i.e.
$V=V(\phi_{1},\phi_{2})$. The first equation of motion in (2.3) is unaltered,
while the second acquires an additional contribution proportional to the
derivative of $V$ with respect to $\phi_{2}$

$\ddot{\phi}_{2}+3H\dot{\phi}_{2}+2\frac{f_{1}}{f}\dot{\phi_{2}}\dot{\phi_{1}}=-\frac{V_{2}}{f^{2}}\,.$
(4.1)

When $V_{2}$ vanishes identically, i.e. for our original model (2.1), there is
a conserved current and and associated conserved charge

$J^{\mu}=\sqrt{-g}f^{2}\partial^{\mu}\phi_{2}\qquad\text{and}\qquad Q=\int
d^{3}xJ^{0}\,.$ (4.2)

Using the charge density $J^{0}=a^{3}f^{2}\dot{\phi_{2}}\equiv q$ one can
eliminate $\dot{\phi_{2}}$ from the first equation in (2.3), thereby obtaining
the effective equation of motion

$\ddot{\phi}_{1}+3H\dot{\phi}_{1}-\frac{f_{1}q^{2}}{f^{3}a^{6}}+V_{1}=0\,,$
(4.3)

which prompts the definition of a time-dependent ‘effective potential’

$V_{\rm eff}(\phi_{1},a)=V(\phi_{1})+\frac{q^{2}}{2f^{2}a^{6}}\,.$ (4.4)

The main point is that, regardless of the time dependence of $V_{\rm eff}$,
eq. (4.3) can be obtained from a two derivative Lagrangian, and therefore
perturbations in such a background propagate at the speed of light. In other
words the speed of sound is unity: $c_{s}=1$.

Let us now turn our attention to the $V_{2}\neq 0$ case analyzed in [46]. We
define $m_{1}^{2}\equiv V_{11}$ and assume that $m_{1}\gg H$ such that
$\phi_{1}$ is a heavy field, and can therefore be integrated out. We are
interested in slow-roll solutions to this system ($\ddot{\phi}_{1}\approx 0$
and $\ddot{\phi}_{2}\approx 0$) with negligible velocity for the heavy field
$\dot{\phi}_{1}\approx 0$. In this regime, the equation of motion for the
heavy scalar reads

$f\,f_{1}\,\dot{\phi}_{2}^{2}\approx V_{1}\,.$ (4.5)

Solving this equation allows us to integrate out the heavy field, thereby
obtaining an effective action for the light scalar driving quintessence. As we
now show following [47], this action involves higher derivative terms of the
form $(\dot{\phi_{2}})^{2n}$. Let us define

$\phi_{1}\equiv\langle\phi_{1}\rangle(1+\delta)\,,$ (4.6)

where $\langle\phi_{1}\rangle$ is the solution to $V_{1}=0$ when $\phi_{1}$
relaxes to its minimum after the end of the quintessence epoch. During dark
energy domination, the field $\phi_{2}$ is displaced from its minimum, and so
the solution to $V_{1}=0$ is shifted to
$\phi_{1}=\langle\phi_{1}\rangle(1+c_{0})$ with $c_{0}\ll 1$ since the mass of
$\phi_{1}$ is much larger than $H$. Moreover (4.5) implies that $\phi_{1}$
does not sit at the effective minimum of the potential during quintessence
since the velocity of $\phi_{2}$, even if tiny, is non-zero. Hence throughout
the whole dynamics $\delta\ll 1$ and can be expanded in powers of
$\dot{\phi}_{2}^{2}$ as

$\delta=\sum_{n\geq
0}c_{2n}\left(\frac{\dot{\phi}_{2}^{2}}{\Lambda^{4}}\right)^{n}\,.$ (4.7)

One can then solve eq. (4.5) order-by-order in the velocity of the light
field. Expanding in powers of $\dot{\phi}_{2}$ yields

$\begin{split}\bar{f}\bar{f}_{1}\dot{\phi}_{2}^{2}&+c_{2}\langle\phi_{1}\rangle\left(\bar{f}_{1}^{2}+\bar{f}\bar{f}_{11}\right)\frac{\dot{\phi}_{2}^{4}}{\Lambda^{4}}+\dots\\\\[3.0pt]
&=\,\bar{V}_{1}+c_{2}\langle\phi_{1}\rangle\bar{V}_{11}\frac{\dot{\phi}_{2}^{2}}{\Lambda^{4}}+\langle\phi_{1}\rangle\left(c_{4}\bar{V}_{11}+\frac{c_{2}^{2}}{2}\langle\phi_{1}\rangle\bar{V}_{111}\right)\frac{\dot{\phi}_{2}^{4}}{\Lambda^{8}}+\dots\,,\end{split}$
(4.8)

where the dots on both sides stand for neglected
$\mathcal{O}\left({\dot{\phi}_{2}}^{6}\right)$ terms, and we defined the
notation $\bar{f}\equiv f(\langle\phi_{1}\rangle(1+c_{0}))$ and $\bar{V}\equiv
V(\langle\phi_{1}\rangle(1+c_{0}),\phi_{2})$, and similarly for the
corresponding derivatives with respect to $\phi_{1}$. From eq. (4.8) one finds
the expressions for the $c_{2n}$ coefficients. The first three terms take the
form

$\begin{split}&V_{1}\left(\langle\phi_{1}\rangle(1+c_{0}),\phi_{2}\right)=0\,,\\\
&c_{2}=\frac{\bar{f}\bar{f}_{1}\Lambda^{4}}{\langle\phi_{1}\rangle\bar{V}_{11}}\,,\\\
&c_{4}=\frac{2\left(\bar{f}_{1}^{2}+\bar{f}\bar{f}_{11}\right)\bar{V}_{11}-\bar{f}\bar{f}_{1}\bar{V}_{111}}{2\langle\phi_{1}\rangle\bar{V}_{11}^{3}}\,\bar{f}\bar{f}_{1}\Lambda^{8}\,.\end{split}$
(4.9)

The first equation in (4.9) implicitly defines $c_{0}$ and implies that, to
leading order, the heavy field adiabatically follows its $\phi_{2}$-dependent
minimum. One can use the solution to the heavy field equation of motion, eqs.
(4.6), (4.7) and (4.9), to integrate it out and find the effective field
theory for the light scalar degree of freedom at the level of the background.

By ignoring the contribution of the heavy field kinetic term, one can write
the single-field effective action as

$\begin{split}\mathcal{L}_{\rm
eff}=\sqrt{-g}\Bigg{(}&\frac{1}{2}\bar{f}^{2}\dot{\phi}_{2}^{2}-\bar{V}(\phi_{2})+\frac{1}{2}\frac{(\bar{f}\bar{f}_{1})^{2}}{\bar{V}_{11}}\dot{\phi}_{2}^{4}\\\
&+\frac{1}{6}\frac{(\bar{f}\bar{f}_{1})^{2}}{\bar{V}_{11}^{3}}\left[3\left(\bar{f}_{1}^{2}+\bar{f}\bar{f}_{11}\right)\bar{V}_{11}-\bar{f}\bar{f}_{1}\bar{V}_{111}\right]\dot{\phi}_{2}^{6}+\mathcal{O}(\dot{\phi}_{2}^{8})\Bigg{)}.\end{split}$
(4.10)

Thus the low energy action of the light scalar $\phi_{2}$ is of the
$P(X,\phi_{2})$ form where $X\equiv\frac{1}{2}\dot{\phi}_{2}^{2}$.
Interactions between the heavy and the light scalar generically give rise not
only to corrections to the scalar potential and kinetic terms of the light
field, but also to higher derivative terms. It is precisely these that play a
crucial role in the observational signatures of this model. Written in this
manner, it is clear that the model of [46] falls into the class of K-essence
models whose most salient feature is the reduction of the speed of sound for
the scalar perturbations induced by the presence of higher derivative terms.
In fact, the speed of sound can be written as

$c_{s}^{-2}=1+2X\frac{P_{XX}}{P_{X}}=1\,+\,4\frac{\bar{f}_{1}^{2}}{\bar{V}_{11}}\dot{\phi}_{2}^{2}\,+\,4\frac{\bar{f}_{1}^{2}\left((\bar{f}_{1}^{2}+3\bar{f}\bar{f}_{11})\bar{V}_{11}-\bar{f}\bar{f}_{1}\bar{V}_{111}\right)}{\bar{V}_{11}^{3}}\dot{\phi}_{2}^{4}\,+\,\mathcal{O}(\dot{\phi}_{2}^{6})\,.$
(4.11)

Taking the explicit model of [23], where $\phi_{1}=r$, $\phi_{2}=\theta$,
$f(\phi_{1})=\phi_{1}/M_{p}$ and
$V=V_{0}-\alpha\phi_{2}+\frac{1}{2}m^{2}(\phi_{1}-\langle\phi_{1}\rangle)^{2}$
one finds [48, 47]

$c_{s}^{-2}=1+\left(\frac{2\dot{\phi}_{2}}{mM_{p}}\right)^{2}+\frac{1}{4}\left(\frac{2\dot{\phi}_{2}}{mM_{p}}\right)^{4}+\,\dots\,,$
(4.12)

in perfect agreement with the results of [23].

##  5 Q-ball formation in two-field models with a conserved charge

In this paper we have investigated the possible existence of stringy two-field
quintessence models compatible with late-time cosmological observations. Since
our models enjoys a shift symmetry in the $\phi_{2}$-direction, they might
develop instabilities at a non-perturbative level associated with spontaneous
formation of solitonic objects that are charged under this symmetry.
Solitonic, non-topological field configurations of a complex scalar field with
$U(1)$ symmetry are known as Q-balls. These extended objects are the minimum
energy configuration for a given charge. They are completely stable by virtue
of charge conservation. Producing Q-balls during quintessence effectively
screens the dark energy, abruptly stopping the accelerating dynamics. In order
to make our analysis of two-field models more complete, we want to conclude by
studying the constraints imposed by stability against spontaneous Q-ball
formation.

Intuitively, Q-balls exist if the interacting potential is less than the free
field part ${V(\phi)<m^{2}|\phi|^{2}}$. Then a compact, self-sustaining
configuration is energetically favored over a cloud of massive, weakly
interacting particles. Quantitatively, [24] showed that the condition for the
formation of these objects is a minimum of the function $V(\phi)/|\phi|^{2}$
at non-zero values of $\phi\neq 0$. However, the derivation of this condition
in [24] assumes the existence of a true vacuum at $\phi=0$ which is not the
case for our quintessence models. Moreover, in the situation of our interest,
the complex scalar field theory is coupled to gravity in an FRW background.

Due to the absence of a true vacuum, one might say that rather than Q-balls,
we have what [49] called PQ-balls. In particular, a spherical Q-ball solution
for exponential potentials and kinetic couplings has divergent total charge
and energy with densities that have a $1/r^{2}$ decay at infinity. Since the
total charge and energy is infinite, the configurations cannot minimize the
energy for fixed charge. In the quintessence case, the background is not a
true vacuum, but rather an evolving state with finite and constant charge and
energy density, and so the total background charge and energy is already
necessarily divergent. As a consequence, while Q-balls may not be totally
stable in this background, we argue that there are no fundamental obstructions
to localized objects which behave like Q-balls.666This situation is similar to
the one occurring with field theory domain walls coupled to gravity. Within
the gravitational theory, non-perturbative processes may create expanding
Coleman-De Luccia bubbles [50] with a non-zero probability. Local Q-ball
formation in a gravitating background may be seen as a generalization of these
bubbles that includes the extra $U(1)$ charge.

We are now interested in the formation of these objects during the evolution
of the system. An abundance of Q-balls formed anywhere along the trajectory
would not only be observationally problematic, but it would even drastically
change the equation of state for the quintessence field to that of matter or
radiation (depending on the stability mentioned above), as the Q-balls screen
the expanding effects of the classical background solution.

Following [25], we consider the scalars $\phi_{1}$ and $\phi_{2}$ as
homogeneous solutions with classical, local fluctuations. Then we identify the
conditions for the exponential growth of these fluctuations, which signals the
onset of non-linearities and the formation of Q-balls. The equations of motion
for non-homogeneous fields $\Phi_{i}(x,t)=\phi_{i}(t)+\delta\phi_{i}(x,t)$ in
an FRW background are given by

$\begin{split}&3H\dot{\Phi}_{1}-\Box\Phi_{1}+ff_{1}(\partial\Phi_{2})^{2}+V_{1}=0\,,\\\
&3Hf^{2}\dot{\Phi}_{2}-f^{2}\Box\Phi_{2}-2ff_{1}\left(\partial\Phi_{1}\cdot\partial\Phi_{2}\right)=0\,,\end{split}$
(5.1)

where $\Box=-\partial_{t}^{2}+a^{-2}\nabla^{2}$. For the homogeneous part
$\phi_{i}(t)$ the equations of motions reduce to (2.3), and for the
fluctuations we find

$\begin{split}&3H\delta\dot{\phi}_{1}-\Box\delta\phi_{1}-\left(f_{1}^{2}+ff_{11}\right){\dot{\phi}_{2}}^{2}\delta\phi_{1}+V_{11}\delta\phi_{1}-2ff_{1}\dot{\phi}_{2}\delta\dot{\phi}_{2}=0\,,\\\
&3Hf^{2}\delta\dot{\phi}_{2}-f^{2}\Box\delta\phi_{2}+2\left(ff_{11}-f_{1}^{2}\right)\dot{\phi}_{1}\dot{\phi}_{2}\delta\phi_{1}+2ff_{1}\left(\dot{\phi}_{2}\delta\dot{\phi}_{1}+\dot{\phi}_{1}\delta\dot{\phi}_{2}\right)=0\,.\end{split}$
(5.2)

The Q-ball forming solutions can be written as777A more general time and field
dependence would not change the main features of the solutions.

$\delta\phi_{i}=\delta\phi_{i}^{(0)}e^{\Omega t+ikx}\,,$ (5.3)

with $\Omega\in\mathbb{R}^{+}$ and $\delta\phi_{i}^{(0)}\neq 0$. The equations
of motion for such exponentially growing fluctuations reduce to

$\begin{split}&3H\Omega\delta\phi_{1}+\left(\Omega^{2}+\frac{k^{2}}{a^{2}}\right)\delta\phi_{1}-\left(f_{1}^{2}+ff_{11}\right){\dot{\phi}_{2}}^{2}\delta\phi_{1}+V_{11}\delta\phi_{1}-2ff_{1}\dot{\phi}_{2}\Omega\delta\phi_{2}=0\,,\\\
&3Hf^{2}\Omega\delta\phi_{2}+f^{2}\left(\Omega^{2}+\frac{k^{2}}{a^{2}}\right)\delta\phi_{2}+2\left(ff_{11}-f_{1}^{2}\right)\dot{\phi}_{1}\dot{\phi}_{2}\delta\phi_{1}+2ff_{1}\Omega\left(\dot{\phi}_{2}\delta\phi_{1}+\dot{\phi}_{1}\delta\phi_{2}\right)=0\,,\end{split}$
(5.4)

which combine to

$\begin{split}&\left(3Hf^{2}\Omega+f^{2}\left(\Omega^{2}+\frac{k^{2}}{a^{2}}\right)+2\Omega
ff_{1}\dot{\phi}_{1}\right)\left(3H\Omega+\left(\Omega^{2}+\frac{k^{2}}{a^{2}}\right)+V_{11}-(f_{1}^{2}+ff_{11})\dot{\phi}_{2}^{2}\right)\\\
&+4\Omega\dot{\phi}_{2}^{2}ff_{1}\left((ff_{11}-f_{1}^{2})\dot{\phi}_{1}+\Omega
ff_{1}\right)=0\,.\end{split}$ (5.5)

Figure 3: Evolution of the two-field quintessence system inside the wedge free
of Q-ball formation for universal string moduli and $p=1,2,3$.

Assuming $H\simeq 0$ and $\phi_{1}\simeq\,{\rm const}$, i.e. neglecting the
effect of cosmic expansion and assuming a predominantly circular trajectory of
the axion, this simplifies to (for $f\neq 0$)

$\left(\Omega^{2}+\frac{k^{2}}{a^{2}}\right)\left(\Omega^{2}+\frac{k^{2}}{a^{2}}+V_{11}-(f_{1}^{2}+ff_{11})\dot{\phi}_{2}^{2}\right)+4\Omega^{2}\dot{\phi}_{2}^{2}f_{1}^{2}=\Omega^{4}+\lambda_{2}\Omega^{2}+\frac{k^{2}}{a^{2}}\lambda_{0}=0\,,$
(5.6)

where

$\lambda_{2}\equiv\frac{k^{2}}{a^{2}}+4\dot{\phi}_{2}^{2}f_{1}^{2}+\lambda_{0}\qquad\text{and}\qquad\lambda_{0}\equiv\frac{k^{2}}{a^{2}}+V_{11}-(f_{1}^{2}+ff_{11})\dot{\phi}_{2}^{2}\,.$
(5.7)

Clearly, if $\lambda_{0}\geq 0$, also $\lambda_{2}\geq 0$, leading to no
solution with $\Omega\in\mathbb{R}^{+}$. $\lambda_{0}<0$ is therefore a
necessary condition for Q-ball formation which reads

$0<\frac{k^{2}}{a^{2}}\,<\,(f_{1}^{2}+ff_{11})\dot{\phi}_{2}^{2}-V_{11}\,,$
(5.8)

where the first inequality just states that the fluctuations are
inhomogeneous. Notice that the condition (5.8) is also sufficient since the
function $\Omega^{4}+\lambda_{2}\Omega^{2}+\frac{k^{2}}{a^{2}}\lambda_{0}$ is
positive for $\Omega\to\infty$ while it is negative for $\Omega\to 0$ if
$\lambda_{0}<0$, implying for continuity a value $\Omega\in\mathbb{R}^{+}$
where it crosses zero. For exponential potentials and kinetic couplings, (5.8)
can be written in terms of $x_{2}$, $y_{1}$ and $k_{i}$ as

$0<\frac{k^{2}}{a^{2}}\,<\,3H^{2}\,\left(4k_{1}^{2}x_{2}^{2}-k_{2}^{2}y_{1}^{2}\right),$
(5.9)

leaving a wedge in parameter space which is free of Q-balls. In particular,
the dark energy models studied in Sec. 3 which involve universal string
moduli, seem to have dynamics free of Q-ball formation. Fig. 3 shows that the
three trajectories (for $p=1,2,3$) with matter domination initial conditions
are predominantly inside the wedge defined by (5.9). Qualitatively similar
considerations hold for the three trajectories which pass through today’s
observations.

To be somewhat more precise, our matter dominated initial conditions
correspond to small perturbations around the unstable $x_{1}=x_{2}=y_{1}=0$
fixed point. In this case the system might initially be in a region in
parameter space where Q-ball formation is possible. However we will show now
that the Q-ball formation time is much larger than the time for the system to
evolve inside the Q-ball free wedge.

The time scale of Q-ball formation can be estimated by the exponent $\Omega$
of the fluctuations. The roots of the quartic eq. (5.6) are given by

$\Omega=\pm\frac{1}{\sqrt{2}}\sqrt{-\lambda_{2}\pm\sqrt{\lambda_{2}^{2}+4\frac{k^{2}}{a^{2}}|\lambda_{0}|}}\,,$
(5.10)

where $\lambda_{0}=-|\lambda_{0}|<0$ to allow for Q-ball formation. Since we
search for real and positive solutions, we select the ‘+’ signs. As expected
from the previous analysis, when $|\lambda_{0}|\to 0$ the solutions go to
zero. On the other hand, for $|\lambda_{0}|\gg k^{2}/a^{2}$, the solutions can
be approximated as $\Omega\simeq\sqrt{|\lambda_{0}|}$. For constant $k_{1}$
and $k_{2}$, we can estimate the Q-ball formation time as

$\Omega\simeq\sqrt{3}H\sqrt{4k_{1}^{2}x_{2}^{2}-k_{2}^{2}y_{1}^{2}}<2\sqrt{3}Hk_{1}x_{2}\qquad\Rightarrow\qquad
t\sim\frac{1}{\Omega}>\frac{1}{2\sqrt{3}Hk_{1}x_{2}}\gg\frac{1}{H}\,,$ (5.11)

for small initial values of $x_{2}$ and $k_{1}=\mathcal{O}(1)$. Let us stress
that these approximations are valid in the physically interesting regime that
we choose for initial conditions and parameters.

However the above analysis is not necessarily accurate enough, since we
assumed $H\simeq\dot{\phi}_{1}\simeq 0$. Reinstating $H$ and $\dot{\phi}_{1}$
using (5.5), eq. (5.6) generalizes to

$\Omega^{4}+\Omega^{3}\left(3H+\lambda_{3}\right)+\Omega^{2}\left(\lambda_{2}+3H\lambda_{3}\right)+\Omega\left(3H\frac{k^{2}}{a^{2}}+\lambda_{3}\lambda_{0}-\frac{4}{M_{p}}\,ff_{1}\dot{\phi}_{2}^{2}\dot{\phi}_{1}\partial_{\phi_{1}}k_{1}\right)+\frac{k^{2}}{a^{2}}\lambda_{0}=0\,,$
(5.12)

where

$\lambda_{3}\equiv
3H+2\frac{f_{1}}{f}\dot{\phi}_{1}=H\left(3-2\sqrt{6}k_{1}x_{1}\right)\,.$
(5.13)

We have already used $\partial_{\phi_{1}}k_{1}=(f_{1}^{2}-ff_{11})$ in the
linear terms for brevity. Interestingly, the new terms do not affect the
$\Omega\to 0$ and $\Omega\to\infty$ limits, and so $\lambda_{0}<0$ still
remains a sufficient condition. The same condition becomes also necessary if
all the coefficients in (5.12) are non-negative for $\lambda_{0}\geq 0$. This
is clearly the case if $k_{1}$ is constant and $\lambda_{3}\geq 0$, i.e. if
$x_{1}\leq\sqrt{3/2}/(2k_{1})$.

Focusing again on the case of two-field quintessence driven by universal
string moduli, where $k_{1}$ is indeed constant and given by (3.11), the
condition for Q-ball formation (5.8) is definitely necessary for
$x_{1}\leq\sqrt{3p}/4$. This implies that when $x_{1}>\sqrt{3p}/4$ the
conditions for Q-ball formation could be satisfied even inside the wedge
defined by (5.9). Let us mention the main features of the trajectories
analyzed in Sec. 3

  * •

For $p=1$, the trajectories with matter dominated initial conditions satisfy
the condition $x_{1}\leq\sqrt{3}/4$ everywhere except for a region close to
the fixed point where $x_{1}=1/\sqrt{3}>\sqrt{3}/4$. Hence at the fixed point
$\lambda_{0}>0$ while $\lambda_{3}<0$, implying that (5.12) will definitely
have a solution $\Omega\in\mathbb{R}^{+}$ for $k\to 0$. This signals that the
evolution of the system for $p=1$ will end with Q-ball formation in the far
future since there is enough time for Q-balls to form close to the fixed
point.

For solutions which pass through today’s observations, $x_{1}\leq\sqrt{3}/4$
also holds everywhere except for two regions: ($i$) close to the fixed point,
implying that also these trajectories will end with Q-ball formation; ($ii$)
in the vicinity of the putative initial conditions $\omega_{\phi_{1}}\simeq 0$
and $\Omega_{\phi_{1}}\simeq 0.55$. However the system will move to
$x_{1}\leq\sqrt{3}/4$ already at $\omega_{\phi_{1}}\simeq-0.3$ and
$\Omega_{\phi_{1}}\simeq 0.525$, and so before Q-balls have enough time to
form. In fact, it can be easily checked that in this region
$|\lambda_{3}|\sim\mathcal{O}(0.1)H$, implying a solution to (5.12) which
scale as $\Omega\sim\mathcal{O}(0.1)\,H$, and so a very long Q-ball formation
time, $t\sim\Omega^{-1}\sim\mathcal{O}(10)/H$.

  * •

For $p=2$ and $p=3$, the condition $x_{1}\leq\sqrt{3p}/4$ is satisfied
everywhere along the trajectories, including the fixed points in the far
future, except for short transients close to these final fixed points.
However, similarly to the $p=1$ case, Q-balls do not have enough time to form
since $|\lambda_{3}|$ remains always small, i.e.
$|\lambda_{3}|\sim\mathcal{O}(0.1)H$. Interestingly, for $p=2$ and $p=3$,
$x_{1}\leq\sqrt{3}/4$ holds around the potential initial conditions,
$\omega_{\phi_{1}}\simeq 0$ and $\Omega_{\phi_{1}}\simeq 0.6$, for the
trajectories which reproduce current data.

Summarizing, we have shown that the obstruction of quintessence dynamics by
Q-ball formation is never a problem for two-field quintessence models from
string theory. Transients where the conditions for exponential growth of non-
linearities are satisfied do exist, but they disappear before Q-balls have
enough time to form. Interestingly, we found trajectories with $p=1$ which end
with Q-ball formation in the far cosmological future.

##  6 Conclusions

The mechanism responsible for the present day accelerated expansion of our
universe is without any doubt the greatest mystery of fundamental physics.
Recent swampland conjectures are expected to shed light on this delicate issue
since their goal is to determine the constraints that quantum gravity imposes
on any consistent effective field theory. In particular, the dS conjecture [9,
10, 11, 12] would seem to rule out both metastable vacua with a positive
cosmological constant and single-field dynamical dark energy models [14, 15] .

Motivated by these considerations, in this paper we investigated the
possibility of realizing multifield quintessence in string theory for two main
reasons: ($i$) two-field models with a curved field space have already been
shown to be able to lead to late-time accelerating solutions even for steep
potentials which are compatible with the dS conjecture [21, 22]; ($ii$) non-
linear sigma models arise very naturally in the 4D supergravity effective
action of string compactifications.

We studied two classes of models depending on whether the quintessence complex
field is a universal modulus, like the axio-dilaton or the overall Kähler
modulus, or a non-universal modulus, like a local mode blowing up a Calabi-Yau
singularity. While if it is not difficult to find accelerating solutions, we
have not been able to obtain multifield trajectories which are compatible with
observations both today and in the past. In fact, if the universe matches
observations in the past, i.e. the system start from matter domination, there
is generically no trajectory which reproduces $\omega_{\phi_{1}}\simeq-1$ and
$\Omega_{\phi_{1}}\simeq 0.7$ in the future. On the other hand, imposing the
system to pass through today’s observable universe, the initial epoch turns
out to be kinetic domination.

These somewhat negative results indicate that embedding multifield
quintessence in explicit string compactifications seems to be very
challenging, especially when compatibility with observations is required.
Hence, from this perspective, dynamical dark energy does not seem a preferred
alternative to the simplest cosmological constant explanation of the current
accelerated expansion of our universe. In this sense, our results might also
be seen as raising doubts about the validity of exisiting dS conjectures.

It is important to stress that in this paper we focused just on two-field
models, while a generic string compactification might feature many more fields
which play an active role during multifield quintessence. While this more
involved scenario would definitely need a more accurate analysis, we expect
our results to still hold qualitatively since the two-field dynamics with
kinetic couplings should already catch the main features of the departure from
a simple single-field geodesic dynamics.

Let us finally point out that the models we have considered feature a
conserved charge due to the underlying axionic shift symmetry. This can lead
to an exponential growth of non-linearities that can end up in Q-ball
formation and a consequent sudden obstraction of the dark energy dynamics.
However we have shown that our models are free from Q-ball formation during
their entire cosmological evolution except for some cases where Q-balls might
form only in the far future when the system approaches a fixed point.

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