Torsion, Gravity Induced Chiral Symmetry Breaking and Cosmological Bounce

Giorgi Tukhashvili

Department of Physics, Princeton University, Princeton, NJ, 08544, USA

In this paper, we elaborate and extend our proposal [1] of the cosmological
bouncing model in which a chiral condensate violates the null energy condition
and induces a bounce. The condensate formation is caused by the gravitational
effects that arise in the effective action when the curvature of space-time is
non-zero. As the chiral condensation is a vacuum effect, the symmetries of the
FRW space-time are not violated. We explicitly calculate the spinor effective
action in the presence of a background gravitational field up to second order
in curvature and with derivative corrections and also derive the initial
conditions of the condensate. In addition, a new interaction term between the
condensate and the scalar field is introduced that might be relevant for
producing the adiabatic perturbations during the contraction phase. The
curvature/energy density stays far from Planckian values throughout this
evolution.

<EMAIL_ADDRESS>
##  1 Introduction

Explaining the isotropy, homogeneity, and flatness of the Universe in a self-
consistent way is the main challenge of modern cosmology. Recently, using
numerical relativity, a novel method has been identified to solve these
problems [2, 3, 4, 5, 6]: rather than invoke exponential expansion, utilize
ultra-slow contraction of space. The matter content to achieve such dynamics
consists of a canonical scalar with negative potential. This mechanism is
robust, meaning that smoothness and flatness are achieved even if the initial
state is far from smooth and flat, and it is rapid, meaning that the goal is
achieved within a few $e$-folds [2, 3, 4, 5, 6]. The most remarkable feature
of this mechanism is its simplicity, as it relies only on one large parameter,
the largeness of which is a natural consequence of its definition.

From the observations, we have overwhelming evidence that the Universe is
expanding today. If one invokes the ultra-slow contraction at the initial
state to smooth and flatten the Universe, then to match the current
observations, the space must stop the contraction and start to expand at some
moment in time. That moment is called the cosmological bounce or simply the
bounce. The singularity theorems by Penrose, Hawking and others [7, 8, 9]
state that unless the matter driving the contraction violates the null energy
condition (NEC), the space will crunch (the scale factor $a\rightarrow 0$) and
singularities will develop. One way to avoid these is to argue that before the
(big) crunch happens, new degrees of freedom associated with quantum gravity
will emerge and prevent the development of singularities. Another way out is
to introduce some form of matter to violate the NEC. The latter is the
foundation for singularity theorems, and its violation might help avoid the
crunching of space and as a result the quantum gravity regime. In this case,
the scale factor, $a$, will decrease to some minimal value and then start to
increase. This paper will refer to such an event as the classical bounce
because the gravitational field stays classical during the evolution as the
energies involved never hit the Planckian regime. Ijjas and Steinhardt [10,
11] showed that a classical bounce could be achieved with a Galileon, a scalar
field, the action of which involves higher derivative interactions; however,
the equation of motion is still of the second order, i.e., no Ostrogradski
instabilities.

In this paper, we elaborate and extend the results of [1] and study the
evolution of the Universe where a scalar field smoothes and flattens the
Universe while the bounce is achieved using spinor condensate. So, the matter
part of our model consists of a scalar field with negative exponential
potential, a Dirac spinor, and a mixing term between these two. No on-shell
particles are present, i.e., the chemical potential is zero. The latter is
sufficient to avoid the inconsistencies in the previous works [12, 13, 14, 15]
that include the direct violation of FRW symmetries and incompatibility of
even a tiny spatial curvature with the physical fermions. The Dirac spinors
source the torsion of space-time, which has to be accounted for by expanding
ordinary Einsteins’ General Relativity into Einstein-Cartan-Sciama-Kibble
theory [16, 17, 18, 19]. In this formalism, torsion can be integrated out. Its
remnants – the four-Fermi interactions [20] can be brought to Nambu-Jona-
Lasinio (NJL) form [21] by including non-minimal couplings between spinor
currents and torsion [15, 22] and by properly choosing their coupling
constants. The coupling constant in front of the NJL potential, $\xi$, has
tuned to be slightly smaller than its critical value (in flat space-time). We
do not have justification for this tuning, which may be considered a downside
of the model. Since the NJL coupling constant is less than its critical value,
the chiral symmetry is not broken, spinor bi-linears have zero vev, and do not
contribute to space-time evolution. The NJL condensation in our model is
induced through the gravitational field. In Appendix B, we calculate the
effective action for the NJL model in the presence of a background
gravitational field up to two orders in the curvature and with the derivative
corrections. This generalizes the results of [23], in which a similar
calculation was done, but only to linear order in the curvature and without
including the condensate derivatives. The calculation is done using both the
Riemann normal coordinates and the Heat kernel method. Both results coincide
in the UV limit. Our result shows that the negative values of the Ricci
curvature can induce chiral symmetry breaking even if the NJL attraction is
insufficient to form the condensates.

The only assumptions we are making about the initial phase are that the space
is contracting and the curvature is not large enough to induce the chiral
symmetry breaking. During this phase, spinor bi-linears have zero vev and do
not contribute to the energy budget. The dynamics is governed by the scalar
field alone, which will take care of smoothing and flattening of the space [2,
3, 4, 5, 6]. After a few $e$-folds, the space will become homogeneous and
isotropic, described by the spatially flat FRW metric. Using the scalar field
to achieve smoothness and flatness is important because we can avoid having to
rely on unjustified high-power spinor bi-linears to do this job [22]. The
Ricci curvature is negative and its absolute value increases as space
contracts. The curvature will eventually reach the critical value at which it
causes chiral symmetry to break. The scalar bi-linear formed from the spinors
acquire a non-zero vev, while the vevs of pseudo-scalar and vector bi-linears
remain zero. Soon after this happens, the scalar field is no longer dominant,
as within the energy budget, the contribution of the condensate becomes
comparable to it. The bounce is what follows. Interestingly the sign of $\xi$
that causes attractive interaction between the spinors also guarantees the
classicality of the bounce.

Compared to [1], the main differences in the model building are the potential
for the scalar field, a new interaction that mixes the scalar and spinor bi-
linear, and the absence of a rigid mass for the spinor. In [1], we used a
potential bounded from below. Here we use a negative exponential potential
that transitions into the Morse potential after the chiral condensate
formation due to the scalar-spinor mixing. All effective field theory (EFT)
limits are respected throughout the evolution.

This paper is organized as follows: in section 2, the action for our model is
described, and the matter content and interactions are explained. The section
3 is dedicated to discussing and demonstrating the curvature-induced NJL
condensation. In section 4, the timeline of the events is discussed, and the
existence of the bounce is shown using the numerical solutions; in addition,
the initial conditions for the fields are derived. The appendix A shows how to
integrate out the torsion and obtain the NJL potential. Finally, the
calculation of the NJL effective action in the presence of the gravitational
field is presented in appendix B.

Conventions through this paper are $(+,+,+)$ from Misner, Thorne and Wheeler
[24]. Explicitly, the Minkowski metric is mostly plus,
$\eta_{\mu\nu}=(-1,1,1,1)$, and the Riemann tensor is defined as
${R^{\rho}}_{\mu\lambda\nu}=+\partial_{\lambda}\Gamma^{\rho}_{\mu\nu}+\cdots$.
The Planck constant will be assumed to have the unit value unless otherwise
stated $\hbar=1$. The Planck mass is defined as $M_{pl}=\left(8\pi
G\right)^{-1/2}\approx 2.5\cdot 10^{18}GeV$, with $G$ being the gravitational
Newtons constant. The Levi-Civita tensor $\epsilon^{0123}=1$. For the gamma
matrices, the Weyl representation is used:
$\gamma^{A}=\left(\begin{array}[]{cc}0&\sigma^{A}\\\ \bar{\sigma}^{A}&0\\\
\end{array}\right)$ and $\gamma_{5}=\left(\begin{array}[]{cc}-1&0\\\ 0&1\\\
\end{array}\right)$, with $\sigma^{A}=(1,\sigma^{a})$ and
$\bar{\sigma}^{A}=(1,-\sigma^{a})$. Here $\sigma^{a}$ are the Pauli matrices.
The capital Latin letters represent the Local Lorentz indices having values
$0,1,2,3$, while the lowercase Latin letters represent the Local Lorentz
indices having values $1,2,3$.

##  2 Overview of Our Model

This section defines the action our model is based on, and the interactions
and matter content are justified. First, since the spinors are present, the
first-order formalism of General Relativity (GR) must be used. Fundamental
variables of the gravitational field are the one forms: the tetrad
$e^{A}=e^{A}_{\mu}dx^{\mu}$ and the spin connection $\tilde{\omega}_{IJ}$. The
metric tensor is defined using the tetrads:
$g_{\mu\nu}=\eta_{AB}e_{\mu}^{A}e_{\nu}^{B}$, where $\eta_{AB}$ is the
Minkowski metric. The spinor sector consists of a Dirac spinor with one
color/flavor. Then, to justify using the FRW metric during the bounce phase,
we need a field during the initial phase of contraction that enables smoothing
and flattening. We entrust this mission to the scalar field with a negative
exponential potential, as commonly used in models of ultra-slow contraction
[2, 3, 4, 5]. Lastly, there should exist a mixing between the scalar and
spinor. The latter should be responsible for the scalar to condensate energy
transfer and, ideally, causing an effective friction for the scalar field,
which is required to produce the adiabatic perturbations during the
contraction [25, 26]. Once the condensate is formed, the mixing term also
contributes to the potential of the scalar and bounds it from below (See Fig.
3(b)). Our action:

$\displaystyle\mathcal{S}=$
$\displaystyle-\frac{M_{pl}^{2}}{4}\epsilon_{ABMN}e^{A}\wedge
e^{B}\wedge\tilde{R}^{MN}-\frac{i}{2\cdot 3!}\epsilon_{ABMN}e^{A}\wedge
e^{B}\wedge
e^{M}\wedge\Big{(}\overline{\Psi}\gamma^{N}\tilde{D}\Psi-\tilde{D}\overline{\Psi}\gamma^{N}\Psi\Big{)}$
$\displaystyle+\sqrt{\frac{\xi}{6}}\epsilon_{IJKL}\tilde{D}e^{I}\wedge
e^{J}\wedge
e^{K}~{}V^{L}+\left(\frac{1}{4}-\sqrt{\frac{\xi}{6}}\right)\tilde{D}e^{I}\wedge
e_{I}\wedge e_{J}~{}A^{J}$ (2.1)
$\displaystyle+\frac{1}{4!}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge e^{M}\wedge
e^{N}\left[\frac{1}{2}(\partial\phi)^{2}+V(\phi)+F(\phi)\overline{\Psi}\Psi\right].$

The integration sign is implicit in front of each term. The quantities with a
tilde are introduced to distinguish them from torsion-free ones (without a
tilde). The first term is the torsion-modified Einstein-Hilbert, and the
second is the action for the massless Dirac spinor in the presence of the
gravitational field. The Dirac field is assumed to have one color and one
flavor, but generalization to $N$ colors/flavors is straightforward. The third
and fourth terms are the non-minimal couplings between the torsion and spinor
currents. The third line describes a scalar field with the potential $V(\phi)$
coupled minimally to gravity. The very last term is the coupling between the
spinor and the scalar. $F(\phi)$ is not necessarily a linear function, i.e.,
Yukawa. The coupling constants on the second line are chosen so that, after
the torsion is integrated out, they induce chiral symmetry. This choice is not
unique. The covariant derivatives and the currents are defined as:

$\tilde{D}\Psi=\left(d-\frac{1}{8}\tilde{\omega}_{AB}\left[\gamma^{A},\gamma^{B}\right]\right)\Psi;\hskip
14.22636pt\hskip 14.22636pt\hskip
14.22636pt\tilde{D}\overline{\Psi}=\overline{\Psi}\left(\overleftarrow{d}+\frac{1}{8}\tilde{\omega}_{AB}\left[\gamma^{A},\gamma^{B}\right]\right);$
(2.2) $\tilde{D}e^{I}=de^{I}+\left.\tilde{\omega}^{I}\right.{}_{J}\wedge
e^{J}=T^{I}.$ (2.3) $V^{M}\equiv\overline{\Psi}\gamma^{M}\Psi;\hskip
14.22636pt\hskip 14.22636pt\hskip
14.22636ptA^{M}\equiv\overline{\Psi}\gamma_{5}\gamma^{M}\Psi.$ (2.4)

Since, in the absence of a source, the torsion, $T^{I}$, vanishes and we
recover the ordinary Einstein-Hilbert action, it is possible to postulate that
the connection $\tilde{\omega}_{IJ}$ consists of two parts:
$\tilde{\omega}_{IJ}=\omega_{IJ}+C_{IJ}$. The first part is the torsion-free
connection $\omega_{IJ}$, while the second part is the tensor, $C_{IJ}$,
related to the torsion as $T^{I}=\left.C^{I}\right.{}_{J}\wedge e^{J}$. From
now on, the quantities without tilde will be defined wrt the torsion-free
connection $\omega_{IJ}$. Since, by definition, $T^{I}$ and $C_{IJ}$ are
linearly related, we will refer to both as torsion. Once this philosophy is
adopted, if the gravitation (curvature) is turned off, (2) will not just
reduce to the action of a spinor field coupled to a scalar. Additional
interactions arise because torsion stays non-zero even when the curvature
vanishes. The tensor $C_{IJ}$ enters algebraically into the action (2); in
addition, its highest degree is quadratic. This allows us to integrate it out
from (2). The equivalent action takes the following form (see Appendix A):

$\displaystyle\mathcal{S}=$
$\displaystyle-\frac{M_{pl}^{2}}{4}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
R^{MN}-\frac{i}{2\cdot 3!}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
e^{M}\wedge\Big{(}\overline{\Psi}\gamma^{N}D\Psi-D\overline{\Psi}\gamma^{N}\Psi\Big{)}$
$\displaystyle-\frac{\xi}{4!M_{pl}^{2}}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
e^{M}\wedge
e^{N}\left[\big{(}\overline{\Psi}\Psi\big{)}^{2}+\big{(}i\overline{\Psi}\gamma_{5}\Psi\big{)}^{2}\right]$
(2.5) $\displaystyle+\frac{1}{4!}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
e^{M}\wedge
e^{N}\left[\frac{1}{2}(\partial\phi)^{2}+V(\phi)+F(\phi)\overline{\Psi}\Psi\right].$

The action (2) is fully equivalent to (2). Torsion was exchanged for the four-
Fermi interactions. The specific form of these interactions is a consequence
of the choice of coupling constants on the second line of (2) and the Fierz
identity:

$V_{K}V^{K}-A_{K}A^{K}=2\left[\big{(}\overline{\Psi}\Psi\big{)}^{2}+\big{(}i\overline{\Psi}\gamma_{5}\Psi\big{)}^{2}\right].$
(2.6)

Ignoring the mixing with scalar, the spinor content of (2) is equivalent to
the NJL model. The Einstein and scalar equations are obtained by taking the
variation of (2) wrt $e^{A}$ and $\phi$ respectively (diffeomorphism and Local
Lorentz indices can be mapped between each other by using
$e^{A}_{\mu};~{}e_{A}^{\mu}$):

$\displaystyle-
M_{p}^{2}G^{X}_{A}+i\delta^{X}_{A}\overline{\Psi}\gamma^{N}D_{N}\Psi-i\overline{\Psi}\gamma^{X}D_{A}\Psi+\delta^{X}_{A}\frac{\xi}{M_{pl}^{2}}\left[\big{(}\overline{\Psi}\Psi\big{)}^{2}+\big{(}i\overline{\Psi}\gamma_{5}\Psi\big{)}^{2}\right]$
$\displaystyle+\partial_{A}\phi\partial^{X}\phi-\delta_{A}^{X}\Big{(}\frac{1}{2}(\partial\phi)^{2}+V(\phi)+F(\phi)\overline{\Psi}\Psi\Big{)}=0;$
(2.7)
$-\nabla^{2}\phi+\frac{dV}{d\phi}+\frac{dF}{d\phi}\overline{\Psi}\Psi=0.$
(2.8)

In the simplest setup of Einstein-Cartan-Sciama-Kibble theory, if the Dirac
spinor sources the torsion, the spin-spin interaction is $A_{\mu}A^{\mu}$ [17,
20], where $A_{\mu}$ is the axial current of the spinor. Trautman [12] noticed
that if the axial current vector is time-like, then these interactions can
source NEC violation, and the cosmic singularity can be avoided.

The phenomenological implications of torsion were studied in [27, 28, 29]. It
turns out that its presence induces a potential similar to the electromagnetic
field. Within the hydrogen atom, torsion interacting with the electron’s spin
will cause the splitting of spectral lines, similar to the Zeeman effect. In
[28], this fact was used to constrain the coupling constant related to the
torsion. In our notation, their constraint roughly translates to
$\sqrt{\xi}<10^{17}$. A more stringent constraint can be achieved by
considering the interaction between polarized test mass and paramagnetic salt
[30, 28], in this case, $\sqrt{\xi}<10^{14}$. In the numerical calculations
done in section 4.1, $\xi\approx 200$, which easily satisfies these
constraints.

##  3 Curvature Induced Spinor Condensation

Through this paper, the gravitational field is treated classically. To have a
self-consistent description, the spinor bi-linears that interact with gravity
must be averaged wrt the vacuum state. The average of these bi-linears is non-
zero if the spinor number density is itself non-zero, but in this case, unless
these spinors are arranged in a specific way, the symmetries of the FRW space
will not be respected. To get around this problem, the spinor number density
will be assumed to vanish $\braket{\overline{\Psi}\gamma^{0}\Psi}=0$. Instead,
since the four-Fermi interactions are present, the NJL mechanism is invoked to
justify the non-zero VEV (quantum average) of $\braket{\overline{\Psi}\Psi}$.
We will assume
$\braket{\left(\overline{\Psi}\Psi\right)^{2}}=\braket{\overline{\Psi}\Psi}^{2}$
to be a good approximation, even though the equality strictly holds in the
large $N$ limit, with difference scaling as $N^{-2}$ [31].

Since the background space-time is not flat, the corrections from the non-zero
curvature within the effective action need to be accounted for. Appropriate
calculation of the effective action is done in the Appendix B, where the
Riemann normal coordinates are used to show that the effective action up to
two orders in the Riemann curvature is given by the formula (B.3). Even though
these corrections have a quantum nature (proportional to $\hbar$), Planck’s
constant arises from the spinor loop, while the gravitational field is
classical. Also, these new terms should not be understood as corrections to
Einstein-Hilbert. The curvatures in (B.3) act as a background field that can
cause or prevent condensate formation (the latter can be compared to the
Cooper pairs in an external magnetic field). The gap equation with the
curvature corrections:

$\displaystyle-\frac{M_{pl}^{2}}{\Lambda^{2}}\frac{8\pi^{2}}{\xi}u+\frac{\partial
f_{4}}{\partial
u}-\frac{2}{\Lambda^{2}}f_{0}~{}\nabla^{2}u-\frac{1}{\Lambda^{2}}\left(\partial
u\right)^{2}\frac{\partial f_{0}}{\partial
u}+\frac{R}{6\Lambda^{2}}\left(2uf_{1}+u^{2}\frac{\partial f_{1}}{\partial
u}\right)$
$\displaystyle+\frac{2}{\Lambda^{4}}\left(-\frac{1}{120}\nabla^{2}R+\frac{1}{288}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}-\frac{7}{1440}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\frac{\partial
f_{1}}{\partial u}$
$\displaystyle-\frac{2}{\Lambda^{4}}\left(\frac{1}{12}\nabla^{2}R+\frac{1}{36}R_{\mu\nu}R^{\mu\nu}+\frac{1}{144}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\frac{\partial
f_{2}}{\partial u}$ (3.1)
$\displaystyle+\frac{2}{\Lambda^{4}}\left(\frac{1}{10}\nabla^{2}R+\frac{1}{72}R^{2}+\frac{7}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{60}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\frac{\partial
f_{3}}{\partial u}=0.$

The functions $f_{i}$ and the dimensionless variables are defined in (B.4),
(B.15-B.18) and (B.26). The derivative terms are necessary as it is no longer
valid in a dynamic space-time to assume the condensate to be constant. Note
that this equation and (B.3) involve a new parameter
$y\equiv\epsilon/\Lambda$, where $\epsilon$ is the infrared cut-off. The
necessity of IR cut-off can be seen from the fact that the effective action

$\mathcal{S}_{eff}\propto\log\frac{\det{\left(i\not{D}-\sigma\right)}}{\det{\left(i\not{D}\right)}}$
(3.2)

involves the loop of a massless spinor. In the zeroth and first order
curvature part of the (B.3), the IR scale plays no role and can be set to
zero, while in the part proportional to the curvature squared, the IR scale is
necessary for the effective action to be regular at $\sigma=0$, see (3.2). The
necessity of the IR cut-off disappears if one accounts for the effect of
explicit chiral symmetry breaking already present within (2). The explicit
symmetry breaking was ignored while calculating the effective action (B.3). As
the role $\epsilon$ is to partially replace $F(\phi)$ in (B.3), it is fair to
assume these two to be of the same order of magnitude at the time of
spontaneous chiral symmetry breaking.

In the absence of gravity, condensate formation is determined by the ratio
$\xi_{c}/\xi$ where $\xi_{c}\equiv 2\pi^{2}M_{pl}^{2}/\Lambda^{2}$ is the
critical value of the four-Fermi interactions when space-time is flat and
$y=0$. For $\xi_{c}/\xi<1$ the chiral symmetry is spontaneously broken. Once
gravity is introduced into the picture, the story might change. If this ratio
is tuned to be near one, then in the presence of positive Ricci curvature, the
chiral symmetry might be restored even if $\xi_{c}/\xi<1$, while in the
presence of negative Ricci curvature, the chiral symmetry can be broken with
$\xi_{c}/\xi>1$. The value of Ricci curvature at which the chiral symmetry
breaking happens is referred as the critical curvature, $R_{c}$. The critical
curvature depends on the value of NJL coupling, $R_{c}(\xi)$. If this coupling
is taken to be its critical value (in flat space-time), then even a tiny
negative curvature can induce the chiral symmetry breaking,
$R_{c}(\xi_{c})=0$. Figure 1 shows the phase diagrams when the space-time is
spatially flat FRW for different Hubble parameters.

Figure 1: Phase diagrams of the NJL model in an external gravitational field
according to (B.1). Orange, the variation of (B.1) has a non-trivial solution
meaning that the chiral symmetry is broken. Green, the chiral symmetry is
unbroken. If the curvature corrections are ignored, the two phases will be
separated with a vertical line $\xi/\xi_{c}=1$. In both cases, the space-time
is spatially flat FRW. Left: the Hubble parameter $=2/(\kappa^{2}\tau)$ with
$\kappa=10$, $y=10^{-3}$ and $\theta=10^{-3}$; see (4.6) and (4.7). Right: the
Hubble parameter has Trautman ansatz, $2\tau/\kappa^{2}(10^{-2}+\tau^{2})$. In
both cases, the back-reaction of the condensate to space-time and the limits
of the curvature expansion in (B.3) are ignored.

The curvature-induced NJL condensation should not be confused with BCS
condensation by Alexander and Biswas [14]. Similar to [12], in [14], there is
an initial non-zero density of physical particles. In [14], the condensation
catalyst is the shrinking volume, while in our case – the increasing
curvature.

##  4 Chiral Condensate Mediated Cosmological Bounce

In the last section, we showed that the negative Ricci curvature could induce
chiral symmetry breaking in the NJL model even when the four-Fermi coupling
strength is insufficient to form the condensate. In this section, this fact is
used, together with the assumption that the four-Fermi coupling is slightly
less than its critical value, such that, when the curvature is small, the
chiral symmetry is not broken and $\braket{\overline{\Psi}\Psi}=0$.

At this point, we are ready to present the cosmological bounce scenario.
Following [1], the only assumption about the initial state is that space is
contracting; otherwise, the initial conditions are inhomogeneous and
anisotropic, non-perturbatively far from FRW. The universe can be partitioned
into different patches with approximately the same Ricci curvature; see Fig.
2(a). If the Ricci curvature of the patch is positive $(R>0)$, or negative,
but larger than the critical curvature $(0>R>R_{c}(\xi))$, then within this
patch the chiral symmetry is not broken, the spinor bi-linears have zero vev
and the scalar field entirely defines the dynamics. These regions are colored
with the blue $+$ pattern in Fig. 2. If, on the other hand, the Ricci
curvature of the patch is less than the critical value $(R<R_{c}(\xi))$, then
within this patch, the chiral symmetry is broken, and the dynamics are defined
by both, the scalar and the spinor fields. These regions are colored with the
red $\times$ pattern in Fig. 2.

(a)

(b)

Figure 2: Left: Initial state of the universe is partitioned into different
patches with approximately the same curvature. Right: The universe after
undergoing the super-smoothing mechanism [2, 3, 4, 5, 6]. Most of the space,
depicted by the blue + pattern, can be well approximated by the flat FRW
metric.

Within the blue regions, the scalar field controls the dynamics; and by using
the results of [2, 3, 4, 5, 6], these regions will rapidly converge to FRW.
Once this happens these regions shrink very slowly
$a\propto(t^{2})^{1/(2\epsilon_{\phi})}$, as $\epsilon_{\phi}\gg 1$. The
dynamics of the red regions are completely different, and to get the complete
picture, one needs to perform numerical studies similar to [2, 3, 4, 5, 6].
The latter goes beyond the scope of this paper, but to get a glimpse of what
might happen, let’s assume that the space is flat, homogeneous, and
anisotropic. Here are three possible scenarios:

1) The equation of state of the dominant matter $\epsilon<3$; in this case,
the mixmaster behavior will develop, and anisotropy will dominate.
Parametrizing the anisotropies with the Kasner-like metric
$g_{\mu\nu}=(-1,a^{2}e^{2\beta_{1}},a^{2}e^{2\beta_{2}},a^{2}e^{2\beta_{3}})$,
the Ricci scalar gets a new contribution:

$R=6\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\right)+\dot{\beta}_{1}^{2}+\dot{\beta}_{2}^{2}+\dot{\beta}_{3}^{2}.$
(4.1)

This formula shows that the anisotropy contributes positively, and the growth
of this term favors the restoration of the chiral symmetry. Once this happens,
the red region becomes light blue; we have already discussed what occurs next.

2) The equation of state of the dominant matter $\epsilon\gg 3$. Even though
the results of [2, 3, 4, 5, 6] are based on a scalar field with a large
equation of state, the smoothing is likely to happen with any type of matter
with a large equation of state [32], so the same outcome $\rightarrow$
spatially flat FRW after a few $e$-folds is expected.

3) The equation of state of the dominant matter $\epsilon\gtrsim 3$, this case
is more complicated, and even the assumptions we made about the initial state
are insufficient to make any predictions.

In any case, since the equation of state of the dominant matter is
$\epsilon\approx 3$, red regions shrink faster than blue regions. After a
while, Fig. 2(a) evolves into Fig. 2(b), where the blue regions dominate the
space.

The remaining paper will focus solely on the blue regions in Figure 2(b),
which are homogeneous and isotropic (FRW) to a good approximation. Within
these regions, the following system of equations determines the dynamics of
the fields $(H\equiv\dot{a}/a)$:

$\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}+\frac{dV}{d\phi}+\frac{dF}{d\phi}\braket{\overline{\Psi}\Psi}=0;$
(4.2)
$3M_{pl}^{2}H^{2}=-\frac{\xi}{M_{p}^{2}}\braket{\overline{\Psi}\Psi}^{2}+\frac{1}{2}\dot{\phi}^{2}+V(\phi)+F(\phi)\braket{\overline{\Psi}\Psi};$
(4.3) $\displaystyle-\frac{4\xi_{c}}{\xi}u+\frac{\partial f_{4}}{\partial
u}+\frac{2}{\Lambda^{2}}f_{0}\left(\ddot{u}+3H\dot{u}\right)+\frac{1}{\Lambda^{2}}\dot{u}^{2}\frac{\partial
f_{0}}{\partial
u}+\frac{1}{\Lambda^{2}}\left(2H^{2}+\dot{H}\right)\left(2uf_{1}+u^{2}\frac{\partial
f_{1}}{\partial u}\right)$
$\displaystyle+\frac{2}{\Lambda^{4}}\left(\frac{1}{20}\dddot{H}+\frac{7}{20}H\ddot{H}+\frac{47}{60}H^{2}\dot{H}+\frac{1}{5}\dot{H}^{2}+\frac{11}{60}H^{4}\right)\frac{\partial
f_{1}}{\partial u}$
$\displaystyle-\frac{2}{\Lambda^{4}}\left(-\frac{1}{2}\dddot{H}-\frac{7}{2}H\ddot{H}-\frac{29}{6}H^{2}\dot{H}-\frac{19}{12}\dot{H}^{2}+\frac{7}{6}H^{4}\right)\frac{\partial
f_{2}}{\partial u}$ (4.4)
$\displaystyle+\frac{2}{\Lambda^{4}}\left(-\frac{3}{5}\dddot{H}-\frac{21}{5}H\ddot{H}-\frac{17}{5}H^{2}\dot{H}-\frac{37}{30}\dot{H}^{2}+\frac{19}{5}H^{4}\right)\frac{\partial
f_{3}}{\partial u}=0.$

The latter is the gap equation (3) in the FRW space-time and
$u=-2\xi\braket{\overline{\Psi}\Psi}/(\Lambda M_{pl}^{2})$.

###  4.1 Numerical Solutions

The potential for the scalar field is a negative exponential, which allows to
achieve a smooth and flat space. The ansatz of the mixing function, $F$, is
also chosen to be an exponent.

$V(\phi)=-M_{pl}^{2}V_{0}^{2}\exp\left(-\frac{\kappa}{M_{pl}}\phi\right);\hskip
14.22636pt\hskip 14.22636pt\hskip
14.22636ptF(\phi)=F_{0}\exp\left(-n\frac{\kappa}{M_{pl}}\phi\right).$ (4.5)

The parameter $\kappa\equiv M_{pl}/\Lambda_{\phi}$ controls the number of
$e$-folds of contraction of the Hubble radius required to achieve the smooth
and flat state. The large value of $\kappa$ is a consequence of its
definition, and increasing this value enhances both rapidity and robustness.
$F_{0}$ has the dimension of mass, and $n\geq 1$. To solve the equations
(4.2)-(4) numerically, it is convenient to define dimensionless variables and
parameters:

$\varphi\equiv\frac{\phi}{M_{pl}};\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pth\equiv\frac{a^{\prime}}{a};\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt\tau\equiv V_{0}~{}t;\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt{}^{\prime}\equiv\frac{d}{d\tau};$ (4.6)
$\theta\equiv\frac{V_{0}^{2}}{\Lambda^{2}};\hskip 14.22636pt\hskip
14.22636pt\hskip
14.22636pt\bar{F}_{0}\equiv\frac{F_{0}}{2\xi\theta\Lambda};\hskip
14.22636pt\hskip 14.22636pt\hskip 14.22636pt\xi_{c}\equiv
2\pi^{2}\frac{M_{pl}^{2}}{\Lambda^{2}}.$ (4.7)

In terms of the new variables, the system (4.2)-(4) takes the following shape:

$\varphi^{\prime\prime}+3h\varphi^{\prime}+\kappa
e^{-\kappa\varphi}+n\kappa\bar{F}_{0}e^{-n\kappa\varphi}u=0;$ (4.8)
$3h^{2}=-\frac{1}{4\xi\theta}u^{2}+\frac{1}{2}\left(\varphi^{\prime}\right)^{2}-e^{-\kappa\varphi}-\bar{F}_{0}e^{-n\kappa\varphi}u;$
(4.9) $\displaystyle
u^{\prime\prime}+3hu^{\prime}+\left(u^{\prime}\right)^{2}\frac{1}{2f_{0}}\frac{\partial
f_{0}}{\partial u}+\frac{1}{2\theta f_{0}}\frac{\partial f_{4}}{\partial
u}-\frac{2\xi_{c}}{\xi\theta
f_{0}}u+\frac{1}{2}\left(2h^{2}+h^{\prime}\right)\left(2u\frac{f_{1}}{f_{0}}+\frac{u^{2}}{f_{0}}\frac{\partial
f_{1}}{\partial u}\right)$
$\displaystyle+\theta\left(\frac{1}{20}h^{\prime\prime\prime}+\frac{7}{20}hh^{\prime\prime}+\frac{47}{60}h^{2}h^{\prime}+\frac{1}{5}(h^{\prime})^{2}+\frac{11}{60}h^{4}\right)\frac{1}{f_{0}}\frac{\partial
f_{1}}{\partial u}$ (4.10)
$\displaystyle-\theta\left(-\frac{1}{2}h^{\prime\prime\prime}-\frac{7}{2}hh^{\prime\prime}-\frac{29}{6}h^{2}h^{\prime}-\frac{19}{12}(h^{\prime})^{2}+\frac{7}{6}h^{4}\right)\frac{1}{f_{0}}\frac{\partial
f_{2}}{\partial u}$
$\displaystyle+\theta\left(-\frac{3}{5}h^{\prime\prime\prime}-\frac{21}{5}hh^{\prime\prime}-\frac{17}{5}h^{2}h^{\prime}-\frac{37}{30}(h^{\prime})^{2}+\frac{19}{5}h^{4}\right)\frac{1}{f_{0}}\frac{\partial
f_{3}}{\partial u}=0.$

As the background space-time is FRW after a period of slow contraction, the
curvature is entirely determined by the time $\tau$, which has a negative
value and is decreasing. At some moment $\tau_{c}$, the curvature will reach
its critical value $R_{c}$, and the chiral symmetry will break. In our case,
the condensate $u$ is a dynamical field and its evolution from $u=0$ to $u\neq
0$ is not spontaneous. We start the numerical evolution soon after the
symmetry breaking, at
$\tau_{i}\gtrsim\tau_{c}~{}(|\tau_{c}|\gtrsim|\tau_{i}|)$. Before symmetry
breaks, the solutions to the field equations are known analytically; see, for
example, [33] chapter 4.2. These solutions can be used to fix the initial
values of $\varphi$ and $\varphi^{\prime}$:

$\varphi(\tau_{i})=\frac{1}{\kappa}\log\left[\frac{\kappa^{4}}{2}~{}\frac{\tau_{i}^{2}}{\kappa^{2}-6}\right];\hskip
14.22636pt\varphi^{\prime}(\tau_{i})=\frac{2}{\kappa~{}\tau_{i}};$ (4.11)

Before the chiral symmetry breaking, $u$ sits in the minimum of the effective
potential, so $u(\tau)=u^{\prime}(\tau)=0$ for $\tau\leq\tau_{c}$. To set
initial conditions at $\tau=\tau_{i}$ the explicit breaking of the chiral
symmetry needs to be accounted for. The explicit symmetry breaking (scalar-
spinor mixing in (2)) tilts the effective potential and forces condensate to
move away from $u=0$, see Fig 3(a). Ignoring the curvature corrections and
accounting for the effect of explicit symmetry breaking within the effective
potential $(\tau>\tau_{c})$:

$u^{\prime\prime}(\tau)=-2\xi\bar{F}_{0}e^{-n\kappa\varphi}\left[\log\left(\frac{e^{n\kappa\varphi}}{2\xi\theta\bar{F}_{0}}\right)\right]^{-1}\equiv
ddu(\tau).$ (4.12)

Since $\tau_{i}\approx\tau_{c}$, this equation can be approximately integrated
and used to set the initial conditions for $u$:

$u(\tau_{i})=\frac{1}{2}ddu(\tau_{i})~{}(\tau_{i}-\tau_{c})^{2};\hskip
14.22636pt\hskip 14.22636pt\hskip
14.22636ptu^{\prime}(\tau_{i})=ddu(\tau_{i})~{}(\tau_{i}-\tau_{c}).$ (4.13)

To set up the initial condition for $h$, (4.9) can be used, together with
(4.11) and (4.13):

$h(\tau_{i})=-\sqrt{\frac{4}{\kappa^{4}~{}\tau_{i}^{2}}-\frac{\bar{F}_{0}}{3}e^{-n\kappa\varphi(\tau_{i})}u(\tau_{i})-\frac{1}{12\xi\theta}u^{2}(\tau_{i})}$
(4.14)

(a)

(b)

Figure 3: Left: NJL effective potential in the spontaneously broken phase.
Solid blue: without explicit symmetry breaking, the effective potential is
$\mathbb{Z}_{2}$ symmetric. Dashed red: with explicit symmetry breaking, the
potential is tilted, and negative values of $u$ are favored.
Right: Total potential of the scalar field (4.19) at different times
$\tau=\tau_{c},<\tau^{\prime},<\tau^{\prime\prime},<\tau_{B}$. At
$\tau<\tau_{c}$, $V_{tot}$ is just a negative exponent. After the chiral
symmetry breaking, $V_{tot}$ transitions into the Morse potential ($n=2$),
which has an exponentially increasing wall from which the scalar field bounces
and continues to move towards increasing positive values.

The scale standing in front of the negative exponent is chosen in the
following way:

$M_{pl}^{2}V_{0}^{2}=\left(10^{-1}M_{pl}\right)^{4};\hskip 14.22636pt\hskip
14.22636pt\hskip 14.22636pt\rightarrow\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636ptV_{0}=10^{-2}M_{pl}.$ (4.15)

In the numerical calculations $\kappa=1000$. Since the value of $V_{0}$ is
fixed, fixing the value of $\theta$ is equivalent to fixing the values of
$\Lambda$ or $\xi_{c}$. In the calculations $\theta=10^{-3}$ is used, which
corresponds to $\Lambda=M_{pl}/\sqrt{10}$ and $\xi_{c}=2\pi^{2}\cdot 10\approx
200$. The four-Fermi coupling, $\xi$, will be tuned to be just below its
critical value: $\xi/\xi_{c}=1-10^{-5}$. The IR cut-off scale
$\epsilon=y\Lambda$ and explicit chiral symmetry breaking scale
$F_{0}\exp(-n\kappa\varphi(\tau_{i}))$ should be related and within an order
of magnitude of each other:

$y\sim 2\bar{F}_{0}\xi\theta\exp(-n\kappa\varphi(\tau_{i}))\ll 1.$ (4.16)

The last inequality ensures the validity of the approximation we took by
ignoring the explicit symmetry breaking in Appendix B. Explicitly:

$y=10^{-3};\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pt\frac{F_{0}}{\epsilon}\exp(-n\kappa\varphi(\tau_{i}))=10;\hskip
14.22636pt\hskip 14.22636pt\hskip 14.22636pt\rightarrow\bar{F}_{0}\approx
92.38.$ (4.17)

The approximation (4.16) breaks down and the value of
$F_{0}\exp(-n\kappa\varphi)$ becomes trans-Planckian when $\varphi$ becomes
negative. This problem doesn’t exist in [1], where a rigid mass of the spinor
was used. Note that compared to [2, 3, 4, 5], we had to add three additional
parameters, two of them are associated with the NJL part of the action
($\theta;~{}\xi$) and one with the scalar-spinor mixing ($\bar{F}_{0}$). In
this sense, the model is very minimal. We do not count $F(\tau_{i})/\epsilon$
and $n$ as additional parameters, as their values do not affect the model’s
outcome.

(a)

(b)

Figure 4: Left: The Hubble parameter, $h=H/V_{0}$, as a function of
$\tau=tV_{0}$. Right: Derivative of the Hubble parameter as a function of
$\tau$. The violation of the NEC happens at around $\tau\approx 0.012$. At
this moment, the Hubble parameter starts to grow. At around $\tau\approx
0.015$, the derivative of the Hubble parameter reaches its maximum value and
starts to decrease, but before $\dot{H}$ reaches zero, the bounce ($H=0$)
happens.

For the mentioned values of $\kappa$, $\xi$, $y$, and $\theta$, the chiral
symmetry breaking happens at approximately $\tau_{c}=-0.0355$. The choice of
$\tau_{i}$ does affect the outcome of the model. Even though it does not
affect the violation of NEC, it does affect the bounce. We solved the system
(4.8)-(4.1) for hundred $\tau_{i}$ evenly distributed between the interval
$[-0.0119,-0.0109]$ (chosen randomly). We observed the NEC violation for all
values of $\tau_{i}$, but the bounce took place only in five cases. If
$\tau_{i}$ is chosen randomly at different patches of space, those patches
that bounce will dominate the volume as they are the ones expanding. In
contrast, those patches that do not bounce will continue to shrink and
eventually crunch to a point. One of the choices for which the bounce does
happen is $\tau_{i}=-0.01099$. The numerical solutions for the dimensionless
Hubble parameter $h$, its derivative $h^{\prime}$, the condensate $u$, the
Ricci scalar curvature $R$, the scalar field $\varphi$ and the derivative of
the scalar field energy density
$\partial_{\tau}\left(\frac{1}{2}(\varphi^{\prime})^{2}-\exp(-\kappa\varphi)\right)$
are shown on Figs. 4(a), 4(b), 5(a), 5(b), 6(a) and 6(b) respectively.

At the moment of bounce, the velocity of the condensate must be zero,
$\dot{u}=0$. This is indeed the case as in Fig. 5(a). The reason is that
derivatives of the Hubble parameter contain terms proportional to $\dot{u}/H$,
and their signs are such that their large values favor chiral symmetry
restoration. As an example for the dimensionless $\dot{H}$ we have:

$h^{\prime}=-\frac{1}{2}\left(\varphi^{\prime}\right)^{2}-\frac{u^{\prime}}{6h}\left(\frac{1}{2\xi\theta}u+\bar{F}_{0}e^{-n\kappa\varphi}\right).$
(4.18)

As $H\rightarrow 0$, derivatives of the Hubble parameter grow, and the wall of
the effective potential $u$ is climbing becomes steeper. At the moment of
bounce, $\tau_{B}$, $\dot{u}=0$, but $\dot{u}/H$ is finite, so the process is
self-consistent, and singularity is never reached.

(a)

(b)

Figure 5: Left: The condensate, $u$, as a function of $\tau$. After the chiral
symmetry breaking, the condensate evolves with initial velocity defined by the
explicit symmetry breaking (4.13). Soon after this, the chiral symmetry gets
restored, but due to the Hubble anti-friction, the absolute value of $u$
continues to increase. At around $\tau\approx 0.00125$, the chiral symmetry
breaks again and we observe a sharp rise of $u$. As $h\rightarrow 0$ chiral
symmetry gets restored and the wall of the effective potential $u$ is climbing
becomes steeper and steeper, this decelerates $u$; its velocity becomes zero
at the moment of bounce, while $u^{\prime}/h$ is finite. The latter guarantees
finiteness of $h^{\prime}$, as in any other case, it would blow up, see
(4.18).
Right: Ricci scalar curvature as a function of $\tau$. Its Maximum value is
reached at around $\tau\approx 0.015$ and is $R\approx
0.04\Lambda^{2}=0.004M_{pl}^{2}$. This value is several orders below the
Planck mass, so calling the bounce classical is justified, as the quantum
gravity regime is never reached. It is also $\ll\Lambda^{2}$, so ignoring
higher order curvature corrections in (B.3) is justified too.

(a)

(b)

Figure 6: Left: The scalar field $\varphi=\phi/M_{pl}$ as a function of
$\tau$. The dynamics is consistent with $\phi$ moving under the influence of
the (4.19) potential. After the chiral symmetry breaking, the negative
exponent effectively becomes a Morse potential, so the scalar field, $\phi$,
which was moving from positive to negative values, finds an exponential wall
and bounces from it. This happens at around $\tau\approx 0$. Then $\phi$
continues to move along the exponential plateau. After the bounce, it stops at
some positive value due to the Hubble friction (not shown here).
Right: Change of the scalar field energy density, $\rho_{\phi}$, as a function
of $\tau$. The negative values of $\rho_{\phi}^{\prime}$ follow from the fact
that, as $\phi$ is climbing the exponential wall, it is decelerating. After it
reaches the top of the wall and its velocity changes sign, $\phi$ starts to
slide down and $\rho_{\phi}^{\prime}$ becomes positive again. At the same time
as $h\rightarrow 0$, the Hubble anti-friction is decreasing, so
$\rho_{\phi}^{\prime}$ approaches zero from positive values.

The scalar field effectively moves through the following potential:

$V_{tot}=M_{pl}^{2}V_{0}^{2}\Big{(}-e^{-\kappa\varphi}-\bar{F}_{0}u~{}e^{-n\kappa\varphi}\Big{)}.$
(4.19)

When the chiral symmetry is unbroken, $u=0$, and $V_{tot}$ coincides with the
negative exponential potential. Once the chiral symmetry breaks, $V_{tot}$
effectively becomes a Morse potential ($n=2$). Fig. 3(b) shows $V_{tot}$ at
different moments of the evolution. After the symmetry breaks, the total
potential of the scalar becomes bounded from below and exponentially
increasing for negative values of $\phi$. $\phi$, which before the chiral
symmetry breaking was moving on a negative exponential potential from positive
to negative values, after the chiral symmetry breaking encounters an
exponential wall, bounces from it and continues to move on a plateau see Fig.
6(a). After the bounce, the field $\phi$ will decelerate due to the Hubble
friction and will freeze at some positive value of $\phi$. Before the chiral
symmetry breaking, the energy density of the scalar field, $\rho_{\varphi}$,
is increasing due to the Hubble anti-friction. From Eq. (4.8) it follows:

$\rho_{\varphi}^{\prime}=\partial_{\tau}\left(\frac{1}{2}(\varphi^{\prime})^{2}-\exp(-\kappa\varphi)\right)=-3h\left(\varphi^{\prime}\right)^{2}-n\kappa\bar{F}_{0}\varphi^{\prime}e^{-n\kappa\varphi}u.$
(4.20)

The first term on the RHS of (4.20) is positive during the contraction and
becomes zero at the moment of bounce, while the sign of the second term
depends on the sign of $\varphi^{\prime}$. After chiral symmetry breaking,
while the scalar is climbing the exponential wall, its velocity is negative
and decreasing. Since $u$ is also negative, the second term contributes
negatively. As a result $\rho_{\varphi}$ decreases, see Fig 6(b). One can
argue that the scalar-condensate mixing causes effect of Hubble friction
(deceleration of $\varphi$), which might be relevant in the production of the
adiabatic perturbations [25, 26]. Keep in mind that, in our case, the
deceleration happens not during the slow contraction but after that and just
before the bounce, so the deceleration of the scalar field by itself might not
be sufficient to achieve that goal.

During the bounce, the Hubble parameter $h=0$ and the total energy density
$\rho=0$, the most reliable candidate to control the validity of the effective
theory is the Ricci scalar curvature. From Fig. 5(b), we see that the maximum
value of Ricci curvature during the evolution is $R_{max}\approx
0.04\Lambda^{2}=0.004M_{pl}^{2}$, which is small enough for both the curvature
expansion in (B.3) to be valid and gravity to stay in the classical regime.

##  5 Discussion and Unanswered Questions

In this paper, we have extended our proposal [1] that a chiral condensate
might induce a cosmological bounce. We explicitly showed that, under certain
conditions, the Nambu-Jona-Lasinio condensation can be triggered with a
gravitational field when the absolute value of the Ricci curvature exceeds a
certain value. The NJL condensation is a vacuum effect and preserves the
symmetries of space-time. This is a significant improvement compared to
earlier works in the literature that attempted to use a dense gas of spin-
aligned fermions to induce a bounce, which can destroy the FRW symmetries that
must be preserved. Using the numerical solutions, it was demonstrated that
this condensate can act as the source of NEC violation and ultimately lead to
a cosmological bounce. We also addressed several points not discussed in [1];
these include the derivation of the initial conditions for the condensate and
derivation of the NJL effective action in the presence of a gravitational
field. Compared to [1], a different potential for the scalar field was used.
The fact that the mechanism works in both cases hints at its robust nature. We
have yet to address what might happen after the bounce or the perturbative
stability of the solutions. This will be the subject of future works.

The four-Fermi interactions that cause condensation create an attractive force
between the spinors. Gravity always creates attractive force, so including
gravitational effects should always enhance the condensation process.
According to the analysis and arguments of section 3, this is not the case, as
a positive Ricci curvature can restore the chiral symmetry even when NJL
coupling constant exceeds its (flat space) critical value. One of the
mysteries that will be explored in future work is explaining how gravity can
act to prevent chiral symmetry breaking.

## Acknowledgements

I would like to express my gratitude to Paul Steinhardt for his invaluable
support and guidance throughout this project. I would like to thank Greg
Gabadadze, Anna Ijjas, Otari Sakhelashvili, Richard Woodard, David Shlivko,
Stephon Alexander, Andrew Sullivan, Alejandro Cárdenas-Avendaño for
discussions and comments and Thibault Damour for correspondence. This work is
supported by the Simons Foundation grant number 654561.

##  Appendix A Integrating out the Torsion

First of all, let’s split the action (2) into torsion dependent,
$\mathcal{S}_{T}$, and torsion independent, $\mathcal{S}_{0}$, parts:

$\mathcal{S}=\mathcal{S}_{0}+\mathcal{S}_{T};$ (A.1)
$\displaystyle\mathcal{S}_{0}=$
$\displaystyle-\frac{M_{pl}^{2}}{4}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
R^{MN}-\frac{i}{2\cdot 3!}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
e^{M}\wedge\Big{(}\overline{\Psi}\gamma^{N}D\Psi-D\overline{\Psi}\gamma^{N}\Psi\Big{)}$
$\displaystyle+\frac{1}{4!}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge e^{M}\wedge
e^{N}\left[\frac{1}{2}(\partial\phi)^{2}+V(\phi)+F(\phi)\overline{\Psi}\Psi\right];$
$\displaystyle\mathcal{S}_{T}=$
$\displaystyle-\frac{M_{pl}^{2}}{4}\epsilon_{ABMN}e^{A}\wedge e^{B}\wedge
C^{MK}\wedge{C_{K}}^{N}-\frac{1}{4!}\epsilon_{ABMN}{\epsilon_{IJ}}^{NK}e^{A}\wedge
e^{B}\wedge e^{M}\wedge C^{IJ}A_{K}$
$\displaystyle+\sqrt{\frac{\xi}{6}}\epsilon_{IJKL}{C^{I}}_{M}\wedge
e^{M}\wedge e^{J}\wedge
e^{K}~{}V^{L}+\left(\frac{1}{4}-\sqrt{\frac{\xi}{6}}\right){C^{I}}_{K}\wedge
e^{K}\wedge e_{I}\wedge e_{J}~{}A^{J}.$ (A.2)

Expressions without the tilde are defined wrt $\omega^{AB}$ and satisfy the
following Cartan’s structure equations:

$De^{A}=de^{A}+\left.\omega^{A}\right.{}_{B}\wedge e^{B}=0;$ (A.3)
$0=DDe^{A}=\left.R^{A}\right.{}_{B}\wedge e^{B}.$ (A.4)

Relation between the different spin connections and curvature two forms:

$\tilde{\omega}^{AB}=\omega^{AB}+C^{AB};\hskip 14.22636pt\hskip
14.22636pt\hskip 14.22636pt\tilde{R}=R^{AB}+DC^{AB}+C^{AK}\wedge{C_{K}}^{B}.$
(A.5)

Since $C^{AB}$ is a difference between the connections, it is a tensor.
Variation of the $\mathcal{S}_{T}$ wrt to $C$, yields an algebraic equation:

${C^{X}}_{MN}-{C^{X}}_{NM}+\delta^{X}_{M}{C^{K}}_{NK}-\delta^{X}_{N}{C^{K}}_{MK}+\frac{1}{M_{pl}^{2}}\sqrt{\frac{2\xi}{3}}\Big{(}\delta^{X}_{M}V_{N}-\delta^{X}_{N}V_{M}-{\epsilon^{X}}_{MNK}A^{K}\Big{)}=0$
(A.6)

Tracing this equation gives:

${C^{K}}_{MK}=-\frac{1}{M_{pl}^{2}}\sqrt{\frac{3\xi}{2}}V_{M}.$ (A.7)

The following ansatz is chosen for $C$:

${C^{X}}_{MN}={\epsilon^{X}}_{MNK}P^{K}+\delta^{X}_{N}Q_{M}-\eta_{MN}Q^{X}.$
(A.8)

Note that this is not the most general ansatz for $C$. There should also be a
3-index term [27]. From (A.6), we see that nothing can source the 3-index
term, so we set it to zero from the beginning. Using (A.8) into (A.7) gives:

$Q_{M}=-\frac{1}{M_{pl}^{2}}\sqrt{\frac{\xi}{6}}V_{M}.$ (A.9)

After plugging this into (A.6):

$P_{M}=\frac{1}{M_{pl}^{2}}\sqrt{\frac{\xi}{6}}A_{M}.$ (A.10)

Using (A.9) and (A.10) into (A.8) and plugging the latter into (A), the
torsion-dependent part of the action can be expressed in terms of spinor
currents:

$\mathcal{S}_{T}=-\frac{1}{4!}\epsilon_{ABIJ}e^{A}\wedge e^{B}\wedge
e^{I}\wedge
e^{J}\left[\frac{\xi}{2M_{pl}^{2}}\left(V_{M}V^{M}-A_{M}A^{M}\right)\right].$
(A.11)

The latter can be further simplified with the help of the Fierz identity
(2.6). The final expression for the torsion-dependent part:

$\mathcal{S}_{T}=-\frac{\xi}{4!M_{pl}^{2}}\epsilon_{ABIJ}e^{A}\wedge
e^{B}\wedge e^{I}\wedge
e^{J}\left[\big{(}\overline{\Psi}\Psi\big{)}^{2}+\big{(}i\overline{\Psi}\gamma_{5}\Psi\big{)}^{2}\right]$
(A.12)

does not involve any spinor current vectors, just the scalar and the pseudo-
scalar bi-linears.

##  Appendix B Fermion Loop in External Gravitational Field

This appendix aims to calculate the spinor loop corrections due to the
Gravitational field’s presence up to the second order in the Riemann curvature
tensor. The Riemann Normal Coordinates are used to achieve this goal, but in
the third part of this appendix, the same calculation is done using the more
rigorous Heat Kernel method, and the two results are compared.

###  B.1 Riemann Normal Coordinates

Given a curved space-time, one can always find inertial frames at a given
event $\mathcal{P}_{0}$. Free particles move along straight lines within these
frames, meaning $\Gamma^{\rho}_{\mu\nu}=0$. This all is true only locally. The
geodesics coming out of $\mathcal{P}_{0}$ can be used to define the
neighboring points. This is the main idea in defining the Riemann normal
coordinates. The Riemann normal coordinates $x^{\mu}$ can be chosen in a way
that $x^{\mu}(\mathcal{P}_{0})=0$ and
$g_{\mu\nu}(\mathcal{P}_{0})=\eta_{\mu\nu}$. In addition if $\mathcal{P}$ is a
point in the neighborhood of $\mathcal{P}_{0}$, then
$x^{\mu}(\mathcal{P})=up^{\mu}$ where $p^{\mu}=dx^{\mu}/du(\mathcal{P}_{0})$
is the tangent vector at $\mathcal{P}_{0}$ to the geodesic
$\mathcal{P}_{0}\mathcal{P}$ and $u$ is the proper time (assuming
$\mathcal{P}_{0}\mathcal{P}$ is time-like) [34]. Using these properties, one
can expand the metric at $\mathcal{P}$ in “powers” of the Riemann curvature
tensor [35, 24, 36]:

$\displaystyle g_{\mu\nu}(\mathcal{P})=$
$\displaystyle~{}\eta_{\mu\nu}-\frac{1}{3}R_{\mu\alpha\nu\beta}x^{\alpha}x^{\beta}-\frac{1}{6}\nabla_{\gamma}R_{\mu\alpha\nu\beta}x^{\alpha}x^{\beta}x^{\gamma}$
(B.1)
$\displaystyle+\left(-\frac{1}{20}\nabla_{\gamma}\nabla_{\delta}R_{\mu\alpha\nu\beta}+\frac{2}{45}R_{\alpha\mu\beta\lambda}{R^{\lambda}}_{\gamma\nu\delta}\right)x^{\alpha}x^{\beta}x^{\gamma}x^{\delta}+\mathcal{O}\Big{(}\left(\partial
g\right)^{5}\Big{)}.$

From now on, I will suppress $\mathcal{P}$ from the left-hand side. Having the
expression for the metric (B.1), one can evaluate the Christoffel gamma, the
spin connection, the tetrad, the determinant of the metric, etc., as a series
in the Riemann curvature tensor. Obtaining these expressions is a tedious but
straightforward task and will not be shown here due to their length.

Calculation of the effective action for spinors involves evaluation of the
functional trace $(\text{Tr})$ from the spinor two-point function $S$:

$\text{Tr}S(x,z,s)=\int d^{4}x\sqrt{-g}~{}\lim_{x\rightarrow
z}~{}\text{tr}S(x,z,s).$ (B.2)

Here tr stands for the trace over the Dirac, flavor, color, etc. indices.
Since $S(x,z,s)$ depends only on the difference $x-z$, and since this
procedure involves taking the limit $x\rightarrow z$, using Riemann normal
coordinates to evaluate the effective action is perfectly justified.

Let’s focus on the spinor part of the action and ignore the terms that
explicitly break the chiral symmetry by making the usual assumptions. The
relevant part of the action:

$\mathcal{S}_{NJL}=\int
d^{4}x\sqrt{-g}\left[i\overline{\Psi}\gamma^{\mu}\nabla_{\mu}\Psi+\frac{\xi}{M_{pl}^{2}}\Big{(}\left(\overline{\Psi}\Psi\right)^{2}+\left(i\overline{\Psi}\gamma_{5}\Psi\right)^{2}\Big{)}\right],$
(B.3)

with $\gamma^{\mu}=e^{\mu}_{A}\gamma^{A}$. By introducing the auxiliary
fields:

$\sigma=-\frac{2\xi}{M_{pl}^{2}}\overline{\Psi}\Psi;\hskip 14.22636pt\hskip
14.22636pt\hskip
14.22636pt\pi=-\frac{2\xi}{M_{pl}^{2}}\left(i\overline{\Psi}\gamma_{5}\Psi\right),$
(B.4)

one can rewrite this action as:

$\mathcal{S}_{NJL}=\int
d^{4}x\sqrt{-g}\left[\overline{\Psi}\Big{(}i\gamma^{\mu}\nabla_{\mu}-\sigma-i\pi\gamma_{5}\Big{)}\Psi-\frac{M_{pl}^{2}}{4\xi}\Big{(}\sigma^{2}+\pi^{2}\Big{)}\right].$
(B.5)

The chiral symmetry allows us to set $\pi=0$. After integrating out the spinor
fields, the effective action is obtained:

$\int\mathcal{D}\sigma\mathcal{D}\pi\mathcal{D}\overline{\Psi}\mathcal{D}\Psi\exp\Big{(}i~{}\mathcal{S}_{NJL}\Big{)}=\int\mathcal{D}\sigma\mathcal{D}\pi\exp\Big{(}i~{}\mathcal{S}_{eff}\Big{)};$
(B.6) $\mathcal{S}_{eff}=\int
d^{4}x\sqrt{-g}\left[i\int_{0}^{\sigma}ds~{}\text{tr}S(x,x,s)-\frac{M_{pl}^{2}}{4\xi}\sigma^{2}\right].$
(B.7)

A few words need to be said about the scalar $\sigma$. When the space-time is
uniform and static, so must be $\sigma$, but since one is dealing with a
dynamical space-time $\partial\sigma\neq 0$. For now, let’s (wrongly) assume
that $\sigma$ is uniform and static. The corrections involving non-zero
$\partial\sigma$ will be accounted for later.

The spinor Greens function in a curved space-time satisfies the following
equation:

$\big{(}i\gamma^{\mu}\nabla_{\mu}-s\big{)}S(x,z,s)=\frac{1}{\sqrt{g}}\delta^{(4)}(x-z).$
(B.8)

Let’s define an auxiliary function:

$\big{(}i\gamma^{\mu}\nabla_{\mu}+s\big{)}F(x,z,s)=S(x,z,s),$ (B.9)

satisfying:

$\sqrt{-g}\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{1}{4}R-s^{2}\right)F(x,z,s)=\delta^{(4)}(x-z).$
(B.10)

At this point, one can use the Riemann normal coordinates to expand the LHS in
the series of Riemann curvatures and solve this equation iteratively in the
Fourier space. After a lengthy calculation we arrive [37, 38]:

$\displaystyle F(p)=$
$\displaystyle-\frac{1}{p^{2}+s^{2}}-\frac{1}{12}\frac{R}{(p^{2}+s^{2})^{2}}+\frac{2}{3}\frac{R^{\mu\nu}p_{\mu}p_{\nu}}{(p^{2}+s^{2})^{3}}$
$\displaystyle+\frac{i}{2}\frac{\nabla_{\alpha}R~{}p^{\alpha}}{(p^{2}+s^{2})^{3}}-\frac{i}{12}\frac{\nabla_{\alpha}\mathcal{R}^{\alpha\beta}~{}p_{\beta}}{(p^{2}+s^{2})^{3}}-2i\frac{\nabla_{\gamma}R_{\alpha\beta}~{}p^{\alpha}p^{\beta}p^{\gamma}}{(p^{2}+s^{2})^{4}}$
$\displaystyle-\frac{1}{(p^{2}+s^{2})^{3}}\left(\frac{3}{20}\nabla^{2}R+\frac{1}{144}R^{2}+\frac{2}{45}R_{\mu\nu}R^{\mu\nu}+\frac{1}{15}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+\frac{1}{128}\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}\right)$
$\displaystyle+\frac{1}{(p^{2}+s^{2})^{4}}\left(\frac{4}{5}\nabla^{2}R_{\alpha\beta}+\frac{7}{5}\nabla_{\alpha}\nabla_{\beta}R+\frac{1}{6}RR_{\alpha\beta}+\frac{8}{5}R_{\alpha\rho}R^{\rho}_{\beta}-\frac{4}{5}R^{\mu\nu}R_{\mu\alpha\nu\beta}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\frac{8}{15}{R^{\mu\nu\rho}}_{\alpha}R_{\mu\nu\rho\beta}-\frac{1}{4}\nabla_{\alpha}\nabla_{\rho}{\mathcal{R}^{\rho}}_{\beta}+\frac{1}{32}\mathcal{R}_{\mu\alpha}{\mathcal{R}^{\mu}}_{\beta}\right)p^{\alpha}p^{\beta}$
$\displaystyle-\frac{1}{(p^{2}+s^{2})^{5}}\left(\frac{24}{5}\nabla_{\gamma}\nabla_{\delta}R_{\alpha\beta}+\frac{16}{15}R_{\mu\alpha\nu\beta}{{{R^{\mu}}_{\gamma}}^{\nu}}_{\delta}+\frac{4}{3}R_{\alpha\beta}R_{\gamma\delta}\right)p^{\alpha}p^{\beta}p^{\gamma}p^{\delta},$

where:

$\mathcal{R}_{\mu\nu}\equiv R_{AB\mu\nu}\left[\gamma^{A},\gamma^{B}\right].$
(B.11)

Using the auxiliary function $F(p)$, the effective action can be written as:

$\mathcal{S}_{eff}=\int
d^{4}x\sqrt{-g}\left[i\int_{0}^{\sigma}ds~{}s\int\frac{d^{4}p}{(2\pi)^{4}}~{}\text{tr}F(p)-\frac{M_{pl}^{2}}{4\xi}\sigma^{2}\right].$
(B.12)

From this point, one can proceed as follows. First, symmetrize the integrals
by making the replacements (odd powers of $p$ do not contribute to the
integration):

$p^{\alpha}p^{\beta}\rightarrow\frac{p^{2}}{4}\eta^{\alpha\beta};\hskip
14.22636pt\hskip 14.22636pt\hskip
14.22636ptp^{\alpha}p^{\beta}p^{\gamma}p^{\delta}\rightarrow\frac{p^{4}}{4(4+2)}\Big{(}\eta^{\alpha\beta}\eta^{\gamma\delta}+\eta^{\alpha\gamma}\eta^{\beta\delta}+\eta^{\alpha\delta}\eta^{\beta\gamma}\Big{)}.$
(B.13)

After which the Wick rotation, $p_{0}\rightarrow ip_{0E}$, is done, and the
momentum integrals are cut at the UV scale $p_{E}^{2}=\Lambda^{2}$ and IR
scale $p_{E}^{2}=\epsilon^{2}$. Then the final integration wrt $s$ is
performed. The flat space invariant cut-off is used because the curvature is
treated as a perturbation over the flat space. The necessity of the IR scale
is discussed in section 3. Finally, the effective action involving corrections
up to the second order in curvature is obtained:

$\displaystyle S_{eff}^{CC}=\frac{\Lambda^{4}}{16\pi^{2}}\int
d^{4}x\sqrt{-g}\left[-\frac{M_{pl}^{2}}{\Lambda^{2}}\frac{4\pi^{2}}{\xi}u^{2}+f_{4}+\frac{R~{}u^{2}}{6\Lambda^{2}}f_{1}\right.$
$\displaystyle~{}~{}~{}~{}~{}+\frac{2}{\Lambda^{4}}\left(-\frac{1}{120}\nabla^{2}R+\frac{1}{288}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}-\frac{7}{1440}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\left(f_{1}+\log\left(\frac{1}{y^{2}}\right)\right)$
$\displaystyle~{}~{}~{}~{}~{}-\frac{2}{\Lambda^{4}}\left(\frac{1}{12}\nabla^{2}R+\frac{1}{36}R_{\mu\nu}R^{\mu\nu}+\frac{1}{144}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)f_{2}$
(B.14)
$\displaystyle~{}~{}~{}~{}~{}\left.+\frac{2}{\Lambda^{4}}\left(\frac{1}{10}\nabla^{2}R+\frac{1}{72}R^{2}+\frac{7}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{60}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)f_{3}\right],$

with:

$f_{1}\equiv\frac{1}{1+u^{2}}-\frac{y^{2}}{y^{2}+u^{2}}-\log\left(\frac{1+u^{2}}{y^{2}+u^{2}}\right);\hskip
14.22636pt\hskip
14.22636ptf_{2}\equiv\frac{1}{2(1+u^{2})^{2}}-\frac{y^{4}}{2(y^{2}+u^{2})^{2}};$
(B.15) $f_{3}\equiv\frac{1}{3(1+u^{2})^{3}}-\frac{y^{6}}{3(y^{2}+u^{2})^{3}};$
(B.16)
$f_{4}\equiv(1-y^{2})u^{2}+\log\left(1+u^{2}\right)-y^{4}\log\left(1+\frac{u^{2}}{y^{2}}\right)-u^{4}\log\left(\frac{1+u^{2}}{y^{2}+u^{2}}\right).$
(B.17) $u\equiv\frac{\sigma}{\Lambda};\hskip 14.22636pt\hskip 14.22636pt\hskip
14.22636pty\equiv\frac{\epsilon}{\Lambda}.$ (B.18)

The first line of (B.1) was also obtained in [23] (with $y=0$), while the
second, third, and fourth lines, i.e., the curvature2 corrections, are new.

###  B.2 Derivative Corrections to the NJL Effective Action

To capture the derivative corrections to the NJL effective action, the
algorithm developed by M. K. Gaillard [39] is used. Her metric is mostly
minus, so the formulas below have been modified to fit our conventions. The
goal is to calculate corrections up to the second order in derivatives, and
since these derivatives act on a scalar, the curvature of the space-time will
be ignored. The spinor action is once again the starting point:

$\int d^{4}x\overline{\Psi}\Big{(}i\not{\partial}-\sigma\Big{)}\Psi.$ (B.19)

After integrating out the spinor field, the one-loop effective Lagrangian is:

$\mathcal{L}\propto\int
d^{4}p~{}\text{tr}\log\Big{(}p^{2}+\hat{M}^{2}-i\widehat{\not{D}M}\Big{)},$
(B.20)

where:

$\hat{M}=\sum_{n=0}^{+\infty}\frac{i^{n}}{n!}\Big{(}\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\sigma\Big{)}\frac{\partial^{n}}{\partial
p_{\mu_{1}}\cdots\partial
p_{\mu_{n}}}=\sigma+i\partial_{\mu}\sigma\frac{\partial}{\partial
p_{\mu}}-\frac{1}{2}\partial_{\mu}\partial_{\nu}\sigma\frac{\partial^{2}}{\partial
p_{\mu}\partial p_{\nu}}+\mathcal{O}\Big{(}\partial^{3}\sigma\Big{)};$ (B.21)
$\widehat{\not{D}M}=\sum_{n=0}^{+\infty}\frac{i^{n}}{n!}\Big{(}\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\not{\partial}\sigma\Big{)}\frac{\partial^{n}}{\partial
p_{\mu_{1}}\cdots\partial
p_{\mu_{n}}}=\not{\partial}\sigma+i\partial_{\mu}\not{\partial}\sigma\frac{\partial}{\partial
p_{\mu}}+\mathcal{O}\Big{(}\partial^{3}\sigma\Big{)}.$ (B.22)

Note that the presence of imaginary unit $i$ in front of the last term in
(B.20) is essential; without it, the kinetic term for $\sigma$ one gets as a
result of this procedure would have the wrong sign when $u=\sigma/\Lambda\ll
1$. This imaginary unit is missing in the original formula (4.21) in [39]. The
reason is that while transitioning from formula (4.7) to (4.14), using (4.8)
and (4.10-4.12), the imaginary unit is dropped, and then the typo/mistake
propagates. After recovering it, one indeed arrives to (B.20). The presence of
the imaginary unit can also be justified from the following reasoning:
dropping all $\partial/\partial p$ corrections in (B.21) and (B.22) one has:

$p^{2}+\hat{M}^{2}-i\widehat{\not{D}M}=p^{2}+\sigma^{2}-i\not{\partial}\sigma\rightarrow-\partial^{2}+\sigma^{2}-i\not{\partial}\sigma,$
(B.23)

which does coincide with the “square” of the Dirac operator:

$\left(i\not{\partial}-\sigma\right)\left(i\not{\partial}-\sigma\right)^{\dagger}=-\left(i\not{\partial}-\sigma\right)\left(i\not{\partial}+\sigma\right)=-\partial^{2}+\sigma^{2}-i\not{\partial}\sigma.$
(B.24)

The integrand in (B.20) has to be understood in the following way:

The second logarithm has to be Taylor expanded up to the second order in
$\partial\sigma$. After performing symmetrization of the momentum integral,
Wick rotating the $p_{0}$ and imposing IR and UV cut-offs:

$\frac{1}{\Lambda^{4}}\mathcal{L}\propto~{}f_{4}+\frac{1}{\Lambda^{2}}\left(\partial
u\right)^{2}f_{0},$ (B.25)

with:

$f_{0}\equiv
f_{1}+4f_{3}+8u^{2}\left(\frac{1}{(1+u^{2})^{3}}-\frac{y^{4}}{(y^{2}+u^{2})^{3}}\right).$
(B.26)

In (B.25), the first term is the usual loop contribution, while the second
term is the kinetic correction to it. We are after the latter since the first
term is already present in (B.1). The next order corrections to (B.25) would
involve terms like $\left(\partial\sigma\right)^{4}$,
$\partial^{2}\sigma\left(\partial\sigma\right)^{2}$ and
$\partial\sigma\partial\sigma\partial\partial\sigma$, as well as
$R\left(\partial\sigma\right)^{2}$ and
$R_{\mu\nu}\partial^{\mu}\sigma\partial^{\nu}\sigma$ when the curvature of the
space-time is taken into account. These terms will be ignored in the current
analysis.

###  B.3 Comparison With the Heat Kernel Results

Combing the results of the two previous subsections, one arrives at the final
NJL action involving both the kinetic and the curvature corrections:

$\displaystyle S_{eff}=\frac{\Lambda^{4}}{16\pi^{2}}\int
d^{4}x\sqrt{-g}\left[-\frac{M_{pl}^{2}}{\Lambda^{2}}\frac{4\pi^{2}}{\xi}u^{2}+f_{4}+\frac{1}{\Lambda^{2}}\left(\partial
u\right)^{2}f_{0}+\frac{R~{}u^{2}}{6\Lambda^{2}}f_{1}\right.$
$\displaystyle~{}~{}~{}~{}~{}+\frac{2}{\Lambda^{4}}\left(-\frac{1}{120}\nabla^{2}R+\frac{1}{288}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}-\frac{7}{1440}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\left(f_{1}+\log\left(\frac{1}{y^{2}}\right)\right)$
$\displaystyle~{}~{}~{}~{}~{}-\frac{2}{\Lambda^{4}}\left(\frac{1}{12}\nabla^{2}R+\frac{1}{36}R_{\mu\nu}R^{\mu\nu}+\frac{1}{144}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)f_{2}$
(B.27)
$\displaystyle~{}~{}~{}~{}~{}\left.+\frac{2}{\Lambda^{4}}\left(\frac{1}{10}\nabla^{2}R+\frac{1}{72}R^{2}+\frac{7}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{60}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)f_{3}\right].$

Setting the IR scale $y=0$, and going in the limit
$\Lambda\rightarrow+\infty$, the action (B.3) simplifies significantly:

$\displaystyle S_{eff}=\frac{1}{16\pi^{2}}\int
d^{4}x\sqrt{-g}\left[-M_{pl}^{2}\frac{4\pi^{2}}{\xi}\sigma^{2}+2\hbar\sigma^{2}\Lambda^{2}-2\hbar\left(\frac{1}{2}\sigma^{4}+\frac{1}{2}\left(\partial\sigma\right)^{2}+\frac{R\sigma^{2}}{12}\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.\left.-\frac{1}{120}\nabla^{2}R+\frac{1}{288}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}-\frac{7}{1440}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\log\left(\frac{\Lambda^{2}}{\sigma^{2}}\right)\right].$
(B.28)

In the last formula, $\hbar$ was recovered to highlight the quantum nature of
the relevant terms, and only those were kept that diverge as
$\Lambda\rightarrow+\infty$.

Terms proportional to $\hbar$ can also be calculated using the Heat Kernel
method. Let’s follow the papers by Barvinsky et al. [40, 41]. The loop
contribution can be written as the logarithm from the functional determinant
of the Dirac operator, i.e.:

$\log\det\Big{(}g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{1}{4}R+i\gamma^{\mu}\left(\nabla_{\mu}\sigma\right)-\sigma^{2}\Big{)}.$
(B.29)

According to [40, 41]:

$\log\det\Big{(}g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+\hat{P}-\frac{1}{6}R\Big{)}=-\int\frac{d\tau}{\tau}K(\tau),$
(B.30)

where $K(\tau)$ is the Heat Kernel and $\tau$ is the proper time. The lower
limit of $\tau$ corresponds to the UV limit in momentum. In the limit
$\tau\rightarrow 0$ (Schwinger-DeWitt), the Heat Kernel can be Taylor expanded
in powers of $\tau$:

$\displaystyle K(\tau)=$ $\displaystyle\frac{1}{(4\pi)^{2}\tau^{2}}\int
d^{4}x\sqrt{-g}~{}\text{tr}\left[1+\tau\hat{P}+\tau^{2}\left(\frac{1}{2}\hat{P}^{2}-\frac{1}{120}\nabla^{2}R+\frac{1}{768}\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}+\frac{1}{180}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\right].$
(B.31)

Note that in [40, 41], the term $\nabla^{2}R$ is neglected as it’s a total
derivative. This term can be recovered by consulting [42]. Comparing (B.29)
with (B.30) yields:

$\hat{P}\equiv-\frac{1}{12}R-\sigma^{2}+i\gamma^{\mu}\nabla_{\mu}\sigma.$
(B.32)

After plugging this into (B.3):

$\displaystyle K(\tau)=$ $\displaystyle\frac{1}{(4\pi)^{2}\tau^{2}}\int
d^{4}x\sqrt{g}~{}\left[1-\tau\left(\sigma^{2}+\frac{1}{12}R\right)+\tau^{2}\left(\frac{1}{2}\sigma^{4}+\frac{1}{2}(\partial\sigma)^{2}+\frac{1}{12}\sigma^{2}R\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-\frac{1}{120}\nabla^{2}R+\frac{1}{288}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}-\frac{7}{1440}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\right]\text{tr}\mathds{1}.$
(B.33)

In the Heat Kernel regularization, the loop integrals are finite, and the UV
divergences manifest themselves in the lower limit of the proper time
integrals. Cutting the proper time integral at $\tau=\Lambda^{-2}$ in the
lower limit and at $\tau=\sigma^{-2}$ in the upper limit and keeping only the
terms that diverge as $\Lambda\rightarrow+\infty$ one finally obtains:

$\displaystyle\log\det\Big{(}g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{1}{4}R+i\gamma^{\mu}\left(\nabla_{\mu}\sigma\right)-\sigma^{2}\Big{)}=$
$\displaystyle\frac{1}{(4\pi)^{2}}\int
d^{4}x\sqrt{-g}~{}\Bigg{[}\Lambda^{2}\sigma^{2}-\left(\frac{1}{2}\sigma^{4}+\frac{1}{2}(\partial\sigma)^{2}+\frac{1}{12}\sigma^{2}R-\frac{1}{120}\nabla^{2}R\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\frac{1}{288}R^{2}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}-\frac{7}{1440}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\log\left(\frac{\Lambda^{2}}{\sigma^{2}}\right)\Bigg{]}\text{tr}\mathds{1}.$

This formula is identical to the $\hbar$ part of (B.3).

Given the function $F(p)$, one can guess that the next order curvature
corrections to it, symbolically, must have the following form:

$F(p)\sim\frac{R^{n}}{10^{n}(p^{2}+s^{2})^{(n+1)}}+\cdots.$ (B.34)

The “of order” factor $~{}10^{-n}$ can be observed from formulas (4.47)-(4.49)
in [41]. At the level of effective action, these corrections translate as:

$\Delta S_{eff}\sim\Lambda^{4}\int
d^{4}x\sqrt{-g}~{}\frac{1}{10^{n}}~{}\frac{R^{n}}{\Lambda^{2n}}~{}q_{n}\left(\frac{\sigma^{2}}{\Lambda^{2}}\right).$
(B.35)

For $n>2$, the functions $q_{n}$ are polynomials of their argument when this
argument is small; and satisfy $q_{n}(0)=0$. The main difference with the case
we studied in detail is that the linear and quadratic curvature corrections
are additionally multiplied by the $\log(\sigma^{2}/\Lambda^{2})$, so when
$\sigma\ll\Lambda~{}(u\ll 1)$ they are less suppressed.

From the analysis above, it follows that several hierarchies take place when
expanding the effective action as a series in curvature. When $u\ll 1$, the
smallness of $R/\Lambda^{2}$ is not necessary for the curvature truncation we
made in (B.1) to work. The latter is a feature of the four-dimensional space-
time. For the solutions presented in section 4.1, both conditions:
$R/\Lambda^{2}\ll 1$ and $u\ll 1$, are satisfied, so ignoring terms of a
higher degree in curvature is perfectly justified.

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