#  HUPD-2403 Non-thermal particle production in Einstein-Cartan gravity with
modified Holst term and non-minimal couplings

Tomohiro Inagaki 1,2,3<EMAIL_ADDRESS>and Naoki Yoshioka 1222na-
<EMAIL_ADDRESS>

1Graduate School of Advanced Science and Engineering, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan
2Information Media Center, Hiroshima University, Higashi-Hiroshima 739-8521,
Japan
3Core of Research for the Energetic Universe, Hiroshima University, Higashi-
Hiroshima 739-8526, Japan

###### Abstract

Non-thermal fermionic particle production is investigated in Einstein-Cartan
modified gravity with a modified Holst term and non-minimal couplings between
the spin connection and a fermion. By using the auxiliary field method, the
theory is rewritten into a pseudoscalar-tensor theory with Einstein-Hilbert
action and canonical kinetic and potential terms for a pseudoscalar field. The
introduced field is called Einstein-Cartan pseudoscalaron. If the potential
energy of the Einstein-Cartan pseudoscalaron dominates the energy density of
the early universe, it causes inflationary expansion. After the end of
inflation, the pseudoscalaron develops a large value and the non-minimal
couplings destabilize the vacuum. Evaluating the non-thermal fermionic
particle production process, we obtain the mass and the helicity dependences
of the produced particle number density. We show the model parameters to
enhance the preheating and reheating processes.

##  1 Introduction

The extension of general relativity (GR) from a geometrical perspective is one
of the candidates for solving cosmological problems. In $F(R)$ theories, the
gravitational action is replaced by a function of the curvature, $R\rightarrow
F(R)$, and it has been shown to describe well various cosmological phenomena
[1, 2, 3, 4]. For example, the idea explains the inflationary expansion of the
universe, and the gravitational interaction produces particles necessary to
reheat the universe [5, 6]. Through the conformal transformation, the
additional degree of freedom in the modified action can be represented as a
dynamical scalar field that plays a role in the inflaton. After the end of the
inflation, the interaction between the inflaton and the matter converts the
inflaton energy into the matter and reheats the universe [7]. Whether the
dominant reheating process is perturbative or non-perturbative particle
production depends on the structure of the interaction.

Recently [8], in the metric-affine gravity [9] where the metric $g_{\mu\nu}$
and the affine connection ${\Gamma^{\mu}}_{\nu\rho}$ are independent
variables, it is shown that a dynamical pseudoscalaron from the modification
of geometrical quantity are obtained due to the existence of the Holst term
$\epsilon R=\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}/\sqrt[]{-g}$ [10,
11, 12] which consists of the antisymmetric part of the affine
connection(torsion) ${\Gamma^{\mu}}_{\nu\rho}-{\Gamma^{\mu}}_{\rho\nu}$.
Consequently, that pseudoscalaron can be used as the inflaton [13]. Thus, even
in Einstein-Cartan (EC) gravity (metric-compatible metric-affine gravity) [14,
15, 16, 17, 18], the inflation can be realized through the pseudoscalar field
[19, 20] since there exists torsion. By introducing an auxiliary pseudoscalar
field, the Euler-Lagrange equation about affine connection gives the algebra
equation regarding the torsion. Since, in this case, the torsion is
represented by the metric, the auxiliary pseudoscalar field, and the matter
coupling to the affine connection, integrating out torsion yields the
effective metric theory as the Palatini $f(R)$ gravity [2]. In this effective
metric theory, one can get the pseudoscalaron and the interaction between the
pseudoscalaron and the matter therefore can realize the inflation and the
reheating of the universe.

In this paper, we discuss the non-thermal particle production in EC gravity
with the modified Holst term. Since fermions naturally couple to torsion in EC
gravity, they are considered as matter fields. Additionally, the natural
extension of the kinetic term of the fermion [21] is considered. This
extension introduces the coupling between torsion and fermion, and
dimensionless parameters $\alpha,\beta$ [22]. To discuss the non-thermal
fermionic particle production, we follow a way of previous work about
fermionic preheating [23, 24, 25, 26, 27]. In the discussion, it is assumed
that the fermionic field operator is composed of (anti) particles. By this
assumption, one of the new parameters $\alpha$ must vanish. Numerical
calculations eventually reveal that particle production occurs when the value
of $\beta$ is large, and its behavior depends on the mass and helicity of the
particle.

The overview of this paper is as follows. The section 2 reviews the Einstein-
Cartan pseudoscalaron with matter fields and introduces the $(\epsilon R)^{2}$
model [13] effective for inflation. In Sec. 3, we introduce the extension of
the kinetic term of the fermion. From this extension in Einstein-Cartan
pseudoscalaron theory, the equation of motion(EoM) of fermions is non-trivial
and thus non-thermal particle production occurs even in the FRLW universe. In
Sec. 4, several results of numerical calculations about non-thermal particle
production are exhibited. By these results, it can be concluded that more
larger value of $\beta$ contributes to more particle production. Also, it is
observed that the behavior of the number density is very different depending
on whether the mass of the fermion is lighter, heavier or intermediate
compared to the inflaton mass. In Sec. 5, we apply the particle production to
reheat the universe. Finally, a discussion and summary of this paper are
presented in Sec. 6.

Calculations in this paper are based on the following notations.
$m_{p}=\sqrt[]{1/8\pi G_{N}}$ is the planck mass and $m_{\phi}$ is the
inflaton mass. $G_{N}$ means the gravitational Newton constant. The gamma
matrix is defiend by $\\{\gamma_{i},\gamma_{j}\\}=2\eta_{ij}$. $\eta_{ij}$ is
the minkowski metric $\eta=\text{diag}(-,+,+,+)$ 333 The definition of gamma
matrice is  $\gamma^{\mu}=-i\begin{pmatrix}0&\sigma^{\mu}\\\
\overline{\sigma}^{\mu}&0\end{pmatrix},$
$\displaystyle\sigma^{\mu}=(1,\bm{\sigma}),$
$\displaystyle\overline{\sigma}^{\mu}=(1,-\bm{\sigma}),$
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}.$ Charge conjugate
matrix $\mathcal{C}$ is  $\mathcal{C}=i\gamma^{2}\gamma^{0}.$ .
$\epsilon_{\mu\nu\rho\sigma}$ is Levi-Civita antisymmetric symbol
$\epsilon^{0123}\equiv 1$. We use the natural units($\hbar=c=1$). The
symmetrization $A_{\\{i,j\\}}$ and the anti-symmetrization $A_{[i,j]}$ are
respectively $\frac{1}{2}(A_{ij}+A_{ji})$ and $\frac{1}{2}(A_{ij}-A_{ji})$.
Greek indices mean the coordinate of spacetime while Roman indices mean that
of local Minkowski spacetime. $\hat{\mathcal{O}}$ means a q-number.

##  2 Einstein-Cartan pseudoscalaron inflation from modified Holst term

EC gravity is a metric-compatible metric-affine theory [9, 28], where the
metric and the connection are independent variables. Fundamental conditions in
this theory are the tetrad hypothesis and the metric compatible condition,

$\displaystyle\partial_{\mu}e^{i}_{\nu}+{\omega^{i}}_{j\mu}e^{j}_{\nu}-{\Gamma^{\alpha}}_{\nu\mu}e_{\alpha}^{i}=0,$
(1) $\displaystyle\nabla_{\mu}g_{\nu\rho}=0,$ (2)

where, $e^{i}_{\mu}$ represents the tetrad which satisfies
$g_{\mu\nu}=e^{i}_{\mu}e^{j}_{\nu}\eta_{ij}$. The inverse of $e^{i}_{\mu}$
serves as a basis component of local Minkowski spacetime,
$e^{\mu}_{i}e^{\nu}_{j}g_{\mu\nu}=\eta_{ij}$. ${\Gamma^{\mu}}_{\nu\rho}$ is
the affine connection and ${\omega^{ij}}_{\mu}$ is the gauge field of local
Lorentz transformation that is called spin connection. Thus, the covariant
derivative $\nabla$ of spacetime is defined by
$\nabla_{\mu}A^{\nu}\equiv\partial_{\mu}A^{\nu}+{\Gamma^{\nu}}_{\alpha\mu}A^{\alpha}$,
and the covariant derivative $D$ of local Minkowski spacetime is defined by
$D_{\mu}B_{i}=\partial_{\mu}B_{i}+{{\omega_{i}}^{j}}_{\mu}B_{j}$. By Eq. (1),
the curvature tensor ${R^{\mu}}_{\nu\rho\sigma}$ and the strength of local
Lorentz transformation ${R^{ij}}_{\mu\nu}$ are connected by

$\displaystyle{R^{\mu}}_{\nu\rho\sigma}$
$\displaystyle={\partial_{\rho}\Gamma^{\mu}}_{\nu\sigma}-\partial_{\sigma}{\Gamma^{\mu}}_{\nu\rho}+{\Gamma^{\mu}}_{\alpha\rho}{\Gamma^{\alpha}}_{\nu\sigma}-{\Gamma^{\mu}}_{\alpha\sigma}{\Gamma^{\alpha}}_{\nu\rho}$
$\displaystyle=e^{\mu}_{i}e_{\nu
j}(\partial_{\rho}{\omega^{ij}}_{\sigma}-\partial_{\sigma}{\omega^{ij}}_{\rho}+{\omega^{i}}_{k\rho}{\omega^{kj}}_{\sigma}-{\omega^{i}}_{k\sigma}{\omega^{kj}}_{\rho})$
$\displaystyle=e^{\mu}_{i}e_{\nu j}{R^{ij}}_{\rho\sigma}.$ (3)

In this theory, the existence of the antisymmetric part of the affine
connection(torsion) is not prohibited. Thus, the connections $\Gamma$,
$\omega$ and the curvature scalar $R$ can be generally separated into
torsionless and torsionful parts,

$\displaystyle{\Gamma^{\mu}}_{\nu\rho}={\Gamma^{\diamond\mu}}_{\nu\rho}+{K^{\mu}}_{\nu\rho},$
(4) $\displaystyle{\omega^{ij}}_{\mu}={\omega^{\diamond
ij}}_{\mu}+{K^{ij}}_{\mu},$ (5) $\displaystyle
R=R^{\diamond}+T-2\nabla^{\diamond}_{\mu}T^{\mu},$ (6)

with

$\displaystyle{T^{\rho}}_{\mu\nu}\equiv{\Gamma^{\rho}}_{\mu\nu}-{\Gamma^{\rho}}_{\nu\mu},$
(7)

and

$\displaystyle
T=\frac{1}{4}T^{\rho\mu\nu}T_{\rho\mu\nu}-\frac{1}{2}T^{\rho\mu\nu}T_{\mu\nu\rho}-T^{\mu}T_{\mu},$
(8)

where the script $\diamond$ indicates the torsionless parts which consist of
metric and tetrad, ${\Gamma^{\diamond\mu}}_{\nu\rho}$ is Levi-Civita symbol
and ${\omega^{\diamond
ij}}_{\mu}\equiv-e^{j\nu}\nabla^{\diamond}_{\mu}e^{i}_{\nu}$. The torsionful
parts consist of the contorsion, ${K^{\mu}}_{\nu\rho}$, defined by
${K^{\mu}}_{\nu\rho}\equiv\frac{1}{2}({T^{\mu}}_{\nu\rho}+{T_{\nu\rho}}^{\mu}+{T_{\rho\nu}}^{\mu})$
and ${K^{ij}}_{\mu}\equiv e^{i}_{\rho}e^{j\nu}{K^{\rho}}_{\nu\mu}$.

In our research, we start from the action with a function of the Holst term,
$\epsilon R\equiv\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}/e$,

$\displaystyle S=\int
ed^{4}x\Big{\\{}\frac{m^{2}_{p}}{2}R+\frac{m^{2}_{p}}{4}H(\epsilon
R)+\mathcal{L}_{\text{matter}}\Big{\\}},$ (9)

where $e=\text{det}(e^{i}_{\mu})$. Since the modification of the curvature
scalar $R$ can not be represented by a dynamical scalar field [8, 13, 20],
$F(R)$ regime is not considered. Introducing an auxiliary field $\chi$, we
obtain the action

$\displaystyle S=\int
ed^{4}x\Big{\\{}\frac{m^{2}_{p}}{2}R+\frac{m^{2}_{p}}{4}H^{\prime}(\chi)\epsilon
R-V(\chi)+\mathcal{L}_{\text{matter}}\Big{\\}},$ (10)

where $V(\chi)=\frac{m^{2}_{p}}{4}(H^{\prime}(\chi)\chi-H(\chi))$ and
$H^{\prime}(\chi)=\frac{dH(\chi)}{d\chi}$. $\chi$ is a pseudoscalar field. The
action (9) is reproduced by substituting the Euler-Lagrange equation with
respect to $\chi$ into Eq. (10). The Cartan equation is obtained as the Euler-
Lagrange equation with respect to ${\omega^{ij}}_{\mu}$,

${T^{\mu}}_{ij}-T_{i}e^{\mu}_{j}+T_{j}e^{\mu}_{i}+H^{\prime}(\chi)\epsilon^{\mu\alpha\beta\gamma}e_{\delta[i}e_{j]\alpha}{T^{\delta}}_{\beta\gamma}/e={S^{\mu}}_{ij}+\epsilon^{\mu\alpha\beta\gamma}e_{\alpha
i}e_{\beta j}(\partial_{\gamma}H^{\prime}(\chi))/e,$ (11)

where $T_{\mu}\equiv{T^{\nu}}_{\mu\nu}$ is the torsion vector and
${S^{\mu}}_{ij}\equiv\frac{2}{m^{2}_{p}}\frac{\partial{\mathcal{L}}_{\text{matter}}}{\partial{{\omega}^{ij}}_{\mu}}$
is the spin density. Thus, the torsion is rewritten in terms of tetrad
$e^{i}_{\mu}$, matter field and auxiliary field $\chi$. The effective metric
theory is obtained by inserting the solution Eq. (11) into Eq. (10). For the
vacuum (${S^{\mu}}_{mn}=0$), it becomes

$S=\int
ed^{4}x\Big{\\{}\frac{m^{2}_{p}}{2}R^{\diamond}-\frac{3m^{2}_{p}}{4}\frac{\partial_{\mu}H^{\prime}\partial^{\mu}H^{\prime}}{(1+H^{\prime
2})}-V(\chi)\Big{\\}}.$ (12)

We introduce the pseudoscalaron $\phi$ by the redefinition,
$H^{\prime}(\chi)=\text{sinh}\Bigl{(}\frac{\sqrt[]{2}(\phi+\delta)}{\sqrt[]{3}m_{p}}\Bigr{)}$,
and obtain

$S=\int
ed^{4}x\Big{\\{}\frac{m^{2}_{p}}{2}R^{\diamond}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\chi(\phi))\Big{\\}},$
(13)

where $\delta$ is a constant to impose $V(\phi=0)=0$. For example, in the
$(\epsilon R)^{2}$ model [13]

$H(\epsilon R)=b\epsilon R+c(\epsilon R)^{2},$ (14)

we obtain the pseudoscalaron with a potential,
$V(\phi)=\frac{m^{2}_{p}}{16c}\Bigl{(}\sinh\Bigl{(}\frac{\sqrt[]{2}(\phi+\delta)}{\sqrt[]{3}m_{p}}\Bigr{)}-b\Bigr{)}^{2}$.
The constant $\delta$ is fixed to satisfy
$\text{sinh}\Bigl{(}\frac{\sqrt[]{2}\delta}{\sqrt[]{3}m_{p}}\Bigr{)}=b$. As is
shown in Fig.1, a certain value of parameters $b,c$ can realize the potential
with a plateau.

Figure 1: The potential of $\phi$ in the $(\epsilon R)^{2}$ model with
parameters
$b=-320,m_{\phi}=\sqrt[]{\frac{d^{2}V}{d\phi^{2}}}|_{\phi=0}=\sqrt{\frac{(1+b^{2})}{12c}}\sim
2.76*10^{13}\text{GeV}$

###  2.1 Pseudoscalaron inflation in FLRW universe

We consider the homogeneous and isotropic universe described by the FLRW
metric,

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}).$ (15)

where $x^{\mu}=(t,x,y,z)$ represents the cosmological time and the conformal
space, and $a(t)$ is the scale factor. By introducing the conformal time,
$d\eta=a^{-1}dt$, the spacetime is represented by

$\displaystyle ds^{2}=a^{2}(\eta)(-d\eta^{2}+dx^{2}+dy^{2}+dz^{2}),$ (16)

The dot and the prime represent the derivative with respect to the
cosmological time $t$, $\dot{()}=\frac{d}{dt}()$, and the conformal time
$\eta$, $()^{\prime}=\frac{d}{d\eta}()$, respectively. The metric (16) is used
for the analysis in Sec. 4. In the metric (15), the scale factor is developed
through the Friedmann-Robertson equations,

$\displaystyle H^{2}=\frac{\rho}{3m^{2}_{p}},$ (17)
$\displaystyle\dot{H}=-\frac{1}{2m^{2}_{p}}(\rho+p),$ (18)

where $H$ is the Hubble parameter defiend by $H=\frac{\dot{a}}{a}$, and $\rho$
and $p$ respectively denote the energy density and the pressure of the matter.
From Eqs. (17) and (18), we derive

$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{6m^{2}_{p}}(\rho+3p).$ (19)

When the scalar field $\varphi$ distributes homogeneously, the energy density
and the pressure are described as
$\rho=\frac{1}{2}\dot{\varphi}^{2}+V(\varphi)$ and
$p=\frac{1}{2}\dot{\varphi}^{2}-V(\varphi)$. The accelerated expansion takes
place for $V(\varphi)>\dot{\varphi}^{2}$.

If the potential $V(\varphi)$ has a plateau, the scalr field starting from the
plateau induces an inflationary expansion. To solve the horizon and flatness
problems encountered in the expanding universe, the total e-folding number
$N_{e}$, defined as $N_{e}=\log(a_{f}/a_{i})$, should exceed $50\sim 60$.
Here, $a_{f}$ represents the value of the scale factor at the end of
inflation, while $a_{i}$ represents the value of the scale factor at the start
of inflation. To obtain the number, we often employ the slow-roll inflation
scenario. In the slow-roll approximation, the end of inflation is fixed by the
slow-roll parameters $\varepsilon$ and $\eta$, defined by
$\varepsilon=\frac{m^{2}_{p}}{2}\Bigl{(}\frac{1}{V}\frac{dV}{d\varphi}\Bigr{)}^{2},\eta=m^{2}_{p}\frac{1}{V}\frac{d^{2}V}{d\varphi^{2}}$.

The quantum fluctuations of $\varphi$ induce the curvature perturbation, $P$,
during inflation. In terms of the conformal momentum space, a component of the
fourieor decomposition of the curvature perturbation is represented as
$P(k)=P_{r}k^{n_{s}}$. Applying the slow-roll approximation, the amplitude
$P_{r}$ of the curvature perturbation can be derived by
$P_{r}=\frac{V(\varphi)}{24\pi^{2}m^{4}_{p}\varepsilon(\varphi)}$. The
spectral index $n_{s}$ is derived by
$n_{s}=1-6\varepsilon(\varphi)+2\eta(\varphi)$. The scalar-tensor ratio $r$ is
calculated by $r=16\varepsilon(\varphi)$.

The potential has a plateau in the $(\epsilon R)^{2}$ model with
$b=-320,m_{\phi}=\sqrt[]{\frac{d^{2}V}{d\phi^{2}}}|_{\phi=0}=\sqrt{\frac{(1+b^{2})}{12c}}\sim
2.76*10^{13}\text{GeV}$ (Fig.1), and the slow-roll inflation scenario can be
adopted . We assume that the pseudoscalar field regarded as an inflaton
dominates the energy density of the early universe. Consequently, we obtain
values such as $n_{s}\sim 0.969,r\sim 0.003,P_{r}\sim 2.1*10^{-9},N_{e}\sim
60$ that agree with the observation regarding the Cosmic Microwave Background
(CMB). Below, we adapt this model to the non-thermal particle production after
the end of inflation. In our analysis, we consider that the particle
production starts at $\phi\sim
0.94m_{p},\frac{d\phi}{dt}\sim-0.293m_{p}m_{\phi}$ where the slow-roll
parameter $\varepsilon(\phi)$ becomes unity.

###  2.2 The dynamics of background field after the end of inflation

After the end of inflation, the oscillating inflaton $\phi$ dominates the
energy density of the universe. The potential is approximated to be
$V\sim\frac{m^{2}_{\phi}}{2}\phi^{2}$ during the particle production. The
energy density of the inflaton and the scale factor are fixed by the Friedmann
equations (17) and (18). Since the contribution to the pressure is cancelled
between the kinetic and the potential energy, $\rho_{\phi}$ follows the
$\dot{\rho_{\phi}}+3H\rho_{\phi}=0$. The solution of this equation with (17)
is given by

$\displaystyle\rho_{\phi}=m^{2}_{p}m^{2}_{\phi}\Bigl{(}\frac{\sqrt[]{3}}{2}m_{\phi}t+A\Bigr{)}^{-2}.$
(20)

From the Eqs. (17) and (20), the scale factor is derived as

$\displaystyle
a(t)=A^{-\frac{2}{3}}\Bigl{(}\frac{\sqrt[]{3}}{2}m_{\phi}t+A\Bigr{)}^{\frac{2}{3}}.$
(21)

The relation between cosmic time and conformal time is determined by
$dt=ad\eta$ and $a(t=0)=a(\eta=0)=1$,

$\Bigl{(}\frac{\sqrt[]{3}}{2}m_{\phi}\eta+3A\Bigr{)}=3A^{\frac{2}{3}}\Bigl{(}\frac{\sqrt[]{3}}{2}m_{\phi}t+A\Bigr{)}^{\frac{1}{3}}.$
(22)

We assume that the oscillating part of the inflaton can be factored out,

$\displaystyle\phi(t)/m_{p}$ $\displaystyle=CH(t)/m_{\phi}\sin(m_{\phi}t+B)$
$\displaystyle=\frac{C}{\sqrt[]{3}\Bigl{(}\frac{\sqrt[]{3}}{2}m_{\phi}t+A\Bigr{)}}\sin(m_{\phi}t+B).$
(23)

It should be noted that Eq. (23) satisfies the EoM of the inflaton
$\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}\simeq\ddot{\phi}+3H\dot{\phi}+m^{2}_{\phi}\phi=0$.
Arbitrary constants $A$, $B$ and $C$ are fixed by the initial values of $\phi$
and $\dot{\phi}$,

$\displaystyle\rho_{\phi}|_{t=0}/m^{2}_{p}m^{2}_{\phi}$ $\displaystyle=A^{-2}$
$\displaystyle=\Bigl{(}\frac{1}{2}{\dot{\phi}^{2}}+\frac{1}{2}m^{2}_{\phi}{\phi^{2}}\Bigr{)}|_{t=0}/m^{2}_{p}m^{2}_{\phi},$
(24) $\cot(B)=\dot{\phi}/\phi|_{t=0}+\frac{\sqrt[]{3}}{2A},$ (25)
$\phi|_{t=0}=\frac{C}{\sqrt[]{3}A}\sin B.$ (26)

We employ these formula as a simple background for the universe after the end
of inflation.

##  3 A model of the non-minimal couplings to fermion

In this paper, we extend the fermionic field with non-minimal couplings
$\alpha$ and $\beta$ [21],

$\displaystyle\mathcal{L}_{\text{matter}}$
$\displaystyle=-\frac{1}{2}(\overline{\psi}(1-i\alpha-i\beta\gamma^{5})\gamma^{\mu}(\partial_{\mu}+\frac{1}{4}{\omega^{ij}}_{\mu}\gamma_{ij})\psi+\text{h.c.})-m_{\psi}\overline{\psi}\psi,$
(27)

where the Dirac conjugate $\overline{\psi}$ is defined by
$\overline{\psi}=i\gamma^{0}\psi^{\dagger}$ and h.c. means the hermitian
conjugate. The spin density is given by

$\displaystyle{S^{\mu}}_{ij}$
$\displaystyle=\frac{2}{m^{2}_{p}}\frac{\partial\mathcal{L}_{\text{matter}}}{\partial{\omega^{ij}}_{\mu}}$
$\displaystyle=-\frac{1}{m^{2}_{p}}(\epsilon_{ijkl}A^{l}e^{\mu
k}-(e^{\mu}_{i}(\alpha V_{j}+\beta A_{j})-e^{\mu}_{j}(\alpha V_{i}+\beta
A_{i}))).$ (28)

$A_{i}\equiv\bar{\psi}i\gamma^{5}\gamma_{i}\psi$ and
$V_{i}\equiv\bar{\psi}i\gamma_{i}\psi$ denote the axial vector and the vector
current, respectivly. Performing the partial integration, the Lagrangian
density (27) can be decomposed into

$\mathcal{L}_{\text{matter}}=\mathcal{L}^{\diamond}_{\text{matter}}+\frac{1}{8}\hat{T}^{\mu}A_{\mu}-\frac{1}{2}T^{\mu}(\alpha
V_{\mu}+\beta A_{\mu}),$ (29)

where the torsionless part is

$\displaystyle\mathcal{L}^{\diamond}_{\text{matter}}=-(\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi+\frac{1}{2}\overline{\psi}\omega^{\diamond
ijk}\eta_{ik}\gamma_{j}\psi+\frac{i}{4}\epsilon_{ijkl}\omega^{\diamond
ijk}\overline{\psi}\gamma^{5}\gamma^{l}\psi)-m_{\psi}\overline{\psi}\psi.$
(30)

Since non-minimal coupling parameters don’t appear in the torsionless part
(30), these parameters only contribute to the interaction between the torsion
and the fermion.

By solving Eq. (11), the torsion is represented by the inflaton and the
fermion,

$\displaystyle
T_{ijk}=-\frac{2}{3}\eta_{i[j}\eta_{k]l}T^{l}+\frac{1}{6}\epsilon_{ijkl}\hat{T}^{l},$
(31)

with

$\displaystyle
T^{l}=\frac{\text{sech}^{2}(X(\phi))}{2}\Bigl{(}\frac{3}{2m^{2}_{p}}(\alpha
V^{l}+(\beta+\sinh{X(\phi)})A^{l})+3\sinh{X(\phi)}\partial^{l}(\sinh{X(\phi)})\Bigr{)},$
(32) $\displaystyle\hat{T}^{l}$ $\displaystyle\equiv\epsilon^{lijk}T_{ijk}$
$\displaystyle=-3~{}\text{sech}^{2}(X(\phi))\Bigl{(}\frac{1}{m^{2}_{p}}((-1+\sinh{X(\phi)}\beta)A^{l}+\sinh{X(\phi)}\alpha
V^{l})-2\partial^{l}(\sinh{X(\phi)})\Bigr{)},$ (33)

where we write $X(\phi)=\frac{\sqrt[]{2}(\phi+\delta)}{\sqrt[]{3}m_{p}}$.
Inserting the solution (31) into (10), we obtain the effective metric action,

$S=\int
ed^{4}x\Big{\\{}\frac{m^{2}_{p}}{2}R^{\diamond}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)+\mathcal{L}^{\diamond}_{\psi}+f_{\mu}(\phi)A^{\mu}+g_{\mu}(\phi)V^{\mu}+\mathcal{L}_{\text{4-fermi}}\Big{\\}},$
(34)

with

$\displaystyle f_{\mu}=\frac{\sqrt[]{3}}{2\
\sqrt[]{2}}\partial_{\mu}\Bigl{(}\frac{\phi}{m_{p}}\Bigr{)}\Bigl{\\{}\text{sech}(X(\phi))+\beta\tanh(X(\phi))\Bigr{\\}},$
(35) $\displaystyle g_{\mu}=\frac{\sqrt[]{3}\alpha}{2\
\sqrt[]{2}}\partial_{\mu}\Bigl{(}\frac{\phi}{m_{p}}\Bigr{)}\tanh(X(\phi)),$
(36)

and four-fermion interactions,

$\displaystyle\mathcal{L}_{\text{4-fermi}}=-\frac{9~{}\text{sech}^{2}(X(\phi))}{16m^{2}_{p}}\Bigl{(}\alpha^{2}V^{2}+2(\alpha\beta-\frac{\alpha~{}\text{sinh}(X(\phi))}{3})VA+(\beta^{2}-\frac{2\beta\text{sinh}(X(\phi))}{3}+\frac{1}{3})A^{2}\Bigr{)},$
(37)

where $A^{2}$, $VA$ and $V^{2}$ are defined by $A^{2}=A^{\mu}A_{\mu}$,
$VA=V^{\mu}A_{\mu}$ and $V^{2}=V^{\mu}V_{\mu}$.

We assume that the leading contribution to the fermionic particle production
comes from $\phi\overline{\psi}\psi$ interactions in Eq. (34). Thus, we apply
the formalism developed in the previous works of the fermionic preheating [23,
24, 25, 26, 27].

###  3.1 The EoM in FLRW universe

For discussing the fermionic non-thermal particle production after the end of
inflation, we derive the EoM of the classical fermionic field $\psi$ and the
Heisenberg operator $\hat{\psi}$. In the spatially flat and homogeneous FLRW
metric (16), the tetrad is given by $e^{i}_{\mu}=a\delta^{i}_{\mu}$ from its
definition $g_{\mu\nu}=\eta_{ij}e^{i}_{\mu}e^{j}_{\nu}$, and components
regarding the torsionless spin connection are $\eta_{ik}\omega^{\diamond
ijk}=3\frac{\mathcal{H}}{a}\delta^{j0}$ and $\epsilon_{ijkl}\omega^{\diamond
ijk}=0$. Thus, Eq. (30) becomes

$\displaystyle\mathcal{L}^{\diamond}_{\text{matter}}=-\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{3\mathcal{H}}{2a}\overline{\psi}\gamma_{0}\psi-
m_{\psi}\overline{\psi}\psi,$ (38)

where $\mathcal{H}$ is defined by $\mathcal{H}=\frac{a^{\prime}}{a}$.
Therefore, the Lagrangian density of $\psi$ is denoted as

$e\mathcal{L}_{matter}=a^{4}\Big{\\{}-\frac{1}{a}\overline{\psi}\delta^{\mu}_{i}\gamma^{i}\partial_{\mu}\psi-\frac{3\mathcal{H}}{2a}\overline{\psi}\gamma_{0}\psi-
m_{\psi}\overline{\psi}\psi+\frac{1}{a}\delta^{\mu}_{i}(f_{\mu}A^{i}+g_{\mu}V^{i})\Big{\\}},$
(39)

where $f_{\mu}(\phi)$ and $g_{\mu}(\phi)$ are functions of the inflaton
defined by Eqs. (35) and (36). Rescaling the fermion as
$a^{\frac{3}{2}}\psi\rightarrow\psi$, one can finally obtain the Lagrangian
density,

$\displaystyle
e\mathcal{L}_{matter}=-\overline{\psi}\delta^{\mu}_{i}\gamma^{i}\partial_{\mu}\psi-
m_{\psi}a\overline{\psi}\psi+\delta^{\mu}_{i}(f_{\mu}A^{i}+g_{\mu}V^{i}).$
(40)

The inflaton $\phi$ after the end of inflation is assumed to be a homogeneous
field, $\partial_{x}\phi=\partial_{y}\phi=\partial_{z}\phi=0$. The EoM of
$\psi$ is

$\displaystyle-\gamma^{\mu}\partial_{\mu}\psi-
m_{\psi}a\psi+i(g_{0}\gamma^{0}+f_{0}\gamma^{5}\gamma^{0})\psi=0.$ (41)

We note that $\gamma^{\mu}$ in Eq. (41) is the gamma matrices in the local
Lorentz frame. Since the Heisenberg operator $\hat{\psi}$ also satisfies the
identical equation,

$\displaystyle-\gamma^{\mu}\partial_{\mu}\hat{\psi}-m_{\psi}a\hat{\psi}+i(g_{0}\gamma^{0}+f_{0}\gamma^{5}\gamma^{0})\hat{\psi}=0,$
(42)

we can obtain the EoM of the spinors $u_{s}(\bm{k},\eta)$ and
$v_{s}(\bm{k},\eta)$ by the decomposition of the operator $\hat{\psi}$,

$\displaystyle\hat{\psi}=\int\frac{d^{3}\bm{k}}{\sqrt[]{2}(2\pi)^{3}}\sum_{s}[u_{s}(\bm{k},\eta)\hat{a}_{s}(\bm{k})e^{i\bm{k}\bm{x}}+v_{s}(\bm{k},\eta)\hat{b}^{\dagger}_{s}(\bm{k})e^{-i\bm{k}\bm{x}}],$
(43)

where $\hat{a}_{s}(\bm{k}),\hat{b}_{s}(\bm{k})$ indicate (anti) particle
annihilation operator. They satisfy the anti-commutation relations,

$\displaystyle\\{\hat{a}_{s}(\bm{k}),\hat{a}^{\dagger}_{r}(\bm{l})\\}=(2\pi)^{3}\delta^{3}(\bm{k}-\bm{l})\delta_{rs},$
(44)
$\displaystyle\\{\hat{b}_{s}(\bm{k}),\hat{b}^{\dagger}_{r}(\bm{l})\\}=(2\pi)^{3}\delta^{3}(\bm{k}-\bm{l})\delta_{rs},$
(45) $\displaystyle\\{\hat{a}_{s}(\bm{k}),\hat{a}_{r}(\bm{l})\\}=0,$ (46)
$\displaystyle\\{\hat{b}_{s}(\bm{k}),\hat{b}_{r}(\bm{l})\\}=0.$ (47)

We consider the charge conjugate of fermion,
$\mathcal{C}\overline{\hat{\psi}}^{T}$, represented by
$\mathcal{C}=i\gamma^{2}\gamma^{0}$, which interchanges the particle and the
antiparticle. Then, we obtain the relation between $u_{s}(\bm{k},\eta)$ and
$v_{s}(\bm{k},\eta)$,

$\displaystyle
u_{s}(\bm{k},\eta)=\mathcal{C}\overline{v_{s}(\bm{k},\eta)}^{T}.$ (48)

From Eq. (42) and its charge conjugate with (43) and (48),
$v_{s}(-\bm{k},\eta)$ should satisfy

$\displaystyle(-\gamma^{0}\partial_{0}-i\gamma\bm{k}-am_{\psi}+i(g_{0}\gamma^{0}+f_{0}\gamma^{5}\gamma^{0}))v_{s}(-\bm{k},\eta)=0,$
(49)
$\displaystyle(-\gamma^{0}\partial_{0}-i\gamma\bm{k}-am_{\psi}+i(-g_{0}\gamma^{0}+f_{0}\gamma^{5}\gamma^{0}))v_{s}(-\bm{k},\eta)=0.$
(50)

Since $g_{0}$ is proportional to $\alpha$, these equations are satisfied for
$\alpha=0$.

We define the spinor $u_{s}(\bm{k},\eta)$ as

$u_{s}(\bm{k},\eta)=\begin{pmatrix}u^{+}_{s,k}(\eta)\xi_{s,\bm{k}}\\\
u^{-}_{s,k}(\eta)\xi_{s,\bm{k}}\end{pmatrix},$ (51)

where $s$ indicates the spin direction and $\xi_{s,\bm{k}}$ describes the
eigen-spinor of helicity,

$\bm{\sigma}\bm{k}\xi_{s,\bm{k}}=sk\xi_{s,\bm{k}},$ (52)
$\xi_{s,-\bm{k}}=-i\sigma_{2}\xi^{\ast}_{s,\bm{k}}.$ (53)

The equation (42) is rewritten as

$\displaystyle\begin{aligned} u^{\pm}_{s,k}(\eta)^{\prime}=\pm
i(ks+f_{0})u^{\pm}_{s,k}(\eta)-iam_{\psi}u^{\mp}_{s,k}(\eta).\end{aligned}$
(54)

Performing the time derivative, we obtain the EoMs of the amplitude of the
spinor, $u^{+,-}_{s,k}(\eta)$,

$\displaystyle\begin{aligned}
u^{\pm}_{s,k}(\eta)^{\prime\prime}=-((ks+f_{0})^{2}+a^{2}m^{2}_{\psi}\mp
if_{0}^{\prime})u^{\pm}_{s,k}(\eta)-ia^{\prime}m_{\psi}u^{\mp}_{s,k}(\eta).\end{aligned}$
(55)

The EoM (42) guarantees the relation $|u^{+}_{s,k}|^{2}+|u^{-}_{s,k}|^{2}=2$
with the anti-commutation relations (44)-(47), the assumption (51) and the
canonical anti-commutation relation
$\\{\psi(\eta,\bm{x}),\psi^{\dagger}(\eta,\bm{y})\\}=\delta^{3}(\bm{x}-\bm{y})$.

Below, we evaluate the particle production for $\alpha=0$.

###  3.2 The number density of fermion

From the Lagrangian density (40), we can define the Hamiltonian operator
$\hat{H}_{\psi}$,

$\displaystyle\hat{H}_{\psi}=\frac{1}{a}\int
d^{3}x(\overline{\hat{\psi}}\gamma^{I}\partial_{I}\hat{\psi}+m_{\psi}a\overline{\hat{\psi}}\hat{\psi}-f_{0}(\phi)\overline{\hat{\psi}}i\gamma^{5}\gamma^{0}\hat{\psi}),$
(56)

where the script $I$ denotes the space components, $I\in\\{x,y,z\\}$, and the
factor $a^{-1}$ comes from the fact that the Hamiltonian is the generator of
the translation regarding the cosmic time $t$, and the relation
$dt=a(\eta)d\eta$. By inserting (43) into (56), the Hamiltonian becomes

$\displaystyle\hat{H}_{\psi}=\frac{1}{a}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\sum_{s}[E^{s}_{k}(\hat{a}^{\dagger}_{s}(\bm{k})\hat{a}_{s}(\bm{k})-\hat{b}_{s}(\bm{k})\hat{b}^{\dagger}_{s}(\bm{k}))+F^{s}_{k}\hat{b}_{s}(-\bm{k})\hat{a}_{s}(\bm{k})+F^{s\ast}_{k}\hat{b}^{\dagger}_{s}(-\bm{k})\hat{a}^{\dagger}_{s}(\bm{k})],$
(57)

where the coefficients $E^{s}_{k}$ and $F^{s}_{k}$ are

$\displaystyle
E^{s}_{k}=(ks+f_{0})(1-|u^{+}_{s,k}|^{2})+am_{\psi}\text{Re}(u^{+}_{s,k}u^{-\ast}_{s,k}),$
(58) $\displaystyle
F^{s}_{k}=-(ks+f_{0})u^{+}_{s,k}u^{-}_{s,k}+am_{\psi}(-u^{+}_{s,k}u^{+}_{s,k}+u^{-}_{s,k}u^{-}_{s,k}).$
(59)

Since they satisfy
$|E^{s}_{k}|^{2}+|F^{s}_{k}|^{2}=\omega^{s2}_{k}=a^{2}m^{2}_{\psi}+(ks+f_{0})^{2}$,
the Hamiltonian operator (57) is diagonalyzed by introducing the time-
dependent annihilation operators, $\hat{a}_{s}(\bm{k},\eta)$ and
$\hat{b}_{s}(\bm{k},\eta)$,

$\hat{H}_{\psi}=\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\sum_{s}\frac{\omega^{s}_{k}}{a}(\hat{a}^{\dagger}_{s}(\bm{k},\eta)\hat{a}_{s}(\bm{k},\eta)-\hat{b}_{s}(\bm{k},\eta)\hat{b}^{\dagger}_{s}(\bm{k},\eta)).$
(60)

The Bogoliubov transformation connects the operators
$\hat{a}_{s}(\bm{k}),\hat{b}^{\dagger}_{s}(-\bm{k})$ and
$\hat{a}_{s}(\bm{k},\eta),\hat{b}^{\dagger}_{s}(\bm{k},\eta)$,

$\displaystyle\begin{pmatrix}\hat{a}_{s}(\bm{k},\eta)\\\
\hat{b}^{\dagger}_{s}(\bm{k},\eta)\end{pmatrix}=\begin{pmatrix}\alpha_{s}(k,\eta)&\beta_{s}(k,\eta)\\\
-\beta^{\ast}_{s}(k,\eta)&\alpha^{\ast}_{s}(k,\eta)\end{pmatrix}\begin{pmatrix}\hat{a}_{s}(\bm{k})\\\
\hat{b}^{\dagger}_{s}(-\bm{k})\end{pmatrix},$ (61)

where $\alpha_{s}(k,\eta)$ and $\beta_{s}(k,\eta)$ are the Bogoliubov
coefficients defined by

$\displaystyle|\alpha_{s}(k,\eta)|^{2}=\frac{\omega^{s}_{k}+E^{s}_{k}}{2\omega^{s}_{k}},$
(62)
$\displaystyle|\beta_{s}(k,\eta)|^{2}=\frac{\omega^{s}_{k}-E^{s}_{k}}{2\omega^{s}_{k}}.$
(63)

From the definition of the Bogoliubov coefficients and the anti-commutation
relations (44)-(47), the anti-commutation relations for the operators,
$\hat{a}_{s}(\bm{k},\eta),\hat{b}^{\dagger}_{s}(-\bm{k},\eta)$, are derived to
be

$\displaystyle\\{\hat{a}_{s}(\bm{k},\eta),\hat{a}^{\dagger}_{r}(\bm{l},\eta)\\}=(2\pi)^{3}\delta^{3}(\bm{k}-\bm{l})\delta_{rs},$
(64)
$\displaystyle\\{\hat{b}_{s}(\bm{k},\eta),\hat{b}^{\dagger}_{r}(\bm{l},\eta)\\}=(2\pi)^{3}\delta^{3}(\bm{k}-\bm{l})\delta_{rs},$
(65) $\displaystyle\\{\hat{a}_{s}(\bm{k},\eta),\hat{a}_{r}(\bm{l},\eta)\\}=0,$
(66) $\displaystyle\\{\hat{b}_{s}(\bm{k},\eta),\hat{b}_{r}(\bm{l},\eta)\\}=0.$
(67)

We redefine the Hamiltonian operator so that a minimum of its expectation
value is zero,

$\hat{H}_{\psi}=\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\sum_{s}\frac{\omega^{s}_{k}}{a}(\hat{a}^{\dagger}_{s}(\bm{k},\eta)\hat{a}_{s}(\bm{k},\eta)+\hat{b}^{\dagger}_{s}(\bm{k},\eta)\hat{b}_{s}(\bm{k},\eta)).$
(68)

We consider the vacuum state defined by
$\hat{a}_{s}(\bm{k})(,\hat{b}_{s}(\bm{k}))|0\rangle_{\eta_{0}}=0$ at
$\eta=\eta_{0}$ . Under the state, the expectation value of the number
operators are developed

$\displaystyle\Bigl{\langle}0\Bigl{|}_{\eta_{0}}\hat{a}^{\dagger}_{s}(\bm{k},\eta)\hat{a}_{s}(\bm{k},\eta)\Bigr{|}0\Bigr{\rangle}_{\eta_{0}}$
$\displaystyle=\Bigl{\langle}0\Bigl{|}_{\eta_{0}}\hat{b}^{\dagger}_{s}(\bm{k},\eta)\hat{b}_{s}(\bm{k},\eta)\Bigr{|}0\Bigr{\rangle}_{\eta_{0}}$
$\displaystyle=|\beta_{s}(k,\eta)|^{2}.$ (69)

The behavior of the expectation value of the number operator for the particle
and the antiparticle is equivalent. Even if the expectation value of
Hamiltonian $\langle\hat{H}_{\psi}(\eta)\rangle_{\eta_{0}}$ is zero at
$\eta_{0}$, it is not necessary to be so at $\eta>\eta_{0}$. This means a non-
thermal particle production. From the conditions of the amplitude of the
spinor,

$\displaystyle|u^{+}_{s,k}|^{2}+|u^{-}_{s,k}|^{2}=2,$ (70)
$\displaystyle\langle\hat{H}_{\psi}(\eta_{0})\rangle_{\eta_{0}}=0,$ (71)

the initial values of the amplitude of the spinor are derived,

$\displaystyle
u^{+}_{s,k}(0)=\sqrt[]{1-\frac{(ks+f_{0}(\phi(0)))}{\omega^{s}_{k}(0)}},$ (72)
$\displaystyle
u^{-}_{s,k}(0)=\frac{m_{\psi}a(0)}{\omega^{s}_{k}(0)}\Bigl{\lparen}1-\frac{(ks+f_{0}(\phi(0)))}{\omega^{s}_{k}(0)}\Bigr{\rparen}^{-\frac{1}{2}}.$
(73)

We define the quantities to observe whether the particle production occurs.
The expectation value of total number density is given by

$\displaystyle\Bigl{\langle}0\Bigl{|}_{\eta_{0}}\frac{\hat{N}^{s}_{particle}}{a^{3}V}\Bigr{|}0\Bigr{\rangle}_{\eta_{0}}$
$\displaystyle\equiv\frac{1}{a^{3}V}\Bigl{\langle}0\Bigl{|}_{\eta_{0}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\hat{a}^{\dagger}_{s}(\bm{k},\eta)\hat{a}_{s}(\bm{k},\eta)\Bigr{|}0\Bigr{\rangle}_{\eta_{0}}$
$\displaystyle\equiv\frac{1}{2\pi^{2}a^{3}}\int
dkk^{2}\Bigl{\lparen}\frac{1}{2}-\frac{E^{s}_{k}}{2\omega^{s}_{k}}\Bigr{\rparen},$
(74)

where $V$ is the volume of conformal space. The number density regarding the
conformal momentum space is

$n^{s}_{m_{\psi},k}\equiv\Bigl{\lparen}\frac{1}{2}-\frac{E^{s}_{k}}{2\omega^{s}_{k}}\Bigr{\rparen}.$
(75)

It vanishes at $\eta=\eta_{0}$ and must not exceed unity at anytime by Pauli
blocking. The total number of (anti) particles is defined by

$\displaystyle N^{s}_{\psi}\equiv\int
dk\frac{k^{2}n^{s}_{m_{\psi},k}}{2\pi^{2}}.$ (76)

We evaluate the energy density,

$\displaystyle\rho_{m_{\psi}}=2\Sigma_{s}\int dk\rho^{s}_{m_{\psi},k},$ (77)
$\displaystyle\rho^{s}_{m_{\psi},k}=\frac{k^{2}\omega^{s}_{k}n^{s}_{m_{\psi},k}}{2\pi^{2}a^{4}},$
(78)

where the factor 2 comes from the degree of freedom of the particle and the
antiparticle.

##  4 Analytical and numerical results

In this section, an analytical implication regarding the behavior of the
number density and some numerical results are exhibited. The former suggests
the behavior for particles lighter than the inflaton. We will attempt a
specific application of the latter results in Sec. 5.

###  4.1 Massless limit

We can analytically evaluate the particle production for a simple case. Here,
we consider the massless limit, $m_{\psi}=0$, which has several differences
from the massive particle.

First, the initial conditions (72) and (73) are not appropriate for the
massless limit. The initial conditions of $u^{+}_{s,k}$ and $u^{-}_{s,k}$ are
divided into four cases. For $f_{0}(0)>0$, the initial conditions become

$\displaystyle u^{+}_{\uparrow,k}(0)=0,$ (79) $\displaystyle
u^{-}_{\uparrow,k}(0)=\sqrt[]{2}e^{i\theta},$ (80)
$u^{+}_{\downarrow,k}(0)=\begin{cases}\sqrt[]{2}e^{i\psi},&\text{$k>|f_{0}(0)|$}\\\
0,&\text{$k<|f_{0}(0)|$}\end{cases}$ (81)
$u^{-}_{\downarrow,k}(0)=\begin{cases}0,&\text{$k>|f_{0}(0)|$}\\\
\sqrt[]{2}e^{i\psi},&\text{$k<|f_{0}(0)|$}\end{cases}$ (82)

where $\theta$ and $\psi$ are arbitrary phases. For $f_{0}(0)<0$, these are
given by

$\displaystyle u^{+}_{\downarrow,k}(0)=\sqrt[]{2}e^{i\psi},$ (83)
$\displaystyle u^{-}_{\downarrow,k}(0)=0,$ (84)
$u^{+}_{\uparrow,k}(0)=\begin{cases}0,&\text{$k>|f_{0}(0)|$}\\\
\sqrt[]{2}e^{i\theta},&\text{$k<|f_{0}(0)|$}\end{cases}$ (85)
$u^{-}_{\uparrow,k}(0)=\begin{cases}\sqrt[]{2}e^{i\theta},&\text{$k>|f_{0}(0)|$}\\\
0.&\text{$k<|f_{0}(0)|$}\end{cases}$ (86)

Second, the number densities are represented by

$\displaystyle n_{\uparrow,k}(\eta)$
$\displaystyle=\frac{1}{2}(1-\text{sign}(k+f_{0})(1-|u^{+}_{\uparrow,k}(\eta)|^{2})),$
(87) $\displaystyle n_{\downarrow,k}(\eta)$
$\displaystyle=\frac{1}{2}(1-\text{sign}(k-f_{0})(1-|u^{-}_{\downarrow,k}(\eta)|^{2})).$
(88)

Third, the 1st order differential equations (54) reduce to

$\displaystyle u^{\pm}_{s,k}(\eta)^{\prime}=\pm
i(ks+f_{0}(\eta))u^{\pm}_{s,k}(\eta).$ (89)

From Eq. (89), the time derivative of $|u^{\pm}_{s,k}(\eta)|^{2}$ vanishes,

$|u^{\pm}_{s,k}(\eta)|^{2\prime}=0.$ (90)

Because of a damped oscillation of the inflaton $\phi$, $|f_{0}|$ becomes
smaller than $k$ after a sufficient amount of time. The signature of
$k+sf_{0}$ is positive and Eqs. (87) and (88) are simplified to

$\displaystyle n_{\uparrow,k}$
$\displaystyle=\frac{1}{2}(1-(1-|u^{+}_{\uparrow,k}|^{2}))$
$\displaystyle=\frac{1}{2}|u^{+}_{\uparrow,k}|^{2}_{\eta=0},$ (91)
$\displaystyle n_{\downarrow,k}$
$\displaystyle=\frac{1}{2}(1-(1-|u^{-}_{\downarrow,k}|^{2}))$
$\displaystyle=\frac{1}{2}|u^{-}_{\downarrow,k}|^{2}_{\eta=0}.$ (92)

Thus, the number densities are fixed by the initial values of
$|u^{\pm}_{s,k}|^{2}_{\eta=0}$. The non-thermal excitations strongly depend on
the initial conditions at the massless limit. The hericity of the produced
particle is determined by the initial condition $f_{0}(0)$ (Table.1).

| $k<|f_{0}(0)|$ | $k>|f_{0}(0)|$
---|---|---
$f_{0}(0)>0$ |  $n^{\uparrow}_{k}=0$ and $n^{\downarrow}_{k}=1$ | $\times$
$f_{0}(0)<0$ |  $n^{\uparrow}_{k}=1$ and $n^{\downarrow}_{k}=0$ | $\times$
Table 1: particle production in massless limit

From the numerical simulation, it is observed that lighter particles have a
similiar property. If the reheating starts at $\phi^{\prime}(0)=0$, the
production of lighter particles hardly occur due to the property in Tab. 1
with $f_{0}(0)\propto\phi^{\prime}(0)=0$.

###  4.2 Numerical results

Since it is difficult to solve Eq. (55) analytically, we evaluate the behavior
of the number density through numerical calculations. Here, we set
$\rho_{\psi}(0)=0$ and employ the $(\epsilon R)^{2}$ model for the dynamics of
the inflaton governing the evolution of the universe. As is shown in Sec. 2,
initial values are fixed at $\phi(0)\sim 0.940m_{p}$ and
$\frac{d\phi}{d\eta}(0)\sim-0.293m_{p}m_{\phi}$, and the inflaton dynamics is
determined by the values $\phi(0)$, $\phi^{\prime}(0)$. We solve the Eq. (55)
to obtain $n^{s}_{m_{\psi},k},N^{s}_{\psi}$ and $\rho_{m_{\psi}}$. Numerical
calculations in our research are performed by Julia and an algorithm specified
by Verner9 solver in the package DifferentialEq.jl.

####  4.2.1 Heavy and light fermion

We evaluate the time evolution of the number density $n^{s}_{m_{\psi},k}$ for
the helicity up and down particles with a certain conformal momentum $k$.
Fig.2 clearly shows that the non-minimal coupling $\beta$ contributes to the
non-thermal particle production. Because of the interaction between the
pseudoscalaron and the pseudovector current
$\overline{\psi}i\gamma^{5}\gamma^{\mu}\psi$ in Eq. (34), the amount of the
produced particles depend on the helicity.

(a) $m_{\psi}=4m_{\phi}$, $\bm{\beta=100}$, $k=10m_{\phi}$

(b) $m_{\psi}=4m_{\phi}$, $\bm{\beta=0}$, $k=10m_{\phi}$

Figure 2: The time evolution of $n^{s}_{m_{\psi},k}$ with $k=10m_{\phi}$ and
$m_{\psi}=4m_{\phi}$.

In Fig.3, we draw the distribution of $n^{s}_{m_{\psi},k}$ for
$m_{\psi}=4m_{\phi}$ as a function of the conformal momentum $k/m_{\phi}$ for
different $\beta$ at $\eta=20m^{-1}_{\phi}$. The upper limit of the conformal
momentum for the non-thermal particle production is observed to be higher as
$\beta$ increases. Thus, a larger $|\beta|$ can lead to a higher conformal
momentum excitation. The extreme oscillations in Fig.3 are confirmed to be
unaltered by the initial values of the inflaton and the solvers.

After the end of of the particle production, the total number of the produced
particles remains at a certain value (Fig.4). In Figs. 3 and 4, different
behavior is observed between up and down particles. Both up and down particles
are produced for $m_{\psi}=4m_{\phi}$. However, Fig.5 shows that extremely
heavy particles ($m_{\psi}=10m_{\phi}$) are not produced.

We also examine the number density distribution for lighter particles
production. Fig.6 shows the number density distribution for
$m_{\psi}=0.01m_{\phi}$ at $\eta=100m^{-1}_{\phi}$. It is observed that the
higher conformal momentum particles can be produced with increasing $\beta$,
as is mentioned in Sec. 4-1. Compared with Fig.3, the property in Tab. 1 is
almost confirmed for the lighter particles.

From Eq. (54), the helicity is inverted when the sign of $f_{0}$ is reversed.
A dominant contribution to $f_{0}$ comes from the second term in Eq. (35).
Thus, the up and down particles are exchanged and the aforementioned numerical
results are nearly inverted for up and down with the sign of $\beta$ flipped.

Figure 3: The distribution of $n^{s}_{m_{\psi},k}$ as a function of the
conformal momentum $k/m_{\phi}$ at $m_{\phi}\eta=20$ ($m_{\psi}=4m_{\phi}$).

Figure 4: The time evolution of $N^{s}_{\psi}$ ($m_{\psi}=4m_{\phi}$).

Figure 5: The distribution of $n^{s}_{m_{\psi},k}$ as a function of the
conformal momentum $k/m_{\phi}$ at $m_{\phi}\eta=20$($m_{\psi}=10m_{\phi}$).

Figure 6: The distribution of $n^{s}_{m_{\psi},k}$ as a function of the
conformal momentum $k/m_{\phi}$ at
$m_{\phi}\eta=100$($m_{\psi}=0.01m_{\phi}$).

####  4.2.2 Intermediate-mass particle

For intermediate-mass particles, $m_{\psi}=0.1m_{\phi}\sim 0.5m_{\phi}$, we
observe an alternative property that does not appear in light and heavy
particles. A fermionic field with a higher conformal momentum is gradually
generated and the Boltzmann-type distribution is established. According to the
distribution of the number density (Fig.7) for $m_{\psi}=0.3m_{\phi}$ at
$\eta=110{m_{\phi}}^{-1}$, particles with greater conformal momentum are
excited than that in the case of light and heavy particle.

We also evaluate the number density distribution in the non-expanding universe
with a constant scale factor. The higher conformal momentum excitation is not
observed in Fig. 8 for the non-expanding case. It means that the higher
conformal momentum excitation is due to the expansion of the universe.

Figure 7: The distribution of $n^{s}_{m_{\psi},k}$ as a function of the
conformal momentum $k/m_{\phi}$ at $m_{\phi}\eta=110$($m_{\psi}=0.3m_{\phi}$).

Figure 8: The distribution of $n^{s}_{m_{\psi},k}$ as a function of the
conformal momentum $k/m_{\phi}$ at $m_{\phi}\eta=110$($m_{\psi}=0.3m_{\phi}$)
in non-expanding case.

##  5 Cosmological consequences

In this section, we apply the particle production to the reheating phenomena
in the early universe. In the standard history of the universe, the energy
density of the matter field must exceed the energy density of the inflaton
after the end of inflation. Therefore, we examine if the condition
$\rho_{\psi}\gg\rho_{\phi}$ can be achieved with the non-thermal and thermal
particle production.

###  5.1 Preheating

The preheating is the thermal process of the universe due to the non-thermal
particle production before the reheating. As we show in Chapter 4, a larger
$|\beta|$ makes higher conformal momentum particles excited. From Fig.9,
${\rho}_{\psi}/{\rho}_{\phi}$ grows as $\beta$ increases. The condition
${\rho}_{\psi}/{\rho}_{\phi}\gg 1$ can be achieved for sufficiently large
$\beta$.

However, it is necessary to discuss thermalization due to the decay of
produced particles into relativistic particles to define the reheating
temperature. Thus, the results presented in this section only indicate that
the energy can be sufficiently transferred from the inflaton to the matter.
Further analyses are required to estimate the reheating temperature.

Figure 9: The time evolution of the ratio of energy density
$(\rho^{\text{up}}_{\psi}+\rho^{\text{down}}_{\psi})/\rho_{\phi}$
($m_{\psi}=1m_{\phi}$).

###  5.2 Reheating

We adapt the perturvative calculations of the zero temperature quantum field
theory to the fluctuation of $\phi$ after a long time from the end of
inflation. If the energy density of the inflaton $\rho_{\psi}$ transfers to
that of relativistic matter $\rho_{r}$ through the decay with the decay rate
$\Gamma$ and $\rho_{r}|_{\eta=0}=0$, the energy density, $\rho_{r}$, is
estimated to be

$\displaystyle\rho_{r}(t_{r})=\frac{27}{25}\Gamma^{2}m^{2}_{p},$ (93)

at the moment for $\rho_{\phi}=\rho_{r}$. From the Stefan-Boltzman law
$\rho_{r}\propto T^{4}$, the reheating temparature $T_{R}$ is proportional to
$\sqrt[]{\Gamma}$.

In Einstein-Cartan pseudoscalaron model with non-minimal couplings to fermion,
the decay of the $\phi$ into the fermions due to the interaction term

$\displaystyle f_{\mu}A^{\mu}$ $\displaystyle\sim\frac{\sqrt[]{3}}{2\
\sqrt[]{2}m_{p}}\frac{1+\beta
b}{\sqrt[]{1+b^{2}}}\overline{\psi}i\gamma^{5}\gamma^{\mu}\psi\partial_{\mu}\phi$
$\displaystyle\equiv
c_{\phi\overline{\psi}\psi}\overline{\psi}i\gamma^{5}\gamma^{\mu}\psi\partial_{\mu}\phi,$
(94)

dominates the particle production. The decay rate
$\Gamma_{\phi\overline{\psi}\psi}$ from this interaction is found to be

$\displaystyle\Gamma_{\phi\overline{\psi}\psi}$
$\displaystyle=\frac{|c_{\phi\overline{\psi}\psi}|^{2}m_{\phi}m^{2}_{\psi}}{2\pi}\sqrt[]{1-\Bigl{(}\frac{2m_{\psi}}{m_{\phi}}\Bigr{)}^{2}}$
$\displaystyle=\frac{3m_{\phi}m^{2}_{\psi}(1+b\beta)^{2}}{16\pi
m^{2}_{p}(1+b^{2})}\sqrt[]{1-\Bigl{(}\frac{2m_{\psi}}{m_{\phi}}\Bigr{)}^{2}}.$
(95)

At $\beta=0$ the result (95) reproduces the one derived in the previous work
[13]. Thus, even in thermal particle production, the effect of non-minimal
coupling to fermion is significant for $|\beta b|\gg 1$. A reheating
temperature is tuned by non-minimal coupling, $\beta$. For $m_{\phi}\gg
m_{\psi}$ and $|b\beta|\gg 1$, reheating temperature is estimated as

$\displaystyle
T_{R}=\Bigl{(}\frac{30*27}{25\pi^{2}g^{\ast}}\Bigr{)}^{\frac{1}{4}}\sqrt{\frac{3}{16\pi}}\beta\Bigl{(}\frac{m_{\psi}}{m_{\phi}}\Bigr{)}^{1/2}m_{\phi},$
(96)

where $k_{B}$ is the Boltzmann constant and $g^{\ast}$ shows the physical
degree of freedom.

##  6 Summary and Discussion

In this paper, we have investigated the non-thermal fermionic particle
production in Einstein-Cartan gravity with modified Holst term and non-minimal
couplings to fermion.

In EC gravity, the only imposed condition is metric compatiblity,
$\nabla_{\mu}g_{\nu\rho}=0$. Thus, the antisymmetric part of the affine
connection
${T^{\mu}}_{\nu\rho}\equiv{\Gamma^{\mu}}_{\nu\rho}-{\Gamma^{\mu}}_{\rho\nu}$(i.e.,
torsion) can have a non-vanishing value. The existence of torsion introduces
invariant scalar components which do not exist in GR. One of these quantities
is the Holst term $\epsilon
R=\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}/\sqrt[]{-g}$. The derivative
of the affine connection in this term introduces a dynamical degree of freedom
through the EoM. Following the auxiliary field method, the EoM for the affine
connection becomes an algebraic equation for the torsion. After solving the
EoM of the torsion, one can get the effective metric action with the dynamical
pseudoscalaron $\phi$. The potential energy of the pseudoscalaron can induce
inflation in EC gravity. In this paper, we utilize the $(\epsilon R)^{2}$
model, which is consistent with the CMB observations.

The coupling between the matter fields and the affine connection in the
original action yields an interaction between the matter fields and the
inflaton in the effective metric action. In this study, we have employed a
theory with non-minimal couplings to fermion where two parameters denoted as
$\alpha$ and $\beta$ are introduced. This extension introduces interactions
between the inflaton and the fermionic field $\psi$ described in Eq. (34).
Since the inflaton $\phi$ has a large value after the end of inflation, the
interactions between the $\phi$ and the $\psi$ destabilize the vacuum. Through
the instability, the non-thermal particle production occurs. Since the
parameter $\alpha$ must be zero to satisfy the condition that
$\mathcal{C}\overline{\psi}^{T}$ represents a field where particles and
antiparticles are exchanged, we have shown that the non-minimal coupling
$\beta$ in Eq. (34) contributes to the non-thermal particle production after
the end of inflation. In this paper, only $\phi\overline{\psi}\psi$ terms in
Eq. (43) that introduce linear term with respect to $\psi$ to the EoM of
$\psi$ is focused.

We have examined how many particles are produced due to the existence of
$\beta$ through numerical calculations and observed several properties. First,
a larger value of $\beta$ leads to the excitation of particles with higher
conformal momentum, and the excitation persists over time. It is also observed
that fermions much heavier than the inflaton are hardly excited. Second, there
is generally a difference in the amount of produced particles between helicity
up and down particles. It should be noted that the produced numbers of
particles and antiparticles are identical, and the total spin of the universe
is conserved. Third, for the lighter mass fermion, $m_{\psi}\lessapprox
0.1m_{\phi}$, the difference in the number of created particles between
helicities becomes more pronounced. This property is also suggested
analytically at the massless limit. If the initial value of
$\frac{d\phi}{d\eta}$ is small enough, light particles are not excited non-
thermally. Eventually, for the intermediate-mass fermion
$0.1m_{\phi}\lessapprox m_{\psi}\lessapprox 0.5m_{\phi}$, the number density
is exponentially supressed for a higher conformal momentum like a Boltzmann
distribution. Therefore, particles with higher conformal momentum can be
excited than for heavier and lighter fermions. This property is not observed
in a non-expanding universe.

We have applied the particle production to the reheating and the preheating of
the universe. From the consistency with the CMB observations, we set a model
paremeter $b\sim-320$ in our analysis. In the preheating era, sufficiently
large values of $\beta$ have the potential to make the energy density of
$\psi$ dominant over that of the inflaton. In the reheating era, we derive the
formula of the reheating temperature $T_{R,\beta}$ with non-minimal coupling.
About $|\beta b|$ times larger reheating temperature is predicted than that
for the minimal coupling [13]. Thus, the contribution of $\beta$ is important
in both eras.

Future research directions include further development of analytical and
numerical discussions and applications to phenomenology. Due to the complexity
of the function of the inflaton coupling to fermion as given in Eq. (35), the
dynamics of fermions during inflation and the backreaction to the evolution of
the universe during particle production are not considered. To discuss a
realistic universe, consideration of these two aspects can not be avoidable.
Moreover, analytical solutions in some limits are necessary to verify the
numerical results. Potential applications of our research include the
production of the dark matter and the matter-antimatter asymmetry. The
particle production investigated in our research is induced by gravitational
effect. Heavy fermions that could be dark matter may be produced after the end
of inflation. Our asymmetric helicity production has a possibility to describe
the matter-antimatter asymmetry through particle production of fermions with
lepton numbers [29, 30]. Our analysis can be applied to Majorana fermion which
directly induces lepton number asymmetry [31].

## References

  * [1] Alexei A. Starobinsky.  A New Type of Isotropic Cosmological Models Without Singularity.  Phys. Lett. B, 91:99–102, 1980.
  * [2] Thomas P. Sotiriou and Valerio Faraoni.  f(R) Theories Of Gravity.  Rev. Mod. Phys., 82:451–497, 2010.
  * [3] Shin’ichi Nojiri and Sergei D. Odintsov.  Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models.  Phys. Rept., 505:59–144, 2011.
  * [4] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou.  Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution.  Phys. Rept., 692:1–104, 2017.
  * [5] Lev Kofman, Andrei D. Linde, and Alexei A. Starobinsky.  Reheating after inflation.  Phys. Rev. Lett., 73:3195–3198, 1994.
  * [6] Lev Kofman, Andrei D. Linde, and Alexei A. Starobinsky.  Towards the theory of reheating after inflation.  Phys. Rev. D, 56:3258–3295, 1997.
  * [7] Hayato Motohashi and Atsushi Nishizawa.  Reheating after f(R) inflation.  Phys. Rev. D, 86:083514, 2012.
  * [8] Gianfranco Pradisi and Alberto Salvio.  (In)equivalence of metric-affine and metric effective field theories.  Eur. Phys. J. C, 82(9):840, 2022.
  * [9] Friedrich W. Hehl, J. Dermott McCrea, Eckehard W. Mielke, and Yuval Ne’eman.  Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance.  Phys. Rept., 258:1–171, 1995.
  * [10] Philip C. Nelson.  Gravity With Propagating Pseudoscalar Torsion.  Phys. Lett. A, 79:285, 1980.
  * [11] R. Hojman, C. Mukku, and W. A. Sayed.  PARITY VIOLATION IN METRIC TORSION THEORIES OF GRAVITATION.  Phys. Rev. D, 22:1915–1921, 1980.
  * [12] Soren Holst.  Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action.  Phys. Rev. D, 53:5966–5969, 1996.
  * [13] Alberto Salvio.  Inflating and reheating the Universe with an independent affine connection.  Phys. Rev. D, 106(10):103510, 2022.
  * [14] E. Cartan.  Sur les variétés à connexion affine et la théorie de la relativité généralisée. (première partie) (Suite).  Annales Sci. Ecole Norm. Sup., 41:1–25, 1924.
  * [15] T. W. B. Kibble.  Lorentz invariance and the gravitational field.  J. Math. Phys., 2:212–221, 1961.
  * [16] Dennis W. Sciama.  The Physical structure of general relativity.  Rev. Mod. Phys., 36:463–469, 1964.  [Erratum: Rev.Mod.Phys. 36, 1103–1103 (1964)].
  * [17] F. W. Hehl and B. K. Datta.  Nonlinear spinor equation and asymmetric connection in general relativity.  J. Math. Phys., 12:1334–1339, 1971.
  * [18] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester.  General Relativity with Spin and Torsion: Foundations and Prospects.  Rev. Mod. Phys., 48:393–416, 1976.
  * [19] Alessandro Di Marco, Emanuele Orazi, and Gianfranco Pradisi.  Einstein–Cartan pseudoscalaron inflation.  Eur. Phys. J. C, 84(2):146, 2024.
  * [20] Minxi He, Muzi Hong, and Kyohei Mukaida.  Starobinsky Inflation and beyond in Einstein-Cartan Gravity.  2 2024.
  * [21] Laurent Freidel, Djordje Minic, and Tatsu Takeuchi.  Quantum gravity, torsion, parity violation and all that.  Phys. Rev. D, 72:104002, 2005.
  * [22] Will Barker and Sebastian Zell.  Consistent particle physics in metric-affine gravity from extended projective symmetry.  2 2024.
  * [23] Patrick B. Greene and Lev Kofman.  Preheating of fermions.  Phys. Lett. B, 448:6–12, 1999.
  * [24] Patrick B. Greene and Lev Kofman.  On the theory of fermionic preheating.  Phys. Rev. D, 62:123516, 2000.
  * [25] G. F. Giudice, M. Peloso, A. Riotto, and I. Tkachev.  Production of massive fermions at preheating and leptogenesis.  JHEP, 08:014, 1999.
  * [26] Marco Peloso and Lorenzo Sorbo.  Preheating of massive fermions after inflation: Analytical results.  JHEP, 05:016, 2000.
  * [27] Nathan Herring and Daniel Boyanovsky.  Gravitational production of nearly thermal fermionic dark matter.  Phys. Rev. D, 101(12):123522, 2020.
  * [28] Frank Gronwald.  Metric affine gauge theory of gravity. 1. Fundamental structure and field equations.  Int. J. Mod. Phys. D, 6:263–304, 1997.
  * [29] Peter Adshead and Evangelos I. Sfakianakis.  Fermion production during and after axion inflation.  JCAP, 11:021, 2015.
  * [30] Peter Adshead and Evangelos I. Sfakianakis.  Leptogenesis from left-handed neutrino production during axion inflation.  Phys. Rev. Lett., 116(9):091301, 2016.
  * [31] M. Fukugita and T. Yanagida.  Baryogenesis Without Grand Unification.  Phys. Lett. B, 174:45–47, 1986.


# Generative Mechanisms:
The mechanisms that implement codes

David Ellerman
University of Ljubljana

###### Abstract

The purpose of this paper is to abstractly describe the notion of a generative
mechanism that implements a code and to provide a number of examples including
the DNA-RNA machinery that implements the genetic code, Chomsky’s Principles &
Parameters model of a child acquiring a specific grammar given ‘chunks’ of
linguistic experience (which play the role of the received code), and
embryonic development where positional information in the developing embryo
plays the role of the received code. A generative mechanism is distinguished
from a selectionist mechanism that has heretofore played an important role in
biological modeling (e.g., Darwinian evolution and the immune system).

###### Abstract

The purpose of this paper is to abstractly describe the notion of a generative
mechanism that implements a code and to provide a number of examples including
the DNA-RNA machinery that implements the genetic code, Chomsky’s Principles &
Parameters model of a child acquiring a specific grammar given ‘chunks’ of
linguistic experience (which play the role of the received code), and
embryonic development where positional information in the developing embryo
plays the role of the received code. A generative mechanism is distinguished
from a selectionist mechanism that has heretofore played an important role in
biological modeling (e.g., Darwinian evolution and the immune system).

Keywords: generative mechanisms, selectionist mechanisms, genetic code,
Chomsky Principles & Parameters, embryonic development, information theory,
logical entropy and Shannon entropy. Generative Mechanisms:
The mechanisms that implement codes David Ellerman
University of Ljubljana

###### Contents

  1. 1 Introduction
  2. 2 Partitions and codes
  3. 3 The genetic code
  4. 4 The Principles & Parameters mechanism for language acquisition
  5. 5 Embryonic development
  6. 6 Selectionist and Generative Mechanisms
  7. 7 Concluding remarks: The connection to information theory

##  1 Introduction

There are a number of mechanisms which can be abstractly defined and are found
to be implemented in biology. The best-known is the selectionist mechanism of
biological evolution. But that mechanism can be abstractly described and then
implemented in non-biological settings as in ‘evolutionary’ or selectionist
programs on computers. In that mechanism, a wide variety of different options
are randomly generated and then whittled down to a ‘select few’ using some
fitness criterion.

Our purpose in this paper is to abstractly describe another type of mechanism,
a generative mechanism, that is also found to be implemented in biology. A
generative mechanism can be abstractly described using the graph-theoretic
notion of a tree usually pictured upside down with the root at the top and
then the branches going downward to eventually terminate in the leaves.

Definition: A generative mechanism implements a “code” that determines which
branch of a tree is taken as one descends from the root to a specific leaf
that was encoded in the code.

Figure 1: Generative mechanisms can be illustrated by tree diagrams.

For instance, in Figure 1, reading the three-letter binary codes words from
left to right, the code word $011$ is implemented by taking the $0$-branch at
the first branching point and then the $1$-branch at the next two branching
points.111The use of a binary code for an illustration does not imply that
generative mechanisms are limited to binary codes, e.g., the genetic code has
a code alphabet of four letters.

One can think of a code as a hierarchical set of switches. The number of
settings on each switch (aside from neutral) is the number of letters in the
code alphabet. Each switch determines a branching point in the tree as in
Figure 2.

Figure 2: Generative mechanism specified by a hierarchy of switches.

An everyday example of a generative mechanism is the game of 20-questions
where a player tries to traverse an implicit tree of binary branching points
(yes-or-no questions) to determine the hidden answer at a leaf of the tree. By
implementing a code, a generative mechanism navigates down through a diverse
set of possible outcomes, represented by the leaves on the tree, to reach a
specific outcome or message.

##  2 Partitions and codes

Mathematically, a partition on a set represents one way to differentiate the
elements of the set into different blocks. The join with another partition
generates a partition with more refined (smaller) blocks that makes all the
distinctions of the partitions in the join. Starting from a single block
consisting of the set of all possibilities (like the unbranched root of the
tree), a sequence of partitions joined together differentiates all the
elements of the set ultimately into singleton blocks that are the leaves of
the tree. All the (prefix-free) codes of coding theory can be generated in
this way and then the codes are implemented in practice to generate the coded
outcomes.

A partition $\pi=\left\\{B_{1},...,B_{m}\right\\}$ is a set of non-empty
subsets $B_{i}$, called blocks, of a universe set $U$ that are disjoint and
whose union is all of $U$—as in Figure 3.

Figure 3: A partition with six blocks on a universe set $U$.

The general idea of a partition is that the elements in the same block are
considered indistinct from one another while the elements in different blocks
are considered as distinct. Given another partition
$\sigma=\left\\{C_{1},...C_{n}\right\\}$, the join $\pi\vee\sigma$ of the two
partitions is the partition whose blocks are the non-empty intersections
$B_{i}\cap C_{j}$ of the blocks of the two partitions.

Figure 4: Join of $\pi$ and $\sigma$ = partition of non-empty intersections
such as $B_{6}\cap C_{4}$.

With consecutive joins of partitions (always on the same universe set), the
blocks get smaller and smaller until they reach the discrete partition
$\mathbf{1}_{U}$ with the smallest non-empty blocks with are the singletons of
elements of $U$. The least refined partition is the indiscrete partition
$0_{U}=\left\\{U\right\\}$ whose only block is all of $U$ and it represents
the root of the tree that would illustrate the consecutive joins in Table 1.
The indiscrete partition is the identity for the join operation:
$0_{U}\vee\pi=\pi$ for any partition $\pi$.

Partitions | Consecutive Joins (tree) | Codes
---|---|---
$\\{\\{u_{1},u_{2},u_{3},u_{4},u_{5}\\}\\}=\mathbf{0}_{U}$ | $\\{\\{u_{1},u_{2},u_{3},u_{4},u_{5}\\}\\}=\mathbf{0}_{U}$ |
$\\{\\{u_{1}\\},\\{u_{2},u_{3},u_{4},u_{5}\\}\\}$ | $\\{\\{u_{1}\\},\\{u_{2},u_{3},u_{4},u_{5}\\}\\}$ |  $0=$ (code for) $u_{1}$
$\\{\\{u_{1},u_{2},u_{3}\\},\\{u_{4},u_{5}\\}\\}$ | $\\{\\{u_{1}\\},\\{u_{2},u_{3}\\},\\{u_{4},u_{5}\\}\\}$ |
$\\{\\{u_{1},u_{2}\\},\\{u_{3},u_{4},u_{5}\\}\\}$ | $\\{\\{u_{1}\\},\\{u_{2}\\},\\{u_{3}\\},\\{u_{4},u_{5}\\}\\}$ | $100=u_{2},101=u_{3}$
$\\{\\{u_{1},u_{2},u_{3},u_{4}\\},\\{u_{5}\\}\\}$ | $\\{\\{u_{1}\\},\\{u_{2}\\},\\{u_{3}\\},\\{u_{4}\\},\\{u_{5}\\}\\}=\mathbf{1}_{U}$ | $1110=u_{4},1111=u_{5}$

Table 1: Prefix-free codes generated by consecutive partition joins.

A code is called prefix-free or instantaneous if no code word is the beginning
of another code word. All prefix-free code words can be obtained by a sequence
of consecutive partition joins in the following manner. The number of letters
in the code alphabet is the number of blocks in each partition. All the
partitions in the Partitions column of Table 1 (except the indiscrete
partition representing the root) have two blocks with the left blocks
associated with $0$ and the right block associated with $1$. When taking
consecutive joins, once a singleton block appears in the Consecutive Joins
column, it stays a singleton since it cannot be split any further.

The code for each $u_{i}$ in $U$ is generated in the Partitions column by the
sequence of blocks in the binary partitions (ignoring $\mathbf{0}_{U}$)
containing the element $u_{i}$, with each block contributing a $0$ or $1$ to
the code word for the element until it appears in a singleton block in the
Consecutive Joins column of Table 1. Then the code word stops so that code
word cannot be the prefix for the code word for any other element of $U$. For
instance, consider the element $u_{2}$ which appears in the $1$-block in the
first partition $\\{\\{u_{1}\\},\\{u_{2},u_{3},u_{4},u_{5}\\}\\}$ and then in
the $0$-block in the next two partitions (in the Partitions column of the
table). The element $u_{2}$ first becomes a singleton in the third row join so
tracing its history through the blocks generates the code $100=u_{2}$.

Corresponding to the generation of a code by consecutive partition joins as in
Table 1, one can construct a tree diagram where each branching is accomplished
by a partition join and each leaf corresponds to when an element first appears
in a singleton. The tree corresponding to Table 1 is the tree in Figure 2.

##  3 The genetic code

The most famous code is, of course, the genetic code which is prefix-free so
it can be generated by a sequence of partitions. In this case, each partition
has four blocks corresponding to the four code letters U, C, A, and G in the
code alphabet. For the partitions in Figure 5, which correspond to the (non-
indiscrete) partitions in the Partitions column in Table 1, the consecutive
joins give all 64 singletons after three branchings or joins so the amino
acids have 3-letter code words. The code is redundant since there can be
several codes for the same acid.

The circles in Figure 5 trace out the code for Thr4 (one of the code words for
Thr, Threonine) which is ACG = Thr4. Note that the order of the partitions
counts in the consecutive-joins determination of the genetic codes. A
different ordering gives a different code which may not describe the operation
of the DNA-RNA machinery to produce a certain amino acid from a given code
word.

Figure 5: The three partitions that generate the genetic code.

In terms of a tree diagram, the tree would branch four ways at each branching
point and there are three levels, so there are $4^{3}=64$ leaves in the tree.
The generative mechanism associated with the genetic code is the whole DNA-RNA
machinery that generates the amino acid as the output from the code word as
the input. If we abstractly represent the DNA-RNA machinery as that tree with
$64$ leaves, then the given code word tells the machinery how to traverse the
tree to arrive at the desired leaf.

##  4 The Principles & Parameters mechanism for language acquisition

Noam Chomsky’s Principles & Parameters (P&P) mechanism ([5]; [4]) for language
learning can be modeled as a generative mechanism. Again, we can consider a
tree diagram where each branching point has a two-way switch to determine one
grammatical rule or another in the language being acquired.

> A simple image may help to convey how such a theory might work. Imagine that
> a grammar is selected (apart from the meanings of individual words) by
> setting a small number of switches - 20, say - either ”On” or ”Off.”
> Linguistic information available to the child determines how these switches
> are to be set. In that case, a huge number of different grammars (here, 2 to
> the twentieth power) will be prelinguistically available, although a small
> amount of experience may suffice to fix one. [12, p. 154]

And the reference to “20” recalls the game of “20 questions” where the answers
to the questions guides one closer and closer to the desired hidden answer.
Chomsky uses the Higginbotham model to describe a Universal Grammar (UG) as a
generative mechanism.

> Many of these principles are associated with parameters that must be fixed
> by experience. The parameters must have the property that they can be fixed
> by quite simple evidence, because this is what is available to the child;
> the value of the head parameter, for example, can be determined from such
> sentences as John saw Bill (versus John Bill saw). Once the values of the
> parameters are set, the whole system is operative. Borrowing an image
> suggested by James Higginbotham, we may think of UG as an intricately
> structured system, but one that is only partially ”wired up.” The system is
> associated with a finite set of switches, each of which has a finite number
> of positions (perhaps two). Experience is required to set the switches. When
> they are set, the system functions. The transition from the initial state S0
> to the steady state Ss is a matter of setting the switches. [3, p. 146]

In the tree modeling of the P&P approach, the relative poverty of linguistic
experience that sets the switches plays the role of the code that guides the
mechanism from the undifferentiated root state (all switches at neutral) to
the final specific grammar represented as a leaf.

The question about the acquisition of a grammar is a good topic to compare and
contrast a selectionist mechanism with a generative mechanism. What would a
selectionist approach to learning a grammar look like? A child would (perhaps
randomly) generate a diverse range of babblings, some of which would be
differentially reinforced by the linguistic environment (e.g., [20]).

> Skinner, for example, was very explicit about it. He pointed out, and he was
> right, that the logic of radical behaviorism was about the same as the logic
> of a pure form of selectionism that no serious biologist could pay attention
> to, but which is [a form of] popular biology – selection takes any path. And
> parts of it get put in behaviorist terms: the right paths get reinforced and
> extended, and so on. It’s like a sixth grade version of the theory of
> evolution. It can’t possibly be right. But he was correct in pointing out
> that the logic of behaviorism is like that [of naïve adaptationism], as did
> Quine. [6, Section 10]

As noted, Willard Quine adopts essentially the behaviorist/selectionist
account.

> An oddity of our garrulous species is the babbling period of late infancy.
> This random vocal behavior affords parents continual opportunities for
> reinforcing such chance utterances as they see fit; and so the rudiments of
> speech are handed down. [18, p. 73]
>
> It remains clear in any event that the child’s early learning of a verbal
> response depends on society’s reinforcement of the response in association
> with the stimulations that merit the response, from society’s point of view,
> and society’s discouragement of it otherwise. [18, p. 75]

A more sophisticated version of a selectionist model for the language-
acquisition faculty or universal grammar (UG) could be called the format-
selection (FS) approach (Chomsky, private communication). The diverse variants
that are actualized in the mental mechanism are different sets of rules or
grammars. Then given some linguistic input from the linguistic environment,
the grammars are evaluated according to some evaluation metric, and the best
rules are selected.

> Universal grammar, in turn, contains a rule system that generates a set (or
> a search space) of grammars, {G1, G2,…, Gn}. These grammars can be
> constructed by the language learner as potential candidates for the grammar
> that needs to be learned. The learner cannot end up with a grammar that is
> not part of this search space. In this sense, UG contains the possibility to
> learn all human languages (and many more). … The learner has a mechanism to
> evaluate input sentences and to choose one of the candidate grammars that
> are contained in his search space. [17, p. 292]

After a sufficient stream of linguistic inputs, the mechanism should converge
to the best grammar that matches the linguistic environment. Since it is
optimizing over sets of rules, this model at least takes seriously the need to
account for the choice of rules (rather than just assuming the child can infer
the rules from raw linguistic data). Early work (through the 1970s) on
accounting for the language-acquisition faculty or universal grammar (UG)
seems to have assumed such an approach. The problems that eventually arose
with the FS approach could be seen as the conflict between descriptive and
explanatory adequacy.

> It was an intuitively obvious way to conceive of acquisition at the time
> for—among other things—it did appear to yield answers and was at least more
> computationally tractable than what was offered in structural linguistics,
> where the alternatives found in structural linguistics could not even
> explain how that child managed to get anything like a morpheme out of data.
> But the space of choices remained far too large; the approach was
> theoretically implementable, but completely unfeasible. [6, Appendix IX]

In order to describe the enormous range of human language grammars, the range
of grammars considered would make for an unfeasible computational load of
evaluating the linguistic experience. If the range was restricted to make
computation more feasible, then it would not explain the variety of human
languages. Hence the claim is that the P&P generative mechanism gives a more
plausible account of human language acquisition than a behavioral/selectionist
approach.222For more of the mathematical background, see [10]) and the
references therein.

##  5 Embryonic development

The role of stem cells in the development of an embryo into a full organism
can again be modeled as a generative mechanism. The original fertilized egg
becomes the stem cell that is the root of the tree. As illustrated in Figure
6, stem cells come in three general varieties: A) the stem cells that can
reproduce undifferentiated copies of themselves, B) the stem cells that can
reproduce but can also produce a somewhat differentiated cell, and C) a
specialized differentiated cell. Each branching point in a tree has a certain
number of possible leaves or terminal types of cells beneath it in the tree.
In a division (#1) of an A-type cell, each of the resulting A-type cell could
have a full set of leaves beneath it. But when it splits (#2) into another
A-type cell and a B-type cell, then the B-cell has a restricted number of
leaves beneath it. The B-type cells can split (#3) in two, and finally when a
B-type cell gives rise (#4) to a specific C-type of cell, that is a terminal
branch, i.e., a leaf, in the tree.

[Attribution: Peter Znamenskiy, CC BY-SA 3.0

$<$http://creativecommons.org/licenses/by-sa/3.0/$>$, via Wikimedia Commons]

Figure 6: Stem cell division and differentiation A: stem cell; B: progenitor
cell; C: differentiated cell;

1: symmetric stem cell division; 2: asymmetric stem cell division; 3:
progenitor division; 4: terminal differentiation.

The codes that inform the progress through the tree are not fully understood,
but apparently the positional epigenetic information in the developing embryo
provides the information about the next development steps. In general terms,

> [t]hat model harks back to the “developmental landscape” proposed by Conrad
> Waddington in 1956. He likened the process of a cell homing in on its fate
> to a ball rolling down a series of ever-steepening valleys and forked paths.
> Cells had to acquire more and more information to refine their positional
> knowledge over time — as if zeroing in on where and what they were through
> “the 20 questions game,” according to Jané Kondev, a physicist at Brandeis
> University. [2]

Again, the reference to the game of 20-questions reveals the common generative
mechanism of traversing a tree from the root to a specific leaf.

##  6 Selectionist and Generative Mechanisms

There is a long tradition in biological thought of juxtaposing selectionism,
associated with Darwin, with instructionism, associated with Lamarck ([16];
[14]). In an instructionist or Lamarckian mechanism, the environment would
transmit detailed instructions about a certain adaptation to an organism,
while in a selectionist mechanism, a diverse variety of (perhaps random)
variations would be generated, and then some adaptations would be selected by
the environment but without detailed instructions from the environment. The
discovery that the immune system was a selectionist mechanism [13] generated a
wave of enthusiasm, a “Second Darwinian Revolution” [7], for selectionist
theories [8].

In his Nobel Lecture [15], Niels Jerne even tried to draw parallels between
Chomsky’s generative grammar and selectionism. One of the distinctive features
of a selectionist mechanism is that the possibilities must be in some sense
actualized or realized in order for selection to operate on and differentially
amplify or select some of the actual variants while the others languish,
atrophy, or die off. In the case of the human immune system, ”It is estimated
that even in the absence of antigen stimulation a human makes at least
$10^{15}$ different antibody molecules—its preimmune antibody repertoire.” [1,
p. 1221]. In Chomsky’s critique of a selectionist theory of universal grammar,
he noted the computational infeasibility of having representations of all
possible human grammars in order for linguistic experience and an evaluation
criterion to perform a selective function on them. The analysis of Chomsky’s
P&P theory as a generative mechanism instead suggests that the old
juxtaposition of “selectionism versus instructionism” is not the most useful
framing for the study of biological mechanisms.

The discovery of the genetic code and DNA-RNA machinery for the production of
amino acids powerfully showed the existence of another biological mechanism, a
generative mechanism, that is quite distinct from a selectionist mechanism.
The examples of Chomsky’s P&P theory of grammar acquisition and the role of
stem cells in embryonic development provide more evidence of the importance of
generative mechanisms.

To better illustrate these two main candidates for biological mechanisms, it
might be useful to illustrate a selectionist and a generative mechanism in
solving the same problem of determining one among the $8=2^{3}$ options
considered in Figure 1. The eight possible outcomes might be represented as:
$|000\rangle,|100\rangle,|010\rangle,|110\rangle,|001\rangle,|101\rangle,|011\rangle,|111\rangle.$

In the selectionist scheme, all eight variants are in some sense actualized or
realized in the initial state $S_{0}$ so that a fitness criterion or
evaluation metric (as in the FS scheme) can operate on them. Some variants do
better and some worse as indicated by the type size in Figure 7.

Figure 7: A selectionist determination of the outcome $|010\rangle$.

The ”unfit” options dwindle, atrophy, or die off leaving the most fit option
$|010\rangle$ as the final outcome.

With the generative mechanism, the initial state $S_{0}$ (the root of the
tree) is where all the switches are in neutral, so all the eight potential
outcomes are in a ”superposition” (between left and right) state indicated by
the plus signs in the following Figure 8.

Figure 8: A generative determination of the outcome $|010\rangle$.

The initial experience or first letter in the code sets the first switch to
the $0$ option which reduces the state to
$|000\rangle+|001\rangle+|010\rangle+|011\rangle$ (where the plus signs in the
superposition of these options indicate that the second and third switches are
still in neutral). Then subsequent experience sets the second switch to the
$1$ option and the third switch to the $0$ option. Thus, we reach the same
outcome $|010\rangle$ as the final outcome in the two models but by quite
different mechanisms. Note that the generative mechanism ‘selects’ a specific
outcome but that does not make it a ‘selectionist’ mechanism.

There is another way to contrast a selective and generative mechanism. In
logic [9], there are two principal lattices, the lattice of subsets of a
universe U and the lattice of partitions on $U$. For $U=\\{a,b,c\\}$, the two
lattices are pictured in Figure 9.

Figure 9: The two basic logical lattices.

A selective mechanism to determine $a$, $b$, or $c$ from $U$ would start with
all the actual elements (top of the subset lattice) and then use a fitness or
evaluation criterion to narrow the set of actualities down to the selected
one.

A generative mechanism starts with the indiscrete partition
$\mathbf{0}_{U}=\\{\\{a,b,c\\}\\}=\\{U\\}$ (at the bottom of the partition
lattice) where all the elements are only potential outcomes not distinguished
from each other. Then distinctions are made, as represented by consecutive
partition joins, until an element is fully distinguished (i.e., appears as a
singleton). A coding scheme to determine each element is given in Table 2.

Partitions | Consecutive Joins | Codes
---|---|---
$\\{\\{a,b,c\\}\\}=\mathbf{0}_{U}$ | $\\{\\{a,b,c\\}\\}=\mathbf{0}_{U}$ |
$\\{\\{a\\},\\{b,c\\}\\}$ | $\\{\\{a\\},\\{b,c\\}\\}$ | $0=a$
$\\{\\{a,b\\},\\{c\\}\\}$ | $\\{a\\},\\{b\\},\\{c\\}\\}=\mathbf{1}_{U}$ | $10=b,11=c$

Table 2: Coding scheme for $a$,$b$, or $c$.

The tree diagram for the Table 2 code is given in Figure 10.

Figure 10: Tree diagram for the code scheme of Table 2.

##  7 Concluding remarks: The connection to information theory

Finally, it should be mentioned that there is an intimate connection between
the tree diagrams representing generative mechanisms and information theory
([19]; [11]). One can imagine a marble rolling down from the root to one of
the leaves where its path was probabilistically determined at each branching
point by a set of probabilities. The simplest assumption on a binary tree is a
half-half probability of the marble taking each branch. From each leaf, there
is a unique path from the root to the leaf and the product of the
probabilities along that path gives the probability of reaching that leaf.
With those assumptions for the tree of Figure 1, each leaf has probability
$1/2^{3}=1/8$. Then the Shannon entropy of that probability distribution
$p=(p_{1},\ldots,p_{m})$ is:

$H(p)=\sum_{i}p_{i}\log_{2}(1/p_{i})=8\times
1/8\times\log_{2}(1/(1/8))=\log_{2}(2^{3})=3$

and the logical entropy is:

$h(p)=1-\sum_{i}p_{i}^{2}=1-8\times(1/8)^{2}=1-1/8=7/8.$

In this simple example, the Shannon entropy is the average number of binary
partitions (bits) needed to distinguish all the leaves on the tree
(”messages”). The logical entropy always has the interpretation that on two
independent trials, there will be different outcomes. In this case, that is
the probability that on two independent rolls of a marble, it will end up at
different leaves. Since all the leaves are equiprobable, it is simply the
probability that the second marble took a different path (than on the first
trial), i.e., $1-1/8=7/8$.

With the same half-half branching probabilities for Figure 10, the leaf
probabilities are $\Pr(a)=1/2$ and $\Pr(b)=\Pr(c)=1/2\times 1/2=1/4$. Then the
Shannon entropy is:

$H(p)=1/2\times\log_{2}(1/(1/2)+2\times
1/4\times\log_{2}(1/(1/4))=1/2+(1/2)\times 2=3/2$

which is, in this simple case, the average length of the code words for the
leaves. The logical entropy is:

$h(p)=1-(1/2)^{2}-2\times(1/4)^{2}=1-1/4-1/8=5/8$

which is always the probability that on two rolls, the marble will end up at
different leaves.

Tracing down a code tree from the root to the desired message generates the
code for that message on the sending side of a communications channel, and
then implementing the received code on the receiving side of the channel will
generate the received message. In the case of the biological examples (genetic
code, generative grammar, and embryonic development), the creation of the
codes is a matter of deep evolutionary history, so the focus of study is
usually on how those generative mechanisms implement codes to give specific
outcomes.

## References

  * [1] Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts, and J.D. Watson. 1994. Molecular Biology of the Cell. New York: Garland.
  * [2] Cepelewicz, Jordana. 2019. “The Math That Tells Cells What They Are.” Quanta Magazine (blog). March 13, 2019. https://www.quantamagazine.org/the-math-that-tells-cells-what-they-are-20190313/ .
  * [3] Chomsky, Noam. 1986. Knowledge of Language: Its Origin, Nature, and Use. New York: Praeger.
  * [4] Chomsky, Noam. 1995. The Minimalist Program. Cambridge: MIT Press.
  * [5] Chomsky, Noam, and Howard Lasnik. 1993. “The Theory of Principles and Parameters.” In Syntax: An International Handbook of Contemporary Research, edited by J. Jacobs, A. von Stechow, W. Sternefeld, and T. Vennemann, 506–69. Berlin: de Gruyter.
  * [6] Chomsky, Noam, and James McGilvray, eds. 2012. The Science of Language: Interviews with James McGilvray. Cambridge UK: Cambridge University Press.
  * [7] Cziko, Gary. 1995. Without Miracles: Universal Selection Theory and the Second Darwinian Revolution. Cambridge: MIT Press (A Bradford Book).
  * [8] Dennett, Daniel. 1995. Darwin’s Dangerous Idea: Evolution and the Meanings of Life. New York: Touchstone.
  * [9] Ellerman, David. 2014. An Introduction of Partition Logic. Logic Journal of the IGPL. 22 (1): 94–125.
  * [10] Ellerman, David. 2016. “Four Ways from Universal to Particular: How Chomsky’s Principles-and-Parameters Model Is Not Selectionist.” Journal of Applied Non-Classical Logics 26 (3): 193–207. https://doi.org/10.1080/11663081.2016.1214803.
  * [11] Ellerman, David. 2017. “Logical Information Theory: New Foundations for Information Theory.” Logic Journal of the IGPL 25 (5 Oct.): 806–35. https://doi.org/10.1093/jigpal/jzx022.
  * [12] Higginbotham, James. 1982. “Noam Chomsky’s Linguistic Theory.” Social Research 49 (1): 143–57.
  * [13] Jerne, Niels. 1955. The natural selection theory of antibody formation. Proc. National Academy of Sciences U.S.A. 41, 849.
  * [14] Jerne, Niels 1967. Antibodies and learning: Selection versus instruction. In The neurosciences: A study program. G. C. Quarton, T. Melnechuk and F. O. Schmitt eds., New York: Rockefeller University Press: 200-5.
  * [15] Jerne, Niels 1985. The Generative Grammar of the Immune System. Science. 229 (4718): 1057-59.
  * [16] Medawar, Peter B. 1960. The Future of Man: Reith Lectures 1959. London: Methuen.
  * [17] Nowak, M. A., and N. L. Komarova. 2001. Towards an Evolutionary Theory of Language. Trends in Cognitive Sciences. 5 (7 July): 288-95.
  * [18] Quine, Willard Van Orman. 2013. Word & Object (New Edition). Cambridge: MIT Press.
  * [19] Shannon, Claude E. and Warren Weaver 1964. The Mathematical Theory of Communication. Urbana: University of Illinois Press.
  * [20] Skinner, B. F. 1976. About Behaviorism. New York: Vintage Books.