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Physics Letters B 745 (2015) 97–104

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Fermi-bounce cosmology and the fermion curvaton mechanism

Stephon Alexander

, Yi-Fu Cai

, Antonino Marcianò

Center for Cosmic Origins and Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA

Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada

Center for Field Theory and Particle Physics & Department of Physics, Fudan University, 200433 Shanghai, China

a r t i c l e i n f o a b s t r a c t

Article history:

Received

23 February 2015

Received

in revised form 6 April 2015

Accepted

14 April 2015

Available

online 20 April 2015

Editor:

M. Trodden

A nonsingular bouncing cosmology can be achieved by introducing a fermion ﬁeld with BCS condensation

occurring at high energy scales. In this paper we are able to dilute the anisotropic stress near the

bounce by means of releasing the gap energy density near the phase transition between the radiation

and condensate states. In order to explain the nearly scale-invariant CMB spectrum, another fermion

ﬁeld is required. We investigate one possible curvaton mechanism by involving one another fermion ﬁeld

without condensation where the mass is lighter than the background ﬁeld. We show that, by virtue of

the fermion curvaton mechanism, our model can satisfy the latest cosmological observations very well

in which the amplitude of the power spectrum of primordial curvature perturbation is determined by a

ratio between the masses of two fermion species.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Nonsingular bouncing cosmologies can resolve the initial sin-

gularity

and horizon problems of the hot Big-Bang theory. These

types of cosmological scenarios appear in many theoretical con-

structions

[1–21]. We refer to [22–24] for recent reviews of various

bouncing cosmologies. Most nonsingular bounce models are based

on the matter ﬁelds with integer spins, i.e. the bosonic sector of

the universe. However, the fundamental particles that make up the

macroscopic world are dominated by fermion ﬁelds, and it is inter-

esting

to investigate their effects in the early universe. Recently, it

was found in [26,25] that a nonsingular bounce can be achieved by

means of a fermion ﬁeld with a condensate state in the ultraviolet

(UV) regime.

this model, Einstein gravity is extended to have topologi-

cal

terms, which present gravitational interactions with the Dirac

fermions, and torsion, which leads to Four-Fermion current densi-

ties

and therefore effectively contribute to a negative energy den-

sity

that evolves as ∼ a

. In the infrared (IR) regime, the fermion-

ﬁeld

is dominated by its mass term which generates a matter-like

contraction preceding the nonsingular bounce. Thus, this model

nicely supports the paradigm of the matter-bounce scenario [27,

E-mail addresses: stephon.alexander@dartmouth.edu (S. Alexander),

yifucai@physics.mcgill.ca (Y.-F. Cai), marciano@fudan.edu.cn (A. Marcianò).

28] that is necessary for generating a nearly scale-invariant power

spectrum of primordial perturbations in the contracting phase.

While

the bounce model realized by this nontrivial cosmologi-

cal

fermion ﬁeld can realize the matter-bounce scenario and pro-

vide

a framework of generating power-spectrum of observational

interest, it shares a common issue that exists in a large class of

bouncing cosmologies. That is, their contracting phases are not sta-

ble

against the instability to the growth of the anisotropic stress,

which is known as the famous Belinsky–Khalatnikov–Lifshitz (BKL)

instability issue [29]. In the context of scalar ﬁeld cosmology,

this issue can be solved if there exists a phase with a steep and

negative-valued potential which dominates over the anisotropies

in the contracting phase [30–33]. A concrete realization of such a

new matter-bounce by using scalar ﬁelds was constructed in [34]

(see

[35,36] for extended studies and [24] for a recent review).

the present work we take a close look at the cosmological

implication of a Dirac Fermion ﬁeld and show that the new matter-

bounce

scenario can be achieved in this model due to the gap en-

ergy

released during the transition from a regular massive fermion

state to the state of a Four-Fermion condensation. As the scale fac-

tor

decreases, the vacuum expectation value of the fermion bilinear



ψψ grows as ∼ a

−3

and evolves to a critical value that trig-

gers

the condensation phase transition. Then, the value of 

ψψ

is locked at the surface of a phase transition and therefore so is

the scale factor. As a result, alarge amount of gap energy density

which dominates anisotropic stress near the bounce.

http://dx.doi.org/10.1016/j.physletb.2015.04.026

0370-2693/

SCOAP

98 S. Alexander et al. / Physics Letters B 745 (2015) 97–104

Based on this scenario, we study a new curvaton mechanism by

considering another ﬂavor of a fermion ﬁeld in which the mass is

much lighter than the background one. Fluctuations of this fermion

ﬁeld, originating from a quantum vacuum state, can form a scale-

invariant

spectrum during the matter-like contracting phase. These

ﬂuctuations automatically dominate over the curvature perturba-

tion

at large length scales since those of the background ﬁeld are

suppressed by its mass term. Therefore, our two-ﬂavor fermion

ﬁeld model provides a fermion curvaton mechanism for generating

a scale-invariant power spectrum of primordial curvature pertur-

bation

which can explain the CMB temperature anisotropies [37].

Moreover, by studying primordial tensor perturbations, we ﬁnd

that, by virtue of this mechanism, there exists a large parameter

space for the tensor-to-scalar ratio to be consistent with the latest

experiments, such as the BICEP2 data [38].

The

paper is organized as follows. In Section 2, we brieﬂy re-

view

the bouncing cosmology by means of the fermion condensate.

Then, in Section 3 we present the detailed study on the theoretical

constraint from primordial anisotropies. To address the instability

issue arisen from primordial anisotropies, in Section 4 we analyze

the cosmological implication of the gap energy density restored

in the fermion ﬁeld and numerically show that this part of con-

tribution

can give rise to a period of contraction. Afterwards, in

Section 5 we study the primordial gravitational waves generated in

this model. We conclude with a discussion in Section 6. Through-

out

the paper we take the sign of the metric to be (+, −, −, −)

and deﬁne the reduced Planck mass through κ ≡ 8π G = 1/M

2. The bounce cosmology with fermion condensate

We start by brieﬂy reviewing the cosmology of the fermion

condensate model. Consider a universe ﬁlled with a fermion ﬁeld,

one can write down the action as

S = S

+ S

, (1)

where the Einstein–Hilbert action is expressed in terms of the

mixed-indices Riemann tensor R

μν

= F

μν

[



ω(e)] (F

μν

being the

ﬁeld-strength of the metric-compatible Lorentz-connection

(e))

2κ



x|e|e

μν

, (2)

the Dirac action S

on curved space–time reads



x|e|



ψγ



∇

ψ −m

ψψ



+h.c., (3)

and ﬁnally the interacting part of the theory is:

Int

=−ξκ



x|e| J

, (4)

which only involve the axial vector current J

of the ψ fermionic

species. The term in Eq. (4) can be either accounted as a phe-

nomenological

term, or can be derived from a gravitational theory

with torsion, such as the Kibble–Holst–Einstein–Cartan theory —

see e.g. Ref. [25], and references therein, for a derivation of the

interaction term in (4). In both cases, ξ can be treated as a free

parameter [25].

Varying

the action with respect to the vierbein can yield the

following energy–momentum tensor

μν

ψγ

(μ



∇

ν)

ψ +h.c. − g

μν

. (5)

Using comoving coordinates for a Friedmann–Lemaître–Robertson–

Walker

(FLRW) universe, we can solve the Euler–Lagrange equa-

tions

of the system using the ansatz for the fermion ﬁeld ψ =

(ψ

, 0, 0, 0), and ﬁnd that

ψψ =

. (6)

Using the Fierz identities, we can then write the ﬁrst Fried-

mann

equation taking into account the contributions due to the

fermionic ﬁeld:

+ξ

. (7)

One can see the ﬁrst term in the above expression corresponds

to the regular phase of the fermion ﬁeld which is dominated by

the mass term. The second term appears when the fermion ﬁeld

enters the condensate state with ξ<0. When applied to the cos-

mological

background, the fermion condensate state can effectively

contribute to a negative energy density and hence cancels the en-

ergy

densities from regular matter ﬁelds. Therefore, a nonsingular

bouncing solution is achieved.

be explicit, for a universe ﬁlled with a single fermion ﬁeld ψ

as described above, one can derive the solution of the scale factor

of the metric as:

a =



κm

(t −t

)

−

κ n



. (8)

In this solution, one can see that the universe shrinks to the mini-

mal

size when t =t

, i.e. a

=(−

ξκn

)

1/3

, which is real if ξ<0.

To end this section, we would like to comment that in the usual

Minkowski space–time, the fermionic ﬁelds in quantum ﬁeld the-

ory

do not carry negative energy based on the Fermi statistics,

otherwise the energy density of these ﬁelds cannot be bounded

from below. In a cosmological background, by taking into account

the high-order fermionic interactions, the energy density of the

fermionic ﬁelds is able to decrease effectively at high energy scales.

However, the energy density in our model is bounded from be-

low

due to the presence of the bouncing behavior. Thus, one ex-

pects

that the bouncing solution can guarantee the stability of this

model. Note that, this does not conﬂict with the argument of no

negative energy in ﬂat space–time. As is well known, the limit

of ﬂat space–time is equivalent with the limit of weak gravita-

tional

ﬁeld with a vanishing Newtonian constant G. In this limit,

it is evident that the interaction term in (4) vanishes and hence

the theory reduces to a free ﬁeld fermion action. In this regard,

the model under consideration is well behaved in the ﬂat-space

limit.

3. Constraint from the anisotropic instability

A general challenge for bouncing cosmologies is to ensure that

the contracting phase is stable against the instability to the growth

of anisotropic stress, whose energy density grows as a

−6

. This

is the famous BKL instability issue of any cosmological mod-

els

involving a contracting phase. In particular, for the matter-

bounce

cosmology, the background energy density scales as a

−3

and hence, one needs to introduce a mechanism to suppress the

growth of this unwanted anisotropies in the contracting phase. In

the model of fermionic bounce, the energy density contributed by

the fermionic condensate evolves also as a

−6

but with a negative

sign. In this case, whether the model is stable depends on the ini-

tial

condition one chooses, namely, it is marginally stable if initially

the contribution of the anisotropic stress is much lower than that

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