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The cosmological bounce scenario, as originated by standard model matter fields, is a viable alternative to inflation that has not yet been ruled out experimentally. The concept of the bounce is that the cosmological dynamics emerges from a pre-Big Bang universe, which solves the cosmological singularity. Almost scale-invariant perturbations are generated during the (matter) contracting cosmological phase — see e.g. Ref. [1] — thus satisfying CMB observations [2]. A scale-invariant power spectrum can be also recovered for curvature perturbations in a matter-filled emergent universe, as in [3]. A possible origin of the bounce can be recovered within the context of the Einstein-Cartan-Holst-Sciama-Kibble theory (ECHSK) [4-7], accounting for gravity with a topological Holst term and non-minimally coupled fermions. In the first-order formalism, one must allow for a torsionful part of the spin-connection, as gravity is coupled to fermion fields. In this framework, the torsion field does entail propagating degrees of freedom and can be integrated-out as an auxiliary field. By virtue of the torsion-fermion coupling, new effective four-fermion interactions arise. A sort of effective Nambu-Jona-Lasinio (NJL) model can emerge, but from a very different dynamical origin than the one addressed in the QCD case. Depending on the sign of the fermion-torsion coupling, this can provide a new attractive or repulsive contribution to the energy conditions. In the repulsive phase, with a certain choice of the torsion-fermion coupling, such a term can contribute to the energy-momentum tensor, triggering a bounce at a related critical energy density scale. Possible bounce cosmology scenarios were investigated in Refs. [8-11]. A detailed analysis of the phenomenological consequences, at late cosmological times, of theories with fermions that violate the Null Energy Condition (NEC) was provided in [12].
A typical issue for bounce scenarios in cosmology is the generation of large
O(1) anisotropic terms in the energy-momentum tensor, which is incompatible with the cosmological isotropy of the CMB. An approach to solve this problem requires the introduction of a new exotic fluid, the Ekpyrotic fluid, provided with an energy-density steeply scaling with the Universe scale factor, i.e.ρEk=ρ0a−n withn>6 . The Ekpyrotic fluid does dominate during the Bounce critical scale, suppressing anisotropic contributions to the background dynamics. However, both the classical Bounce and the Ekpyrotic mechanism are based on a classical analysis. During the Bounce stage, quantum corrections are expected to be large, and potentially they may completely change and wash-out the Bounce dynamics, along with the Ekpyrotic classical solution to the anisotropies' problem.This highly motivates the analysis of quantum corrections to the Ekpyrotic mechanism in some specific Bounce cosmology models. In this study, we analyze the one-loop quantum corrections to the four-fermion term generated by torsion, in a Friedmann-Lemaître-Robertson-Walker (FLRW) background. We show that quantum corrections induce extra corrections to the classical bare energy-momentum, which was previously not considered. We find that these new radiative terms theoretically have two healthy consequences:
i) they favor the classical Bounce, generating contributions that violate the NEC;ii) they generate new terms mimicking the effect of the Ekpyrotic fluid. This result implies that the four-fermion curvaton mechanism can induce the Bounce without introducing any new exotic matter field. This hypothesis is theoretically appealing, as it avoids adding any particle to the theoretical framework that cannot be accommodated in the Standard Model of particle physics, or in any Grand Unified extensions of it. -
We follow the theoretical framework and the conventions introduced in Refs. [4-21]. This makes us consider a generalization of the Einstein-Hilbert action with a topological term, the Holst action for gravity in the Palatini formalism, which allows to couple gravity to chiral fermions. The theory can be coupled to a Dirac field
ψ , and to the related field¯ψ=(ψ∗)Tγ0 . The Dirac action involves the Dirac matrices,γI withI=0,⋯,3 andγ5 . The action for pure gravity can be cast in terms of the gravitational fieldgμν=eIμeJνηIJ , whereeIμ is the tetrad/frame field (with inverseeμI and determinant e), and the Lorentz connectionωIJμ . The curvature ofωIJμ , namelyFIJμν=2∂[μωIJν]+[ωμ,ων]IJ,
is the triadic projection of the Riemann tensor.
The total action involves the Einstein-Cartan-Holst (ECH) action — namely the Palatini-formulated action for gravity plus fermions that also includes the topological term à la Holst, which resembles the
θ -term for gauge fields — with a non-minimal component of the covariant Dirac action. Note that, in absence of the gravitational Holst topological term, the whole theory provided with torsion and minimally-coupled fermions is referred to in the literature as the Einstein-Cartan-Sciama-Kibble theory (see e.g. Refs. [4-6]). In the Palatini first order formalism, the ECH action casts (see e.g. [11]),SHolst=12κ∫Md4x|e|eμIeνJPIJ KLF KLμν(ω).
(1) In Eq. (1),
κ=8πGN is the square of the reduced Planck length,ϵIJKL denotes the Levi-Civita symbol, while the tensor in the internal indicesPIJ KL=δ[IKδJ]L−ϵIJ KL/(2γ) involves the Barbero–Immirzi parameterγ and is invertible forγ2≠−1 . The Dirac action for massless fermions readsSDirac=12∫d4x|e|LDirac , withLDirac=ı2¯ψ(1−ıαγ5)γIeμI∇μψ+h.c..
(2) In Eq. (2),
α denotes a real parameter called non-minimal coupling. Forα=γ , we may recover the Einstein-Cartan action, namelySECH=SGR+SDirac . An additional term is also present, which reduces to the Nieh-Yan invariant (see e.g. Ref. [20]) when the second Cartan structure equation is satisfied. Within the framework of the Holst action in Eq. (1), the minimal coupling can be recovered in the limitα→±∞ . Phenomenologically allowed values of the parametersα andγ are recovered from the four fermion axial-current Lagrangian (7), from measurements of lepton-quark contact interactions [14, 22].The presence of fermions induces a torsional part of the connection to enter the non-minimal ECH action. Torsion is non-dynamical in this theory, and can be integrated out once the second Cartan structure equation is solved. This requires to introduce the contortion tensor
CIJμ , which is defined by(∇μ−˜∇μ)VI=C JμIVJ, with˜∇μ denoting the covariant derivative compatible with the tetradeIμ andVJ a vector in the internal space. The Cartan equation expresses the contortion tensorCIJμ in terms of the fermions' currents and the tetrad, namelyeμICμJK=κ4γγ2+1(βϵIJKL JL−2θηI[JJK]),JL=¯ψγLγ5ψ.
(3) The coefficients appearing in Eq. (3) are related to the free parameters of the non-minimal ECH theory,
β=γ+1/α andθ=1−γ/α . Deploying the solution in (3), the non-minimal ECH action recasts in terms only of the metric compatible variables, further including a novel interaction term that captures the new physics:SECH=SGR+SDirac+SInt.
(4) In Eq. (4), the Einstein-Hilbert action involves the mixed-indices Riemann tensor
RIJμν=FIJμν[˜ω(e)] , namelySGR=12κ∫Md4x|e|eμIeνJRIJμν,
(5) the Dirac action
SDirac on curved space-time recasts in terms of the metric compatible variablesSDirac=ı2∫Md4x|e| ¯ψγIeμI˜∇μψ+h.c.,
(6) while the interacting term reads
SInt=−ξκ∫Md4x|e|JLJMηLM,
(7) with the coefficient
ξ being a function of the fundamental parameters of the theory, namelyξ:=316γ2γ2+1(1+2αγ−1α2).
(8) In few words, the net effect of having coupled fermions to gravity in the first order formalism means to have recovered, in the metric-compatible formalism, an extra four fermion term. That means that the structure of General Relativity remains untouched, but that the matter part of the Einstein equations acquires at tree-level a self-interaction term, the coupling constant of which,
ξ , can be either positive or negative. Furthermore, we shall comment that the choice of real values ofγ , which allows to retain only real-valued effective potentials, ensures that the reality condition is automatically implemented on the gravitational field.For completeness, we show in this section the energy-momentum tensor components, i.e.
Tferμν=14¯ψγIeI(μı˜∇ν)ψ+h.c.−gμνLfer.
(9) -
Only a complete treatment of the vast literature that hinges on the matter bounce scenario would require per se a sizable review. For the purpose of our analysis, we will devote this section to provide an introductory review of the argument, focusing on the cases of scalar matter fields, and then of fermion fields. Then, we will summarize a few crucial elements of the ekpyrotic models, in preparation for our analysis and the discussion of the novelty of our framework.
It is rather important also to emphasize that matter bounce cosmologies may achieve a nearly scale-invariant power spectrum, and they are characterized by a slightly red tilt of the scalar perturbations and a small tensor-to-scalar ratio. Furthermore, a positive running of the scalar index may provide in this scenario few distinctive predictions, an enlightening element to be kept in mind, as it distinguishes from the scenario our analysis hinges on, and the predictions that arise in one-field inflation and ekpyrotic models [23]. Thus, this paradigm provides the concrete chance to falsify a matter bounce scenario with forthcoming observations.
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A bounce scenario can be achieved either by modifying in an effective way the Einstein equations, or by assuming a matter field content that violates the null energy conditions (NEC). We can revive the main directions along which the bounce scenario is attained, fitting them into the following classes:
● String Cosmology:
a resolution of the Big Bang singularity is notably provided by string theory, through the picture of a gas of strings that evolve in a space-time with compactified dimensions on a circle of radius R — specifically, one should deal with weakly coupled
N=(4,0) superstrings compactified to 4D. The thermal properties of the gas should then explain the emergence of a bounce. The one-loop partition functionZ(R) is finite and fulfills thermal duality (dubbed T-duality)Z(R)=Z(R2cR),
(10) with
Rc being of the order of the string length [24]. At the critical pointR=Rc , thermal string states start to become massless and to condensate. For time-dependent temperatures, the time-slices on which the condensates appear form space-like branes, or S-branes. As a byproduct of T duality, and of the formation of the string condensate, a maximal temperatureTmax is reached at the critical point. Thus, the string gas cools both forR>>Rc , a regime where the energy of the strings is mostly concentrated in their momentum, andR<<Rc , a regime where the energy of the strings is concentrated in windings around compactified dimensions. Therefore, one can imagine a process for which the string gas starts at a cold temperature, in the winding regime, and then gradually increases, until it reachesTmax , at which a phase transition takes place to the momentum regime. Hence, the temperature starts to decrease again. The thermodynamical properties of the string gas then imply the dynamical features of the cosmological evolution. The conservation of the thermal entropy of matter fields in a co-moving volume (in four-dimensional homogeneous space-time), which readsS=a3ρ+pT∼(aT)3,
(11) with a scale factor and p pressure density, then implies that
aT is constant. A bouncing cosmology follows, in which T increases until it reaches the maximal valueTmax , and then decreases again. At the same time, the scale factor a decreases until it reaches a minimal value, forT=Tmax , at which a bounce of the universe happens. Perturbation of the action at second order around the FLRW solution enables to compute the evolution of the cosmological perturbations from the pre-bounce contracting phase. This was achieved for instance in [25], due to the presence of the S-brane at the bounce instantiating a transition toward the expanding post-bounce phase.● Loop Quantum Cosmology (LQC):
a theory inspired and motivated by the non-perturbative quantization methods of Loop Quantum Gravity [26] is Loop Quantum Cosmology (LQC) [27]. Within this framework, a bounce picture [28] emerges at a critical Planckian energy density
ρc≃ρPl , which determines the energy scale of the bounce, namelyH2=8πG3ρ(1−ρρc),
(12) H denoting the Hubble parameter. In this picture, the continuity equation is not deformed by the curvature scale. For perturbations that can be described as long-wavelength Fourier modes, the separate universe approximation [29] applies, and the effective Friedmann equations in each patch provides the usual form of the long-wavelength Mukhanov-Sasaki equation.
●
f(R) and extended gravity:modified gravity theories that belong to the class of
f(R) extension of the Einstein-Hilbert action, defined by the actionS=12κ∫d4x√−gf(R),
(13) have been invoked either to explain dark matter and/or dark energy, or to provide a consistent ultraviolet completed theory of quantum gravity, with higher order curvature terms in the action that become relevant when approaching the Planck scale [30]. Furthermore,
f(R) gravity allows to mimic several bouncing scenarios. Specifically, a change of variables allows to unveil the dynamics off(R) theories: forgμν→˜gμν=f′gμν=ϕgμν,
(14) and
ϕ→˜ϕ,withd˜ϕ=√32κdϕϕ,
(15) the theory recasts
S=∫d4x√−g[˜R2κ−12(∂˜ϕ)2−U(˜ϕ)],
(16) with
U(˜ϕ)=(Rf′−f)/(2κf′2) . This enables an analysis of the cosmological perturbations over the pre-bounce phase, as affected byf(R) theories.● Effective Field theory:
As a violation of NEC, effective field theory scenarios have been also envisaged to achieve bouncing cosmologies. A particular instantiation can be provided by ghost condensates scalar fields, which are employed to generate the bounce [31-34]. Introducing both a Horndeski-type operator and a dynamical ghost condensate operator, non-singular bounce cosmological models can be obtained, with a Lagrangian expressed in the Kinetic Gravity Braiding (KGB) form by
L=K(ϕ,X)+G(X),
(17) K(ϕ,X)=[1−g(ϕ)]X+βX2M4Pl−V(ϕ),
(18) G(X)=γXM3Pl,
(19) where
X=gμν(∂μϕ)(∂νϕ)/2 is the regular kinetic term for the scalar fieldϕ ,β , andγ are real-valued parameters,g(ϕ) is a function of the scalar field, andgμν∇μ∇ν d'Alembertian operator, with∇μ as the gravitational covariant derivative. As soon as for (even for a short time)g(ϕ)>>1 , ghost condensation phase starts, giving rise to a non-singular bounce. Forβ>0 , theβX2 term stabilizes at high energy scales the kinetic energy.● Fermi Bounce:
Both the non-minimal ECH and the ECHSK actions entail of four-fermion term that can trigger the instantiation of a matter bounce. This term, corresponding to a fermionic superconductive scenario, directly originates from torsion. Nonetheless, effective higher order operators can be phenomenologically introduced, as in [35], even if fermions are coupled to gravity cast in a metric compatible form. For a fermion of mass m, one can consider a potential of the form
V(ˉψψ)=V0+mˉψψ−λ(ˉψψ)2,
(20) with
V0 contribution to the cosmological constant andλ a dimensionfull coupling constant. For positive values ofλ , the fermion system can induce a bounce scenario. Cosmic evolution can be studied resorting to a time-reversed description of the expanding universe. One can start considering the epoch over which matter energy density dominates over the fermion energy density, but the latter is still positive. Because for an expanding universeˉψψ∝1/a3 , over reversed time direction the value of the fermion bilinear increases, and the fermion energy density becomes negative, while the other form of energy density, due to radiation (∝a−4 ) and dust (∝1/a3 ), redshift away and approach zero. A bounce then takes place when the negative fermion energy density equals in absolute value the sum of the other matter energy density contributions.● Quintom bounce:
The Quintom scenario intertwines among the quintessence and the phantom paradigms, combined into a unique scenario for dark energy. Within this framework, the system is characterized by a dynamically evolving equation of state
p=wρ , which crosses over the cosmic evolution the critical valuew=−1 , corresponding to the cosmological constant. The phases along which the system evolves are characterized by a phantom-energy-like behavior whenw<−1 , and by a quintessence-like behavior forw>−1 .As shown in [36], quintom models entailing dark energy that deal with ideal gases and scalar fields, requires the presence of at least two degrees of freedom, as summarized in the Lagrangian
L=12∂μϕ∂μϕ−∂μφ∂μφ−V(ϕ,φ),
(21) where potential can be specialized to have the Coleman-Weinberg form
V(ϕ,φ)=14λϕ4(ln|ϕ|v−14)+116λv4,
(22) as in [37]. In contrast, a quintom model involving a fermion field was studied in [12]. The decay of the quintom energy may then trigger avoidance of the Big Bang and Big Rip singularity. Specifically, Quintom models for inflation can be constructed, as in [37], in which the bounce replaces the Big Bang singularity [38].
Below, we address a specific model of bouncing cosmology, to encompass the elements that are necessary to a direct comparison with the results we are about to derive.
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Ekpyrotic cosmologies appeared within the framework of string theory, such as to allow a bounce interconnecting the Big Crunch and Big Bang singularities. Singularities correspond here to brane collisions in higher-dimensions [39-41]. This is thus a viable realization of the matter cosmological bounce scenario.
The basic ingredients of the most straightforward version of the ekpyrotic scenario coincide with those ones of inflation, i.e., a scalar field rolling down a potential
V(ϕ) . However, as opposed to inflation, here the potential is steep and negative. This is a crucial feature that induces a slower contraction than inflation. Indeed, instead of an exponentially growing scale factor and a corresponding almost constant Hubble radius, which set a quasi de Sitter geometry, in this scenario the scale factor is almost constant, and the Hubble radius rapidly shrinks, a situation that corresponds to an approximately flat space-time.The ekpyrotic potential is generically composed by three parts:
1. a steep and negative part, down which the potential rolls with sizable kinetic energy, providing a scaling solution with attractor. Over this phase, the large-scale density fluctuations are generated once the modes of the perturbed field exit the horizon. The so-called fast-roll conditions hold, namely
ϵ≡1MPl(VV,ϕ)2<<1,η≡1−V,ϕϕVV2,ϕ<<1.
(23) Usually, an approximated potential that satisfies these conditions is
V(ϕ)≃−V0exp(−√2kϕMPl),
(24) in which k is a real parameter that satisfies
k<<1 . The scaling behavior then ceases as soon as (23) are no-longer valid;2. to avoid that a large negative vacuum energy is left, at the end of the ekpyrotic phase one usually assumes that the potential has a minimum, thus rising it back up to positive values;
3. a further part is left unconstrained by the specific features of the ekpyrotic scenario.
On a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, specified in comoving coordinates by
ds2=−dt2+a2dx⋅dx,
(25) with
a=a(t) scale factor, the first Friedmann equation and the evolution of the scalar field read, respectively,3H2M2Pl=12˙ϕ2+V(ϕ),
(26) ¨ϕ+3H˙ϕ=−V,ϕ.
(27) Specifying the potential as in (24), a scaling solution is found:
a(t)∼(−t)k,H=kt;
(28) ϕ(t)=√2kMPlln(−√V0M2Plk(1−3k)t).
(29) At
k<<1 , the scalar field entails large values of both the kinetic and the potential energy term. Nonetheless, the resulting total energy density is small, since the two contributions almost cancel. Furthermore, the scaling solution is an attractor and retains the equation of state of a very stiff fluid, withw=p/ρ=2/(3k)−1>>1.
(30) This is a very remarkable property, as it entails that
ρϕ∼a−2/k , which fork<<1 implies a more relevant blueshift than any other contribution to the Friedmann equation, either curvature (∼a−2 ), or matter (∼a−3 ), or radiation (∼a−4 ), or anisotropy (∼a−6 ). Thus, this term will be predominant with respect to anisotropies, while approaching the bounce. -
Because non-dynamical torsion provides an interaction term, we may reconstruct at one-loop the quantum effective action of the matter part of the theory. For simplicity, to calculate the these terms, we assume the background to be FLRW, which in conformal coordinates reads
ds2=a2(−dη2+dx⋅dx),
(31) with
a=a(η) scale factor andη conformal time.The four-fermion vertex is
V4=−ıξκγμγ5⊗γμγ5.
(32) The massless Feynman propagator — for a detailed analysis involving massive fermions, we refer to Refs. [22, 42] — in position space between the points x and
˜x is denoted asıSF(x,˜x) , and readsıSF(x,˜x)=(a˜a)−D−12Γ(D/2−1)4πD/2ıγa∂a1ΔxD−2++(x,˜x),
(33) having denoted
˜a=a(˜η) , andΔxD−2++(x,˜x)=(‖x−˜x‖2−|η−˜η|2)D−2,
(34) in which the two spacetime points are labelled with
x={η,x} and˜x={˜η,˜x} . Eq. (33) is anstraightforward result to be calculated, which is easier than the estimate of the massless scalar propagator, because of the conformal nature of massless fermions in any dimension — see e.g. [43, 44]. Assuming fermion matter content to be massless around the bounce is a viable approximation, given that the critical energy density at bounce is considerably above the condensate phase of the Higgs. In this preliminary study, we disregard to include Yukawa couplings of fermionic species to the Higgs, leaving a refined analysis to a forthcoming analysis [45].The sign of the coupling constant
ξ , expressed in Eq. (8) in terms of the bare parameters of the ECHSK theory,α andγ , individuates two phases at the cosmological level. Forξ>0 , the theory is in a Fermi liquid phase, with a tree-level repulsive interaction that has been studied as a source of dark energy [21], or to drive inflation [46]. Instead, the super-conductive phase corresponds to the branch of valuesξ<0 , for which a bouncing scenario has been taken into account [10, 11, 13, 15, 16, 35]. Therefore, we will proceed taking into account negative values ofξ in what follows. In Sec. 5, we come back to the issue of the quantum stability of the bounce, and analyze the running of theξ parameter around the bounce. -
We start considering the 2-point and the 3-point 1-loop contributions to the effective action, derived from the evaluation of the diagrams in Fig. 1a and Fig. 1b. For the former one, derived from the Wick contraction of two pairs of axial currents, one has to calculate the tensorial contribution
Πμν2 , which is definedFigure 1. a) One-loop correction to the four-fermions self-interaction term; b) one loop correction to the six-points self-interaction term; c) one loop correction to the eight-points self-interaction term; d) one loop correction to the 2n-points self-interaction term.
Πμν2=−Trx′[γμγ5SF(x,˜x)γνγ5SF(˜x,x)],
(35) where we have the tensorial structure is the one specified in (32), the fermion propagator
SF(x,˜x) is the one defined in (33),Trx′ denotes the trace over the internal spinorial indices, and integration over spacetime pointsx′ . The massless propagator in (33) can be expanded according toSF(x,˜x)=γaSa(x,˜x) , which allows to factorize the tensorial structure from the integral, i.e.Πmr2=−4(ηamηbr−ηrmηab+ηbmηar)×(−ıξκ)2∫√−gd4xSa(x,˜x)Sb(˜x,x).
(36) This amounts to calculate two types of contribution:
ˉΠab2=−(ξκ)2∫√−gd4xSa(x,˜x)Sb(˜x,x),
(37) ˉΠ2=−(ξκ)2∫√−gd4xSa(x,˜x)Sa(˜x,x).
(38) For the former, in generic D dimensions, we find
ˉΠab2=−(ξκ)2a−(D−1)∫dD˜x[Γ(D/2−1)]216πD×(ı∂a1ΔxD−2++(x,˜x))(ı˜∂b1ΔxD−2++(˜x,x))=−(ξκ)2a−(D−1)∫dD˜x[Γ(D/2−1)]216πD×1ΔxD−2++(˜x,x)∂a˜∂b1ΔxD−2++(x,˜x).
For
D=4 , the above formulas yield the expressionsˉΠab2=−(ξκ)2a−3∫d4˜x116π41Δx2++(˜x,x)×∂a˜∂b1Δx2++(x,˜x)=(ξκ)22π4a3∫d4ζζaζbζ8=−(ξκμ2)24π4a3∫d4Ωnanb=−(ξκμ2)216π2a3ηab,
(39) μ representing the sliding scale, and having used the fact that∫d4Ωnanb=π22ηab . In the renormalization group (RG) flow of the coupling constantξ , we shall retain this dependence, according to the hierarchyμ2⩽1/(ξκ) .For the quantity in (38), the expression recasts as
ˉΠ2=−(ξκ)2a−(D−1)∫dD˜x[Γ(D/2−1)]216πD×(ı∂a1ΔxD−2++(x,˜x))(ı˜∂a1ΔxD−2++(˜x,x))=−(ξκ)2a−(D−1)∫dD˜x[Γ(D/2−1)]216πD×1ΔxD−2++(˜x,x)∂a˜∂a1ΔxD−2++(x,˜x).
For
D=4 , making use of∂21Δx2++=4ıπ2δ4(x−˜x),
(40) we finally obtain
ˉΠ2=ı(ξκ)24π3a3∫d4˜x1Δx2++(˜x,x)δ4(x−˜x)=0.
(41) It is trivial to realize that the three-point contribution vanishes, because of the properties of the gamma matrices, i.e.
Πμνρ3=−Tr˜x,x′[γμγ5SF(x,˜x)γνγ5SF(˜x,x′)γνγ5SF(x′,x)]=0.
The four-point function is recovered from the expression
Πμ1μ2μ3μ44=−Trx2,x3,x4[γμ1γ5SF(x1,x2)×γμ2γ5SF(x2,x3)γμ3γ5SF(x3,x4)γμ4γ5SF(x4,x1)].
The one-loop corrections arising from the two-point and four-point functions, as depicted in Fig. 1a and Fig. 1c (and accounting for the symmetry degeneration factor of the internal lines), leads to the form of the effective potential
Veff=V0+ξκ(¯ψγ5γaψ)2(1+ξκμ2π2a3)+(ξκ)4ln(ξκμ)8π2a3[(¯ψγ5γaψ)2]2+O((ξκμ2)6).
(42) The constant
V0 represents a shift of the energy density, required to avoid negative values, which would be inconsistent with the Einstein equations. This is a short way to ensure the theoretical consistency of this approach. Differently, in a fully general approach that takes into account also the inclusion of hypercharge fields, positive energy densities would be provided by the energy density of radiation. It is also worth noticing that the vacuum polarization diagram in Fig. 1b cannot contribute to the energy density, because of the tensorial structure of the vertex in Eq. (32).From the conservation of the chiral current, which is implied by the assumption of having massless fermions, it follows that in (42), each axial current redshifts as the volume factor. Isotropy in the background requires the spatial component to be vanishing. This can be verified for slowly varying backgrounds, by expressing the axial current in terms of the scalar and pseudoscalar bilinears, and the vector current, i.e., by using the Pauli-Fierz identity
(¯ψγ5γaψ)(¯ψγ5γaψ)=(¯ψψ)2−(¯ψγ5ψ)2+(¯ψγIψ)(¯ψγIψ).
The vector current can then be verified to posses only non-vanishing temporal components — see e.g., Ref. [47] — by contracting the fermion Feynman propagator with the Dirac matrices, and extracting the dimensional regularized part of the coincident limit. Therefore, we can recast the quadratic terms entering the effective potential by means of the expression
(¯ψγ5γaψ)2=(ψ†γ5ψ)2=Q25a6,withQ5∼q5μ3.
(43) These arguments immediately lead to the conclusion that an Ekpyrotic quantum effect is generated at one-loop, considering the 2-point contribution to the effective action. The bounce is not spoiled by this effect. With a natural choice of the conformal factor normalized at the bounce, the two-point contribution to the one-loop correction retains a numerical factor subdominant with respect to the three-level contribution. At the same time, around the bounce, the extra term (to the energy potential) washes out anisotropies, owing to the dependence
ρ(2)Ek=ρ0a9withρ0=(Q5ξκμ)2256π4.
(44) The four-point amplitude then contributes to the effective potential with a dumping factor that redshifts as
a15 , and thus enhances the Ekpyrotic behavior. We can further verify that generic n-point contributions do not spoil this effect. -
Calculations can be carried out for the
2n -point contributions to the effective one-loop action (see Fig. 1d). One should indeed consider all the possible amplitudes arising fromΠμ1…μ2nn=−Trx1,...x2n−1[γμ1γ5SF(x,x1)…γμ2nγ5SF(x2n−1,x)].
(45) For large n, these terms behave as
Πμ1…μ2nn=(tensorial structure)×(ξκ)2na32π2(2μ)2n(2n−1)!,
thus entailing the asymptotic contribution to the potential at
2n -orderV(2n)eff∼(2ξκ)2na32π2(2μ)2n[(¯ψγ5γaψ)2]n.
Summing for the potential, we find
Veff∼V0+ξκ(¯ψγ5γaψ)2+∑n(−1)n(ξκ)2na32π2(2μ)2(n−2)[(¯ψγ5γaψ)2]n∼V0+ξκ(¯ψγ5γaψ)2+μ4a32π2[1+(ξκ)22μ2(¯ψγ5γaψ)2]2.
(46) The potential can be finally recast in order to explicitly the scale factor dependence:
Veff∼V0+ξκQ25a6+μ4a32π2[1+(ξκ)22μ2Q25a6]2∼V0+ξκμ6a6+μ4a32π2[1+(ξκμ2)22a6q25]2.
It follows that the bounce cannot be spoiled in this scenario, but that extra contributions arise that enhance the quantum Ekpyrotic mechanism. With respect to the standard ekpyrotic framework involving scalar fields, illustrated at the end of Sec. 3.2, it is remarkable that the equivalent of the stiff ekpyrotic fluid is provided in this picture by the loop corrections to the dynamics of the very same field that sources the bounce. A detailed comparison is then achieved by recalling that for, a scalar field with potential approximated as in Eq. (24), the energy density scales as
ρ(ϕ)Ek∼1a2m.
(47) This behavior must be now confronted with our result in Eq. (44), hence providing as a specific choice for the parameter of the effective potential in Eq. (24) the value
m=2/9 . We have dubbed this behaviour as quantum ekpyrotic, to emphasize its origin. -
Consistency requirements with CMB observables impose that the bounce must take place at critical energies not below the GUT scale. This sets the values of the critical energy density to be
ρc≃(ξκ)−2≃M4GUT . Correspondingly, we might wonder that the sliding scales could approach the valueμ≃MGUT at the bounce, and eventually induce a change of sign of the the renormalized value ofξ . Our estimate shows that this is not the case, as the (predominant in the one-loop calculation)2 -point correction, namelyξren.=ξ(1+ξκμ2π2+O(ξκμ2)2),
(48) maintains the sign at lowest approximation. The two-point contribution yields indeed only an order
O(1/10) numerical suppression factor at the bounce. This conclusion can be strengthened looking at the calculation of the2n -point amplitude contribution. In general, the numerical factor driving RG flow ofξ can be summing the coupling constant dependence of the vertex-operators that enter the2n -point function, and the coefficients of the fermion Feynman propagators. In the calculations, we have already shown that while deriving the effective potential, it is then easy to convince ourselves that the higher order coefficients ofξren. will actually decrease, as soon as more interaction vertices are accounted for in the one-loop contribution to the effective action, and that the sum can be obtained, without encountering a Landau pole at the bounce. -
We have shown that a quantum Ekpyrotic scenario emerges from the calculation of the one-loop quantum effective action. The lowest-order vertex interaction is already sufficient to produce this remarkable effect. This result has been achieved assuming chiral symmetry unbroken at the energy density scale, at which the bounce takes place, a scenario accomplished in several unification theories, and naturally envisaged in the non-condensate phase of the Higgs. An important step in testing this result would be to reproduce this analysis involving anisotropic metrics. The technical limitation in dealing with quantum field theory on Bianchi backgrounds forbids to pursue this goal at the present.
-
The results we obtained on the stability of the bounce and the occurrence of a quantum ekpyrotic scenario hinge on some straightforward but lengthly calculations, the details of which are reported in this section.
-
To recover the four-point function
Πμ1μ2μ3μ44 we shall calculate the trace over eight Dirac matrices. Projecting on the Clifford algebra indices providesΠm1m3m5m74=−4[ηm1m2ηm3m4ηm5m6ηm7m8−ηm1m3ηm2m4ηm5m6ηm7m8+ηm1m3ηm2m5ηm4m6ηm7m8−ηm1m3ηm2m5ηm4m7ηm6m8+ηm8m3ηm2m5ηm4m7ηm6m1−ηm8m2ηm3m5ηm4m7ηm6m1+ηm8m3ηm2m4ηm5m7ηm6m1−ηm8m3ηm2m5ηm4m7ηm6m1+ηm8m3ηm2m5ηm4m6ηm7m1−ηm3m2ηm8m5ηm4m6ηm7m1+ηm3m2ηm5m4ηm8m6ηm7m1−ηm3m2ηm5m4ηm6m7ηm8m1+ηm8m2ηm5m4ηm6m7ηm1m3−ηm8m2ηm4m6ηm5m7ηm1m3+ηm8m2ηm4m6ηm5m1ηm7m3]×(−ıξκ)4∫√−gd4x2∫√−gd4x3∫√−gd4x4×Sm2(x1,x2)Sm4(x2,x3)Sm6(x3,x4)Sm8(x4,x1).
(49) The multiple integral term reshuffles as
Im2m4m6m8:=3∏h=1∫√−gd4xh+14∏s=1Sm2s(xs,x|s+1|4)=132π8a33∏h=1∫d4xh+14∏s=1∂(s)m2s1Δ2++(xs,x|s+1|4),
(50) where
|…|p denotes modulo p and∂(s) for the derivative with respect to them2s component of thexs coordinate. We further recast the integral through a series of passages:3∏h=1∫d4xh+14∏s=1∂(s)m2s1Δ2++(xs,x|s+1|4)=∫d4x2∂(1)m21(x1−x2)2×∫d4x3∂(2)m41(x2−x3)2∫d4x4∂(3)m61(x3−x4)2∂(4)m81(x4−x1)2=−24∫d4x2(x1−x2)m2(x1−x2)4∫d4x3(x2−x3)m4(x2−x3)4∫d4x4(x4−x3)m6(x4−x3)4×(x4−x1)m8(x4−x1)4=8π2ηm6m8∫d4x3∫d4x2(x2−x1)m2(x2−x1)4(x2−x3)m4(x2−x3)4×∫d|y||y−(x3−x1)|3=4π4ηm2m4ηm6m8∫d4z(∫d|y||y−z|3)2=2π6ηm2m4ηm6m8∫d|z||z|=π6ηm2m4ηm6m8ln(1ξκμ2),
where in the last hand-side we have used as ultraviolet regulator
(ξκ)−12 . Combining all the terms together, we find the contribution to the effective potential at4 -point order. -
The
2n -point functionΠμ1˙μ2nn requires the knowledge of the trace over4n gamma matrices. In general, one can show thatTr[γm1…γm4n]=4[(−1)|P|ησ(m1)σ(m2)…ησ(m2n−1)σ(m2n)],
where
σ denotes all the possible non-cyclic permutations and P denotes their order.The integral
Im2⋯m4n can be easily calculated at leading orderIm2⋯m4n:=2n∏h=2∫√−gd4xh2n∏s=1Sm2s(xs,x|s+1|4)=142nπ4na32n∏h=2∫d4xh2n∏s=1∂(s)m2s1Δ2++(xs,x|s+1|4)=1a3(4π2)2nμ2(n−2)(2n−1)!.
(51)
