Inflation from fermions with curvature-dependent mass

# Inflation from fermions with curvature-dependent mass

D. Benisty Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel    E.I. Guendelman Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View Circle, Stella Maris, Long Island, The Bahamas    E. N. Saridakis Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece Department of Astronomy, School of Physical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China Chongqing University of Posts & Telecommunications, Chongqing, 400065, P.R. China    H. Stoecker Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany Fachbereich Physik, Goethe-Universitat, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany    J. Struckmeier Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany Fachbereich Physik, Goethe-Universitat, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany    D. Vasak Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany
###### Abstract

A model of inflation realization driven by fermions with curvature-dependent mass is studied. Such a term is derived from the Covariant Canonical Gauge Theory of gravity (CCGG) incorporating Dirac fermions. We obtain an initial de Sitter phase followed by a successful exit, and moreover, we acquire the subsequent thermal history, with an effective matter era, followed finally by a dark-energy epoch. This behavior is a result of the effective ‘weakening’ of gravity at early times, due to the decreased curvature-dependent fermion mass. Through the perturbation level, we obtain the scalar spectral index and the tensor-to-scalar ratio, which are in agreement with the Planck observations. The efficiency of inflation from fermions with curvature-dependent mass, at both the background and perturbation level, reveals the capabilities of the scenario and makes it a good candidate for the description of nature.

###### pacs:
98.80.-k, 04.50.Kd, 98.80.Cq

## I Introduction

The developments in cosmology have been influenced to a great extent by the idea of inflation Starobinsky:1979ty (); Kazanas:1980tx (); Starobinsky:1980te (); Guth:1980zm (); Linde:1981mu (); Albrecht:1982wi (); Blau:1986cw (), which provides an attractive scenario for the solution of the fundamental puzzles of the old Big Bang paradigm, such as the horizon and the flatness problems. Additionally, inflation was proved crucial in providing a framework for the generation of primordial density perturbations Mukhanov:1981xt (); Guth:1982ec (). Although the inflationary scenario is very attractive, it has been recognized that a successful implementation requires special restrictions on the underlying dynamics. The inflationary mechanism can be achieved in several different ways, considering primordial scalar fields Lidsey:1995np (); Bassett:2005xm () or geometric corrections into the effective gravitational action Nojiri:2003ft ().

On the other hand, it is known that Hamiltonian formulations of a theory may have theoretical advantages. One such framework is the covariant canoninal gauge theory of gravity (CCGG) Struckmeier:2017vkf (). The CCGG ensures by construction that the action principle is maintained in its form by requiring that all transformations of a given system are canonical. The imposed requirement of invariance of the original action integral with respect to local transformations in curved spacetime is achieved by introducing additional degrees of freedom, namely the gauge fields Struckmeier:2017vkf (). In these lines, in Benisty:2018ufz (); Benisty:2018efx (); Benisty:2018fgu (); Vasak:2018lhz () quadratic Riemann theories by the covariant Hamiltonian approach were formulated, which were shown to lead to inflationary models Myrzakulov:2014hca () based on the correspondence between the metric affine (Palatini) formalism and the metric formalism Benisty:2018fgu (); Benisty:2018efx ().

The covariant Hamiltonian formulation was recently implemented to include Dirac fermions Struckmeier:2018psp (). This covariant Hamiltonian incorporates additional terms with a new dependence for fermions, namely the effective mass depends linearly on the curvature through

 meff=m0+ξ6κ2R, (1)

where is the usual fermion rest mass, is the coupling to the Ricci scalar with dimensions , and . For , the covariant Hamiltonian approach for fermions reduces to that of standard Dirac fermions.

The curvature-dependence mass of the fermions is expected to be significant in regimes where the curvature is significant, such as close to black holes Struckmeier:2018psp () or new description for Neutrino Onofrio:2013iya (). However, one can deduce that this curvature-dependent correction can be important also in the early universe, where it is known that the Ricci scalar acquires large values. Hence, it would be interesting to investigate the effect of the fermion curvature-dependent mass in the early universe. In particular, we desire to examine whether it can drive a successful inflation. Indeed, as we will see, such a novel coupling can both drive a successful inflation at the background level, accompanied by a successful exit and the subsequent thermal history of the universe, but it can also be very efficient at the perturbation level, giving rise to a scalar spectral index and a tensor-to-scalar ratio in agreement with observations.

The plan of the work is the following: In Section II we present fermionic cosmology with curvature-dependent mass, extracting the equations of motion. In Section III we apply the scenario in the early universe, obtaining the inflation realization at the background level. In Section IV we examine the perturbation-related observables such as the spectral index and the tensor-to-scalar ratio. Finally, in Section V we summarize our results.

## Ii The model

In this section we construct the scenario of fermionic cosmology with curvature-dependent mass. We first briefly review the basics of the covariant Hamiltonian incorporation of fermions in curved spacetime, and then we present the Lagrangian of the model, extracting the equations of motion.

### ii.1 Spinors in curved spacetime

Fermionic sources in general relativity were studied in detail in Ellis:1999sf (); Ribas:2005vr (); Samojeden:2010rs (), especially for cosmological applications Myrzakulov:2010du (); Chimento:2007fx (); Ribas:2016ulz (); Ribas:2007qm (); Hossain:2014zma (); Basilakos:2015yoa (); Geng:2017mic (). The tetrad formalism was used to combine the gauge group of general relativity with a spinor matter field. The tetrad and the metric tensors are related through

 gμν=eaμebνηab,a,b=0,1,2,3, (2)

with Latin indices referring to the local inertial frame endowed with the Minkowski metric , while Greek indices denote the (holonomic) basis of the manifold.

The spinor field Lagrangian in curved torsion-free space-time reads as

 Lf=i2[¯ψΓμDμψ−(Dμ¯ψ)Γμψ]−meff¯ψψ, (3)

where is the adjoint spinor field and is the fermionic rest mass, as defined in (1). In curved space time the Dirac matrices are replaced by their generalized counterparts , which satisfy the extended Clifford algebra . Thus, the ordinary derivatives are replaced by their covariant versions

 Dμψ=∂μψ−Ωμψ. (4)

Furthermore, the metric compatibility condition implies that the spin connection is given by

 Ωμ=14gβν[¯Γναμ−eνj∂μejα]ΓβΓα, (5)

with the Christoffel symbols. The original CCGG formulation assumes that the affine and spin connections are fields independent of the metric (i.e. the metric-affine or the Palatini formulation) Struckmeier:2018psp (), however in this work we use the metric compatibility derived from the action, that allows for the substitution (5).

The action of fermions in the gravitational background of general relativity is then

 S=∫d4x√−g[12κ2(R−2Λ)+Lf], (6)

where is the Ricci scalar and the cosmological constant. Note that concerning the fermions we have assumed a minimal coupling with gravity, while their inertial mass is given by (1). Here, rather then applying the complete CCGG action, we simplify the analysis is order to illuminate the impact of the effective spinor mass in the conventional Einstein-Hilbert theory in the metric formulation.

### ii.2 Field equations

Let us now extract the field equations of the action (6). Variation with respect to the spinor field yields the generalized Dirac equations:

 iΓμDμψ−meffψ=0iDμ¯ψΓμ+meff¯ψ=0. (7)

Moreover, variation with respect to the metric leads to the field equations

 1κ2Gμν=Tμν(f)+ξ3κ2[Gμν¯ψψ+□μν(¯ψψ)]+gμνΛκ2, (8)

where is the Einstein tensor, and . In the above equations is the kinetic part of the spinor fields, given by

 −gμν{i2[¯ψΓλDλψ−Dλ¯ψΓλψ]−meff¯ψψ}. (9)

Equation (8) can be re-written as

 1κ2(1−ξ3¯ψψ)Gμν=i4[¯ψΓ(μDν)ψ−D(ν¯ψΓμ)ψ] −gμν{i2[¯ψΓλDλψ−Dλ¯ψΓλψ]−meff¯ψψ} −ξ3κ2□μν(¯ψψ)+gμνΛκ2. (10)

Finally, note that the phase invariance of the spinor field

 ψ→eiθψ,¯ψ→e−iθ¯ψ, (11)

through the Noether’s theorem leads to the conserved current

 jμ;μ=0, (12)

where , which proves convenient in simplifying our analysis.

## Iii Background evolution

In this section we apply the above fermionic model at a cosmological framework, focusing on early-time universe, and in particular on the inflationary realization. As usual we neglect standard matter, i.e. we incorporate only the fermions with curvature-dependent mass. We consider the Friedman-Lemaitre-Robertson-Walker (FLRW) homogeneous and isotropic metric

 ds2=−dt2+a(t)2(dx2+dy2+dz2), (13)

with the scale factor , and thus through (2) the tetrad components are found to be

 eμ0=δμ0,eμi=1a(t)δμi. (14)

Moreover, the covariant version of the Dirac matrices are

 Γ0=γ0,Γi=1a(t)γi, (15)

while the spin connection becomes

 Ω0=γ0,Ωi=12˙a(t)γiγ0. (16)

The Dirac equation (7) then becomes

 ˙ψ+32Hψ+imeffγ0ψ=0, (17)

and similarly for , with

 meff=m0+ξκ2(˙H+2H2), (18)

and the Hubble function. For an isotropic and homogeneous universe the spinor field is exclusively a function of time, namely

 ψ=⎛⎜ ⎜ ⎜ ⎜⎝ψ0(t)eiϕ(t)000⎞⎟ ⎟ ⎟ ⎟⎠, (19)

with the phase of the state, where we have considered the fermions to be at the minimal state. Introducing this ansatz into the generalized Dirac equation (17) yields the relation between the phase of the fermion and the effective mass, namely

 ddtϕ(t)=−meff. (20)

An elegant way to extract the solution for the spinor field is by making use of the conserved current (12). For the FLRW metric the conserved current becomes

 1√−g∂μ(√−gjμ)=0⇒1a3∂t(a3j0)=0, (21)

which then leads to a dust-like behavior for the absolute value of the spinor field, i.e.

 ¯ψψ=ψ0(t)2=Ca3, (22)

with the particle number density . The quantity is thus the total gas energy density.

Let us now turn to the field equations (II.2). Inserting the FLRW metric (13) they give rise to the Friedmann equations

 −H2ξψ20+3H2−2Hξψ0˙ψ0−Λ−κ2m0ψ20=0, (23)
 (3−2Cξa3)H2+(Cξ3a3+2)˙H +Cκ22a3(H−2m0+2ϕ˙ϕ)−Λ=0. (24)

Combining (22) with (23) we obtain the convenient form

 3H2κ2+2~ξH2κ2a3=ρf0a3+Λ, (25)

with the definitions and . Finally, since , from Eq. (25) we deduce that the curvature-dependent fermion mass term induces an effective Newtonian constant

 GeffGN=3a33a3+2~ξ. (26)

We proceed by investigating the inflation realization in the above construction. From now on we set . For small scale factors, such that

 a3≪ρf0Λ=m0CΛ≡s,a3≪~ξ, (27)

the modified Friedman equation (25) accepts the de Sitter solution , with

 Hinf=√ρf02~ξ. (28)

Thus, we observe that the inflation phase is driven by the effectively dust-like fermionic component. However, after a suitable inflationary expansion the scale factor increases significantly and the approximation (27) breaks down, leading to a decrease of the Hubble function and thus to a successful inflationary exit. In particular, after these intermediate times the dust-like terms of the Friedmann equation (25) will dominate. Finally, at late-times, the cosmological constant term will dominate in (25) and the Hubble constant asymptotically approaches the value

 H∞=√Λ3. (29)

In order to show the above behavior in a transparent way we solve the Friedmann equation (25) numerically. Fig. 1 shows the inflationary realization, followed by a successful exit, the subsequent effective dust matter epoch, and the final appearance of an effective dark-energy era. This thermal history of the universe is in agreement with observations and is one of the main results of the present work.

We now present a convenient way to extract the above results. In particular, we introduce an “effective potential” through the relation (i.e. the “kinetic energy” and the potential energy add up to zero), which using the Friedmann equation (25) leads to

 V(a)=−a2Λ(s+a32~ξ+3a3). (30)

The effective potential is presented in Fig. 2. The initial point incorporates time scales beyond Planck time, and thus our discussion starts at point , which corresponds to the inflationary initial scale factor (which is calculated in the next section). The potential contains one minimal point and one maximal point , with the scale factors

 a3B,C=112(3s−10~ξ±√3s−50~ξ√3s−2~ξ), (31)

where the sign corresponds to point . One can easily see that point is the point where inflation ends, and the universe enters the effective dust epoch of Fig. 1. This phase holds up to point , after which the universe enters into the late-time, dark energy regime.

We close this section by examining the behavior of the effective Newton’s constant from (26), as well as the effective (curvature-dependent) fermion mass from (18), which using the Friedmann equation (25) becomes

 meffΛ=3~ξ[12a6+a3(14~ξ+3s)+8~ξs]2(3a3+2~ξ)2+m0sρf0. (32)

In Fig. 3 we depict their normalized evolutions, for various values of . As we observe, at early times the curvature-dependent fermion mass is small, which makes the effective Newton’s constant small too, and this effective “weakening” of gravity is the reason for the inflation realization. In particular, the effective Newtonian constant starts from and results at (we mention that we can choose in order to obtain a small post-inflationary variation of , in order to be in agreement with the Big Bang Nucleosynthesis (BBN) constraints Nesseris:2017vor (); Kazantzidis:2018rnb ()), while starts from and results to , in agreement with the fact that fermionic matter is cold.

## Iv Perturbations

In this section we investigate the perturbations of the above background evolution, and in particular we focus on the inflationary observables such as the scalar spectral index and the tensor-to-scalar ratio . As usual, we introduce the Hubble slow-roll parameters , Martin:2013tda (); Bamba:2016wjm (), which in our case using the Friedmann equation (25) read

 ϵ(a)=3s2(a3+s)−3~ξ3a3+2~ξ, (33) η(a)=32−6~ξ3a3+2~ξ. (34)

Inflation ends when , and as mentioned above using the “effective” potential one can see that , with the scale factor value at the minimal point of Fig. 2. Thus, if the initial scale factor is then the number of -folding is

 N=log(afai). (35)

Using the slow-roll parameters, one can calculate the values of the scalar spectral index and the tensor-to-scalar ratio respectively as Nojiri:2019kkp (); Dalianis:2018frf ()

 r=16ϵ=24sa3i+s−48~ξ3a3i+2~ξ (36) ns=1−6ϵ+2η=4−9sa3i+s+6~ξ3a3i+2~ξ. (37)

As we can see, the first slow-roll condition leads to

 0

while the second slow-roll condition yields

 2~ξ3≤a3i≪10~ξ3. (39)

Hence, combining them we obtain the requirement

 23~ξ≲a3i<43~ξ,23~ξ≲s. (40)

Finally, from (36),(37) we can obtain

 r=8(4−ns)3−32~ξ3a3i+2~ξ. (41)

As one can see, by suitably choosing the values of and we obtain and well inside the Planck observed values Akrami:2018odb (). For instance, taking according to (40) that we obtain that , which gives and . This is one of the main results of the present work and reveals the capabilities of the scenario at hand, which can give a solution in agreement with observations, both at background and perturbation levels.

## V Conclusions

In this work we constructed a model of inflation realization, driven by fermions with curvature-dependent mass. We started from the Covariant Canonical Gauge Theory of gravity (CCGG) formulation, which imposes the requirement of invariance with respect to local transformations in curved spacetime through the insertions of gauge fields Struckmeier:2017vkf (). Incorporation of Dirac fermions in such a framework implies that the fermions acquire a curvature-dependent mass. The effect of such a correction is expected to be significant in regimes where the Ricci curvature is large, such is the case in the early universe. In particular, the dark spinor dust is indeed repulsive due to the term, as long as we are close to a singularity, either close to black holes or close to the Big Bang.

We first investigated the scenario at the background level. As we showed, we obtained the inflation realization, with an initial de Sitter phase followed by a successful exit. Furthermore, we acquired the subsequent thermal history, with an effective matter era, followed finally by a dark-energy epoch. This behavior is qualitatively expected, since as we showed the curvature-dependent fermion mass induces a smaller effective Newton’s constant at early times, and this effective “weakening” of gravity is the reason for the inflation realization.

We proceeded to the investigation of the scenario at the perturbation level, focusing on observables such as the scalar spectral index and the tensor-to-scalar ratio. We extracted analytical expressions for their values, and we showed that by suitably choosing the coupling parameter, we can obtain values in agreement with Planck observations. The efficiency of inflation from fermions with curvature-dependent mass at both the background and perturbation level reveals the capabilities of the scenario and makes it a good candidate for the description of nature.

One could try to investigate extensions of the above basic scenario. In particular, in this work we treated the spinor fields classically, assuming them to lie at the minimal state, namely being cold. Nevertheless, incorporation of high-temperature effects, as for example in the case of warm inflation Berera:1995ie (); Kamali:2017zgg (), could lead to the appearance of an additional friction term, and this could further improve the values of the spectral index and the tensor-to-scalar ratio. Another possible generalization is to consider a quadratic curvature dependence, alongside the linear one, in the fermion mass, or even more general functions of the form . Such studies lie beyond the scope of the present work and are left for future projects.

## Acknowledgments

D. B. thank the FIAS for generous support. D.V. thanks the Carl-Wilhelm Fueck Stiftung for generous support through the Walter Greiner Gesellschaft (WGG) zur Foerderung der physikalischen Grundlagenforschung Frankfurt. D.B., E.I.G. and E.N.S are grateful for the support of COST Action CA15117 “Cosmology and Astrophysics Network for Theoretical Advances and Training Action” (CANTATA) of the European Cooperation in Science and Technology (COST). H.S. thanks the WGG and Goethe University for support through the Judah Moshe Eisenberg Laureatus endowed professorship and the BMBF (German Federal Ministry of Education and Research).

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