Quantum-Ordering Ambiguities in Weak Chern—Simons 4D Gravity and Metastability of the Condensate-Induced Inflation

In a series of previous works [1,2,3,4,5,6,7], a rather novel approach to string-inspired cosmology, the so-called Stringy Running Vacuum Model (StRVM), has been developed. The StRVM embeds, in a non-trivial way, the Running-Vacuum-Model (RVM) approach to modern cosmology [8,9,10,11,12] into string theory [13,14]. The RVM is a phenomenologically consistent and innovative cosmological framework, with important predictions on observable deviations from $\mathsf{\Lambda }$CDM [7,15,16,17,18], especially in the modern era, where it may also provide alleviations of the cosmological tensions [19,20], features which are fully shared by (variants of) the StRVM [7]. Moreover, the RVM approach describes a smooth evolution of the Universe from the inflationary to the modern eras, explaining also the thermodynamic properties of the Universe [21,22,23].

In the StRVM there are highly non-trivial and diverse rôles played by the antisymmetric tensor field of the massless multiplet of the underlying string model [13,14,24,25,26], from driving inflation to the generation of matter-antimatter asymmetry in this Universe, during the early postinflationary epochs.1 An important feature, which by the way is shared between the StRVM and the works in Refs. [29,30,31] on string-dilaton cosmologies, is the linear time dependence of the (3+1)-dimensional dual of the antisymmetric-tensor field strength after string compactification. This dual field is a pseudoscalar, and is commonly known as a KR, or gravitational, or string-model-independent, axion, $b\left(x\right)$, to be distinguished from the string-compactification-induced axions, $a^{\left(i\right)}\left(x\right)$, $i=1,\cdots N$ (axion species), which depend on the specific string model [26,32]. In [29,30,31], such linearly dependent KR axions characterize static Einstein Universes with positive constant curvature (perfect-fluid de Sitter solutions). Although our approach and situation in this article will be different from that discussed in [29,30,31], nonetheless in our previous works [1,2,3,4] we have demonstrated, as already mentioned, the importance of such a linear dependence in cosmic time of the KR axion for the evolution of the string-inspired Universe in a Friedman-Lemaitre-Robertson-Walker (FLRW) expanding-Universe spacetime background, specifically the existence of an inflationary phase, as well as the post-inflationary dominance of matter over antimatter in the Cosmos, via the scenario advocated in [33,34,35,36].

As regards the former property, inflation in this model is induced [1,2,3,4,5,6] by the condensation of primordial Gravitational-Wave (GW) tensor perturbations, which in turn lead to a non trivial condensate of the anomalous gravitational Chern-Simons (gCS) terms. As for the generation of matter-antimatter asymmetry in the (early) post-inflationary epoch of this Universe, as discussed in [1,2,3,4,5,6], the presence of a constant $\dot{b}$ (where the dot denotes cosmic time derivative) during inflation, implies that the cosmic KR axion background remains undiluted at the end of inflation. At that epoch, chiral fermionic matter is generated through the decay of the metastable (as we shall show here) vacuum. Such fermions are characterized by their own gravitational and chiral anomalies in the gauge sector. In the scenario of [1,3,4], the fermion-induced chiral anomalies cancel the primordial ones, the latter assumed to exist in the early-eras of the compactified string Universe due to the Green-Schwarz mechanism [37], introduced for anomaly cancellation in (the higher-dimensional) string theories [13,14,24,25]. This cancellation implies that during the early radiation era, that succeeds inflation, the KR background axion field b scales with the scale factor of the universe as $a^{-3}\sim T^3$, where T is the Universe temperature. Under such scaling, it is possible to show, then, that in theories with right-handed neutrinos one can have the generation of a Lepton asymmetry (Leptogenesis) according to the mechanism of [34,35], which for the short time scales of the process resembles the situation advocated in [33], for a Leptogenesis induced by an approximately constant $\dot{b}$ background. This Leptogenesis can then be communicated to the baryon sector [36] via, say, Baryon-minus-Lepton-number preserving sphaleron processes [38,39,40,41,42] in the standard model sector (Baryogenesis [43,44,45,46]), thus leading to an explanation of the observable matter-antimatter asymmetry in the Universe [1], in the framework of this string-inspired CS gravity model.

The presence of anomalous gCS terms is therefore an important feature of our approach, which has not been considered in [29,30,31]. Such terms couple to the KR axion, as a consequence of a Bianchi identity of a modified field strength of the antisymmetric tensor field of the massless gravitational string multiplet [26,32]. The presence of the pertinent modifications is dictated by the appropriate Green-Schwarz counterterms that are added to the effective action in order to potentially cancel gauge and gravitational anomalies [37] in string theory [13,14,24,25]. In the approach of [1,2,3,4,5], (3+1)-dimensional gravitational anomalies are assumed not to be canceling. Moreover, only fields from the massless gravitational string multiplet are assumed to appear as external lines in the pertinent Feynman graphs of the effective theory that describes the primordial Universe. Supergravity is assumed to have been broken dynamically in this scenario at the pre-inflationary epoch, with the partners having obtained masses of order of the reduced Planck mass $M_{\mathrm{Pl}}=2.4\times {10}^{18}$ GeV and thus decoupled during the inflationary epoch of interest to us here [3,4]. Therefore, for our purposes, we deal with effective low-energy gravitational theories emerging from string theory, after compactification to (3+1)-dimensions, involving graviton, dilaton and antisymmetric tensor fields, which are at most quadratic in derivatives, or, equivalently, $\mathcal{O}\left({\alpha }^{\prime }\right)$ effective actions, where ${\alpha }^{\prime }=M_s^{-2}$ is the Regge slope of the string, and $M_s$ the string mass scale. The latter is treated as a phenomenological parameter in our approach, which is in general different from the four-dimensional reduced Planck scale ${\kappa }^{-1}=M_{\mathrm{Pl}}$, where $\kappa =\sqrt{8\pi \mathrm{G}}$ denotes the (3+1)-dimensional gravitational constant, with $\mathrm{G}$ Newton’s constant. For our purposes, following [1,2], we shall assume that the dilaton field is constant, which can be self-consistently arranged in string theory by an appropriate choice of the dilaton potential (possibly generated by string loops). Under this assumption, the $\mathcal{O}\left({\alpha }^{\prime }\right)$ effective action we shall be dealing with is a Chern-Simons gravitational theory of the form discussed in [47,48].2

In the presence of a sufficient amount of chiral primordial gravitational waves (GW), the gravitational CS terms condense, leading to an effective potential which is linear in the KR axion, thus driving inflation, as in the so-called axion-monodromy situation encountered in strings [51]. However, there is an important difference between conventional-string axion-monodromy scenarios for inflation, and our condensate-induced inflation. In our case, the CS condensate depends [1,3,4,6] on the fourth power of the Hubble parameter $H_I$ during inflation, which varies very mildly with the cosmic time. This has, as a consequence, the non-linear dependence of the vacuum energy density on $H_I^4$, which dominates and leads to a running-vacuum-model (RVM) type of inflation [9,10,21,22,23]. In fact, the RVM form of the pressure and energy densities of the corresponding cosmic fluid, that is their dependence on even powers of the Hubble parameter, of which terms of order $H^2$ and $H^4$ are the dominant ones, persist during the post-inflationary period as well, until the current epoch [1]. It is for this reason that the abovedescribed cosmological model is termed Stringy Running Vacuum Model (StRVM) [1,3,4].3

Phenomenology requires that the slow-roll parameters of this inflationary model fall into the range of values inferred from the plethora of the currently available cosmological data [52,53]. A linear axion alone cannot yield optimal fits to the data, as we shall discuss in this article. Nonetheless, one may have periodic modulations of the axion potentials in string theory due, e.g., to non-perturbative instanton effects of appropriate gauge groups, that exist during the RVM inflation. Such modulations do yield consistent results with the data, provided one fixes appropriately the (target-space) energy scales of the instantons, which in our approach are treated as phenomenological, string-model independent parameters. In addition to target-space gauge-group instantons in string theories, there are also world-sheet instantons which couple to the compactification axions [32,51], but not to the KR action. These world-sheet instantons also contribute appropriate periodic modulations in the multiaxion potential, which contribute to the slow-roll inflationary parameters. In the current work we shall treat all such modulations as being characterized by phenomenological scales. We stress, however, that in principle one should start from a consistent microscopic string theory model and study in detail the world-sheet or target-space gauge instanton effects. This is a highly non-trivial task, and at present, as far as we know, the situation is incomplete. For our purposes, therefore, we shall follow the aforementioned bottom-up approach, by means of which we shall determine the energy scales of the non-perturbatively-induced modulations of the axion potential phenomenologically, by requiring that they yield agreement with the cosmological (inflationary) data [52,53,54].

Another important novel aspect of the model, which we shall discuss here, concerns the microscopic origin of the imaginary parts of the primordial-GW-induced CS condensate. As we shall demonstrate, the latter are associated with quantum effects of the effective Hamiltonian of the GW, that is, they vanish classically. Specifically, we shall show that the pertinent imaginary parts are proportional to quantum commutators of chiral GW creation and annihilation operators. Hence, it appears that it is the quantum aspects of the chiral GW that lead to an effective violation of unitarity, and the metastability of the condensate-induced inflationary vacuum.

It is important to stress that the presence of imaginary parts in the gravitational CS condensate appears to be quantum-operator-ordering dependent. Par contrast, the real part of the condensate is independent of such orderings. In a symmetric ordering, such parts vanish. However, we argue in the current article, that choosing ordering schemes in which they are present has physical significance, in that such schemes imply instabilities of the inflationary vacuum and a finite life-time, as indicated in the classical dynamical-system approach [6]. Matching the resulting finite life-time with the phenomenologically acceptable 50–60 e-foldings [52,54], will lead, as we shall see, in adopting a specific ordering scheme, and also a constraint on the magnitude of the string scale $M_s$ as compared to $M_{\mathrm{Pl}}$, in agreement with the findings of the classical analysis of [6].

Before concluding the introduction with the outline of the article, we feel placing our work in perspective of the more general effective framework of Chern-Simons gravity and its applications [47,48], in the presence of contorted geometries [55,56]. As is well known, in the context of string theories [13,14,24,25], the field strength of the antisymmetric tensor spin-one field of the string gravitational multiplet, ${\mathcal{H}}_{\mu \nu \rho }$, plays the rôle of a totally antisymmetric torsion in the low energy effective field theory, up to and including fourth-order derivative terms [26,49,50]. Although in our present work the torsion interpretation will not play a rôle, nonetheless for completeness we shall make some comments in this direction, which will hopefully constitute future research avenues in this context.

First, of all, as discussed in some detail in [57], where we refer the interested reader, the KR axion in string theory, which is dual in (3+1)-dimensions to the antisymemtric-tensor field-strength three form, ${\mathcal{H}}_{\mu \nu \rho }$, can be thought of as completely analogous to the Barbero-Immirzi parameter [58,59] of loop quantum gravity [60,61,62], when the latter is promoted to a (pseudoscalar) field [63,64,65,66], in analogy to the case of the instanton angle in Quantum Chromodynamics (QCD) [67]. According to [57], the string effective action of the StRVM, in the presence of CS gravitational anomalous terms, is linked to the so-called Nieh-Yan invariant [68,69,70,71,72], which is a topological torsional invariant, in contrast to the Holst term [73] $\int d^{4}x{\epsilon }^{\mu \nu \rho \sigma }{\widehat{R}}_{\mu \nu \rho \sigma }\left(\overline{\mathsf{\Gamma }}\right)$ ($\overline{\mathsf{\Gamma }}$ denotes the generalised torsionful Christoffel connection), which alone is not a topological invariant quantity in the presence of torsion [74,75]. The Nieh-Yan term is defined as where $T_{\nu \rho }^{\mu }$ is the torsion tensor, which in the context of the Einstein-Cartan theory is a non-propagating field, since the corresponding action contains non-derivative terms of the torsion tensor. Thus, the Nieh-Yan term is indeed a total derivative even in the presence of torsion and replaces [74,75] the Holst term [73]. In our string-theory case, the torsion, as already mentioned, has a single totally antisymmetric component, proportional to the antisymemtric-tensor field strength $T_{\mu \nu \rho }\propto {\mathcal{H}}_{\mu \nu \rho }$. The BI can be considered as a constant coefficient $\beta$ of the torsional invariant term (1).

The promotion of the BI parameter to a field would naively be equivalent to considering adding to the effective gravitational action in the presence of torsion a term (1) but with a coordinate-dependent coefficient $\beta \left(x\right)$, which is viewed as a field variable, integrated over in the path-integral (with a measure $\mathcal{D}\beta \left(x\right)$): where in the last equality the proportionality factor takes into account numerical coefficients that appear in the precise relation between the torsion and the KR field strength [57]. As discussed in previous works [1,26,57], path-integrating over (the non-propagating field) $\mathcal{H}$ in the low-energy string-inspired effective action, would produce a dynamical propagating field $\beta \left(x\right)$, with canonically normalised kinetic terms, which would lead [57] to the result of [63,64,65].

However, the above procedure would imply a Bianchi identity constraint ${\epsilon }^{\mu \nu \rho \sigma }{\partial }_{\mu }$${\mathcal{H}}_{\nu \rho \sigma }=0$, which does not take into account the Green-Schwarz (GS) counterterms required in string theory for cancellation of gauge and gravitational anomalies in the extra-dimensional target spacetime of strings [37]. The GS counterterms lead to appropriate modifications in the definition of ${\mathcal{H}}_{\mu \nu \rho }$, by appropriate gravitational and (non-Abelian) gauge CS three forms [26], the gauge terms pertaining to the appropriate gauge groups that characterise the underlying string theory model. The modified ${\mathcal{H}}_{\mu \nu \rho }$ obeys a Bianchi-identity constraint [1], which, when implemented in the path integral via a pseudoscalar Lagrange multiplier field [1,26,32,57], leads to the emergence of a dynamical KR axion field, $b\left(x\right)$, dual to the modified field strength ${\mathcal{H}}_{\mu \nu \rho }$, which is analogous to the BI field. Thus, in the string case, the promotion of the BI parameter accompanying the Nieh-Yan invariant (1) to a dynamical pseudoscalar field cannot be done solely via (2), as in [63,64,65,66], but by adding instead to the low-energy, string-inspired, effective gravitational field-theory action an appropriate combination of topological invariants, the Nieh-Yan invariant and the gravitational (without torsion) Chern-Simons term, with the BI field appearing as its coefficient [57]: where $\widetilde{(...)}$ denotes the dual Riemann tensor, defined in the next Section 2 (cf. (8)), $\kappa$ is the gravitational constant in (3+1)-dimensions, and ${\alpha }^{\prime }$ the Regge slope of the string. The pseudoscalar (axion-like) field $b\left(x\right)$ is the so-called string-model independent axion [32]. The reader should notice that the fields $b\left(x\right)$ (3) and BI $\beta \left(x\right)$ (2) share the standard axionic shift symmetry, that is, the corresponding action terms are invariant under the shift of the fields by constants.

Another aspect of the CS theories is the coupling of the axion fields $b\left(x\right)$ (or, in general the axions arising from string compactification $a\left(x\right)$) to mixed anomalies, that is gravitational but also gauge CS terms, the latter being of the form where the dual of the gauge field strength is defined in (6), in Section 2 below.

In our current paper, such gauge anomalous terms shall not be considered, the reason being that we are interested in the inflationary epoch of the StRVM, which is not characterised by gauge fields. As discussed in [1,3] in the StRVM, the latter are generated at the end of the inflationary period, as a result of the decay of the unstable running vacuum. In the primordial eras one could at most have non-perturbative instanton-type configurations of appropriate non-Abelian gauge fields, which are integrated out in a path integral to yield periodic modulation terms in the axion potentials, as we shall discuss in Section 5.1. The latter can contribute positively to the correct phenomenology of the inflationary slow-roll parameters, leading to agreement with the data [52,53], but lead to no further physical effects. Par contrast, in the post-inflationary period, such chiral gauge anomalies can play a crucial rôle in several respects [1,2,3,4]. For instance, during the QCD epoch, color-group instanton effects can lead, via chiral anomalies that survive the post-inflationary eras of the StRVM, to non-perturbative generated masses of the KR (or other) axions, which can thus play the rôle of dark matter with mass of the order of the QCD axion [2]. In general, in string models, instantons of the appropriate non-Abelian gauge groups can lead to massive compactification axions, which can play the rôle of even ultralight DM [76]. In addition, Abelian electromagnetic U(1) chiral-anomaly terms (4) coupled to axions can exist at late epochs of the post-inflationary StRVM Universe [1], and can lead to interesting effects, via axion-photon couplings. At such epochs, the effects of the KR field torsion could also be significant in leading to the generation of cosmic magnetic fields from galactic dynamo instabilities in the Universe, in similar spirit to what happens [77,78,79,80] in generic torsional Einstein-Cartan theories [81]. Moreover, on the related front of the BI parameter of the loop quantum gravity [60,61,62], which, as we conjectured above [57], appears to be linked to the StRVM, one should make the important reamark that the BI parameter may be constrained via photon-axion conversions and axion mixing in the presence of (generic) torsion, since such phenomena are found to depend on the BI parameter [82]. In our case, there are additional constraints on the magnitude of the KR axion background (which plays the rôle of the BI field, as discussed above) at late eras of the Universe evolution, which arise from the characteristic cosmic evolution of the KR field and its rôle in Leptogenesis [1,33,34,35,83]. All these avenues of research are interesting ones to pursue, and we hope to come back to such issues in future publications.

The structure of this article is the following: in the next Section 2, we shall review briefly the emergence of a linear axion potential in the context of our string-inspired effective theory, as a result of the (real part of the) gCS condensate induced by chiral primordial GW. In the subsequent Section 3, as part of the original material of this work, we shall demonstrate-by means of formal path integral methods- the rôle of the presence of N sources of GW as a multiplicative factor in front of the real part of the CS condensate induced by the GW due to a single source. This has been conjectured in [5], using hand-waving arguments, but here we shall prove it rather rigorously, within the assumptions made in our perturbative (weak) quantum gravity approach. In Section 4, we shall demonstrate in detail the presence of imaginary parts of the gCS condensate, in generic quantum-ordering schemes, and show that such parts are linked to quantum commutators of the pertinent chiral GW. These imaginary parts of the effective gravitational action lead to a finite lifetime of the inflationary period. The latter is independent of the number of sources of GW, N, as it is determined by the imaginary parts of the GW Hamiltonian due to a single source. We shall then estimate the magnitude of these imaginary parts, and thus the life time of the inflationary vacuum. Compatibility with the inflationary phenomenology [52,53,54] will then constrain the ratio of the energy scales $M_s/M_{\mathrm{Pl}}$ in this string-inspired cosmological model. Section 5 deals with the derivation of the slow-roll parameters of a variant of the StRVM, in which periodic modulations in the axion potential are included, which are induced by non-perturbative stringy instanton effects. The inclusion of such modulations yields better fits of the StRVM to the cosmological inflationary data. We shall first consider in Section 5.1 target-space-gauge-group-instanton-induced periodic modulations of the KR axion potential, which can lead to the correct slow-parameter inflationary phenomenology, upon appropriate fixing of the relevant energy scales. In Section 5.2, we shall discuss briefly the rôle of world-sheet instantons that characterize other axions that arise from string compactification, which are in general present together with the KR axion in string theories (the KR axion does not couple to world-sheet instantons). Finally, conclusions and outlook will be given in Section 6. Some conjectures on the microscopic origin of the observed acceleration of the Universe in the current-era, within the context of GW condensation, are also given briefly at the end of the section. Technical aspects of our approach are discussed in several Appendices. Appendix A deals for completeness with a technical discussion concerning the necessary conditions for obtaining a consistent variational principle of the CS gravity, by means of the addition of appropriate boundary terms. This discussion also justifies our use of the GW as proper solutions of the gravitational equations. In Appendix B we construct the perturbed CS action to second order in GW perturbations around an arbitrary spacetime background, which will give the full graviton propagator to this order. It is then shown that, in the context of a FLRW background, the latter is equivalent to the propagation of two independent scalar modes. Our study contains a careful inclusion of the boundary terms.

The CS gravitational theory [26,47,48] introduces a linear coupling of the (pseudo)scalar field b, with the so-called mixed anomaly term, that is a specific linear combination of the gauge CS term $\mathbf{F}\tilde{\mathbf{F}}$ and the gCS term, $R_{CS}$ (throughout this article Greek indices denote (3+1)-dimensional spacetime indices):4 where A is a coupling constant, to be discussed further below. The ⋯ denote the “kinetic” terms of the gauge fields, which will not be important for our discussion until Section 5. The quantity ${\mathbf{F}}_{\mu \nu }$ is a (non-Abelian, in general) gauge field strength, with its dual, where ${\epsilon }_{\mu \nu \alpha \beta }=\sqrt{-g\left(x\right)}{\widehat{ϵ}}_{\mu \nu \alpha \beta }$ is the covariant Levi-Civita tensor, and ${\widehat{ϵ}}_{\mu \nu \rho \sigma }$ denotes the Minkoswki space-time Levi-Civita totally antisymmetric symbol, with the convention ${\widehat{ϵ}}_{0123}=1$, etc.

The gCS term $R_{CS}$ in (5) is given by: with the symbol $\widetilde{(\cdots )}$ denoting the dual of the Riemann tensor, defined as to be contrasted with the Hodge-dual which uses the Minkowskian Levi-Civita symbol ${\widehat{ϵ}}_{\mu \nu \rho \sigma }$. In what follows we shall alternatively make use of both duals, exploring the identity

The mixed anomaly $R_{CS}+\mathbf{F}\tilde{\mathbf{F}}$ is a total derivative [84,85]. Consequently, the theory (5) is shift symmetric, i.e., invariant under global transformations of the pseudoscalar field

The gauge anomaly term ${\mathbf{F}}^{\mu \nu }{\tilde{\mathbf{F}}}_{\mu \nu }$ is topological, that is independent of variations of the metric tensor, so it does not contribute to the gravitational (Einstein) equations of motion. Par contrast, the gCS term $b{R}_{CS}$ is characterized by a non trivial variation with respect to the metric tensor, namely the Cotton tensor $C_{\mu \nu }$ [47]: Variation of the action with respect to the metric and the axion field5 yields the following equations of motion [26,47] where $T_{\mu \nu }^{\left(b\right)}$ is the stress energy-momentum tensor associated with the kinetic term of a matter field,

The property of the Cotton tensor [47] $C_{\mu \nu }^{;\mu }=-{\displaystyle \frac{1}{4}}\left({\partial }_{\nu }b\right)R_{CS}$, where ; denotes gravitational covariant derivative (with respect to a torsion-free connection), implies that the naive covariant conservation of the axion-matter stress tensor fails,

This implies the non conservation of the naive axion stress tensor, which is physically interpreted as indicating a non trivial exchange of energy between the axion matter and the gravitational field. The $R_{CS}$ term is CP violating,6 and, as such, it vanishes for spherically symmetric or isotropic and homogeneous spacetime backgrounds. This is the case when a Friedman-Lemaitre-Robertson-Walker (FLRW) spacetime background is considered, which means that in such a case the modified Lagrangian is equivalent to the Einstein—Hilbert of General Relativity. Consequently, in that case, the scalar field is minimally coupled to gravity and the cosmological evolution is governed by a stiff equation of state $w_{\mathrm{stiff}}^b=+1$.

However, when chiral GW are produced, through non-spherically symmetric coalescence of primordial black holes or collisions of domain walls, the gCS term becomes non-trivial because different helicities of the tensorial perturbations propagate in a different way [86,87,88,89]. This difference to the wave equations for the left and right-handed polarizations arises because of the presence of the Cotton tensor (12) in the gravitational Equation (13), leading to gravitational-wave birefringence of cosmological origin.

In [6] the GW were treated quantum-field theoretically (in a weak-gravity setting) by applying the process of second quantization [89], i.e., by promoting the perturbations to operators through the definition of the corresponding creation and annihilation operators. In this sense, the gCS term becomes also an operator, ${\widehat{R}}_{CS}$, which we calculate up to second order in the tensorial perturbations. The gCS operator backreacts onto the effective Lagrangian through its vacuum expectation value (vev), $\langle R_{CS}\rangle \equiv \langle 0|{\widehat{R}}_{CS}|0\rangle$, where $|0\rangle$ denotes the appropriate gravitational ground state of the system, and the symbol $\widehat{(...)}$ is used to denote quantum operators. If $\langle R_{CS}\rangle$ acquires a constant value, thus having the form of a (translationally invariant) gravitational condensate, then in this case a linear potential for the KR axion arises, which breaks the shift symmetry (11) of the effective gravitational theory (5). We argued in [6], by making use of a dynamical-system approach, that the linear-axion potential (17) leads to inflation. A linear-axion potential also characterizes axion monodromy in conventional string/brane theory [51], but in our case its origin its entirely different, given that it arises as a result of the formation of the GW-induced condensate of the gravitational anomaly term.

We now come to the string-inspired CS theories [26], which we shall restrict our attention upon for the remainder of this article. In particular, we shall concentrate on the stringy running vacuum model (StRVM) of cosmology [1,3,4], which, as already mentioned, is based on the assumption that in the primordial Universe only fields from the massless bosonic gravitational multiplet of the string appear as external fields. In such models, the gauge fields $\mathbf{F}$, together with other matter excitations, are assumed absent, and generated only at the end of the inflationary period, due to the decay of the metastable ground state (vacuum) of the StRVM [1,3]. The action in this model, which we shall restrict ourselves on in the next couple of sections, is described by (5) upon setting $\mathbf{F}=0$, that is:

In this class of models the coupling A is determined by string theory considerations [26], and is given by

Since the StRVM (5) is viewed as a low-energy string effective gravitational theory, it is natural to consider that the UV cutoff $\mu$ of the momenta of the pertinent low-energy excitations, including the (quantum) graviton modes, is provided by the string mass scale [5], so that the transplanckian conjecture is satisfied [90,91].

The detailed analysis of [6] has then demonstrated that the GW-induced gCS condensate is a complex quantity. Its real part (to leading order in the assumed small quantity $|{\kappa }^2\dot{b}|\ll 1$) has been estimated to be:

Notably, the negative sign in front of the quantity on the right-hand side of the above equality guarantees a positive sign for the effective cosmological constant. The quantity $H_I$ is the Hubble rate at inflation, whose upper bound has been imposed by the Planck Collaboration data [52] to have the order of magnitude

We next recall [5,6] that for the actual value of the condensate in the inflationary epoch, we have to multiply the result (21) with the number ${\mathcal{N}}_I$ of all kinds of sources of gravitational waves during inflation, thus obtaining as a final result:

A detailed proof of this feature, within the path-integral approach of weak quantum gravity, will be given in the next Section 3.

In the analysis of [6] an estimate of the order of magnitude of $\dot{b}$, which remains approximately constant during the entire inflation era, has also been obtained as a consequence of the imposition of appropriate initial conditions in the dynamical-system approach: confirming in this way the assumptions made in [1,3,4,5], including the assumption on the smallness of $|{\kappa }^2\dot{b}|$ that lead to (21).

We stress at this stage that the approximate constancy of the rate of change of the KR axion background in our StRVM approach is due to the presence of a gCS condensate $\langle R_{CS}\rangle =\mathrm{const}.\ne 0$. Indeed, if we recall that the gCS-Hirzebruch term in (5) is a total derivative [26,85] where ; denotes covariant derivative in the curved background, and make the following plausible approximation in our homogeneous and isotropic FLRW inflationary cosmology with an (approximately) constant Hubble parameter H: [1,3] then, from the equations of motion for the b-field stemming from the action (5), we obtain (ignoring the gauge terms, as in the model of [1,3]) where □ denotes the covariant D’Alembertian. For $H=\mathrm{constant}$, and $\langle R_{CS}\rangle =\mathrm{constant}$, one obtains from (27) that

Equivalently, on using (26), we may also write the b equation as: which also accepts as a solution a constant $\dot{b}$ axion background:7

Thus, Equation (24), which in [6] has been obtained as a consistent condition in the dynamical-system approach to linear-axion inflation, constitutes, in view of (27), (28), (29), a highly non-trivial consistency check of the StRVM approach to inflation via primordial-gravitational-wave-induced CS condensates, which in this way is mapped into a dynamical evolution of a single-(axion) field system with a linear potential. The reader should therefore notice the appearance of linear in time KR axion backgrounds in this StRVM system, as happened in the string-inspired cosmological models of [29,30,31], but here the origin of this dependence is different.

Moreover, in view of (23), (24), the gCS condensate is of $\mathcal{O}\left(H_I^4\right)$, which, as already mentioned in the introductory section of the article, makes the corresponding energy density having a form that is encountered in the RVM approach to cosmology [7,9,10,11,15,16,17,18,19,21,22,23], that is being a function of even powers of the Hubble parameter, as dictated by the underlying general covariance of the formalism.

In the context of the dynamical-system approach to inflation, the analysis of [6] also provides a restriction on the magnitude of the string scale $M_s$, compared to $M_{\mathrm{Pl}}$: which is consistent with the transplanckian censorship hypothesis [4,90,91].

From (19), (20), (22),(24) and (31), we therefore obtain for (23):

Finally, an estimate of the number of sources of GW during the inflationary epoch, ${\mathcal{N}}_I$, in terms of the corresponding number ${\mathcal{N}}_S$ during the stiff-axion era, which, in the StRVM Cosmology, precedes the inflationary era [1,3,4], is also provided as [6] which lies in the range given in [5], stemming from the assumption of the constancy of the CS condensate during inflation, which thus avoids exponential dilution [1,3]. For our purposes we may take without loss of generality that ${\mathcal{N}}_S=\mathcal{O}\left(1\right)$.

We proceed in the next section to a detailed discussion on the emergence of the enhancement (32) of the value of the real part of the gCS condensate by the number of GW sources, following a path integral approach in the framework of (gauge fixed) weak quantum gravity of GW perturbations.

We now turn our attention to the path integral formulation which, as we shall see, can indeed lead to a better physical insight on the StRVM approach to Cosmology, albeit there are still several formal questions that remain unanswered, especially the ones related to the still wide-open issue of gauge invariance in quantum gravity [94,95,96].

The full path integral of the StRVM theory reads, where $S[b,g]$ is given by (18). This, of course, is a formal expression, since both the measure and the action of quantum gravity are still not known. However, since in our case we are dealing with an effective low-energy gravitational theory obtained from strings [13,14,24,25,49,50], the above quantum theory is valid only at its semiclassical approximation. In general, when assuming the semiclassical approximation, the partition function becomes a functional of the background fields, where $W[g,b]$ denotes the effective action of the background fields, which obeys: at appropriate saddle points of the path integral.

Upon ignoring any perturbation of the fields, the functional $W[g,b]$ can be easily identified with the effective action, $W[g,b]=S^{\left(0\right)}[g,b]$, where $S^{\left(0\right)}[g,b]$ is the action in the background spacetime without perturbations. However, upon considering variations of the metric about the stationary points (36), as we did previously in the form of GW [6], we obtain where $S^{\left(0\right)}[b,g]$ is the unperturbed background action (18), and $S^{\left(2\right)}[b,g,h]$ denotes the action up to second order in the metric variations, $h_{\mu \nu }$, with the latter treated as weak quantum variables (fluctuations). This order in the expansion about the stationary point suffices for the purposes of this work. Then, the path integral has to be of the following form: where b and g act as background fields, while $S^{\left(2\right)}[b,g,h]$ is the action up to second order in the metric perturbations and ${\mathcal{Z}}_h$ denotes the path integral of the GW perturbations,

From now on we focus on the FLRW spacetime background, in the presence of chiral GW perturbations. In such a case, h is an abbreviation for the two independent modes, $h_L$ and $h_R$ of the GW (cf. Section 4). These perturbations correspond to a GW induced by some source. In this case, $S^{\left(2\right)}[b,g,h]$ denotes the action of the GW, which has the form,

In the above expression, $S^{\left(2\right)}[b,g,h]{|}_{A=0}$ denotes the terms of the gravitational action (5), perturbed to second order, in the absence of gCS and chiral gauge anomalies, which is derived in Appendix B (cf. Equation (A80)). The quantity ${\mathcal{O}}_h={R_{\mu \nu \rho \sigma }}^{*}{R}^{\mu \nu \rho \sigma }$, with ${R_{\mu \nu \rho \sigma }}^{*}{R}^{\nu \mu \rho \sigma }$ expressed up to second order in the (weak) GW perturbations, and we used (9), (10). Then, the path integral reads, implying that the effective action, upon assuming metric perturbations, $W[b,g]$, differs from $S^{\left(0\right)}[b,g]$. In order to find $W[b,g]$, we take the functional derivative (assuming that the variations with respect to the background fields commute with the measure of the GW path-integration) where is the vev of the composite operator, ${\mathcal{O}}_h$, which contains the metric perturbations. Such a vev could depend only on the background metric and in the case of FLRW background only on even powers of the Hubble parameter, H, as implied by diffeomorphism invariance of the theory [9].

We now proceed to incorporate a number of GW sources $\mathcal{N}>1$ in the path integral. This is introduced by treating the GW coming from various sources as “an ideal gas of non-interacting GW perturbations”—in the sense that the partition function of the $\mathcal{N}$ sources of GW is constructed as a product of $\mathcal{N}$ partition functions of the individual GW:

This explains, within the path integral formulation, the linear superposition of GW effects assumed in [5].

Then, following the same procedure as in the single GW case above, we have, where by the symbol $\langle \cdots \rangle$ we have defined the following vev of a composite operator $\mathcal{B}[\cdots ,h_i]$ (for every GW perturbation $h_i$) which obviously depends only on the background fields, and as such we denote it simply as $\langle \mathcal{B}[\cdots ,h]\rangle$ or $\langle {\mathcal{B}}_h\rangle$, i.e., we omit the index referring to each GW perturbation, and instead we use a generic h. Therefore, Equation (45) reads,

Introducing the notion of the proper number density n of GW sources in the spacetime background corresponding to the metric $g_{\mu \nu }$: and using the identity (10), we define the following quantity,

We next make the assumption on the approximate constancy of the gCS condensate during inflation (which is confirmed by explicit calculations in our case [1,3,4,5,6])

As we have already shown in previous works [5], the introduction of the sources is crucial in order to satisfy (50) but also not to dilute the condensate during inflation [1,5], while respecting (20). Assuming that these conditions are valid (i.e., assuming a sufficiently large number of sources), indeed an effective linear potential for the axion is introduced. In this interpretation, such a linear potential comes from a “re-classicalization” process of the quantum perturbations around the classical background. The term “re-classicalization” is used here to stress that, upon its formation, the condensate $\langle R_{\mu \nu \rho \sigma }{}^{*}R^{\mu \nu \rho \sigma }\rangle$ acts as a “classical” source for the axion, affecting in this way its already classical background, and producing an effective linear potential for the (pseudo-) scalar field.

The reader should then observe that any variation of the condensate with respect to the background fields vanishes. As a consequence, a linear potential for the axion arises leading to inflation, as discussed in detail in [6]. We recall from our previous analysis in that work that the number of GW sources is not constant throughout the evolution of the primordial universe, but it necessarily increases abruptly as we pass from the stiff to the RVM inflationary epoch (33). This is an essential behavior to guarantee the onset of inflation in our approach. The nature of the GW sources depends on the underlying microscopic model. There are various mechanisms for GW production during the pre-inflationary era, ranging from non-spherically-symmetric merging of primordial black holes to annihilation/non-spherically-symmetric collapse of domain walls [3].

Using (50) we now apply a mean-field Hartree-Fock (HF) approximation [97,98,99] for the evaluation of the source path integral (44). According to the HF treatment, the gCS term is split into a condensate term and quantum fluctuations about it (denoted by the symbol $:\cdots :$) [1,3]: where we used the identity (10), and the definition (49), with the property (50) in mind. On account of (51), the corresponding variational equations with respect to the b-axion and graviton fields, obtained from (44), read: The normal-ordered ($:\cdots :$) terms of the composite operators bring in new UV divergencies, which in general may lead to background-dependent terms in both equations (52). Such terms can be arranged to vanish by an appropriate choice of counterterms. This lies at the heart of our mean-field HF treatment, which entails ignoring quantum fluctuations about the gCS condensate. Moreover, from our analysis in Appendix B (cf. Equation (A79)), we observe that in the Traceless-Transverse (TT) gauge of GW (A85), the third term on the right-hand-side of the b-axion Equation (52) vanishes, in a self consistent way to second order in the graviton perturbation, in which the corrections ${\mathcal{S}}^{\left(2\right)}$ are gauge invariant, as discussed in Appendix B.

On account of (50), it is evident that an (approximate) linear-axion contribution to the effective stress tensor arises during the inflationary period (cf. Equation (23)), which is equivalent to an approximate positive-cosmological-constant (de-Sitter type) contribution, since, for the duration of inflation, the b field remains of the same order of magnitude, as explained in [1,3,4,5,6].

At this stage we would like to make some brief but important remarks regarding the limitations of our results. The above analysis was somewhat formal and did not deal with the important issue of gauge invariance of the quantum gravitational path integral. The estimates (23) of the gravitational CS condensate in our approach [1,6] have been performed in a specific gauge (TT gauge (A85) of GW) within the framework of weak quantum gravity. Unfortunately, due to the lack of knowledge of the full theory of quantum gravity, we cannot demonstrate gauge invariance of our condensate at a full non-perturbative level. Such a task would require the application of methods as in [94,95,96,100], which, in the current state of affairs of our model, lies well beyond the scope of the current article. Our aim here was to provide evidence for the existence of a complex non-zero condensate of the gravitational CS anomaly term in the effective action (5) during inflation, and estimate the life time of the associated unstable vacuum. Hopefully, when a fully UV completion of the model is at hand, which in our case implies a specific string model of phenomenological relevance to particle physics, our estimates will not change much quantitatively.

The reader should then notice that in this HF form, the equations (52) include the (classical) background parts, associated with the respective variations of the $S^{\left(0\right)}$ part of the effective action (not involving GW corrections), plus non-trivial averages of the quadratic GW corrections, such as the third term on the right-hand-side of the graviton equation in (52), which expresses the quantum average of the contribution of the second order GW perturbations to the GW stress-energy tensor. The latter is not gauge invariant. From dimensional considerations, we expect such terms to yield corrections to the Ricci and scalar-curvature parts of the Einstein tensor, plus a potential vacuum-energy-like contribution. If we restrict ourselves to terms of second-order in derivatives, for consistency with our discussion so far, such vacuum energy contributions in a FLRW background are expected to be of order $\mathcal{O}\left(H^2\right)$, where H is the Hubble parameter. Such terms would be subdominant at early eras of the Universe, like the RVM inflation we are interested in here, as compared to the $\mathcal{O}\left(H^4\right)$ terms coming from the gCS condensate ${\displaystyle \frac{A}{2}}{g}^{\mu \nu }{\langle R_{\mu \nu \rho \sigma }{}^{*}R^{\mu \nu \rho \sigma }\rangle }_{\mathcal{N}}$ appearing in the graviton Equation (52). However, their presence would complete the description of the energy density of the vacuum in this StRVM approach, which would thus be in agreement with the considerations of [1,3]. In this work we shall not proceed further to evaluate such terms, which will be the subject of a forthcoming publication. We may conjecture though that, if the above terms are indeed of order $H^2$, then one might encounter similar contributions to the vacuum energy density in modern eras, from appropriately condensing sources of GW in the current epoch, in analogy to the inflationary era. Such terms, of order $\mathcal{O}\left(H_0^2\right)$ (where $H_0$ denotes today’s value of the Hubble parameter), would thus provide a contribution to today’s dark energy, according to generic RVM features [7,9,10,11], which the StRVM would share.

In the next section we shall evaluate the imaginary parts of the gCS condensate, thereby proving the metastability of the inflationary vacuum, and provide an estimate of the associated duration of inflation. On comparing with cosmological data [52,53,54], then, we shall constrain the string scale $M_s$ of the StRVM.

In order to estimate the gravitational anomaly condensate, we use canonical quantization techniques. Applying such methods for GWs in an FLRW background, we are also able to investigate the quantum effects that show up regarding the behavior of the composite operator of the Hirzebruch signature,

As we shall show, this operator may contain imaginary parts, which are absent in the classical theory. In our approach these imaginary parts play an important rôle in inducing metastability of the inflationary quantum vacuum, and thus a finite duration of inflation in the StRVM. Such a metastability is in agreement with the swampland criteria of de-Sitter-like inflation in string theory [101,102,103,104,105,106].8

The method of quantizing GWs followed in this work lies on the fact that the tensorial metric perturbations have two degrees of freedom. Then, one can show that their evolution is equivalent with that of two independent scalar modes (see Appendix B for a detailed analysis in the case of a background with a minimally coupled scalar field). In this way, the gauge-fixing issues encountered in the quantization of gauge theories, such as the theory of the graviton field, can be bypassed by identifying the two scalar modes and quantize them independently using appropriate mode expansions [107]. The case of the 4D Chern-Simons gravity is similar. The graviton in this theory also contains two degrees of freedom [47], and the quantization procedure, as we shall demonstrate below, can be applied in a similar fashion [87,88,89,108]. The main difference occurs because of the derivative couplings arising due to the Chern-Simons interaction and especially due to the presence of the Levi-Civita tensor. Due to the latter, such derivative couplings contain odd powers of spatial derivatives, thereby inducing a dependence on the direction of propagation, which explains the parity-violating nature of $R_{CS}$, but also a form of derivatives with respect to the conformal time, implying ordering ambiguities upon quantization. As we shall discuss below, this implies the presence of (quantum-operator-ordering dependent) imaginary parts.

For a spatially-flat FLRW spacetime, where $\alpha \left(t\right)$ denotes the scale factor, the tensor perturbations (GWs) have the following form:

We can express $h_{ij}$ in the linear polarization basis [109], expressed as: where the polarization tensors are defined through: where $(e_1\left(\overrightarrow{k}\right),e_2\left(\overrightarrow{k}\right),e_3\left(\overrightarrow{k}\right))$, with $e_3\left(\overrightarrow{k}\right)=\overrightarrow{k}/\left|\overrightarrow{k}\right|$, form a right-handed orthogonal triad of unit vectors. In the presence of the gCS coupling, it proves more appropriate to expand the perturbation (56) with respect to the helicity basis tensors ${ϵ}_{ij}^{L,R}$, as [89], where ${ϵ}_{ij}^{L,R}$ are defined as follows:

The polarization tensors obey the following normalization: where $\lambda =L,R$ denotes Left and Right polarization. We can then obtain the following property: with $l_R=+1$, $l_L=-1$, while also [87], where $\theta$ denotes the polar angle of the arbitrary vector $\overrightarrow{k}$. Moreover, by construction, the following relations hold:

Expanding the gravitational perturbations in Fourier space and helicity basis, we obtain:

We may now show in an analytical way the derivation of the Chern-Simons part of the action, starting from the relative interaction term:

Up to second order in $h_{ij}$, $R{}_{\mu \nu \rho \sigma }{}^{*}R^{\nu \mu \rho \sigma }$ is expressed as [89]: where the prime denotes differentiation with respect to the conformal time $\eta$. Keeping only the antisymmetric part with respect to the indices i and j, performing partial differentiations and using the cyclic permutation symmetry of ${\widehat{\epsilon }}^{ijk}$, we obtain, that is, a total divergence, $R{}_{\mu \nu \rho \sigma }{}^{*}R^{\nu \mu \rho \sigma }={\partial }_{\mu }{J}^{\mu }$, with $J^{\mu }=\sqrt{-g}{\mathcal{J}}^{\mu }$ and ${\mathcal{J}}^{\mu }$ the covariant current in (25). Thus, combining with the part of the Einstein-Hilbert action given in (A87) (cf. Appendix B), we obtain, where $S_B$ denotes boundary terms,

Substituting (67) into (71), we obtain the classical action of GW in Fourier space, where, with

The above considerations are classical. Canonical quantization of GW perturbations in our context has been discussed in detail in [6], following and extending the analysis in [88,89]. Basically, by representing the tensor perturbation $h_{ij}$ in (67) as: where $a\left(\eta \right)$ is the scale factor of the expanding inflationary Universe, the quantization of $h_{ij}$ is achieved by considering the Fourier transforms of the “scalar” field operators [6,89] where we have introduced two sets of creation and annihilation operators, ${\alpha }_{\overrightarrow{k}}^{\pm }$ and $b_{\overrightarrow{k}}^{\pm }$, for which ${\left({\alpha }_{\overrightarrow{k}}^{-}\right)}^{†}={\alpha }_{\overrightarrow{k}}^{+}$ and ${\left(b_{\overrightarrow{k}}^{-}\right)}^{†}=b_{\overrightarrow{k}}^{+}$, obeying the commutation relations:

At this point we turn our attention to the quantization issues of the anomaly term (69). In the helicity basis (59), and choosing, without loss of generality, the $z-axis$ as the direction of GW propagation, the CS term has the following structure [6,88,89]:

The above expression makes clear our earlier assertion that the gCS anomaly term is non-trivial only in the presence of chiral (i.e., left-right asymmetric) GW perturbations.9

At this stage we make some important remarks regarding the canonical quantization of the gCS theory, and in particular the anomaly term (81). The issue is the known ordering ambiguities that occur in a quantum theory when we replace the classical quantities, such as $h_{L\left(R\right)}\left(x\right)$ in the above equation, by quantum operators.10

In general, we may present various orderings of quantum operators, ${\widehat{\varphi }}_i$, as: where $w_{\mathrm{P}}$ denote the weights of the various permutations P of the operators appearing on the right-hand-side of (82).

Using the Fourier decomposition for each mode separately, we can arrive at the following relation for the operator form of (81), upon applying a given quantum-ordering scheme (82): where, for the sake of generality, we assumed different ordering schemes (provided by the different weights $w_{{\mathrm{P}}_i}$, $i=1,\cdots ,4$) for each product of operators appearing in (83), with the weights satisfying